Keywords

1 Introduction

HIV is one of the leading causes of death globally, and it is a significant public health challenge for all around the world (Joint United Nations Programme on HIV/AIDS 2012). In 2017, approximately 37 million people were living with HIV (PLWH), and this number is increasing with 5,000 new infections every day. To decrease the number of new infections and eventually eliminate HIV as a public health threat, the Joint United Nations Programme on HIV/AIDS (UNAIDS) prepared a new strategy intending to eliminate HIV by 2030. Although some targets such as the number of AIDS-related deaths and the number of new infections have fallen from their peaks, there is still a gap between targets to accomplish this goal and global realities (UNAIDS 2018). Moreover, significant variation exists for these statistics among regions. For example, Sub-Saharan Africa is the most severely affected region by HIV epidemic, and it has the most robust reductions in the AIDS-related mortality and the new infections while HIV epidemic is expanding in the Eastern Europe and several countries of Asia. (UNAIDS 2018; UNAIDS 2012; Piot and Quinn 2013). Overall, there is a need for further research and better strategies in the area of HIV prevention to reach UNAIDS goals and eliminate HIV.

HIV risk groups can be defined as subpopulations that have a higher risk for transmission and/or acquisition of HIV. In the low prevalence countries, risk groups are responsible for the majority of HIV infections. People who inject drugs (PWID, also known as IDU), sex workers (SW), transgender people, prisoners, and gay men and other men who have sex with men (MSM) and their sexual partners are among these key populations (UNAIDS 2017). The HIV prevalence among MSM in capital cities is 27 times higher than that in the general population, and recent data show that there is a rising trend in MSM HIV prevalence. PWIDs are among the population groups that are most severely affected by HIV, and the risk for acquiring HIV is 23 times higher for this group (UNAIDS 2012; UNAIDS 2018).

The recommended treatment for HIV is known as antiretroviral therapy (ART). In 2017, around 22 million, which is 59% of PLWH have access to ART globally (UNAIDS 2018; UNAIDS 2018). Health care organizations and governments have been endeavoring to fight against the disease, and there are some promising signs like advancing ART coverage in recent years. However, considering years of significant efforts, a large percentage of PLWH or people at risk for HIV have yet to have access to prevention, care, and treatment, and there is still no cure (UNAIDS 2018).

A reduction in HIV incidence has been a top priority to control the disease (Piot and Quinn 2013). It requires both to keep tracking of data available and use them in estimations for the future of the epidemic. In addition to this, we need to understand HIV behavior over time and evaluate the prevention efforts in a way that make the outcomes as useful as possible. With the significant increase in the modeling studies about HIV prevention in the last decade, it is apparent that mathematical models have become valuable and vital tools in analyzing the spread of infectious diseases, prevent and control these diseases. Models can be used for analyzing and testing theories, answering specific questions, figuring out the transmission characteristics, and key parameters from data for many infectious diseases (Hethcote 2000). Similarly, there has been a wide variety of mathematical models of HIV to understand disease dynamics and offer better prevention strategies.

In this study, we briefly discussed key methods of mathematical modeling for infectious diseases that included both deterministic and stochastic models. We described and focused on mainly three modeling techniques: (i) Bernoulli process models, (ii) dynamic compartmental models, and (iii) Markov models. We conducted a thorough literature search and summarized critical examples for each model. Although we have not included agent-based simulation in the modeling methods section, we included studies applied agent-based simulation in the literature review table. Our goal is to summarize main infectious disease modeling methods, to list HIV-related modeling studies chronologically, and then for each selected study, to present the study objectives, the model type, populations and interventions included in the study as well as critical insights. Although this list of studies is not comprehensive, it provides a summary of significant studies, and it covers the majority of high-impact research on HIV prevention.

The remainder of this paper is organized as follows. In Sect. 2, we present three main types of modeling methods which are often employed for infectious diseases. In Sect. 3, we present a table of HIV-related mathematical modeling studies.

2 Modeling Methods

Models reduce the complexity of a system to its essential elements; in other words, they represent a simplification of reality at a sufficient level of detail. From the health care perspective, models create mathematical frameworks that are used for the estimation of the consequences for health care decisions (Caro et al. 2012). Therefore, they are essential tools to utilize in exploring the dynamics of HIV infection. Many mathematical models developed for HIV use individual-level data to attain population-level outcomes such as incidence and prevalence of infection (Sayan et al. 2017).

There are many different modeling approaches in the literature. However, three most widely used methods are included in this review, and these methods are Bernoulli process model, dynamic compartmental model, and the Markov model. Other methodologies, such as agent-based simulation models, stochastic models, and system dynamics, are also employed to analyze and project the dissemination of HIV (Akpinar 2012). While we have not included these techniques in this paper, we included studies that used them in Sect. 3. We conducted a search of 3 electronic databases, including PubMed, Web of Science, and Google Scholar for relevant studies published in English from the earliest data available for the database to January 2019. We used a broad search strategy with appropriate keywords and Medical Subject Heading (MeSH) terms to identify HIV mathematical modeling studies. We included the keywords “HIV,” “AIDS,” “Mathematical Modeling,” “Bernoulli Model,” “Compartmental Model,” “Markov Model,” “Agent-Based Simulation” with Boolean operators ‘OR’ and ‘AND.’ From the articles and conference papers obtained through the electronic search, we screened based on the abstracts and titles, and we included HIV studies if they were original analyses; included a mathematical model as its main methodology and reported the model results. When we encountered two similar studies, we preferred to include more recent study. We summarized the selected studies in Table 1, where they are grouped based on the modeling technique, and they are ordered chronologically within each group.

2.1 Bernoulli Process Model

Bernoulli process model is based on the assumption that there is a certain probability of transmission each time an HIV infected person engages in some of HIV risk behaviors such as unprotected sex while each type of behavior is considered an independent event. Each event has a small fixed probability of transmission, which is called HIV infectivity, and it accumulates with the repetition of the related risk behavior (Pinkerton and Abramson 1998).

The cumulative probability of transmission for multiple contacts with an infected person is calculated using the following equation. In this equation, α represents infectivity, while n is the number of contacts (Pinkerton and Abramson 1998).

$$ P = 1 - \left( {1 - \alpha } \right)^{n} $$
(1)

Today, among people living with HIV, about 9.4 million people do not know their HIV status (UNAIDS 2018). HIV may not show any symptoms for several years. Thus, in many cases, people do not know whether their partner is HIV infected or not. To include this uncertainty to the model, the following equation is proposed and the prevalence of infection, denoted with π, is added as a coefficient that reflects the probability of selecting an infected partner (Pinkerton and Abramson 1998).

$$ P = \pi \left[ {1 - \left( {1 - \alpha } \right)^{n} } \right] $$
(2)

Infectivity can be affected by various factors. Protected sexual intercourse is one of the prevention methods, and it reduces the infectivity of HIV. To provide a cumulative probability for a more complex situation, we can include m different sexual partners with ki protected, and ni unprotected sexual contacts, respectively. The model becomes following general form for the situation with multiple partners (Pinkerton and Abramson 1998).

$$ P = 1 - \prod\limits_{{{\dot{I}} = 1}}^{m} {\left\{ {1 - \pi \left[ {1 - \left( {1 - \alpha_{n} } \right)^{{n_{i} }} \left( {1 - \alpha_{k} } \right)^{{k_{i} }} } \right]} \right\}} $$
(3)

2.2 Dynamic Compartmental Model

Infectious diseases have been analyzed using dynamic compartmental models since 1927 (Akpinar 2012). In this type of models, populations divided into subgroups, which are called compartments. The main idea behind the construction of the model is that an infected person comes across with healthy individual and transmit the disease. There are many types of compartmental models such as SI, SIR, SIRS, SEIR, MSEIR, etc. selected based on different characteristics of infectious disease.

SIR compartmental model has three different compartments. In this model, S represents Susceptible, I represents Infected, and R shows the Recovered (or removed) (Fig. 1). For SIR model, we have transitions from S to I, which involves disease transmission and from I to R, which shows that infected patients recover from the disease. These movements occur at some defined rates known as infection rate and recovery or removal rate. To define the model with differential equations, a closed population that has no births and deaths, no migration is taken into account (Keeling and Rohani 2011).

Fig. 1.
figure 1

SIR model diagram (Keeling and Rohani 2011).

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SIR model is mathematically defined in the following equations where β is the transmission rate, and γ is removal or recovery rate (Keeling and Rohani 2011).

$$ \frac{dS}{dt} = - \beta SI $$
(5)
$$ \frac{dI}{dt} = - \beta SI - \gamma I $$
(6)
$$ \frac{dR}{dt} = \gamma I $$
(7)

2.3 Markov Model

Markov model is known as a stochastic process that has Markovian property. Markovian property is based on the assumption that the conditional distribution of any future state depends only on the present, and it is shown in the equation as follows (Ross 2007).

$$ P\left\{ {X_{n + 1} = j |X_{n} = i, X_{n - 1} = i_{n - 1} ,..,X_{1} = i_{1} , X_{0} = i_{0} } \right\} = P\left\{ {X_{n + 1} = j |X_{n} = i} \right\} = P_{ij} $$

We assume that \( X_{n} \) is state i at time n and \( P_{ij} \) represents the probability that process will move from state i to state j then, this equation defines a model for all states i, j and all n that are greater or equal to 0. (Ross 2007).

Many clinical situations can be described in terms of the conditions that individuals can be in, how they can move among such states, and how likely such moves are (Fig. 2). These correspond to states, transitions, and transition probabilities, respectively. State transition models (STM) are well suited to the decision problems in these situations, and they are found reasonable when the decision problem can be described in terms of states and interactions between individuals are not necessary. Markov models are one of the STM modeling approaches, and they are based on cohort level. Thus, Markov modeling approach presents transparency, efficiency, and ease of debugging. They are recommended if a manageable number of health states are sufficient to describe all relevant characteristics of the problem. These states help biological/theoretical understanding of the disease and reflect the disease process with transitions. Interventions such as screening, diagnostics, and treatment can be included in the model to evaluate their effects (Siebert et al. 2012).

Fig. 2.
figure 2

Markov model diagram (Siebert et al. 2012)

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3 HIV Mathematical Modeling Studies

Table 1. Literature review
Full size table

4 Conclusion

Many mathematical models have been developed and implemented in estimating and controlling infectious diseases. HIV is one of such diseases with an extensive literature of modeling studies that aimed at the prediction of HIV in different countries and target populations, the evaluation of prevention and treatment strategies in terms of their cost-effectiveness, the optimal allocation of HIV budgets and other many applications. This study summarizes 28 papers about HIV modeling that covers between 1991 and 2018. Different modeling methodologies applied in these studies have been presented, and these methods included the three most common techniques, which are the Bernoulli model, a dynamic compartmental model, and the Markov model. The majority of the papers focus on the economic analysis of prevention strategies, and mathematical modeling is employed to estimate the number of infections averted under different prevention strategies. Determining the incidence and prevalence of the disease in a specific location or for a target population is another popular study objective through understanding the spread of HIV. These studies often analyzed risk groups and their risky behaviors and estimated the health and economic outcomes to inform the decision-makers about the future of epidemic, possible interventions, and their consequences. There are some advantages and limitations of modeling methods used in the studies. Bernoulli model is a static approach while estimations from a compartmental model continue over some time due to its dynamic nature. However, compartmental models assume people in the same compartments are acting and responding to the disease in the same way. Markov models complement the HIV progression process by adding the stochastic framework into the problem. Agent-based simulation systems are useful and displays system in a more realistic way, however, they are data-intensive and cannot be generalized to other regions or populations. Although each methodology has some advantages and disadvantages, they are essential to understand the disease dynamics, and they provide us the opportunity to predict the future of the disease and find appropriate prevention and intervention methods.