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A systematic literature review on mathematical models of humanitarian logistics.

1. Introduction

2. research methodology, 3. research in humanitarian logistics, 3.1. facility location problems, 3.1.1. deterministic models, median problem.

Covering Problem

• Set Covering Problem

P -Centre Problem

Other Models on FLPs

3.1.2. non-deterministic models, stochastic programming approach, robust optimization and other non-deterministic approaches, 3.2. relief distribution, 3.2.1. deterministic models, 3.2.2. non-deterministic models, the stochastic programming model for relief distribution, robust optimization and others, 3.3. mass evacuation, 3.3.1. public and private transport evacuation model, 3.3.2. urban area evacuation model, 4. future research direction, 5. conclusions, author contributions, data availability statement, acknowledgments, conflicts of interest.

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Hezam, I.M.; Nayeem, M.k.; Lee, G.M. A Systematic Literature Review on Mathematical Models of Humanitarian Logistics. Symmetry 2021 , 13 , 11. https://doi.org/10.3390/sym13010011

Hezam IM, Nayeem Mk, Lee GM. A Systematic Literature Review on Mathematical Models of Humanitarian Logistics. Symmetry . 2021; 13(1):11. https://doi.org/10.3390/sym13010011

Hezam, Ibrahim M., Moddassir k. Nayeem, and Gyu M. Lee. 2021. "A Systematic Literature Review on Mathematical Models of Humanitarian Logistics" Symmetry 13, no. 1: 11. https://doi.org/10.3390/sym13010011

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During the last three decades, scientists in formal and natural sciences have been proposing models of language competition. Such models could prove instrumental in informing efforts made towards preserving the world’s linguistic diversity but have yet to gain significant interest among linguists. This situation could be due to a lack of overlap between the concepts and methods used in those models and those used by linguists. In an effort towards promoting interdisciplinary dialogue on the topic of language competition, this study describes the concepts and methods used in mathematical models of language competition and assesses whether these concepts and methods are becoming more similar over time to those used by linguists. To this end, studies that proposed mathematical models of language competition were systematically retrieved and analysed. Change over time in those models was first assessed concerning the way they are specified, including the parameters they contain. Next, it was checked whether models were increasingly fitted to empirical data. Finally, change in the disciplines covered by the journals where those models were published was evaluated. Results show that overall, models have been including few sociolinguistic parameters, have been relying little on empirical data, and have been mostly published in journals covering the fields of mathematics and physics. However, the last years have seen an important turnaround along each of these three axes. A common language seems to be emerging between fields regarding mathematical models of language competition, which should prove instrumental in informing efforts made towards preserving the world’s linguistic diversity.

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Introduction.

Since at least the 1950s, linguists have been playing a leading role on the language preservation scene. They were the first to sound the alarm about the perplexing pace at which languages are becoming extinct worldwide (Dorian, 1981 ; Hale et al., 1992 ; Krauss, 1992 ). They have been at the forefront of efforts made towards preserving the world’s linguistic diversity, notably within international organisations such as the United Nations Educational, Scientific and Cultural Organization (UNESCO) (Grenoble and Whaley, 2006 ; Moseley, 2010 ). Finally, linguists have continued to save moribund languages from oblivion through extensive fieldwork and documentation (Crystal, 2000 ; Seifart et al., 2018 ).

Recently, scientists in formal and natural sciences have also started to show interest in the problem of language death, albeit from a different perspective. Starting in the early 1990s, they have been proposing mathematical models that aim to establish the mechanisms through which languages go extinct (Seoane and Mira, 2017 ). Knowledge gained from such models could be tapped into to help to slow down the rate of language extinction worldwide but have yet to gain significant interest among linguists and other researchers in the humanities and social sciences. This situation could be due to a lack of overlap between the concepts and methods used in those models and those used by linguists. In fact, researchers interested in the question already suggested that it is “because models have not sought to engage with the intellectual framework used by linguists” that “the real influence of mathematical modelling has been severely limited in the field of language revitalization” (Fernando et al., 2010 , p. 49). In an effort towards promoting interdisciplinary dialogue on the topic of language competition, this study aims to describe the concepts and methods used in mathematical models of language competition, assess how models have changed over time, and determine whether there is any common language that is emerging between the different fields of science in modelling how languages become extinct.

To achieve this aim, a systematic review of the literature on mathematical models of language competition was performed. Information was retrieved from relevant studies and organised along three axes. The first axis concerns the methods used for modelling language competition. This point is important because mathematical models of language competition were initially developed based on methods that already existed in formal and natural sciences, but not necessarily in linguistics. However, research on models of language competition could provide practitioners involved in language preservation more practical tools if it relied more on methods that are common to the different disciplines involved. Below, the methods on which models were built are characterised concerning the level of analysis (macroscopic, mesoscopic, microscopic), the form of the model (equation vs. simulation-based), the linguistic composition of the population being modelled, and the parameters included.

The second axis concerns the use of data for validating models. Mathematical models of language competition are often purely theoretical and do not take empirical data as input. Conversely, linguistic analyses of language endangerment are typically data-driven. In the second part of the results section, studies on language competition are analysed concerning whether they validate their models against empirical data and whether there is any trend towards greater use of such data. Also, for each study that fitted its model to empirical data, a detailed account of the languages considered is provided, alongside the regions (or countries) covered.

The third axis concerns the way that research on mathematical models of language competition is communicated among different disciplines. Poor communication has long been recognised as an impediment for successful collaboration between disciplines (Bracken and Oughton, 2006 ). If publications on language competition appear predominantly in journals covering formal and natural sciences, linguists and social scientists are less likely to take note of them. On the other hand, mathematical models of language competition might more easily reach linguists and social scientists if they are published in multidisciplinary journals or journals covering the fields of linguistics or social sciences. Furthermore, an increase in the number of publications on mathematical models of language competition in journals covering the fields of linguistics and social sciences could be an indication of a certain appropriation of such models by linguists and social scientists. The third part of the results section therefore provides an analysis of the disciplines covered by the journals in which models of language competition were published.

Other reviews have already covered some of the studies that are reviewed here (Gong et al., 2014 ; Schulze and Stauffer ( 2006a , 2006b ); Seoane and Mira, 2017 ; Solé et al., 2010 ; Vogt, 2009 ; Wang and Minett, 2005 ). This study differs from those on at least three points. First, the focus of most of the previous reviews was broader than the focus of the present review. Those reviews discussed studies that used mathematical models to solve different problems allying languages and population dynamics, including—but not limited to—language competition. In contrast, our review concentrates specifically on language competition, which allows a more in-depth discussion of the relevant literature. Second, none of the previous reviews employed a systematic approach at retrieving and analysing articles. As a result, they could not provide a completely balanced and objective picture of the literature, which this study aims at doing. Third, the goal of the previous reviews was primarily to describe the main findings of the studies they covered. The present review mainly focuses on the methodological innovations that have taken place in the field over the years, which we hope will help us achieve our aim formulated above.

Search strategy

Methods follow the Prisma statement for reporting systematic reviews (Moher et al., 2009 ). Relevant articles were retrieved using the search engines of Arxiv, Scopus and Web of Science. Articles must have included in their title the word “language” in combination with at least one of the following words: death, competition, extinction, endangerment, shift, disappearance, invasion, revitalization, coexistence, survival, or preservation. To avoid finding too many articles from psycholinguistics—which are not relevant for our purposes—articles must not have included in their title the following words: competence, teaching, learning, processing, disorder, acquisition and comprehension. We further specified that the word “model” should appear in either the title, abstract or keyword list. Truncation was used to allow for different words that share the same root and meaning. To avoid overseeing any important work, articles that cited two seminal studies in the field were searched using the Scopus search engine. These two studies are those of Baggs and Freedman ( 1990 ) and Abrams and Strogatz ( 2003 ). The reference lists of selected articles were also checked, and experts in the field were consulted for missing articles. Eligible studies must have been research articles, thus excluding book chapters, and must have been written in English. Searches were performed on the 25th of May 2020 and refreshed on the 27th of July 2020. The exact word strings used in searches are provided in Table S1 of the supplementary materials.

Article selection and data extraction

Figure 1 shows how studies were selected for review. The different search streams provided a total of 821 records, of which 624 remained after duplicates were removed. The first round of screening was performed based on titles and abstracts. A total of 134 articles with mathematical models of language competition as a topic was selected for a complete assessment. At this stage, more articles were excluded if they assessed the sensitivity or qualitative properties of existing models without proposing a new one (model analyses), if they were not published in the form of a research article, or if they did not have sufficient information about the type of model they presented. This resulted in 56 studies left for data extraction, to which six were added from the reference lists of the selected studies or based on expert knowledge, resulting in a total of 62 studies selected for analyses.

Article selection process.

We extracted from the 62 selected studies information about authors’ names, year and journal of publication, the level of analysis (macroscopic, mesoscopic, microscopic), the composition of the population concerning its linguistic groups, the parameters included in the models, and the languages and region or country of interest covered by the data (in case empirical data were used). Model parameters were considered as being the same if they denoted similar concepts despite being called differently. For example, the parameter “Status”, which is sometimes used to refer to the socioeconomic position of the speakers of a given language, was considered as equivalent to the parameter “Prestige”. A full account of the parameters included in the analyses and what they were called in the different studies is given in Table S2 of the supplementary materials.

Analytical framework

Evolution of methods.

The way that methods evolved over the years is accounted for using four characteristics. The first one refers to the level of analysis. We distinguish between the macroscopic, mesoscopic and microscopic levels. Studies on models of language competition, and mathematical models of social processes in general, commonly distinguish between the macroscopic and microscopic approaches (Castellano et al., 2009 ; Stauffer and Schulze, 2005 ). Macroscopic models describe population processes from an aggregate level. Results translate mean outcomes and do not account for the variability among individuals. An example of an early macroscopic model is provided by Abrams and Strogatz ( 2003 ), who use a differential equation to describe change over time in the proportion of speakers among two competing languages and apply their model to different situations of language competition around the world. Microscopic models, on the other hand, describe population processes considering each individual separately. This approach confers certain advantages compared to the macroscopic approach. More specifically, microscopic models allow to explore the full range of an outcome instead of just its mean. These models further allow for stochasticity, which can play an important role among smaller populations. Finally, they allow to explicitly consider interactions between individuals. An example of an early microscopic model is provided by Castelló et al. ( 2006 ), who define a population of agents speaking either language A or B, or both (bilingual agents). Agents interact with each other and acquire a second language under the influence of repeated contact with the speakers of another language, or abandon an already known language due to a lack of interaction with the speakers of that language. Studies less often consider mesoscopic models as a separate approach. In practice, mesoscopic models are similar to microscopic models in that both consider each individual separately and allow for stochasticity and interactions. However, in mesoscopic models, the unit of analysis is not the speaker like in microscopic models, but rather groups of speakers, or even entire languages. An example of an early mesoscopic model is provided by Schulze and Stauffer ( 2005 ), who define a set of bit-strings where each different string represents a different language. The strings—or languages—can duplicate themselves, which leads to an increase in the number of speakers of that language, they can undergo mutations, which represents the phenomenon of language birth, or they can disappear, which represents language death. This kind of model is usually used to study the world’s current languages distribution concerning their numbers of speakers, i.e., to explain why a few languages are spoken by a large share of the world’s population while most languages are spoken by only a fraction of it.

The second characteristic used to account for the way that models have evolved refers to the type of model or the form of the equation used. Six types were identified. We first consider three types of models that are based on ordinary differential equations: ordinary differential equations models (ODE), Lotka–Volterra models (LV), and reaction-diffusion model (RD). ODEs refer to the simpler form of ordinary differential equations models. These models estimate the change in the proportion of speakers belonging to a language as a function of time and some parameters and can be fitted to data that include only limited information. In these models, the total population is normalised, meaning that these models assume that it is not the absolute size of a linguistic group that matters, but its respective proportion in the whole population. It follows that in an ODE model with two linguistic groups, growth in one group means a decline in the other one. LV models are also based on ordinary differential equations, but additionally include information about group size, allowing the size of a linguistic group to change independently from its proportion in the whole population. Diffusion terms can be added to ordinary differential equations (including to LV models) to allow speakers to spread in space, in which case the model will be referred to as an RD model.

We further distinguish models that were developed using a system dynamics (SD) approach. These models offer the same possibilities as an ordinary differential equations model but are expressed in terms of stocks and flows rather than in terms of equations, which may increase their accessibility to non-mathematicians (Wyburn and Hayward, 2009 ). Furthermore, such models often make explicit some parameters that otherwise remain implicit in ordinary differential equations models.

Studies that rely on ODE, LV, RD and SD models usually model language competition from a macroscopic level. Among those studies that took a mesoscopic or microscopic approach, most relied on agent-based (AB) models or a conversation game (CG) framework, or a combination of both. AB models simulate interactions between speakers according to a set of pre-established rules. Due to the complex nature of these interactions, these models often provide insights that could not be obtained using equation-based models. CG, on the other hand, is a form of game theory that concentrates on decisions made by speakers during conversations. Speakers’ information about the world is imperfect and leads them to make unexpected choices in multilingual settings. This approach was used on its own or in combination with agent-based models, in which case language shift depends on a series of one-on-one interactions. One study could not be assigned to any of the model types, presenting results from a simulation based on a series of equations. It will be referred to as “Other simulations” (OS).

The third characteristic refers to the composition of the population concerning its linguistic groups, i.e., their number and whether bilingualism is considered. Among the studies reviewed here, competition is either considered among two, three or four languages, or among hundreds or even thousands of them. While the former refers to competition in a specific country or a region, the latter refers to competition in the whole world. Bilingualism refers to the explicit modelling of speakers that are fluent in two or more languages.

The fourth and final characteristic refers to the parameters included in each model. All models consider at least one parameter that drives growth in the number of speakers of one language to the expense of the number of speakers of another language. The universe of such parameters is presented in Table 1 . To facilitate analyses, parameters were assigned to three broad categories. The first category includes parameters that do not directly predict the intensity of the shift between languages but instead specify the form in which it occurs, i.e., whether it occurs through a horizontal, vertical or exogenous transmission. Horizontal transmission occurs from adults to adults, vertical transmission (also called intergenerational transmission) occurs from parents to their children, and exogenous transmission occurs via institutions such as schools. The second category refers to geodemographic parameters. These parameters describe how speakers occupy the space, reproduce, migrate and die. The third category refers to sociolinguistic parameters. These include the prestige conferred to speakers of a given language, the language to which speakers identify, and language use itself, e.g., the level of fluency. Some parameters directly influence shift. For example, speakers are more likely to shift towards languages that are spoken by larger numbers of speakers. Other parameters have an indirect influence on shift. For example, groups of speakers with higher birth rates will have higher numbers of speakers not because more births induce more speakers, but because more births allow for more vertical transmission.

Information on data usage includes languages and countries (or regions) covered by the data, for each study. We consider that a study made use of data if it explicitly mentions the languages and country (or regions) covered by the data, and if data minimally reflects the numbers or proportions of speakers of two or more competing languages at two or more points in time, or transitions intensities between languages at different points in time. Conversely, we consider that a study did not make use of data in cases where the model is strictly theoretical, or in cases where data only reflects proportions of speakers at one point in time, which is often the case in studies that aim at reproducing the world’s languages distribution according to their number of speakers. A distinction is made between studies that used data on a specific population for the first time or for a second time or more. These will be referred to as first and second analysis, respectively. Populations are considered different when they either speak a different combination of languages or speak a similar combination of languages in different countries (or regions). For example, several studies considered competition between Welsh and English in Wales. The study that first used data on the combination Welsh/English in Wales is considered as a first analysis. All subsequent studies that used data on the combination Welsh/English in Wales are considered as second analyses.

Discipline coverage

Scopus’ Subject area classifications were used to assign a scientific discipline to each journal in which studies were published (Scopus, 2020 ). These “classifications” include a total of 26 disciplines covering the whole of the health, life, physical and social sciences. Scopus may assign more than one discipline to each journal. We only consider the first of those disciplines, which is also the main one. Journals in which the studies analysed here were published cover eleven of Scopus’ 26 disciplines, namely: Agricultural and biological sciences; Arts and humanities; Biochemistry, genetics and molecular biology; Chemistry; Decision sciences; Economics, econometrics and finance; Engineering; Mathematics; Multidisciplinary; Physics and astronomy; and Social sciences. To facilitate analyses, these eleven categories were merged into six: Biology; Economics; Linguistics and other humanities; Multidisciplinary; Physics and Mathematics; and Social Sciences. The exact correspondences are presented in Table S3 of the supplementary materials.

Number of publications by year

According to our selection, the first mathematical model of language competition was proposed in 1990. It is however during the mid-2000s that this topic started to gain in popularity, mostly as a result of Abrams and Strogatz’ influential study published in 2003. Studies on models of language competition continue to be more or less regularly published to this day, with between zero and eight publications per year since 2005. Information on the number of publications per year can be derived from Figs. 2 and 3 presented below.

Colours indicate whether a study used data or not, and if so, whether it used data on a specific population for the first time (First analysis) or for a second time or more (Second analysis). Populations are defined according to the languages they speak and the region or country they inhabit.

Disciplines are based on Scopus’ Subject area classification. Disciplines were merged into a reduced number of categories to facilitate interpretation (see Table S3 of the supplementary materials).

Table 2 presents a comprehensive overview of how methods for modelling language competition evolved over the last thirty years. Studies are ordered chronologically. References in column 1 are colour-coded to represent the level of analysis. Column three shows the model employed, while column four shows the composition of the population according to its linguistic groups. Column five contains tiles which show the parameters used in each model. Information is also included about data usage, where darker tiles indicate the use of data, and about the journal of publication’s main discipline, in column two. These last two points will the focus of the next two subsections and will not be commented on further here.

Early models of language competition took a macroscopic approach, and this approach remained popular throughout the observation. Most of these models are based on a system of ordinary differential equations, though many studies published between 2005 and 2014 used models of the types Lokta-Volterra, reaction-diffusion or system dynamics. Models that took a microscopic approach started to appear from 2005 onwards. This approach continued to be regularly utilised over time, relying either on agent-based or conversation games models or on both. The mesoscopic approach, on the other hand, was mostly used between 2005 and 2007 in the form of agent-based models. These focused on large numbers of languages as they were mostly used to study the world’s linguistic diversity.

Turning to the parameters included in each model, a vast majority of models considered horizontal shift between speakers of different languages, while about half as many considered vertical shift. A few studies considered exogenous influences on language shift, while a few did not include any shift parameter but rather concentrated on the choice of different languages in different contexts using a conversation game framework. The choice of language shift parameters does not appear to have significantly varied over time or to systematically change according to the type of model used.

Regarding geodemographic parameters, the parameter “number of speakers” was included in an overwhelming number of studies. That is, most studies let shift between linguistic groups depend at least in part on the size of those groups: large groups thus tend to become larger and smaller groups tend to become smaller. Many studies additionally considered the births and deaths of speakers as a factor influencing the size of linguistic groups. Many studies considered the role of space, acknowledging the fact that speakers who are separated by a greater physical distance have less influence on each other than speakers who are closer to each other. Among the studies that considered space, more than half considered movement, a feature often modelled through means of reaction-diffusion models. Few studies considered the role of migration or borders. A small number of studies, all based on the mesoscopic approach, assumed that some speakers have a higher propensity to reproduce than others, i.e., they have higher fitness. Five studies distinguished different phases in the lives of speakers by including ageing in their models. The use of features such as borders, fitness and ageing went from rare to almost inexistent towards the end of the period of observation.

The most commonly included sociolinguistic parameter is prestige, followed closely by loyalty. These two parameters can be considered as two opposing forces influencing language shift: speakers shift to a new language due to its prestige but remain loyal to the already known one for convenience or cultural reasons. These two parameters are to be found in most models that took a macroscopic perspective, as well as in some microscopic models, but are absent from mesoscopic models. The parameter loyalty tends to appear more in models that were proposed from 2012 onwards. The parameter language change, on the other hand, appeared in many mesoscopic models published in the mid-2000s but was almost completely abandoned afterwards. In fact, language change was mainly operationalized in the form of random mutations affecting artificially created languages (e.g., bit-strings) in the context of mesoscopic models. Other sociolinguistic parameters have become more common over time. This is the case of the parameters network topology, homophily, environment, accommodation and competency. These were often included in microscopic models, or in models that took a system dynamics approach. To finish, the parameter similarity was sporadically used throughout the observation, in the context of ordinary differential equations or mesoscopic agent-based models.

Figure 2 displays for each year of interest counts of studies according to whether the models they presented were fitted to data, and if so, whether they used data on a specific population for the first time (first analysis) or for a second time or more (second analysis). A total of 22 studies fitted their models to empirical data. Of these, five did so before 2010, while seventeen did so in 2010 or later. Strikingly, eight of the nine studies published in 2019 and 2020 were fitted to empirical data.

Table 3 lists the languages and regions or countries for which data was used. Panel A lists the languages and the corresponding countries (or regions) that were covered in studies that modelled competition between two languages, while Panel B lists the languages and the corresponding countries (or regions) that were covered in studies that modelled competition between more than two languages. Studies modelled competition among fifteen pairs of languages, five combinations of three languages, and one combination of four languages. Twenty-seven countries or regions were covered. Different studies sometimes modelled competition between similar languages, either in a similar or a different country (or region). The same languages also sometimes appeared in different models in combination with different languages. Notably, eighteen studies considered competition involving the English language. Most studies considered language competition in economically developed countries, including European and North American countries, but also Hong Kong and Singapore.

Figure 3 breaks down the selected studies according to the disciplines covered by the journals in which they were published. Journals covering the fields of physics and mathematics have by far been the most popular outlets for studies on models of language competition, covering 36 publications out of a total of 62. Journals covering the field of biology and journals covering the field of social sciences come second, each with seven publications. Journals covering the field of economics, those covering the field of linguistics and other humanities, and multidisciplinary journals each published four studies on mathematical models of language competition. The dominance of journals covering the fields of physics and mathematics seems to be slowly diminishing: their share went from 75% before 2010 to 40% from that year onwards. Initially, publications in journals covering the field of biology tended to replace journals covering the fields of physics and mathematics. However, the picture is more diversified concerning the later period, with a higher number of publications in journals covering the fields of economics, linguistics and other humanities, and social sciences. We note that one particular journal was relied on heavily as an outlet for publishing models of language competition. Namely, Physica A: Statistical Mechanics and its applications published sixteen of the 62 publications reviewed here. In comparison, the second most common outlet is the International Journal of Modern Physics C , with five publications, while most other journals are represented once or twice (Table S3 , supplementary materials).

In which direction should mathematical models of language competition evolve so that they engage better with the intellectual framework used by linguists, and thus better inform the current efforts made towards preserving the world’s linguistic diversity? As seen above, mathematical models of language competition are appearing more and more in journals covering disciplines outside formal and natural sciences. This section aims at providing an answer to the question raised above by looking into how those studies published in the fields of humanities and social sciences differ from those published in the formal and natural sciences. Then, a discussion is provided as to whether models, in general, are becoming more similar over time to those recently developed in the fields of humanities and social sciences.

Comparison of models across disciplines

Results presented above suggest that models published in journals covering the fields of linguistics and other humanities exhibit properties that are different from models published in journals covering other disciplines. Table 4 shows how models differ between those two groups of disciplines which we, respectively, refer to as Humanities and social sciences and Formal and natural sciences . Those differences can be summarised the following way: compared to models published in journals covering formal and natural sciences, models published in journals covering humanities and social sciences rely more on a microscopic rather than a macroscopic approach, consider more often bilingualism as a separate state, privilege the use of sociolinguistic parameters over geodemographic ones, and are slightly more often based on data.

First, the more common use of the microscopic approach among linguists and social scientists might be explained by the fact this approach allows to model interactions between speakers more explicitly and accommodates a larger number of parameters. Some of the models of this class recently proposed by linguists and social scientists indeed include a large number of parameters, allowing for a high degree of precision (Civico, 2019 ; Karjus and Ehala, 2018 ). These models, furthermore enhanced by the use of data for calibration, allow to unravel the multiple pathways through which language shift can occur, and what can be done about it.

Second, the importance of bilingualism was recognised relatively early by linguists interested in questions relating to language competition (Minett and Wang, 2008 ). This recognition comes from observations made on the field stating that speakers rarely shift directly from one language to another, but instead go through a phase of bilingualism (Appel and Muysken, 2005 ). Increasingly, however, it seems that formal and natural scientists are recognising this fact as more of their models published recently are considering bilingualism (Heinsalu et al., 2014 ; Seoane and Mira, 2017 ).

Third, it speaks for itself that linguists and social scientists rely more on sociolinguistic parameters than formal and natural scientists do. Two parameters appear much more often in publications in journals covering humanities and social sciences: homophily and competence. The former refers to the greater likelihood that unions are formed among people who speak a similar language than among people who speak different languages. This is important since bilingual unions often lead to bilingual children and as noted above, bilingualism is an important vector of language shift (Appel and Muysken, 2005 ). The other parameter, competence, is probably equally important since the degree of fluency of speakers in one language will have a direct impact on the frequency of its usage, and the less a language is used, the more likely it is to become extinct. Other sociolinguistic parameters often encountered in models developed by linguists and social scientists include network typologies, which refers to the fact that not all speakers are equally likely to talk to each other due to, for example, shared interests and acquaintances, and loyalty, which we discuss below.

Fourth, one crucial aspect of the efforts made towards preserving the world’s linguistic diversity consists in making inventories of extant languages in terms of their number of speakers (Ethnologue, 2020 ). Language preservation is, in this sense, a highly applied and quantitative field. Though validation against empirical data is not a prerequisite for mathematical models of language competition to be insightful, it remains an important step that has often not been made, especially in studies published in journals covering the fields of physics and mathematics.

Is the discrepancy between sciences in the techniques used for modelling of language competition due to a lack of engagement among formal and natural scientists with the intellectual framework used by linguists? Part of the explanation for this discrepancy, which was previously raised by Fernando et al. ( 2010 ), could lie in the origins of mathematical models of language competition. Many of these models were first developed to answer questions relating to natural phenomena, for example about the transmission of infectious diseases among living species, the spread of particles in space, or predator-prey dynamics (Kandler and Unger, 2018 ; Prochazka and Vogl, 2018 ). It will not come as a surprise, therefore, that these models focus in the first place on population dynamics and interactions between individuals and their environment—thus on geodemographic parameters—rather than on sociolinguistics phenomena. Obviously, when applying their models to questions relating to language competition, formal and natural scientists included parameters that reflect sociolinguistic realities. For example, early models included, next to population size, the notion of prestige as an important factor influencing shift from one language to another (e.g., Abrams and Strogatz, 2003 ). However, these models often failed to also consider the fact that speakers sometimes resist to shifting to a more prestigious language due to loyalty towards their heritage language, an important factor in efforts made towards language preservation (Thomason, 2015 ). Other early models considered language similarity (e.g., Mira and Paredes, 2005 ) or language change (e.g., Stauffer and Schulze, 2005 ). However, these concepts were often implemented based more on mathematical convenience than on phonological or syntactical properties of the languages involved, making them highly vulnerable to criticism from the linguistic perspective.

Comparison of models over time

The first mathematical model of language competition was developed thirty years ago, but until ten years ago, almost all models were developed by formal and natural scientists. The last ten years have seen more models developed by linguists and social scientists, but these continue to form only a minority of all models. Taken as a whole, have models evolved to resemble more to those recently published by linguists and social scientists?

Results presented above suggested change over time in a few characteristics inherent to models of language competition. Table 5 quantifies this change concerning the year 2010, allowing to divide models into two roughly equal numbers. The characteristics considered are the same as in Table 3 , which was presented in the previous subsection. Change between the two periods is rather minor concerning the approach taken and the consideration of bilingualism as a separate state. However, there are clear increases in the use of sociolinguistic parameters and on the reliance on data. Part of this increase could be explained by the fact that after 2010, linguists and social scientists started to more regularly develop their own models of language competition. As we saw above, models developed by linguists and social scientists tend to differ considerably in the parameters they use and their use of data. And indeed, as shown in Table 4 , the number of publications in journals covering the fields of humanities and social sciences doubled between the period pre-2010 and the period post-2010. Linguists have been increasingly relying on methods developed in the formal and natural sciences over the last years (Bromham, 2017 ) and this trend seems to apply to the field of language competition as well. By doing so, linguists and social scientists may have contributed to changing practices in the field of mathematical modelling of language competition. However, analyses shown in Table S4 of the Supplementary materials suggest that practices have also started to change among formal and natural sciences, as we notice there an increase in the mean number of sociolinguistic parameters over time. We could thus be witnessing the emergence of a common language between fields concerning mathematical models of language competition.

Conclusion: a new wave of research on mathematical models of language competition?

Analysing studies published in the last thirty years on models of language competition, two facets can be distinguished. First, mathematical models of language competition, which were initially developed in the formal and natural sciences, have established themselves as a powerful tool for understanding language competition and language death, but have been slow in reaching the linguistic community. This can be seen in the fact that most models are published in journals covering the fields of physics and mathematics, but also in the fact that those models often lack a complete or accurate representation of the key processes affecting language competition, and in the fact that they are rarely validated against data. These assertions, however, seem to represent the reality of models of language competition less and less well. Since the last ten years, the way language competition is modelled has changed considerably. Models published recently typically consider a greater number of sociolinguistic parameters and are more often validated against data than before. This change is in part because more linguists and social scientists have started to show interest for models of language competition, harnessing their unique set of expertise, but also in part to the fact that formal and natural scientists themselves have started to adopt the concepts and terminology used by linguists. In conclusion, though a gap can still be noticed between disciplines concerning mathematical models of language competition, important steps have been made in recent years towards reducing it. A new wave of research on mathematical models of language competition is forming and is set to prove instrumental in informing efforts made towards preserving the world’s linguistic diversity.

Data availability

The list of all selected articles is provided in Tables S2 and S3 of the supplementary materials .

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Boissonneault, M., Vogt, P. A systematic and interdisciplinary review of mathematical models of language competition. Humanit Soc Sci Commun 8 , 21 (2021). https://doi.org/10.1057/s41599-020-00683-9

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Predictive Mathematical Models of the COVID-19 Pandemic : Underlying Principles and Value of Projections

• 1 Department of Medical Statistics, London School of Hygiene & Tropical Medicine, London, United Kingdom
• 2 Division of Epidemiology & Biostatistics, School of Public Health, University of California, Berkeley
• 3 MRC Centre for Global Infectious Disease Analysis, Abdul Latif Jameel Institute for Disease and Emergency Analytics, and Department of Infectious Disease Epidemiology, Imperial College, London, United Kingdom

Numerous mathematical models are being produced to forecast the future of coronavirus disease 2019 (COVID-19) epidemics in the US and worldwide. These predictions have far-reaching consequences regarding how quickly and how strongly governments move to curb an epidemic. However, the primary and most effective use of epidemiological models is to estimate the relative effect of various interventions in reducing disease burden rather than to produce precise quantitative predictions about extent or duration of disease burdens. For predictions, “models are not crystal balls,” as Ferguson noted in a recent overview of the role of modeling. 1

Nevertheless, consumers of epidemiological models, including politicians, the public, and the media, often focus on the quantitative predictions of infections and mortality estimates. Such measures of potential disease burden are necessary for planners who consider future outcomes in light of health care capacity. How then should such estimates be assessed?

Although relative effects on infections associated with various interventions are likely more reliable, accompanying estimates from models about COVID-19 can contribute to uncertainty and anxiety. For instance, will the US have tens of thousands or possibly even hundreds of thousands of deaths? The main focus should be on the kinds of interventions that could help reduce these numbers because the interventions undertaken will, of course, determine the eventual numerical reality. Model projections are needed to forecast future health care demand, including how many intensive care unit beds will be needed, where and when shortages of ventilators will most likely occur, and the number of health care workers required to respond effectively. Short-term projections can be crucial to assist planning, but it is usually unnecessary to focus on long-term “guesses” for such purposes. In addition, forecasts from computational models are being used to establish local, state, and national policy. When is the peak of cases expected? If social distancing is effective and the number of new cases that require hospitalization is stable or declining, when is it time to consider a return to work or school? Can large gatherings once again be safe? For these purposes, models likely only give insight into the scale of what is ahead and cannot predict the exact trajectory of the epidemic weeks or months in advance. According to Whitty, models should not be presented as scientific truth; they are most helpful when they present more than what is predictable by common sense. 2

Estimates that emerge from modeling studies are only as good as the validity of the epidemiological or statistical model used; the extent and accuracy of the assumptions made; and, perhaps most importantly, the quality of the data to which models are calibrated. Early in an epidemic, the quality of data on infections, deaths, tests, and other factors often are limited by underdetection or inconsistent detection of cases, reporting delays, and poor documentation, all of which affect the quality of any model output. Simpler models may provide less valid forecasts because they cannot capture complex and unobserved human mixing patterns and other time-varying characteristics of infectious disease spread. On the other hand, as Kucharski noted, “complex models may be no more reliable than simple ones if they miss key aspects of the biology. Complex models can create the illusion of realism, and make it harder to spot crucial omissions.” 3 A greater level of detail in a model may provide a more adequate description of an epidemic, but outputs are sensitive to changes in parametric assumptions and are particularly dependent on external preliminary estimates of disease and transmission characteristics, such as the length of the incubation and infectious periods.

In predicting the future of the COVID-19 pandemic, many key assumptions have been based on limited data. Models may capture aspects of epidemics effectively while neglecting to account for other factors, such as the accuracy of diagnostic tests; whether immunity will wane quickly; if reinfection could occur; or population characteristics, such as age distribution, percentage of older adults with comorbidities, and risk factors (eg, smoking, exposure to air pollution). Some critical variables, including the reproductive number (the average number of new infections associated with 1 infected person) and social distancing effects, can also change over time. However, many reports of models do not clearly report key assumptions that have been included or the sensitivity to errors in these assumptions.

Predictive models for large countries, such as the US, are even more problematic because they aggregate heterogeneous subepidemics in local areas. Individual characteristics, such as age and comorbidities, influence risk of serious disease from COVID-19, but population distributions of these factors vary widely in the US. For example, the population of Colorado is characterized by a lower percentage of comorbidities than many southern states. The population in Florida is older than the population in Utah. Even within a state, key variables can vary substantially, such as the prevalence of important prognostic factors (eg, cardiovascular or pulmonary disease) or environmental factors (eg, population density, outdoor air pollution). Social distancing is more difficult to achieve in urban than in suburban or rural areas. In addition, variation in the accuracy of disease incidence and prevalence estimates may occur because of differences in testing between areas. Consequently, projections from various models have resulted in a wide range of possible outcomes. For instance, an early estimate suggested that COVID-19 could account for 480 000 deaths in the US, 4 whereas later models quoted by the White House Coronavirus Task Force indicated between 100 000 and 240 000 deaths, and more recent forecasts (as of April 12) suggest between 60 000 and 80 000 deaths.

A recent model from the Institute of Health Metrics and Evaluation has received considerable attention and has been widely quoted by government officials. 5 On the surface, the model yields specific predictions of the day on which COVID-19 deaths will peak in each state and the cumulative number of deaths expected over the next 4 months (with substantial uncertainty intervals). However, caveats in these projections may not be widely appreciated by the public or policy makers because the model has some important but opaque limitations. For instance, the predictions assumed similar effects from social distancing as were observed elsewhere in the world (particularly in Hubei, China), which is likely optimistic. The projected fatality model was not based on any epidemiological science and depended on current data on the reported prior increasing number of fatalities in each region—data that are widely acknowledged to be undercounted and poorly reported 6 —and did not consider the possibility of any second wave of infections. Although the Institute of Health Metrics and Evaluation is continuously updating projections as more data become available and they adapt their methods, 7 long-term mortality projections already have shown substantial volatility; in New York, the model predicted a total of 10 243 COVID-19 deaths on March 27, 2020, but the projected number of deaths had increased to 16 262 by April 4, 2020—a 60% increase in a matter of days. Some original projections were quickly at the edge of earlier uncertainty bands that were apparently not sufficiently wide.

Models can be useful tools but should not be overinterpreted, particularly for long-term projections or subtle characteristics, such as the exact date of a peak number of infections. First, models need to be dynamic and not fixed to allow for important and unanticipated effects, which makes them only useful in the short term if accurate predictions are needed. To paraphrase Fauci: models do not determine the timeline, the virus makes the timeline.

Second, necessary assumptions should be clearly articulated and the sensitivity to these assumptions must be discussed. Other factors that are already known or thought to be associated with the pandemic, but not included in the model, should be delineated together with their qualitative implications for model performance. Third, rather than providing fixed, precise numbers, all forecasts from these models should be transparent by reporting ranges (such as CIs or uncertainty intervals) so that the variability and uncertainty of the predictions is clear. It is crucial that such intervals account for all potential sources of uncertainty, including data reporting errors and variation and effects of model misspecification, to the extent possible. Fourth, models should incorporate measures of their accuracy as additional or better data becomes available. If the projection from a model differs from other published predictions, it is important to resolve such differences. Fifth, the public reporting of estimates from these models, in scientific journals and especially in the media, must be appropriately circumspect and include key caveats to avoid the misinterpretation that these forecasts represent scientific truth.

Models should also seek to use the best possible data for local predictions. It is unlikely that epidemics will follow identical paths in all regions of the world, even when important factors such as age distribution are considered. Local data should be used as soon as those data become available with reasonable accuracy. For projections of hospital needs, data on clinical outcomes among patients in local settings are likely to enable more accurate conclusions than poorly reported mortality data from across the world.

At a time when numbers of cases and deaths from COVID-19 continue to increase with alarming speed, accurate forecasts from mathematical models are increasingly important for physicians; epidemiologists; politicians; the public; and, most importantly, for individuals responsible for organizing care for the populations they serve. Given the unpredictable behavior of severe acute respiratory syndrome coronavirus 2, it is best to acknowledge that short-term projections are the most that can be expected with reasonable accuracy. Always assuming the worst-case scenario at state and national levels will lead to inefficiencies and competition for beds and supplies and may compromise effective delivery and quality of care, while assuming the best-case scenario can lead to disastrous underpreparation.

Modeling studies have contributed vital insights into the COVID-19 pandemic, and will undoubtedly continue to do so. Early models pointed to areas in which infection was likely widespread before large numbers of cases were detected; contributed to estimating the reproductive number, case fatality rate, and how long the virus had been circulating in a community; and helped to establish evidence that a significant amount of transmission occurs prior to symptom onset. Mathematical models can be profoundly helpful tools to make public health decisions and ensure optimal use of resources to reduce the morbidity and mortality associated with the COVID-19 pandemic, but only if they are rigorously evaluated and valid and their projections are robust and reliable.

Corresponding Author: Nicholas P. Jewell, PhD, Department of Medical Statistics, London School of Hygiene & Tropical Medicine, London, United Kingdom ( [email protected] ).

Published Online: April 16, 2020. doi:10.1001/jama.2020.6585

Conflict of Interest Disclosures: Drs Jewell, Lewnard, and Jewell reported currently having a paid contract with Kaiser Permanente to advise them regarding hospital demand associated with coronavirus disease 2019 cases. No other disclosures were reported.

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Jewell NP , Lewnard JA , Jewell BL. Predictive Mathematical Models of the COVID-19 Pandemic : Underlying Principles and Value of Projections . JAMA. 2020;323(19):1893–1894. doi:10.1001/jama.2020.6585

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• 1 Department of Statistics, People's Hospital of Ningxia Hui Autonomous Region.
• 2 Department of Applied Mathematics, School of Mathematics and Statistics, Xi'an Jiaotong University.
• 3 Department of Epidemiology and Biostatistics, School of Public Health, Xi'an Jiaotong University Health Science Center.
• PMID: 29276213
• PMCID: PMC5911672
• DOI: 10.2188/jea.JE20160203

A mathematical model of the transmission dynamics of infectious disease is an important theoretical epidemiology method, which has been used to simulate the prevalence of hepatitis B and evaluate different immunization strategies. However, differences lie in the mathematical processes of modeling HBV transmission in published studies, not only in the model structure, but also in the estimation of certain parameters. This review reveals that the dynamics model of HBV transmission only simulates the spread of HBV in the population from the macroscopic point of view and highlights several main shortcomings in the model structure and parameter estimation. First, age-dependence is the most important characteristic in the transmission of HBV, but an age-structure model and related age-dependent parameters were not adopted in some of the compartmental models describing HBV transmission. In addition, the numerical estimation of the force of HBV infection did not give sufficient weight to the age and time factors and is not suitable using the incidence data. Lastly, the current mathematical models did not well reflect the details of the factors of HBV transmission, such as migration from high or intermediate HBV endemic areas to low endemic areas and the kind of HBV genotype. All of these shortcomings may lead to unreliable results. When the mathematical model closely reflects the fact of hepatitis B spread, the results of the model fit will provide valuable information for controlling the transmission of hepatitis B.

Keywords: compartmental model; hepatitis B; transmission dynamic.

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Figure 1.. Flowchart of the search process.

Figure 2.. The epidemiological compartments and definitions.…

Figure 2.. The epidemiological compartments and definitions. Susceptible (S), at risk of infection with HBV;…

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Learning mathematical modelling with digital tools: A systematic literature review

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Muhammad Ghozian Kafi Ahsan , Adi Nur Cahyono , Iqbal Kharisudin; Learning mathematical modelling with digital tools: A systematic literature review. AIP Conf. Proc. 16 June 2023; 2614 (1): 040082. https://doi.org/10.1063/5.0126587

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Mathematical modelling is an approach to solving mathematical problems in the real world by using mathematical models to find solutions. Mathematical modelling activity assisted by digital tools makes modelling proccess and activity run smoothly. The aim is to know how digital tools supported mathematical modelling learning. This research use systematic literature review method. The main topic of the discussion is how digital tools prompt mathematical modelling proccess, how digital tools play role for mathematical modelling learning, and what student achieve during and after mathematical modelling learning. This research get several digital tools types, such as Geogebra, MathCityMap, Gizmos, etc. This software provide features to prompt mathematical modelling proccess and learning. Student achievement in mathematics is also affected during use several digital tools for modelling activity. So, its important to know how to teach mathematical modelling using digital tools.

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A systematic literature review of measurement of mathematical modeling in mathematics education context

Eurasia Journal of Mathematics, Science and Technology Education

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Back to Journals » Journal of Multidisciplinary Healthcare » Volume 17

A Systematic Literature Review of Mathematical Models for Coinfections: Tuberculosis, Malaria, and HIV/AIDS

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Authors Inayaturohmat F , Anggriani N , Supriatna AK   , Biswas MHA  

Received 31 October 2023

Accepted for publication 19 February 2024

Published 13 March 2024 Volume 2024:17 Pages 1091—1109

DOI https://doi.org/10.2147/JMDH.S446508

Checked for plagiarism Yes

Review by Single anonymous peer review

Peer reviewer comments 2

Editor who approved publication: Dr Scott Fraser

Fatuh Inayaturohmat, 1 Nursanti Anggriani, 2 Asep K Supriatna, 2 Md Haider Ali Biswas 3 1 Doctoral in Mathematics Study Programme, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang, Indonesia; 2 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Sumedang, Indonesia; 3 Mathematics Discipline, Science, Engineering and Technology School, Khulna University, Khulna 9208, Bangladesh Correspondence: Nursanti Anggriani, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjadjaran, Jalan Raya Bandung Sumedang KM 21 Jatinangor, Kab, Sumedang, 45363, Indonesia, Email [email protected] Abstract: Tuberculosis, malaria, and HIV are among the most lethal diseases, with AIDS (Acquired Immune Deficiency Syndrome) being a chronic and potentially life-threatening condition caused by the human immunodeficiency virus (HIV). Individually, each of these infections presents a significant health challenge. However, when tuberculosis, malaria, and HIV co-occur, the symptoms can worsen, leading to an increased mortality risk. Mathematical models have been created to study coinfections involving tuberculosis, malaria, and HIV. This systematic literature review explores the importance of coinfection models by examining articles from reputable databases such as Dimensions, ScienceDirect, Scopus, and PubMed. The primary emphasis is on investigating coinfection models related to tuberculosis, malaria, and HIV. The findings demonstrate that each article thoroughly covers various aspects, including model development, mathematical analysis, sensitivity analysis, optimal control strategies, and research discoveries. Based on our comprehensive evaluation, we offer valuable recommendations for future research efforts in this field. Keywords: tuberculosis, malaria, HIV, AIDS, coinfection, model, systematics literature review

Introduction

Human immunodeficiency virus (HIV) is responsible for causing acquired immunodeficiency syndrome (AIDS). 1 The transmission of HIV occurs when infected individuals come into contact with certain bodily fluids, such as blood, semen, breast milk, and vaginal secretions. 2 Even though a cure for HIV is not available, the virus can be effectively managed through the use of antiretroviral (ARV) drugs, to control its progression. 3 HIV epidemic significantly influences the prevalence of tuberculosis infections and 4 the virus compromises the immune system, making individuals who are infected highly susceptible to different infectious diseases. 5

Tuberculosis (TB) is transmitted through the air when infected individuals speak, sneeze, or cough, making it an airborne disease. It is caused by Mycobacterium tuberculosis , a bacterium that primarily targets the lungs but can also affect other. 6 After successful treatment, individuals who have been declared cured can still be susceptible to reinfection. Failure to consistently take the prescribed medication for the designated duration can lead to the development of Multi-Drug Resistant Tuberculosis (MDR-TB), where the bacteria become resistant to drugs. 7 Even individuals who are subjected to treatment can still transmit tuberculosis when the virus remains active in their bodies. 8 According to the 2013 global report by the World Health Organization (WHO), approximately one-third of the population is affected by the infection. The 2020 global report stated that around 9.96 million individuals were estimated to have contracted tuberculosis in 2020. 9 Furthermore, between 2000 and 2019, the diagnosis and treatment efforts saved the lives of approximately 63 million people. 10 Among individuals coinfected with HIV and tuberculosis, tuberculosis is the primary cause of mortality. 11 Globally, approximately 10.6 million individuals contracted tuberculosis in 2022, marking an increase from the estimated 10.3 million cases in 2021. There is a possibility of a resurgence in the declining trend observed before the pandemic, anticipated to take place in either 2023 or 2024. 12

Malaria is caused by a parasite called Plasmodium . 13 Common signs of malaria consist of fever, muscle pain, fatigue, and chills. In severe instances, the illness can result in fatalities. The disease is a highly fatal contagious illness solely transmitted by Anopheles mosquitoes. 12 The transmission occurs when female mosquitoes infected with the Plasmodium parasite bite humans, passing on the infection. 14 Female mosquitoes require blood meals to produce eggs. 15 According to World Health Organization (WHO), the count of malaria-endemic countries reporting fewer than 10,000 cases has risen from 26 nations in 2000 to 47 in 2020. Moreover, the number of countries with fewer than 100 native cases has increased from 6 to 26. 16 On a worldwide scale in 2022, an approximate 249 million cases of malaria were reported across 85 countries where malaria is prevalent. The incidence of malaria cases in 2022 stood at 58 per 1000 population at risk. 17 Global initiatives are in progress to eliminate malaria, which involves the development of novel vaccines and the implementation of insecticides to prevent mosquito bites.

Based on this data, the imperative to take action becomes evident to mitigate the transmission risk and curb the dissemination of tuberculosis, malaria, and HIV. 18 As we progress towards controlling and potentially eliminating malaria, HIV, and TB, mathematical models offer a robust framework for assessing the potential impact of interventions. They help identify areas requiring additional empirical research, prioritize crucial policy and research questions, and, most importantly, necessitate improved communication between modelers and those involved in experiments or fieldwork. This collaboration is essential to fine-tune questions, pinpoint critical data, and ensure that analytical work contributes to enhanced policies and effectively controls all three infections. 19 This model divides the population into different compartments based on certain assumptions and characteristics. 20 Model can also serve to depict the occurrence of epidemic events involving interactions between two diseases. Using this model enables the estimation of individuals who are infected. 21 , 22 Subsequently, through the application of optimal controls, effective recommendations can be identified to manage and curb the transmission of the disease.

In recent years, progress has been made in the development of coinfection model for tuberculosis, malaria, and HIV. This Systematic Literature Review is dedicated to compiling existing model related to coinfection of these diseases and evaluating the extent of research and its corresponding outcomes. Ultimately, the objective is to provide valuable recommendations for future investigations in this field.

Materials and Methods

The method employed for articles to be included in this systematic literature review is the Preferred Reporting Item for Systematic Review and Meta-Analyses (PRISMA). 23 PRISMA involves three key stages: identification, screening, and eligibility. In the identification stage, articles pertinent to the research are searched in different databases using specific keywords. During, at the screening stage, all identified articles from multiple databases are consolidated and duplicated entries are removed. The remaining articles are subjected to relevance assessment base on title and abstracts. At this stage, the relevance of the title and abstract in question is adjusted to the keywords used to search for these articles in the previous stage. The article goes to the next stage if the title or abstract contains a combination of these keywords. However, if there is no combination of these keywords, then the article is categorized as irrelevant at this stage. Additionally, the accessibility is checked, and any articles that do not meet the criteria are excluded. Following the screening stage, the articles that pass the initial assessment proceed to the eligibility phase, where a comprehensive evaluation is conducted to ascertain their relevance. At this stage, the relevance of the article in question is adjusted to the research topic of the article to be reviewed, namely mathematical models of tuberculosis, malaria, and HIV/AIDS coinfection in the form of ordinary differential equations. The article goes to the next stage if it is appropriate to the topic. However, if it is inappropriate, the article is categorized as irrelevant at this stage. By following these stages, the selected articles serve as the research material for this systematic literature review, facilitating a thorough exploration of coinfection model for tuberculosis, malaria, and HIV.

This systematic literature review encompasses articles not only focusing on coinfection models involving all three diseases but also combinations of coinfections arising from pairs of distinct diseases. The infectious diseases under investigation in this article include tuberculosis, malaria, and HIV/AIDS. Hence, PRISMA was conducted four times with different keywords. Furthermore, the articles were searched on four databases, namely Dimensions, Scopus, PubMed, and Science Direct. The first PRISMA used keywords: (“Mathematical Model” OR “Mathematical Modelling” OR “Compartmental Model” OR “Transmission Model”) AND (“Coinfection” OR “Co-infection”) AND (“Tuberculosis” OR “TB”) AND (“AIDS” OR “HIV”). The second PRISMA used keywords: (“Mathematical Model” OR “Mathematical Modelling” OR “Compartmental Model” OR “Transmission Model”) AND (“Coinfection” OR “Co-infection”) AND (“Tuberculosis” OR “TB”) AND “Malaria”. Meanwhile, the third used keywords: (“Mathematical Model” OR “Mathematical Modelling” OR “Compartmental Model” OR “Transmission Model”) AND (“Coinfection” OR “Co-infection”) AND “Malaria” AND (“AIDS” OR “HIV”). The last PRISMA used keywords: (“Mathematical Model” OR “Mathematical Modelling” OR “Compartmental Model” OR “Transmission Model”) AND (“Coinfection” OR “Co-infection”) AND (“Tuberculosis” OR “TB”) AND “Malaria” AND (“AIDS” OR “HIV”).

• Research articles published in English language.
• Research articles published between 1986 and 2024.
• Research articles from Dimensions, ScienceDirect, Scopus, and PubMed databases.
• The topic of the research article is the mathematical model of tuberculosis, malaria, and HIV/AIDS coinfection in the form of ordinary differential equations.

Duplication selection employed JabRef, while research topic mapping used VOSviewer. JabRef and VOSviewer were opensource software applications accessible to all users. JabRef enabled users to store their data in a simple text-based file format without being tied to any specific vendor. In contrast, VOSviewer was a software tool designed for creating and visualizing bibliometric networks.

Results and Discussion

• 1. Which disease coinfection model does the article include?
• 2. Is the coinfection model mathematically analyzed?
• 3. What sensitivity analysis method is used in the article?
• 4. What controls are used in optimal control of coinfection model in the article?
• 5. How many compartments does the coinfection model consist of, and what are the compartments identified?
• 6. What are the research results obtained in the article?

Mathematical analysis can encompass various aspects of coinfection models, such as examining their fundamental properties, such as positivity, uniqueness, and invariant regions. This can also involve conducting local and global stability analyses. Additional analyses such as bifurcation may be incorporated. If the article does not discuss one of these analyses, then the article is said to be “No” on mathematical analysis, although all articles explain the formulation of mathematical models.

In models of tuberculosis, malaria, and HIV/AIDS coinfection, interventions in the form of various forms of control are often used to reduce the spread of these diseases. Mathematically, this control is used to create an objective function to minimize the spread of the disease. The solution can be found using optimal control theory. Optimal control theory is a subset of control theory focused on determining a control strategy for a dynamic system throughout a specific time frame, aiming to optimize an objective function. These articles were reviewed to find out what type of controls were used in each article.

Optimal control plays a crucial role in determining the most appropriate and efficient interventions for model. Several articles employ these methods, incorporating different types of prevention and intervention controls. 28 , 31 , 34 , 38 , 39 , 45 , 49 , 58 Certain articles determine the best strategy to reduce infection by altering parameter values. 26 , 33 , 40 However, some articles did not incorporate any control measures in their analysis.

Table 2 illustrates that model with the 36 lowest and 37 highest number of compartments consist of 4 and 21 compartments, respectively. Despite the complexity of their model, the authors conducted mathematical and sensitivity analyses, and utilized optimal control to obtain results.

The presented articles have yielded recommendations for mitigating the transmission of tuberculosis, malaria, and HIV. 24 The integration of Long-Lasting Insecticide-Treated Nets (LLITN), malaria treatment, tuberculosis treatment, Indoor Residual Spraying (IRS), and tuberculosis prevention proves to be effective in reducing the spread of tuberculosis, malaria, as well as their coinfection.

The results obtained for the malaria-HIV coinfection model differ. According to 30 article, the most effective method to diminish malaria-HIV coinfection involves the combination of malaria prevention measures and antiretroviral (ARV) treatment. However, 31 article obtained the result that treating malaria and HIV individually proved to be more effective in reducing infection compared to administering combined treatment. 28 The escalation of HIV/AIDS prevalence due to coinfection with malaria, highlights the significance of treatment in mitigating this interplay, particularly for individuals already affected by AIDS. 26 The mortality rate rises with coinfection and doubles when the infectivity escalates by 30%. 27 The most cost-effective control to inhibit the spread of HIV-Malaria coinfection is prevention. Furthermore, 29 significant reductions or potential eradication of HIV prevalence can be achieved by ensuring high bed-net coverage, a high rate of malaria treatment to effectively minimize the incidence of malaria-HIV coinfection.

The outcomes derived from coinfection model of tuberculosis and HIV exhibit variations. 40 The infections exert a significant impact on the population due to the presence of a hidden population with tuberculosis infection. 33 Tuberculosis and HIV are intricately interconnected with each other, while AIDS is influential on tuberculosis infection. Likewise, 34 the presence of tuberculosis infection can expedite the progression of HIV, potentially leading to more rapid development of AIDS.

In terms of optimal control, 36 article implemented effective isolation measures to restrict the contact and transmission of infections within the population, to eliminate tuberculosis-HIV coinfection. Furthermore, 37 the integration of case findings and prevention treatment failure of tuberculosis proved to be effective in reducing the spread of tuberculosis and HIV. 38 Screening plays a crucial role in managing and containing the transmission of HIV and tuberculosis. 39 Individuals with weakened immune systems are more vulnerable to contracting HIV and tuberculosis. The overall burden arising from coinfection can be minimized by employing well-selected strengths and initiating antiretroviral therapy (ARV). 43 Additionally, maintaining a higher early treatment rate compared to the late treatment rate throughout the entire treatment program is essential. 45 Early detection of HIV and tuberculosis cases and the prompt initiation of treatment can effectively decrease the rate of infection, slow down the progression of HIV infected individuals toward AIDS, and reduce the occurrence of coinfection. 46

The combination of prevention and treatment gives good results from economic and epidemiological perspectives. In addition, 49 vaccination leads to rapid recovery in individuals. 48 Optimal detection and integrated therapy, administered at the appropriate time, also yield superior clinical outcomes. 50 The optimal result can be obtained by combining all the detection or one of tuberculosis or HIV only for a longer period.

To decrease the prevalence of infection and fatalities caused by the disease, it is necessary to implement all control measures collectively and at an optimal level. However, 51 this method carries the potential risk of inducing immune reconstitution inflammatory syndrome (IRIS) in infected individuals. 52 Enhancing awareness through education results in a decline in the cumulative occurrence of new cases of coinfection within a population. 53 The most effective method to minimize infection, maximize the rate of recovery, and control the disease progression is through a combination of vaccination and treatment.

Based on the method employed to search for articles, only one addresses coinfection model involving tuberculosis, malaria, and HIV. However, 58 this model refrains from employing vectors as compartments, and the management of malaria and tuberculosis can potentially decelerate the progression of HIV.

Some coinfection models consider the effects of prevention and intervention in various forms. Prevention is in the form of Net insecticides and mass spraying to reduce mosquito populations. There is also in the form of education and use of condoms or in any form without specifying the prevention. Meanwhile, intervention can take the form of treatment for malaria and tuberculosis or various therapies for HIV/AIDS. Based on these articles, prevention and intervention significantly influence the dynamics of disease spread. Prevention and intervention in various forms can be used as optimal control in reducing the spread of disease in a population.

Mathematical models of coinfection between tuberculosis, malaria, and HIV/AIDS can be used to help in deciding intervention policies that must be implemented to suppress the spread of these diseases. 11 One of the important mathematical models in the spread of infectious diseases is the use of country-specific dynamic models to estimate the incidence and mortality of TB in that period 2020–2022. Estimates for 2020–2022 were generated utilizing a dynamic model tailored to each country, considering the impact of disruptions to tuberculosis diagnosis and treatment caused by the Covid-19 pandemic. Determining the burden of TB during the Covid-19 pandemic and its aftermath poses challenges, and the current approach involves the use of country- and region-specific dynamic models, particularly for many low- and middle-income countries. These models also used by WHO. 52

Article use data relevant to Kogi state of Nigeria to study the coinfection of tuberculosis and HIV. Analytical results supported by numerical simulation prove that educational awareness campaigns and treatment can reduce the burden of tuberculosis, HIV, and the coinfection of tuberculosis and HIV. 40 Article use population and health statistic from the Ministry of Interior, the Ministry of Public Health, Thailand and the World Health Organization to estimate the parameter value of the model. The extended duration of latent TB infection means that newly infected cases do not exhibit clinical symptoms immediately. Consequently, these cases remain unnoticed for a considerable period. To account for this delay, the development of a time-delay differential equation model becomes essential. 26

Global parameter estimates were derived from data collected in sub-Saharan Africa. The complete biological interactions between the malaria parasite and HIV are not yet fully understood. However, it is plausible that coinfection might result in a magnitude increase in the parasite or viral load. Future research endeavors should involve parameter fitting to data. Exploring coinfection at a cellular level would be needed as well.

In conclusion, a comprehensive search was conducted for articles using four different databases and specific sets of keywords. The selection process involved removing duplicated articles, assessing titles and abstracts, checking accessibility, and evaluating the relevance of entire article. Furthermore, the search yielded a total of 761 articles with the specified keywords. After completing the selection process, 35 articles were identified for analysis in systematic literature review. The articles covered different aspects, including model construction, mathematical analysis, sensitivity analysis, optimal control, and research findings. Valuable research and findings were also presented, contributing to the field of tuberculosis, malaria, and HIV coinfection model. Infectious diseases are a global health problem, especially deadly diseases such as tuberculosis, malaria and HIV/AIDS. This can become a more crucial health problem when coinfection occurs between these diseases. Mathematical models of tuberculosis, malaria, and HIV/AIDS coinfection can find the most effective intervention policies in reducing the spread of disease. Several studies on tuberculosis, malaria, and HIV/AIDS coinfections have been carried out in several countries using mathematical models and parameter estimation using data resulting from collaboration with local departments. For example, in Nigeria, Thailand, and Africa. This research produced several optimal solutions to reduce the spread of disease, including educational awareness campaign and treatment. Mathematical models of disease spread can produce recommendations for the government to create policies to reduce the spread of disease. However, there was still potential for further development of coinfection model by considering factors such as lifestyle, hospitalization, traditional treatment, and bacterial or virus evolution. Future research can also carry out parameter fitting estimation methods or use time delay models. Even the research object can be modified, namely from distribution at the human level to distribution at the cellular level.

Acknowledgments

This research was supported and funded by Universitas Padjadjaran Research Grant within the Padjadjaran Postgraduate Excellent Scholarship with contract number 1549/UN6.3.1/PT.00/2023. The author would also like to thank the Directorate of Research and Community Service (DRPM) of Universitas Padjadjaran for the Article Processing Charge (APC) support.

The authors declare no conflicts of interest in this work.

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Table of Contents

• - Background
• - Objectives
• - Conclusion

Introduction

Results and discussion.

• - Mathematical modeling of infectious diseases
• - Deterministic mathematical modeling
• - Stochastic mathematical modeling
• - Deterministic versus stochastic
• - The basic reproductive number

Conflicts of Interest/Competing Interests

Ethics approval, consent to participate, consent for publication, availability of data and material, code availability.

Costa ALP, Pires MA, Resque RL, Almeida SSMS (2021) Mathematical Modeling of the Infectious Diseases: Key Concepts and Applications. J Infect Dis Epidemiol 7:209. doi.org/10.23937/2474-3658/1510209

Review Article | OPEN ACCESS DOI: 10.23937/2474-3658/1510209

Mathematical modeling of the infectious diseases: key concepts and applications, anderson luiz pena da costa 1* , marcelo amanajas pires 2 , rafael lima resque 3 and sheylla susan moreira da silva de almeida 3.

1 Macapa Institute of High Education - IMMES, College of Pharmacy, Brazil

2 Brazilian Center for Research in Physics (CBPF), Brazil

3 Post-Graduation Program in Pharmaceutical Sciences (PPGCF), Federal University of Amapa - UNIFAP, Brazil

The transmission dynamics of infectious diseases is susceptible to changes governed by several factors, whose recognition is critical for the rational development of strategies for prevention and control, as well as for developing health policies. In this context, mathematical modeling can provide useful insights concerning transmission patterns and detection of parameters to mitigate disease in the population.

To didactically present the mathematical modeling of infectious diseases for health students and professionals as a tool in epidemiology.

A comprehensive literature review was conducted with articles obtained from PubMed, Web of Science, and Google Scholar databases with the term infectious diseases mathematical modeling.

There are two main types of models built with a basis on fixed or probabilistic rates that describe individuals' movement in compartments that designate stages in the natural history of the disease. In this sense, deterministic models are non-probabilistic and stochastic models are probabilistic, the first one helps in developing a prospection of possible scenarios in epidemiology, while the second is more applicable in the study of the influence of variables in the transmission dynamics.

The infectious agents are in a constant process of biological evolution, as well as the environment and human conceptions, culture, and behavior, implying a constant transformation in the epidemiological profile of infectious diseases, in which the mathematical modeling can provide support to the decision-making processes concerning epidemiology and public health.

Infectious diseases, Mathematical modeling, Epidemiology, Public health

Infectious diseases are the result of a disharmonious ecological interaction between a microbial infectious agent (bacteria, fungi, parasites, or viruses, except for prions that are infectious proteins) and a host, where the dynamics in this interaction is subjected to the modulatory influence of several factors, such as the environment, the biological properties of the pathogens and the host susceptibilities to disease, as well as the influence of behavioral, cultural, and social patterns that can enhance or mitigate the host's exposure to the disease sources and consequently its transmission in the population [ 1 ].

In this sense, the connection between the above mentioned factors is critical for a better understanding of the transmission patterns to support the development of effective strategies of control, health assistance, and policies [ 2 ]. Highlighting that some infectious diseases have the potential for treatment and eradication in most cases by antimicrobial drugs and vaccines, as well as prevention by proper hygienic-sanitary and prophylactic measures, however, they also present great unpredictability regarding their epidemiological magnitudes in the population due to the biological evolution of infectious agents against therapeutic drugs along with constant social and environmental changes intrinsic to the continuous and unstoppable process of globalization and urbanization [ 3 - 5 ].

In this context, the unpredictability of these diseases represents a factor that can compromise the capacity of the health systems and services to meet the needs of the population, especially considering finite and limited human and economic resources, where the mathematical modeling of the infectious diseases can contribute to the health services by allowing extrapolations of the epidemiological behavior of the infectious diseases, as well as interventions, whose effectiveness can be analyzed considering numerous factors that can influence the dynamics of disease transmission and guide the public health decision-making [ 6 ], such as the study by Hoertel, et al. [ 7 ], Which, through mathematical modeling, analyzed the effects of measures such as lockdown, physical distance, and the use of masks on the COVID-19 cumulative incidence and mortality, bed occupancy, concluding that the preventive measures mentioned would be effective in reducing the speed of the epidemic in France, but not enough to prevent the maximum occupancy of the ICU beds, still emphasizing that without such measures, the magnitude of the pandemic would be much greater in the French population, suggesting the continuity the use of masks and social detachment after periods of lockdown.

Therefore, this work didactically presents mathematical modeling as a tool useful in epidemiology, having as target students and professionals from health sciences.

This work adopted the methodology described by [ 8 ], presenting a text with an educational approach derived from the analysis of the conceptual and experimental articles raised in the PubMed, Web of Science and Google Scholar databases with the boolean operators infectious diseases 'AND' mathematical modeling. The selection criteria were pertinence to the theme and consistency of the information provided, and full availability of the material online, while the exclusion criteria were works published in events and works not fully available online.

This review brings a conceptual understanding of what is a model, which data are used in the mathematical models applied to the infectious disease dynamics, the types of models, and how they are built through a didactic text with a comprehensive approach.

In this topic, the data and the main methods used in the infectious diseases mathematical modeling are presented considering the graphic shape an epidemic can present regarding the number of infected individuals during a time interval, followed by explanations about the basic measure used to assess epidemic risks and the effectiveness of interventions according to mathematical modeling, as well as few strategies that can be adopted to increase the realism of a mathematical representation of an epidemic.

Mathematical modeling of infectious diseases

A mathematical model is an abstract representation of a phenomenon constructed with the use of equations that generate perspectives of the general behavior of an epidemic event, also representing a way to investigate the influence of determinate factors over disease spread, providing a crude general behavior of an epidemic as addressed by epidemic curves, thus allowing predictions about the endurance of an epidemic, its magnitude in the population, and the evaluation of factors that influence the transmission dynamics, and consequently the number of cases. Highlighting it is possible to apply mathematical refinements to the models to enhance their proximity to real data [ 9 , 10 ].

In the epidemiology of infectious diseases, mathematical modeling is a tool of great versatility, that allows the identification of patterns in epidemics, extrapolations of epidemic behaviors along with the effect of interventions such as pharmacological treatment, immunization, quarantine, social distance, and hygiene measures in a dynamic context, presenting low cost and enabling simulations of experiments condemnable ethically in human beings, as well as simulations of experiments that present low economic viability in animal models [ 11 ].

In general terms, the mathematical models applied to the epidemiology of infectious diseases can be classified into two types: 1) The deterministic models considering nonrandom rate flows in a population stratified in compartments; and 2) The stochastic models that consider probabilities in the movements between the compartments of the model, such as the probability of a susceptible individual being infected and the probability of transmitting the disease in the population addressed by a mathematical system [ 12 ].

Deterministic mathematical modeling

The deterministic models of the infectious diseases represent the most practical way for the approximate analysis of how an epidemic will behave in a closed system, in which the population is divided into compartments that describe disease states, where differential equations appearing as derivatives describe the movements between these states by determining variations over time [ 13 ].

In this context, considering a total population N (equation 1) that is initially found in a compartment called susceptible (S equation 2), which after the introduction of a pathogen, gradually moves to the compartment infected (I equation 3), according to the differential equations below:

Equation 1 N = S + I , where

Equation 2 d S s t = − β + 1 ,

Equation 3 d I d t = β S I .

In which the negative sign preceding the infection rate β indicates that the product of S x I decreases among the individuals in compartment S, while the number of infected people grows in the same proportion [ 14 ], as seen in the SI model in Figure 1 (on the left side).

In this model, the onset of infection and the exponential growth in the number of cases are approximate to the observed in the epidemic curves, however, the model cannot represent the natural decline in the number of cases after the system reaches saturation due to the absence of susceptible individuals to support new cases [ 15 ]. In this sense, the SI model is more suitable for infectious diseases that become chronic with no recovery, such as infection by HIV [ 16 ].

In this sense, the addition of a new compartment capable of capture the decline in the epidemic curves due to acquired immunity or death brings to the system more realism, as well as applicability to self-limiting, or treatable infections. Highlighting that the new compartment, the recovered or removed (R equation 7) should increase exponentially regardless of the number of susceptible individuals, depending exclusively on the number of infected multiplied by a recovery constant Ɣ [ 17 ].

In this case, the system of equations becomes:

Equation 4 N = S + I + R ,

Equation 5 d S d t = − β S I N ,

Equation 6 d I d t = β S I N − γ I ,

Equation 7 d R d t = γ I .

As a result of adding this new compartment to the model, the graphic representation becomes capable of describing the essence of the epidemiological behavior of most infectious diseases, as shown in Figure 1 (on the right side), in which the number of susceptible individuals decreases in the same proportion as the number of infected increases, and the number of recovered people increases as the number of infected people declines [ 14 , 15 ].

Highlighting that the deterministic models are not limited to the SI, and the SIR structures, being possible to implement different structures and dynamics in the deterministic systems, which, after refinement, attribute to the model a greater degree of realism in their epidemic representations of the infectious diseases, considering factors such as the incubation period by adding the exposed compartment (E), age stratification, and spatial structures [ 16 - 19 ].

In this sense, Figure 2 shows a hypothetical case considering the compartments Susceptible, Exposed, Infected, and Recovered with two dynamics in the populations of three countries, assuming the flow of individuals between different geographic spaces.

However, although the deterministic models capture the epidemiological behavior essence of the infectious diseases, it does not consider numerous events of random nature that can influence the transmission dynamics, such as environmental factors, and protective behaviors in the susceptible host population through prophylactic measures, rapid and effective responses of the health systems, and the participation of asymptomatic individuals in the transmission dynamics, being the deterministic models the most suitable to simplify the epidemiological behavior of a given infectious disease in a worst-case scenario, but not the most suitable for making decisions in real-time [ 20 ].

Exemplifying the employability of deterministic mathematical models, Ngeleja, et al. [ 21 ] developed a model to assess the role and magnitude of the involvement of human populations, rodents, fleas, and the survival of pathogens in the environment on the spread of bubonic plague, an infectious disease caused by the bacterium Yersinia pestis, which represents a serious public health problem for some African countries, such as Tanzania.

The model by Ngeleja, et al. [ 21 ] adopted the susceptible (S), exposed (E), infected (I), and recovered (R) structure and dynamics in the human population, assuming a return to the susceptibility and reinfection stage; in the flea population a susceptible (S) and Infected (I) structure and dynamics were used since once infected by the pathogen Yersinia pestis, the fleas do not recover from the infection; for the rodent population, the susceptible (S), exposed (E), and infected (I) structure and dynamics were assumed, considering natural death, and interaction/contact between humans, fleas, rodents, and the environment, according to the differential equations below:

Human population (SEIR)

Equation 8 d S H d t = π 1 ψ + ϖ 1 R H = α 1   ( Γ f h I F N 2 + ω 1 A )   S H − μ 1 S H ,

Equation 9 d E d t = π 2 ψ + α 1   ( Γ f r I F N 2 + ω 1 A )   S H − α 2 E H − μ 1 E H ,

Equation 10 d I H d t = α 2 E H − α 3 I H −   ( μ 1 + δ 1 )   I H ,

Equation 11 d R H d t = π 3 ψ 1 + α 3 I H − ϖ R H − μ 1 R H ;

Rodent population (SEI)

Equation 12 d S R d t = κ 1 ψ 3 − γ 1   ( Γ f r I F N 2 + ω 2 A )   S R − μ 3 S R ,

Equation 13 d E R d t = κ 2 ψ 3 + γ 1   ( Γ f r I F N 2 + ω 2 A )   S R − γ 2 E R − μ 3 S R ,

Equation 14 d I R d t = κ 3 ψ 3 + γ 2 E R −   ( μ 3 + δ 3 )   I R ,

Equation 15 d S F d t = ψ 2 s − β   ( ρ Γ h f I H N 1 +   ( 1 − ρ )   Γ r f I R N 3 )   S F − μ 2 S F ,

Equation 16 d I F d t = ψ 2 i + β   ( ρ Γ h f I H N 1 +   ( 1 − ρ )   Γ r f I R N 3 )   S F −   ( μ 2 + δ 2 )   I F ,

Where ψ2s < ψ2i

Pathogens in the environment

Equation 17 d A d t = λ 4 − ω 1 A S H − ω 2 A S R − μ 4 A .

Where the following parameters are added to the differential equations: Γfr, Γfh, Γrf, Γhf are parameters related to the adequate contact rate between flea to rodent, flea to human, rodent to flea, and human to flea; ω1 is the adequate contact rate between pathogens to human, and ω2 is the adequate contact rate between pathogen to rodent ; α1, α2 and α3 represent progressions rates, respectively, susceptible human to exposed, exposed human to infected, and infected to recovery, and ϖ is the progression rate from recovered human to susceptible ; Ɣ1 represent the progression rates of susceptible rodents to exposed, and Ɣ2 the progression rate of exposed rodent to infected; µ1, µ2, µ3 and µ4 are in order, the natural death for human, flea, rodent and pathogen, δ 1 , δ 2 and δ3 are the disease induced rates for human, flea and rodent; ψ1, ψ2s, ψ2i, and ψ3 are respectively the imigrations rate of human, susceptible flea, infected flea, and rodent; π1, π2, π3 are the proportional migrantion that are susceptible, exposed, and recovered; κ1, κ2 and κ3 are the proportional rodent migrants that are susceptible, exposed and infected; and β is the rate that fleas become infected. Highlighting these parameters were from the literature.

In the numerical solution presented by Ngeleja, et al. [ 21 ], In the absence of interventions, the parameter that has the greatest effect on the dynamics of transmission to humans and rodents is the β infectivity rate in the flea population, suggesting that the most effective strategies should focus on vector control, with the infected populations of humans and rodents being negligible parameters, as well as the environment in the transmission dynamics. This result is similar to those from Mbogo, et al. [ 22 ], that developed a comparative analysis of deterministic and stochastic models with the structural dynamics susceptible and infected (SI) for human and vector populations, also observing the relevance of strategies with a focus on vector control to mitigate malaria transmission, however differently, the environment was a parameter with some degree of significance in the malaria transmission to humans.

Stochastic mathematical modeling

The stochastic mathematical models of infectious diseases represent a more realistic approach to epidemics, because they allow the recognition of the initial patterns in an epidemic the analysis of the spatial distribution of case numbers in a given location, and allow estimations about the duration of an epidemic, considering differences among individuals in the population, such as age, sex, as well as social and geographic aspects that impose non-uniformity in the contact between individuals, in addition to environmental factors such as seasonality and transmission pathways [ 23 - 25 ].

These models tend to have complex and sophisticated mathematical formulations, and can be classified into three types: 1) Stochastic differential equations; 2) Markov chains with continuous-time or 3) Markov chains with discrete-time. All the stochastic models assume probabilities in the transition between the compartments of the system, also assuming the presence of a state free of disease, which is different from the deterministic models that assume an equilibrium state, which does not represent the end of an epidemic. In addition, they have great employability in asymptotic analysis, where the main objective is to describe the behavior limits considering the number of infected individuals in a large population [ 23 ].

The models of the type Markov chain assume that each infection will occur independently of the past one in a probabilistic fashion, being possible to obtain from these models predictions about the stochastic risk of a major or a minor outbreak to occur when the number of infectious individuals is too small in a population of susceptible hosts [ 26 ], and the stochastic differential equations consider special probability and diffusion coefficients in the structures defined by the deterministic models considering random processes in the movement of compartments that compose the model [ 27 ].

The stochastic models represent a more realistic approach to model infectious diseases because they contemplate the high degree of uncertainty in the dynamics of transmission, providing a range of possible outcomes of an outbreak considering a large number of variables that influence the epidemic behavior of an infectious disease, however, they tend to be limited regarding the degree of complexity in the formulation of the mathematical system and data interpretation through different methods, such as the use of master equations [ 28 ], itô calculations [ 29 ], as well as other mathematical and statistical approaches based on brownian motion [ 30 ], or markov processes that add stochasticity to differential equations [ 31 ], monte carlo method [ 32 ], the [ 33 ] gillespie's first reaction method, among other methods.

In this sense, as an example of stochastic model development and implications, Legrand, et al. [ 34 ] evaluated the transmission of ebola hemorrhagic fever in the community, hospitals, and due to traditional burial ceremonies using real epidemiological data from epidemics in the Democratic Republic of Conga, and Uganda, respectively in 1995 and 2000. Their model was composed of the compartments susceptible (S), exposed and not infectious individuals in the community (E), Infected and infectious (I), hospitalized individuals (H), dead individuals that remain infectious during traditional funerals (F), and removed individuals by cure or death, as seen in equations 18-23, assuming that all cases were due to human to human transmission, except the first case

Equation 18 d S d t = − 1 N   ( β I S I + β H S H + β F S F ) ,

Equation 19 d E d t = 1 N   ( β I S I + β H S H + β F S F )   -   α E ,

Equation 20 d I d t = α E −   ( γ h θ 1 + γ i ( 1 − θ 1 )   ( 1 − δ 1 )   +   γ d   ( 1 − θ 1 ) δ 1 ) I ,

Equation 21 d H d t = γ d θ 1 −   ( γ d h δ 2 + γ i h   ( 1 − δ 2 ) )   H ,

Equation 22 d F d t = γ d   ( 1 − θ 1 ) δ 1 I + γ d h δ 2 H − γ f F ,

Equation 23 d R d t = γ i   ( 1 − θ 1 ) ( 1 − δ 1 )   I + γ i h   ( 1 − δ 2 )   H + γ f F .

The model presents the following parameters: β I , B H , and B F are coefficients of transmission, respectively, in the community, at the hospital, and during funerals; θ 1 is the percentage of infected cases that are hospitalized; δ 1 and δ 2 are the case-fatality ratios; α is the inverse of the mean duration of the incubation period; Ɣh -1 is the symptom onset to hospitalization; Ɣ dh -1 is the mean duration from hospitalization to death; Ɣ ih -1 is the mean duration from hospitalization to the end of infectiousness for survivors; and Ɣf -1 is the mean duration from death to burial. Highlighting the model's parameter were evaluated using the maximum-likelihood adopting 95% of confidence interval, then sets of parameters generated by latin hypercube sampling were computed assuming twice the difference of log-likelihood values that was X 2 distributed with degrees of freedom with values equal to the number of estimated parameters, then 700 simulations were run by the gillespie's first reaction method, and in each simulation, the partial rank correlation coefficient was computed to analyze the influence of each parameter in the epidemic size.

In this model, Legrand, et al. [ 34 ] found out the community transmission was a significant source of infection in Uganda, while the traditional burial ceremonies played a more important role in the transmission dynamics in the Democratic Republic of Congo, emphasizing that appropriate hospital precautions to avoid transmission between patients and health care workers, as well as precautions with the corpses, are important to decrease the size and duration of epidemics. The authors [ 34 ] also proposed tracing contact interventions as a strategy to identify sources of infection, thus decreasing the transmissibility due to proper isolation and health care of infected and suspicious cases.

Deterministic versus stochastic

In general, both approaches exhibit the same behavior, where the deterministic models can capture the essential patterns of an epidemic; however, it cannot answer the question: what are the margins of error of the estimates for disease peaks? Therefore, to address this question, the stochastic model can be used, considering the minimum and maximum probabilistic range an epidemic can assume.

That is, the deterministic approach provides an overall insight about the disease spreading in a fast way, whereas the stochastic framework provides statistical insights into the transmission events providing a range of possible epidemic scenarios [ 22 ].

Deterministic models tend to present results that do not undergo major changes due to fluctuations in the population, but undergo significant changes if the parameters inserted in the differential equations are modified; in this context, the stochastic models are more responsive to quantitative changes both in the populations and subpopulations, as well as in the modeling parameters, making it important to emphasize that there are several ways of working probabilities in stochastic processes as addressed above, which makes these models too complex and difficult to interpret [ 22 , 30 , 35 ].

Moreover, both deterministic and stochastic models share the challenge in representing the natural history of infectious diseases, considering sources of infection, transmission routes, incubation period, infection and transmissibility periods, treatment, and development of natural immunity, which are parameters that can be implemented in mathematical systems through three ways: i) Using data already described in the literature [ 36 ], ii) Empirically through estimations with basis on epidemiological data [ 37 , 38 ], or iii) Estimated by computer programs using statistical methods such as root mean square error on epidemiological data [ 17 ].

In this sense, when the parameters come from the literature, they can generate results slightly different from a real epidemic because of variability concerning environmental, socio-cultural aspects, the virulence of the pathogens, and resistance-susceptibility profile among the host populations, as well as behaviors with a protective effect to the disease, which can manifest differently in time and space. While the parameters established by analysis of contact rates, incubation time, the prevalence of pathogens in the population and the environment, among other parameters, demand time and statistical treatment, making the mathematical representation more accurate for what is intended to be analyzed, however, also presenting challenges for the development of generalizations valid for other models [ 39 , 40 ]. Therefore, any mathematical modeling study should present calibration and validation considering statistical analyzes of real epidemiological data to assess the accuracy of the models [ 41 ].

The basic reproductive number

In the epidemiology of infectious diseases, the basic reproductive number or R0 (Figure 3) represents a parameter that expresses the typical number of secondary cases produced from a single infected individual, informing the epidemic risks, in which values of R0 greater than 1 indicate a high predictive risk of an epidemic event to occur, while values of R0 smaller than 1 indicate a low risk of the occurrence of an epidemic event [ 42 ]. In this context, the R0 can be obtained from ordinary differential equations considering the relation of the rate of infection (β) by and the rate of recovery (Ɣ) [ 43 , 44 ], usually calculated by next-generation matrixes [ 21 , 22 , 29 , 45 ], by the exponential growth rate method and maximum likelihood method [ 17 ], or by bayesian statistics [ 46 ].

Moreover, the R0 is a parameter used to evaluate the efficacy of interventions such as quarantine, mask-wearing, vaccination, washing hands in hospital sets, among others; where if the intervention decreases the R0 to values smaller than 1, it is considered effective, and uneffective if it does not change the R0 [ 7 , 47 - 49 ].

However, it is important to point out that the value of R0 is a dynamic parameter restricted to the time and space in which a given infection is occurring or occurred [ 50 ] due to social and cultural factors that modulate the social contact rate between individuals, virulence factors from the pathogen, environmental conditions that enable the pathogen survival, treatment availability, as well as susceptibility of the pathogen to the antimicrobial drugs employed in the treatment along with other random factors present in the population [ 51 - 53 ].

Several factors can modulate the interaction of a pathogen with its host population, imposing a constant transformation for both organisms, and in this context, the mathematical modeling of the infectious diseases represents a precious tool for understanding transmission dynamics patterns and how factors, such as the environment, human behavior, and the microbial evolution to the antimicrobial drugs and vaccines can change the epidemiological behavior of infectious disease considering local peculiarities.

Highlighting that addressing local peculiarities is an important aspect for the success of public health policies, programs of disease prevention, and health assistance because in most cases the universal approaches do not consider the social-cultural differences, economic power, human resources, and facilities to deal with infectious diseases epidemic events. Therefore, efforts to translate biological, clinical, environmental, epidemiological, and social data into mathematics, and vice versa represent a low-cost strategy to approach health issues of high complexity in order to search for effective solutions.

This work did not receive funds from any institution.

The authors declare no conflict of interest.

Not applicable.

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Mathematical Model of Interaction of Therapist and Patients with Bipolar Disorder: A Systematic Literature Review

Indah nursuprianah.

1 Doctoral Program of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjajaran, Sumedang 45363, Indonesia

Nursanti Anggriani

2 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Padjajaran, Sumedang 45363, Indonesia

Nuning Nuraini

3 Department of Mathematics, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Bandung 40132, Indonesia

Yudi Rosandi

4 Department of Geopysics, Faculty of Mathematics and Natural Sciences, Universitas Padjajaran, Sumedang 45363, Indonesia

Associated Data

Not applicable.

Mood swings in patients with bipolar disorder (BD) are difficult to control and can lead to self-harm and suicide. The interaction between the therapist and BD will determine the success of therapy. The interaction model between the therapist and BD begins by reviewing the models that were previously developed using the Systematic Literature Review and Bibliometric methods. The limit of articles used was sourced from the Science Direct, Google Scholar, and Dimensions databases from 2009 to 2022. The results obtained were 67 articles out of a total of 382 articles, which were then re-selected. The results of the selection of the last articles reviewed were 52 articles. Using VOSviewer version 1.6.16, a visualization of the relationship between the quotes “model”, “therapy”, “emotions”, and “bipolar disorder” can be seen. This study also discusses the types of therapy that can be used by BD, as well as treatment innovations and the mathematical model of the therapy itself. The results of this study are expected to help further researchers to develop an interaction model between therapists and BD to improve the quality of life of BD.

1. Introduction

In 2018, around 792 million people in the world experienced mental disorders. Mental disorders are the leading causes of disability, illness, and death among the civilian population and the military. Most treatments for mental disorders currently available have limited efficacy, especially in mental disorders whose symptoms vary over a relatively short time scale [ 1 ]. As many as 0.6%, or about 46 million people, suffer from Bipolar Disorder. Bipolar disorder (BD) is the world’s sixth leading cause of disability [ 2 ]. According to the Ministry of Social Affairs of the Republic of Indonesia on its official website, in 2012, 11.6% of Indonesia’s population, which is around 24 million people, experienced mental disorders. Of the 24 million people, only 9% are undergoing medical treatment.

Naturally, everyone will experience an up or down mood. However, the mood changes experienced by patients with bipolar disorder are extreme and tend to be more rapid [ 3 ]. The moods of a BD person, ranging from severe depression to extreme mania and irritability, are often accompanied by difficulties in functioning in society [ 4 ]. Although patients with BD can achieve higher education than the general population, 65% of patients with BD are unemployed [ 5 ]. Patients with BD are people of talent and creative work [ 6 ] and also some of them are extraordinary individuals such as Ernest Hemingway, Charles Dickens, Tchaikovsky, Vincent Van Gogh, and Winston Churchill [ 7 ].

The symptoms of bipolar disorder and related disorders can be found in the Diagnostic and Statistical Manual of Mental Disorders, Fifth Edition (DSM-5). In the DSM-5, symptoms of bipolar disorder and related disorders such as: A new specifier “with mixed features” can be applied to bipolar I disorder, bipolar II disorder, bipolar disorder NED (not elsewhere defined, previously called “NOS”, not otherwise specified), and major depressive disorder (MDD); allow bipolar disorder and other related disorders for certain conditions; and anxiety symptoms are a specifier (called “anxious distress”) added to bipolar disorder and to depressive disorders (but are not part of the bipolar diagnostic criteria) [ 8 ].

In previous studies, the mood phenomena studied were hypomanic and depressed in type 2 BD [ 9 , 10 ]. The model of individual mood changes that are not treated uses the Van der Pol equation approach with negative damped harmonic oscillations [ 10 ]. With some assumptions, such as untreated BD, dysregulation of dopaminergic neurotransmission [ 11 ], dysfunction of mood stabilizer neurons [ 12 ], and abnormalities in circadian rhythm [ 13 ] play an important role in triggering BD. In addition to models of mental health, the Van der Pol equation is also used for several applied sciences that involve spring oscillations, such as in biology, seismology, electrical circuits [ 14 ], and the dynamics of the heart [ 15 ].

The study of interaction models between therapists and patients with BD is important because it is an ongoing topic, but publication is slow. It is hoped that many researchers will be more interested in this study because patients with BD usually have great potential and a high Intelegence Quotient (IQ) [ 16 , 17 ]. This hypothesis is useful for research with various further analyses. In addition, a person with BD who has a high IQ has a high risk of injury. This is because someone who has difficulty controlling their mood has a risk factor for psychological overstimulation as a consequence of a high IQ [ 17 ]. In previous studies, there were very few mathematical formulations that resulted in dynamic analysis between therapists and patients with BD. This can be seen from the literature review conducted in this study. Research shows that mathematical models of bipolar disorder can follow a variety of different systematic approaches, according to the type of mood episode, bipolar type, type of therapy, and family condition.

The method used in this study is Systematic Literature Review (SLR), which is a method for interpreting and evaluating previous research related to a phenomenon explicitly, systematically, and can be done by other researchers [ 18 ]. SLR was first used in the health sector, but now it has been used in other fields, such as management [ 19 ], mathematics [ 20 ] and insurance [ 21 ], information diffusion on social media [ 22 ], and information systems [ 23 ]. The way SLR is related to the mathematical model of interaction between therapists and bipolar disorder is quite interesting. The development of types of therapy and how the therapist interacts with patients with BD so that BD stabilizes more quickly became the initial motivation for this study, and we wanted to know how far research on the mathematical model of bipolar disorder has grown each year. In addition, what kinds of mathematical models of bipolar disorder have been extensively studied? With this knowledge, it is possible to find out how the therapist and patients with bipolar disorder interact and what therapies are good and interesting for future research and development. This research was conducted based on articles that discussed the mathematical model of bipolar disorder and its therapy.

The mathematical model of BD Therapy is a model that describes how BD therapy is carried out [ 24 ], as well as how therapy techniques can improve the mental stability of patients with BD [ 25 ]. The combination of one therapy and another is needed to accelerate the mental stability of BD patients. The development of symptoms in BD has given rise to various types of therapy, according to the risks faced. These dangers, such as chronic depression, can result in self-harm and sanctification [ 26 ]. To overcome the risks of self-harm and sacredness, intervention and protection are used [ 27 ] in addition to drug therapy, of course. Interventions can be carried out by therapists such as psychologists and psychiatrists to carry out therapy, while protection can be carried out by family, friends, and the community around patients with BD. Things that can be done include: building a collaborative and respectful therapeutic relationship in which the therapist seeks to understand the hopelessness of patients with BD and provide accurate empathy, designing an anti-suicide action plan, teaching various skills, maximizing the patient’s social support network, and fighting the stigma of bipolar disorder through acceptance of limitations while still striving to live life to the fullest through medication, an optimistic attitude, and meaningful long-term goals. From these various things, the goal to be achieved is to obtain a mathematical model of the interaction between the therapist and BD patients, with the type of therapy and duration of therapy, so that it can be seen what kind of interaction with the therapist makes BD patients stable faster. The results of the study are expected to provide an overview for future researchers to be able to develop a more accurate model of interaction between therapists and patients with BD.

2. Materials and Methods

2.1. scientific article data.

In this study, we selected and identified literature focusing on journals or proceedings on the mathematical model of BD and interactions with therapists. This material focuses on various types of therapy, such as drug therapy, psychotherapy, and other therapies. The articles used were obtained from several data sources, including Science Direct, Google Scholar, and Dimensions, and were published between 2009 and 2022 in English only. The articles in the literature are in the form of journal articles. These data sources were selected because they are easy to access. In selecting articles from Science Direct, Google Scholar, and Dimensions data sources, the keywords used were the same, namely “Mathematics Model” (MM), “Bipolar Disorder” (BD), “Interaction” (I), and “Therapist” (T). The articles obtained were collected in Mendeley, visualized using VOSviewer software version 1.6.16 is support by the Centre for Science and Technology Studies of Leiden University, and analyzed bibliometrically.

In Table 1 , it can be seen how the selection of selected articles from the data sources of Science Direct, Google Scholar, and Dimensions is done with several keywords. In the first screening process, starting with keyword 1, namely “Mathematical Models” (MM) and “Bipolar Disorder” (BD), scientific publications from Science Direct amounted to 149 articles, Dimensions 2086 articles, and Google Scholar 1960 articles. In the second screening, with the addition of the keyword “therapist” (T) and the conjunction “and”, 6 articles were obtained from Science Direct, Dimensions 305 articles, and 96 Google Scholar articles. The last screening, with the addition of the keyword “Interaction” (I) and the conjunction “and,” obtained scientific publications from Science Direct totaling 5 articles, Dimensions 289 articles, and Google Scholar 76 articles. From this last screening, the articles that can be entered into the identification, screening, and inclusion process are as listed in Figure 1 .

Flowchart of the search strategy.

Number of publications from three databases with three types of keywords.

Description: Mathematical Models: MM; Bipolar Disorder: BD; Therapist: T; Interaction: I.

2.2. Selection of Literature Database

Data for this research was obtained from three data sources (Science Direct, Google Scholar, and Dimensions), then stored in Mendeley, selected by deleting literature in the form of books or topics deemed irrelevant to this research. This selection was carried out to obtain literature that was in accordance with the purpose of this study, in the form of articles that discussed the mathematical model of the interaction between therapists and patients with BD. Each result obtained was checked one by one to get the appropriate database in the form of journal articles. Then, the articles were identified by comparing the duplicate databases from the three data sources. If a duplicate is found, the data is deleted, leaving one version of the same title in the database. The total amount of literature data obtained from the search results consisted of 382 results. After checking and sorting, 52 articles were obtained, which were included in the detailed analysis in this study. The literature review was carried out by mapping the article data obtained, including the development of research on mathematical models of therapist and BD patients’ interactions from 2009 to 2022; types of therapy in BD; interactions of therapists and patients with BD; and the development of mathematical models of BD in several countries, based on the literature used. The search process and strategies to get relevant and quality articles are given in the flowchart in Figure 1 .

2.3. Systematic Data Analysis and Methods

The method used in this study is systematic literature review (SLR), which is an approach to obtain an overview of existing research and trends related to the mathematical model of interaction between therapists and bipolar disorder. In addition, the SLR also identifies, assesses, and interprets findings on a research topic that answer predefined research questions [ 28 ]. In general, the SLR consists of four stages, namely the determination of objectives, initiation and selection of literature, analysis, and plans to represent the results [ 29 ]. In this study, the SLR method was considered based on published articles. The articles obtained were then assessed, identified, and interpreted based on the findings obtained after reading each article, according to the research topic (i.e., therapeutic models as an alternative to BD treatment). However, in the SLR process, a systematic evaluation stage is also required when conducting research, so there is no similarity with previous research. A systematic analysis of data articles was carried out in the following stages: (1) visualization of the article database was carried out on the relationship between article data and the most discussed topics. Visualization was done using VOSviewer software to get the most used quotes; (2) mapping the number of articles each year (from 2009 to 2022), presented in the form of a bar chart, and providing general information on frequently cited research topics; and (3) mapping of the mathematical model of BD and therapists, types of therapy for BD in each country, and the number of articles that have discussed it. This mapping is done by examining the articles one by one, to determine the country and type of therapy discussed, including psychotherapy, drug therapy, and therapist interactions with clients and others.

This section describes the results of data analysis focused on 52 articles, including article data visualization, development of mathematical models of BD, and types of therapy for BD. Based on the database used from three search sources, most of the references published by Scopus were 73.1%, while those cited by Scopus were 11.5%, and other references were 15.4%. The scientific work in this study was mostly published by Scopus. This can be caused because only selected scientific works are published in the form of articles in journals. In this study, unused scientific papers are in the form of theses, doctoral dissertations, book chapters, and proceedings. It is hoped that by taking articles that are mostly published by Scopus, this research can be a reference for further researchers.

3.1. Article Data Visualization

This section provides an overview of the article data visualization obtained using the VOSviewer software. Visualization is done to find out the relationship between the data in the articles. The cluster size in the visualization shown in Figure 2 reflects how many articles cover keywords in the research topic. If the cluster in the visualization is large, it indicates that the word (quote) is present in most of the articles in the database. The line connecting the two clusters indicates that there is a relationship between the two. Furthermore, the distance between the two clusters indicates the strength of the relationship between the two clusters. For clusters that have a tendency to be strongly bound, the distance between the two clusters is shorter [ 30 , 31 ].

( a ) Visualization of the article database: The aggregated network graph of the article database. ( b ) The network graph grouping the keyword “model” with other keywords from the article database.

In Figure 2 a, the cluster associated with the word “model” has the largest size compared to the others. This shows that the word “model” is the most talked about in the database. This word (quote) is followed by the words “emotion” and “therapy,” which rank second and third. In this picture, it can also be seen that research with the words “model” and “therapy” was mostly carried out in the 2015–2017 period, while research with the word “emotion” was carried out in 2014. Furthermore, to further determine the relationship between clusters (in this case, quote words), simply hover over the word quote that you want to see the relationship with other word quotes. From here, the relationship of the word “model” with other words is shown in Figure 2 b, such as with the words “therapy”, “emotion”, “BD”, “psychotherapy”, “mental illness”, “research”, “mental disorder”, etc. From the picture, the distance between the words “model” and “therapy” is greater than “model” with “emotions.” This means that the “therapy” tendency is weaker than the “emotional” one with the “model”. Research on “models” that address the issue of “emotions” is far more than “therapy”. The quote of the word “BD”, even though it is in a small cluster, has a strong relationship with the word “model”. This means that research related to “model” and “BD” is very closely related.

3.2. BD Therapy Mathematical Model Development

The study of the interaction model between therapists and patients with BD is a topic that must continue because it is very much needed by the community. Despite this, publication on the topic has been slow. Although development is still slow, mathematical models of therapy in BD continue to develop around the world. The mathematical model of BD therapy has also received a good response from the government, as evidenced by the increasing support for research on mental health. The development of research shows that the mathematical model of bipolar disorder can follow a variety of different systematic approaches, according to the type of mood episode, bipolar type, type of therapy, and family conditions. In addition, it is also necessary to pay attention to research on the development of therapy for BD and combine it with the mathematical model of BD. The fact that patients with BD usually have great potential and talent as well as a high IQ [ 16 , 17 ] presents its own challenges for researchers so that the development of mathematical models of BD therapy can be carried out optimally so that the quality of life for BD patients increases.

Research conducted on the topic of the mathematical model of interaction of therapist and patients with BD has been published in three databases, namely Science Direct, Google Scholar, and Dimension. However, this research still seems slow. This indicates a novelty on this topic during the period 2009 to 2022. Research published in the form of proceedings, journal articles, and doctoral theses is collected in the literature database. However, what is reviewed in this study is only articles from journals. The development of research on the mathematical model of BD therapy from a database that is collected every year can be seen in Figure 3 .

Publications from 2009 to 2022 based on the collected database.

From Figure 3 , it can be seen that the number of publications on the topic of BD mathematical models and therapists is still very small. The most publications were in 2012, with as many as nine scientific articles. At least in 2013 and 2018, there was one scientific article. Overall, the development of research on this topic tends to decline. This is possible because of the difficulty of research on mental health, especially BD. Therefore, it is a challenge for researchers to conduct this research. Apart from being really needed by the community, the impact of the COVID-19 pandemic has worsened the mental condition of the community.

From previous research, the country that consistently discusses the topic of BD is the USA. Many scientific publications are published in that country. Based on Figure 4 , the top seven countries discussing the topic of BD mathematical models and therapists are the USA, UK, Germany, Switzerland, Canada, Spain, and Greece. The countries with the most scientific publications are the USA (24 publications), the UK (up to 7 publications), Germany (up to 4 publications), Switzerland, Canada, Spain, and Greece (2 publications each). Apart from the countries above, based on the database, there are other countries that publish scientific publications on the same topic, namely France, the Netherlands, South Africa, Norway, Chile, Finland, Austria, Camerun, and Brazil (one publication each).

Country of publishing articles based on the collected database.

Based on the database of the previous 52 articles, the 10 articles with the most citations were classified. The number of citations shows how interesting the information provided in an article is, so that it can be used as a reference in other research articles. The more articles are cited, the more interesting the information they provide, and the more references are made to other topics related to the topic of the article in the next period. Figure 5 shows the 10 articles with the most citations, accompanied by the author and country of origin of the publisher.

Top 10 article citations based on the collected database [ 1 , 8 , 32 , 33 , 34 , 35 , 36 , 37 , 38 , 39 ].

Based on Figure 5 , the most cited authors are Andrews et al. from Switzerland in 2011, with as many as 156 citations. This article discusses major depressive disorder, neurochemical disorders, antidepressant drug therapy, and meta-analysis of studies. In the second place, Millan et al. (2015) from France had as many as 90 citations. This article discusses neuropsychopharmacology, pharmacotherapy, therapy, and psychiatric disorders. The oldest citation was in 2010, namely Stevenson et al., from UK and Hemmeter and Krieg from Switzerland; while the latest citation is Wedge in 2022 from the USA, which is the third highest citation in the database. The fewest quotes are Holmes from UK in 2016 as many as 42 citations. Out of the 10 most-cited quotes, most of them talk about therapeutic problems in mental disorders, be it drug therapy, cognitive therapy, and others; they also discuss the diagnosis and symptoms of mental disorders; two articles discuss BD; and one article discusses computational mathematical models in psychiatry. In addition, the countries with the most citations were the USA and UK, then Switzerland, and finally France and Norway.

A detailed explanation of the research topics discussed in the top 10 articles with the most citations is given in Table 2 . Of the top 10 cited articles, all of which are indexed by Scopus Q1, it can be seen that the biggest discussion of the mathematical topic of BD interaction models and therapists can be used as quotes by other researchers who want to develop this topic in the future. Based on Table 2 , it can also be seen that there are two main discussions, namely therapy and psychiatric diagnosis. Of the top 10 articles, six are about therapy and four are about diagnosis. The therapies discussed can be in the form of drug therapy, cognitive therapy, and therapeutic innovation; while the diagnoses discussed were severe mental symptoms, mood disorders, mental disorders, and BD. Many (words) quotes in an article indicate that the topic is considered interesting enough to be discussed and has a great contribution for future researchers. The discussion on therapy and diagnosis in the 10 most-cited articles are explained in greater depth in the discussion section.

Research topics from the top most-cited articles.

3.3. Types of Therapy in Psychiatry

Linguistically, the meaning of the word therapy is an effort to restore the health of people who are sick or disease treatment. There are several types of therapy for psychiatric diseases, such as drug therapy, psychotherapy, cognitive, and light. The therapy is chosen to treat people with mental disorders and other psychiatric problems. Of the 52 articles selected for review in this study, there were 25 articles on therapy. The types of therapy discussed in this study can be seen in the following Ven diagram:

Based on Figure 6 ., of the 25 articles, it is found that the type of therapy that is widely discussed in the selected articles is drug therapy, also referred to as pharmacotherapy, in as many as 8 articles, followed by psychotherapy in as many as 7 articles. This is consistent with current practice, which is that BD treatment includes at least two types of therapy, including drugs and psychotherapy with a psychologist or psychiatrist. The next type of therapy is cognitive therapy, with six articles. Another two articles are on holistic therapy and music therapy, with one article each. From Figure 7 , it can be seen that most of the articles were published in the USA (seven articles); followed by UK (five articles); France (three articles); Germany (two articles); and then Italy, Russia, Japan, Switzerland, Greece, Spain, South Africa, and Canada (one article each).

Types of therapy in the Database.

Number of articles from the database by type of therapy.

3.4. Methodology Used in Types of Therapy

The methodology used in this type of drug therapy research usually uses experimental methods by taking data before and after therapy is carried out. This method takes a sample of BD individuals who have been treated with drugs for some time. The time span used is usually quite long, so that the effect of the treatment can be seen for each individual sample taken. In addition, monitoring is also carried out during therapy so that psychopathological conditions can be seen [ 40 ]. The long-term effects of drug therapy are also seen, so that the possibility of changing from one drug therapy to another is still monitored by a psychiatrist.

In this type of holistic therapy, one of the methods used is the Clinical-Neuropsychological Perspective, which uses a comprehensive perspective both in terms of clinical and neuropsychological/biological neuropsychology [ 41 ]. The holistic approach views humans as a whole, in the sense of humans with cognitive, affective, and behavioral elements. With this type of therapy, humans cannot stand alone but are closely related to their environment. Treatment of brain disorders with holistic therapy must include aspects of individuality and subjectivity, treating the sequelae of brain damage in clinical neuroscience, which demands a biopsychosocial perspective, both for conceptual and historical reasons.

Several articles conducted a series of design studies for cognitive therapy, including design workshops and prototype testing in people who had previously received cognitive therapy for depression, as well as qualitative interviews and role-playing sessions with cognitive therapists experienced in depression treatment [ 42 ]. In addition, there are also those using different linear and quadratic mixed model analysis methods with random effects for each patient tested. In this method, baseline change is defined as the percentage change in individual symptoms during the first 2, 3, 4, and 5 weeks. Then symptoms from sessions four, five, six, seven, and so on were predicted using different models, with the initial changes added to the model in the last step [ 43 ].

In this type of psychotherapy, methodological and substantive issues are raised in relation to what can be said about evidence-based psychotherapy and its effects. Among the methodological problems are control conditions comparing evidence-based psychotherapy with selective reporting of measures and the lack of evidence that evidence-based psychotherapy has a significant clinical impact [ 44 ]. The success of psychotherapy depends on the nature of the therapeutic relationship between the therapist and client. In selected articles, there are those who use dynamic systems theory to model the dynamics of emotional interactions between therapists and clients, using a very similar approach (physics-science paradigm) to the model and make predictions about the relationship between therapist and client [ 45 ].

Methods for this type of music therapy examine the impact of this therapy on circadian biological rhythms in everyday life. In particular, the vagal tone circadian rhythm was monitored, indexed by heart rate variability (HRV), and the hypothalamic–pituitary–adrenal (HPA) axis, and indexed by the diurnal cortisol profile. Observations were made before and after therapy, with a treatment time span of about 10 weeks. Then, before and after the intervention period, psychological data (48 h HRV, salivary cortisol samples for 2 consecutive days), and observer ratings were collected. As a result, music therapy affects the HPA axis and autonomic regulatory processes [ 46 ].

4. Discussion

In some cases, mood changes occur in a certain pattern. There are several types of mood episodes that occur in bipolar disorder, including manic, depressive, hypomanic, and mixed [ 47 , 48 ]. To achieve a stable condition, patients with BD need to take treatment in the form of drugs and therapy regularly. Otherwise, over time it will get worse [ 49 ]. More than 20% of BD patients (mostly without treatment) end their lives by suicide [ 50 ]. It was also found that 18% of patients with BD had the urge to harm themselves at night, either using sharp weapons or poison [ 51 ]. In addition, the mental stability of patients with BD also affects the family or people living with them [ 52 ]. Mood phenomena that occur in BD need to be studied in depth. This can minimize the instability of patients with BD. The treatments that can be done for BD include psychotherapy [ 53 ], ISRT [ 54 ], food [ 55 ], and drugs [ 56 ]. With the right therapy, it will greatly help the mental stability and safety of patients with BD.

4.1. Research Trends in Mathematical Models of Interaction of BD and Therapists

Based on Figure 2 a and Figure 3 , it can be seen that research on this topic is still very small and its development has decreased. In Figure 2 a, research in 2014 in the form of dark blue dots (dynamic, emotion, depression, cbt group) is still far more than the research in 2019 in the form of yellow dots (mood, mental illness, risk, person). While in Figure 3 , it can be seen that the number of articles published from 2009 to 2022 decreased. This can give new researchers an opportunity to research this topic in more depth, with a higher publication value because of its high novelty. According to Siqi Xue et al. [ 57 ], the COVID-19 pandemic has posed significant challenges to healthcare globally, and individuals with BD are disproportionately affected. Individuals with BD will experience poorer physical and mental health than normal people because several risk factors associated with BD, including impaired social rhythm, risk-taking behavior, substantial medical comorbidities, and common substance use, can be exacerbated by lockdown, social isolation, and decreased preventive and maintenance care in the face of the COVID-19 pandemic. Research on this topic during a pandemic is an additional challenge and could become a new topic linking mitigation strategies for working with individuals with BD in clinical and research contexts with a focus on digital medicine strategies to improve quality and accessibility to services. It is hoped that the development of research on this topic will increase further because it is not affected by environmental conditions due to the COVID-19 pandemic.

In Figure 4 , it can be seen that the development of publications on the topic of the mathematical model of the interaction of BD and therapy is dominated by the USA and European countries. This is closely related to the high number of BD sufferers in the USA and European countries, thus increasing the interest of researchers in these countries in this topic. From the data on the https://ourworldindata.org [ 58 ] website accessed on 17 July 2022, you can see a graph of the total number of patients with BD in the USA, UK, France, Switzerland, and Norway, measured for both sexes and all ages. These figures provide an accurate estimate (beyond reported diagnoses) of the number of patients with BD based on medical, epidemiological, survey, and meta-regression modeling data.

To obtain more article data, it can be done with other data sources such as Crossref, Web of Science, and Microsoft Academic. So, hopefully the analysis and findings can be more specific. In addition, articles that are not only in English but also in other languages such as Spanish, French, German, Russian, Arabic, and others, are also considered to expand the range of articles so that the research findings are more comprehensive.

4.2. Therapeutic Analysis and Psychiatric Diagnosis

In this section, we discuss the results of the analysis obtained from the literature review ( Table 2 ) of the 10 articles with the top citations. The main topics covered in the 10 articles are psychiatric therapy and diagnosis.

Andrews et al. [ 32 ] described major depressive disorder (MDD) and its treatment with antidepressant drugs. Episodes of major depressive disorder (MDD) have five of the following nine symptoms: (1) depressed mood; (2) anhedonia; (3) a significant decrease (or gain) in weight or appetite; (4) insomnia (or hypersomnia); (5) psychomotor retardation (or agitation); (6) fatigue or loss of energy; (7) feelings of worthlessness or guilt; (8) reduced ability to concentrate; and (9) repeated thoughts of death (not just a fear of death), or thoughts or actions of suicide. In this article, MDD is given therapy in the form of anti-depressants, and the effect is seen on the symptoms that arise.

KN Fountoulakis et al. [ 34 ] selected treatment specifically for acute mania, mixed episodes, acute bipolar depression, maintenance phase, psychotic and mixed features, and anxiety; cycles were immediately evaluated with regard to drug efficacy. The drug therapy used for BD includes anticonvulsants, antidepressants, antipsychotics, lithium, mood stabilizers, and others, which are complemented by clinical trials of each of these drugs.

In U.-M. Hemmeter et al. [ 36 ], the acute response to sleep deprivation (SD) is an investigation of the basic neurobiological mechanisms underlying depression and antidepressant treatment. Focusing on the neurobiology of depression and the discovery of the integration of new methods in psychiatric research, such as neuroradiology and neuroendocrinology, a number of new findings have been developed on brain function in depression, which touch on the relationship between SD and depression. The therapy used is anti-depressant drugs along with sleep therapy.

The cognitive therapy discussed in the article by Holmes et al. [ 39 ] focused on a new image (MAPP; Mood Action Psychology Program) targeting mood instability and applied the measurement method in a non-concurrent multiple base design case series of BD. After that, treatment innovation was carried out with the aim of detecting an increase in BD mood stability. These innovations can be in the form of pharmacological or psychological treatments carried out together or individually.

From the several therapies above, it is concluded that drug therapy always accompanies other therapies such as sleep therapy, cognitive therapy, MAPP, and others. This is because a person with BD has a lack of neurotransmitter substances in their brain [ 59 ], so drug therapy is the main therapy and other therapies are complementary therapies. Nonetheless, cognitive behavioral therapy, family-focused therapy, and psychoeducation offer the strongest efficacy in terms of relapse prevention, while interpersonal therapy and cognitive-behavioral therapy may offer more benefit in treating residual depressive symptoms [ 60 ].

The most widely discussed psychiatric diagnosis in the 10 articles above is depression symptoms, which is one of the symptoms of mental disorders [ 61 ]. To diagnose symptoms of depression, you can use the DSM-5 (Diagnosis and Statistical Manual of Mental Disorders) [ 8 ]. In addition to the article Wedge et al., the diagnosis made in mental disorders is to identify biomarkers or neurological signs of mental illness and fluctuating symptoms through neuroimaging and electroencephalography (EEG) [ 1 ]. The most common diagnoses were anxiety disorders (15.8%), followed by depression (6%) and somatoform disorders (5.6%) using the Primary Care Evaluation of Mental Disorders (PRIME-MD) Patient Health Questionnaire (PHQ), or abbreviated PRIME-MD PHQ [ 62 ].

4.3. Research Development in Mathematical Models of Interaction of BD and Therapists

This research has limitations, because it only knows about which countries do the most BD research, what topics are discussed the most, the number of citations in articles, and the tendency for the number of articles to continue to decline every year especially with the COVID-19 pandemic. Although the time period taken is 24 years from 2009 to 2022, the development of this research has a decreasing trend. In this research, there was no discussion of more specific BD problems regarding response to therapy, so it is necessary to study more deeply what things must be considered so that therapy can be chosen that can make mood stability from BD occur fast.

The interesting thing that can be developed from this research is to offer a client therapist model into a more general mathematical model of the interaction between therapist and patient with BD, so that the developed model describes mood stability from BD to be stable fast.

5. Conclusions

In this study, a systematic literature review is presented on the mathematical model of the interaction between bipolar disorder and therapists. There were 370 articles obtained after screening scientific publications from data sources Science Direct, Dimension, and Google Scholar. After going through the selection of duplicates, book chapters, titles, and abstracts, 52 selected articles were obtained. From these 52 articles, they were then selected based on the most words (quotes), the most topics, and the countries with the most publications. There are very few publications on the topic of BD mathematical models and therapists. The most publications were in 2012, with as many as nine scientific articles. At least in 2013 and 2018, there was one scientific article. Overall, the development of research on this topic tends to decline. This is possible because of the difficulty of research on mental health, especially BD.

In this study, bibliometric analysis was also carried out on the data in Table 2 , so that two main discussions were obtained, namely therapy and psychiatric diagnosis. From the data in Table 2 , six articles discuss therapy and four articles diagnosis. The therapies discussed can be in the form of drug therapy, cognitive therapy, and therapeutic innovation, while the diagnoses discussed were severe mental symptoms, mood disorders, mental disorders, and BD.

The use of bibliometric mapping techniques can reveal a general picture of existing themes and their changes over time. Based on bibliometric results, information reveals that the development of publications on the topic of the mathematical model of BD interaction and therapy is dominated by the USA and European countries. This is closely related to the high number of BD sufferers in the USA and European countries, thus increasing the interest of researchers in these countries in this topic.

This research can show that there are very few publications on the mathematical model of the interaction between BD patients and therapists, and that the number tends to decrease every year. This has the author’s attention, as well as a new hope for this research. Nevertheless, this research is expected to continue to grow because the need for mental health knowledge is very much needed, especially during the current COVID-19 pandemic. It is hoped that with SLR and bibliometric, other researchers can see which topics are still vacant so that they can conduct more in-depth research on these topics. In addition, by knowing the topic that is the main discussion, it is hoped that future researchers can conduct research by developing appropriate therapeutic techniques and diagnosing BD so that their quality of life improves and their safety is guaranteed.

The purpose of this research is the success of therapy and to simulate the success of therapy strategies. From this literature study, it has implications for choosing the right therapeutic method, so that BD mood can stabilize fast. The best strategy will be made in determining the best combination between drugs and psychotherapy, as well as making qualitative research into quantitative ones.

Acknowledgments

Thanks to RISTEKDIKTI 2021 through the Doctoral Dissertation Research Grant and tuition fees are funded by Degree By Research LIPI.

Funding Statement

This research is funded by Universitas Padjadjaran with contact number 2064/UN6.3.1/PT.00/2022.

Author Contributions

Conceptualization, I.N. and N.N.; methodology, I.N.; software, I.N.; validation, N.A., N.N., and Y.R.; investigation, N.N.; data curation, N.N. and Y.R.; writing—original, I.N.; writing—review and editing, I.N.; visualization, I.N.; supervision, N.A., N.N., and Y.R.; project administration, N.A.; funding acquisition, N.A. All authors have read and agreed to the published version of the manuscript.

Institutional Review Board Statement

Informed consent statement, data availability statement, conflicts of interest.

No conflict of interest.

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A Literature Review Is Not:

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• a grouping of broad, unrelated sources
• a compilation of  everything  that has been written on a particular topic
• literature criticism (think English) or a book review

So, what is it then?

A literature review is an integrated analysis-- not just a summary-- of scholarly writings that are related directly to your research question.  That is, it represents the literature that provides background information on your topic and shows a correspondence between those writings and your research question.

A literature review may be a stand alone work or the introduction to a larger research paper, depending on the assignment.  Rely heavily on the guidelines your instructor has given you.

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A literature review is important because it:

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• Demonstrates why a topic is significant to a subject area.
• Discovers relationships between research studies/ideas.
• Identifies major themes, concepts, and researchers on a topic.
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• Discusses further research questions that logically come out of the previous studies.

1. Choose your topic, define your question

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A continuous approximation model for the electric vehicle fleet sizing problem

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• Brais González-Rodríguez 1 ,
• Aurélien Froger 2 ,
• Ola Jabali 3 , 4 &
• Joe Naoum-Sawaya   ORCID: orcid.org/0000-0002-4908-225X 1  

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Establishing the size of an EV fleet is a vital decision for logistics operators. In urban settings, this issue is often dealt with by partitioning the geographical area around a depot into service zones, each served by a single vehicle. Such zones ultimately guide daily routing decisions. We study the problem of determining the optimal partitioning of an urban logistics area served by EVs. We cast this problem in a Continuous Approximation (CA) framework. Considering a ring radial region with a depot at its center, we introduce the electric vehicle fleet sizing problem (EVFSP). As the current range of EVs is fairly sufficient to perform service in urban areas, we assume that the EV fleet is exclusively charged at the depot, i.e., en-route charging is not allowed. In the EVFSP, we account for EV features such as limited range, and non-linear charging and energy pricing functions stemming from Time-of-use (ToU) tariffs. Specifically, we combine non-linear charging functions with pricing functions into charging cost functions, establishing the cost of charging an EV for a target charge level. We propose a polynomial time algorithm for approximating this function and prove that the resulting approximation is exact under certain conditions. The resulting function is non-linear with respect to the route length. Therefore, we propose a Mixed Integer Non-linear Program (MINLP) for the EVFSP, which optimizes both dimensions of each zone in the partition. We strengthen our formulation with symmetry breaking constraints. Furthermore, considering convex charging cost functions, we show that zones belonging to the same ring are equally shaped. We propose a tailored MINLP formulation for this case. Finally, we derive upper and lower bounds for the case of non-convex charging cost functions. We perform a series of computational experiments. Our results demonstrate the effectiveness of our algorithm in computing charging cost functions. We show that it is not uncommon that these functions are non-convex. Furthermore, we observe that our tailored formulation for convex charging cost functions improves the results compared to our general formulation. Finally, contrary to the results obtained in the CA literature for combustion engine vehicles, we empirically observe that the majority of EVFSP  optimal solutions consist of a single inner ring.

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Acknowledgements

Aurélien Froger and Joe Naoum-Sawaya would like to acknowledge the financial support of the Department of Elettronica, Informazione e Bioingeneria of Politecnico di Milano that made this collaboration possible. Joe Naoum-Sawaya and Brais González-Rodríguez are also supported by NSERC Discovery Grant RGPIN-2017-03962 and NSERC Discovery Grant RGPIN-2024-04176.

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A mixed integer linear programming model to compute the minimum charging cost up to a target SoC

In this appendix, we present a mixed integer linear programming model that computes the minimum charging cost for a given target SoC \(\bar{r}\in [{r^{\text {min}}},1]\) . To do this, we use the same notation for the \(SoC_{{r^{\text {min}}}}\) and Price functions that in Sect.  3.2 . Thus, we denote by \(\mathcal {B}=\{0, \dots , B\}\) the set of breakpoints of the charging function \(SoC_{{r^{\text {min}}}}\) , \(a_b\) is the charging time of the breakpoint \(b\in \mathcal {B}\) . The Price function is defined by a set of time periods denoted by \(\mathcal {P}=\{1,\dots ,P\}\) , the duration \(\Delta _p\) and the energy cost \(\Gamma _p\) per kWh for each \(p\in \mathcal {P}\) . For convenience, we denote by 0 a dummy time period that precedes the first time period in \(\mathcal {P}\) .

We introduce the following variables. Variable \(x_p\) is the SoC of the vehicle at the end of each time period \(p\in \mathcal {P}\cup \{0\}\) . Variable \(\Psi _p\) indicates how long we charge in time period \(p\in \mathcal {P}\) . We model the function \(SoC_{{r^{\text {min}}}}\) using SOS2 sets with the following continuous variables. Variable \(\alpha _{pb}\) is the weight of breakpoint \(b\in \mathcal {B}\) in function \(c_{{r^{\text {min}}}}\) for the SoC at the end of period \(p\in \mathcal {P}\) .

With this notation, we present model \(\text {CM}(\bar{r})\) below:

CM( \(\bar{r}\) ) minimizes the total charging cost required to charge an empty battery to a target SoC \(\bar{r}\) . Constraints ( 46 ) and ( 47 ) ensure that we charge to a target SoC \(\bar{r}\) from a SoC equal to \({r^{\text {min}}} \) at the beginning of the EV charging time interval. Constraints ( 48 )–( 50 ) model the piecewise linear charging cost function. Constraints ( 51 ) model the charging duration at time period \(p\in \mathcal {P}\) . Finally, constraints ( 52 )–( 54 ) define the domain of the decision variables.

Data for the example

Charging cost functions \(\hat{c}_{0}\) (plain line) and \(c_{0}\) (dashed line)

An example where \(\hat{c}_{r^{\text {min}}} \) is not equal to \(c_{r^{\text {min}}} \)

The functions \(\hat{c}_0\) and \(c_0\) for the instance described in Fig. 11 , are shown in Fig. 12 . In Table  11 we show the charging schedule and its cost for every target SoC that is an x-value of the breakpoints of \(\hat{c}_0\) . Additionally, in italic font between the dashed lines, we show the target SoC at which \(\hat{c}_0\) starts to differ from \(c_0\) . From this point, we observe that the charging schedule for \(c_0\) cannot be obtained by an extension of the charging schedule obtained for the previous point. This also occurs from a target SoC equal to 0.8991. It should be noted that the difference between \(\hat{c}_0(r)\) and \(c_0(r)\) is not monotonically increasing as r increases.

A comparison between Algorithm 1 and a greedy charging algorithm

We refer to the greedy procedure that charges the EV by considering the time periods in non-decreasing order of their energy cost (generally, this order is not chronological) as the greedy charging algorithm. We first show that for a given target SoC, the greedy algorithm builds a charging schedule with a higher cost than Algorithm 1. Considering Example  3.1 , let us assume that the EV should be charged from \({r^{\text {min}}} =0\) to a target SoC equal to 0.82. From Fig.  4 a, we know that this requires a total charging duration equal to 6.6. The question is how to schedule this duration over the three time periods of the price function.

If we apply the greedy charging algorithm, we build the charging schedule incrementally, starting by considering the cheapest time period, that is, period 2. The EV is charged for the whole period 2 (i.e., for a duration equal to 3), and its SoC at the end of it is 0.5273. Because the target SoC is not reached, the charging schedule is expanded by charging during time period 1, which is the second cheapest time period. To get a SoC of 0.82 at the end of period 2, charging the EV for a duration equal to 3.6 in period 1 is required. Note that as soon as the EV is charged for longer than 0.3 during period 1, the initial quantity charged during period 2 decreases (but not the charging time during period 2 that stays equal to 3) because part of the charging is now carried out with a lower charging rate (that is the rate associated with the second segment of the function \(SoC_0\) in Fig.  4 a). The resulting schedule increases the SoC of the EV by 0.6018 and 0.2182 in time periods 1 and 2, respectively. According to \(\tau R(0.6018\Gamma _1 + 0.2182\Gamma _2)\) , the resulting cost of this schedule is 12.2011.

Algorithm 1 builds the charging schedule incrementally in a different way. Similarly to the greedy algorithm, it first starts charging the EV for the whole period 2. The charging schedule is then expanded by charging the EV during time period 1 so that the SoC reached by the end of period 2 is 0.58. This amounts to a charging duration equal to 0.3 during period 1. Then, the EV is charged for a duration of 3.3 during period 3 to reach the target SoC of 0.82. The resulting schedule is due to increasing the SoC of the EV by 0.0527, 0.5273, and 0.24 in time periods 1, 2, and 3. According to \(\tau R(0.0527\Gamma _1 + 0.5273\Gamma _2 + 0.24\Gamma _3)\) , the resulting cost of this schedule is equal to 10.3330.

We now show that the greedy algorithm overestimates the charging cost function for any target SoC in (0.58, 1]. In Table 12 , we show the breakpoints that would be obtained by using the greedy charging algorithm for Example  3.1 . Using the notation of the paper, the value \(\omega _p\) is the time during which the EV charges in time period p . The value \(SoC_0(\omega _{1,p})\) is the SoC of the EV at the end of time period p . The value \(\dfrac{\phi _p(\varepsilon )}{\varepsilon \overline{\lambda }(P)}\) is the cost increase if we charge during time period p to increase the target SoC. The value \(p^{\star }\) is the time period selected by the algorithm to expand the current schedules. Additional charge time is added during the \(p^{\star }\) period to increase the SoC reached by the end of the charging time interval.

Compared to the result obtained using Algorithm 1, in Fig.  13 , we observe that the greedy charging algorithm is sub-optimal, i.e., for a given target SoC that is larger than 0.58, the charging cost is overestimated. Moreover, the function \(\hat{c}^{\text {greedy}}_{0}\) obtained by the greedy charging algorithm is non-convex, whereas the optimal charging cost function \(\hat{c}_{0}\) is convex. What happens is that when the target SoC becomes larger than 0.58, then charging an extra amount of energy in time period 1 is more costly than charging it in time period 3, although the energy price is lower in time period 1. This is due to the fact that charging in time period 1 reduces the quantity of energy charged in time period 2 due to the concavity of function \(SoC_0\) , and this is not accounted for by the greedy algorithm. Specifically, if the target SoC is 1, we observe that the EV should be charged during the whole of period 2 and that this corresponds to charging \((0.7036{-}0.3515)\) =0.3521kWh using Algorithm 1 ( \(\hat{c}_{0}\) ) and \((0.8412{-}0.6310)\) =0.2102kWh using the greedy algorithm ( \(\hat{c}^{\text {greedy}}_{0}\) ).

Charging cost functions \(\hat{c}_{0}\) (black line) and \(\hat{c}^{\text {greedy}}_{0}\) (gray line) in Example  3.1 (color figure online)

Computing the tour length in each zone using an estimator based on [ 3 ]’s theorem

As discussed in Sect.  2.2 , estimating the minimum traveling salesman (TSP) tour length \(L^*\) to visit a set of N points has been investigated extensively in past research. In this respect, a famous result is the theorem introduced by [ 3 ] regarding the asymptotic behavior of \(L^*\) for the Euclidean distance metric. Points are assumed to be uniformly distributed over a service region of area \(A:\lim _{N\rightarrow \infty } L^*/\sqrt{N}=\beta \sqrt{A}\) . This led to the introduction of \(L^*\) estimators of type \(a+b\sqrt{NA}\) with \(a,b\in \mathbb {R}\) , which have been shown empirically to give accurate estimates for both Euclidean and non-Euclidean instances [ 9 , 38 ]. As in Newell and Daganzo’s approximation, Beardwood et al.’s approximation assumes customers to be uniformly distributed with density \(\delta \) across the service region. In our problem, we consider multiple capacitated vehicles (where capacity refers to the battery charge) that are routed from a central depot assuming a ring-radial topology, which is similar to several works in the literature [ 24 ]. However, the Beardwood et al. formula, which approximates a single tour length, is commonly used in districting applications to approximate the routing distance within a district. In such cases, the route is composed of the distance between the depot and the district (i.e., zone), and the routing distances within the district are approximated via Beardwood et al.’s formula (see [ 2 ], for an example). We apply a similar logic in what follows.

In the context of the problem discussed in this paper, the region should be partitioned into sector-shaped zones in the inner ring and trapezoid-shaped zones in the outer rings. In order to derive the length of the vehicle route in a zone taking into account the fixed location of the depot, i.e., the route starts and ends at the depot, we decompose the total route distance traveled into two components that include the linehaul distance and the route length within the zone. For a sector-shaped zone \(j\in \mathcal {J}_0\) , its area is \(\theta _j(l_0L)^2\) , and the number of customers to visit is approximated by \(\theta _j(l_0L)^2\delta \) given that the customers are uniformly distributed with a density \(\delta \) over the service region. Thus, the length of a vehicle route in the inner ring is approximated as \(a+b\sqrt{\theta _j(l_0L)^2\delta \theta _j(l_0L)^2} = a+b\theta _j(l_0L)^2\sqrt{\delta }\) (note that the linehaul distance is equal to 0). Similarly, for a trapezoid-shaped zone ( i ,  j ), its area is \(2w_{ij}l_iL\) and the number of customers to visit is approximated by \(2w_{ij}l_iL\delta \) . Thus, the length of a vehicle route in the outer ring is approximated by \(a+b\sqrt{\big (2w_{ij}l_i\delta \big )2w_{ij}l_i}+2L\sum _{i^\prime =0}^{i-1}l_{i^\prime }=a+2bw_{ij}l_iL\sqrt{\delta }+2L\sum _{i^\prime =0}^{i-1}l_{i^\prime }\) , where \(2L\sum _{i^\prime =0}^{i-1}l_{i^\prime }\) is the linehaul distance. We summarize in Table 13 the similarities and differences between estimating the tour length using [ 13 ] and [ 3 ] approximations. We also refer the reader to [ 31 ] and [ 38 ], which provide empirical evaluations of the accuracy of both approximations, as well as to [ 43 ] that provide validation of continuous approximation models against real-world data.

While CA-EVS has been developed based on the tour length approximation model of [ 13 ], the framework presented in this paper is relatively general and can accommodate other approximation models. In particular, we show next that CA-EVS can be easily modified to use [ 3 ]’s approximation. The only change that is required are Constraints ( 9 ) and ( 10 ) in CA-EVS (and by extension in CA-EVSS) which are replaced by

Constraints ( 28 ) and ( 29 ) in CA-EVSC are also replaced by

Furthermore, when using [ 3 ]’s approximation, we can derive a similar result as Proposition  3.7 with \(\sigma =\lceil \frac{b\pi L^2\sqrt{\delta }}{R(1-{r^{\text {min}}})-a}~\rceil \) . The proof follows the same line as the original proof. The upper bound on CA-EVS can also be computed as described in Sect.  4.3.1 .

Next, we assess the computational performance of the models when [ 3 ]’s approximation is used. In particular, we evaluate CA-EVSS with [ 3 ]’s approximation using the same set of instances that are solved to optimality in Table  8 of Sect.  5.5 (with \({r^{\text {min}}} =0\) ). Specifically, based on [ 3 ]’s theorem, the tour length to visit N customers uniformly distributed in a zone of area A can be estimated by a linear regression model \(a+b\sqrt{NA}\) with \(a,b\in \mathbb {R}\) . [ 38 ] generated 90 instances with 10, 20, and 50 customers uniformly distributed over squares of sizes 1x1, 2x2, and 5x5. In their experiments, setting \(a=1.564\) and \(b=0.904\) leads to a linear regression model with a R-squared value equal to 96% and a mean absolute percentage error (MAPE) of 15.43% (see Figure 3 in [ 38 ]). We assume these same values in our experiments.

In Table  14 , we present a similar analysis as in Table  8 . First, we notice that all nine instances are solved to optimality according to LB and UB. The running times when computing the lower bound by solving CA-EVSC with a convex approximation of \(c_{{r^{\text {min}}}}\) are much lower in Table  14 than those in Table  8 . However, when directly solving CA-EVSS, three instances are solved to optimality, compared to six instances solved to optimality in Table  8 . Thus, computing the lower bound by solving CA-EVSC with a convex approximation has a significant impact on improving the computational performance when [ 3 ]’s approximation is used.

Finally, in Table  15 , we present a comparison of the charging cost and the total cost of the solutions. We see that neither of the models presents strictly better results than the other in terms of the charging cost or the total cost. This is not surprising, given that the approximations are based on different metrics, as well as all the differences summarized in Table  13 .

In summary, this section shows that the framework that we present in this paper is fairly generic and can accommodate different approximation models for the optimal tour length. The computational results in terms of computational performance, number of partitions, and the associated charging costs are however dependent on the approximation. As such, future research may potentially investigate approximations that are best suited for the framework that is presented in this paper.

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González-Rodríguez, B., Froger, A., Jabali, O. et al. A continuous approximation model for the electric vehicle fleet sizing problem. Math. Program. (2024). https://doi.org/10.1007/s10107-024-02141-9

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DOI : https://doi.org/10.1007/s10107-024-02141-9

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