Research Article: A Markov Chain Monte Carlo Approach to Estimate AIDS after HIV Infection

Date Published: July 6, 2015

Publisher: Public Library of Science

Author(s): Ofosuhene O. Apenteng, Noor Azina Ismail, Dimitrios Paraskevis.


The spread of human immunodeficiency virus (HIV) infection and the resulting acquired immune deficiency syndrome (AIDS) is a major health concern in many parts of the world, and mathematical models are commonly applied to understand the spread of the HIV epidemic. To understand the spread of HIV and AIDS cases and their parameters in a given population, it is necessary to develop a theoretical framework that takes into account realistic factors. The current study used this framework to assess the interaction between individuals who developed AIDS after HIV infection and individuals who did not develop AIDS after HIV infection (pre-AIDS). We first investigated how probabilistic parameters affect the model in terms of the HIV and AIDS population over a period of time. We observed that there is a critical threshold parameter, R0, which determines the behavior of the model. If R0 ≤ 1, there is a unique disease-free equilibrium; if R0 < 1, the disease dies out; and if R0 > 1, the disease-free equilibrium is unstable. We also show how a Markov chain Monte Carlo (MCMC) approach could be used as a supplement to forecast the numbers of reported HIV and AIDS cases. An approach using a Monte Carlo analysis is illustrated to understand the impact of model-based predictions in light of uncertain parameters on the spread of HIV. Finally, to examine this framework and demonstrate how it works, a case study was performed of reported HIV and AIDS cases from an annual data set in Malaysia, and then we compared how these approaches complement each other. We conclude that HIV disease in Malaysia shows epidemic behavior, especially in the context of understanding and predicting emerging cases of HIV and AIDS.

Partial Text

Acquired immune deficiency syndrome (AIDS), caused by infection with human immunodeficiency virus (HIV), is one of the most alarming and deadly diseases in human history. The total number of people living with HIV and AIDS in 2013 was 35 million [1]. The spread of AIDS through populations has caused panic and economic disturbance. In the last three decades since the emergence of AIDS, many interdisciplinary scientific efforts have coalesced to model the spread of this disease. In 1989, Hyman and Stanley [2] generated mathematical models based on the underlying transmission mechanisms of HIV/AIDS that were used to understand and anticipate its spread in different populations. In 1991, Romieu et al. [3] presented work demonstrating how to model the spread of HIV/AIDS in Mexico City; the goal of their work was to provide a conceptual framework to help understand the transmission dynamics of infection and give a reasonable estimation of the short-term cumulative number of AIDS cases. Mathematical modeling of the spread of HIV/AIDS has become even more useful in the modern era of AIDS research. In 2011, Nyabadza [4] presented a simple deterministic HIV/AIDS model that applied ordinary differential equations to the current South African situation and considered the use of condoms, sexual partner acquisition, behavior change, and treatment; their results suggested that HIV/AIDS could be controlled through these measures. Naresh [5] calculated the spread of the AIDS epidemic with immigration among HIV-infected individuals, and the findings revealed a constant flow of immigrating susceptible individuals and individuals infected with HIV. Merli [6] presented an exploration of the implications of patterns of sexual behavior for the spread of HIV in China; this model reflected the uncertainty surrounding key parameters, and the analyses used showed a range of possible outcomes. In 1999, Kakehashi formulated a mathematical model to describe the spread of HIV/AIDS among adult commercial sex workers in Japan that was used to analyze the effect of introducing HIV-infected commercial sex workers into a population without HIV [7]. De Arazoza and Lounes (2002) outlined how a non-linear model could be used to develop an epidemic with contact tracing, specifically in Cuba. These authors suggested that to control the spread of HIV/AIDS, the target group must be in contact with individuals who carry HIV [8]. In 2008, Kim [9] formulated a simple continuous model for the transmission of HIV, although this model failed to consider the demographic parameters that have a significant impact on modeling the spread of HIV. Furthermore, most of these previous models have serious drawbacks. For instance, most of these models failed to demonstrate how the impact of AIDS causes the death of HIV-infected individuals. These models also typically describe changes in time and are therefore referred to as ‘dynamic’ models, where time is the independent variable. Similar work was conducted by Haario et al. (2006) [10], who proposed various strategies to combine two quite powerful ideas in the Markov chain Monte Carlo method (MCMC), adaptive Metropolis samplers and delayed rejection, to study the spread of algae.

We present the simplest HIV disease models where individuals classified as a sexually active population are divided into four classes: susceptible, S(t); infected, (HIV) I(t); pre-AIDS cases who did not progress to AIDS after HIV infection, A1(t); and AIDS cases who have AIDS after HIV infection at time t, A2(t). HIV can be transmitted to a susceptible person when they come into contact with an infected person via the appropriate transmission routes. In 2003, Rao [14] formulated a model for individuals who did or did not develop AIDS after the HIV epidemic in India. Unlike the model from this report [14], our model assumed that γ is the rate at which an individual will fully move from A1(t) class to A2(t) class, which is a very significant indicator of when an intervention should be introduced. We assumed that the infected individuals are capable of having children that are either infected with HIV or will not have HIV. However, the susceptible class has a recruitment rate equivalent to the birth rate, b, which is independent of vertical transmission. Moreover, this model assumes that infected newborn babies enter the HIV class at the rate of b(I + A1 + A2), for which we assume that I, A1, and A2 are sexually active, and πb(I + A1 + A2) are individuals who are infected and enter the HIV stage. The portion π of these individuals is assumed to be infected with HIV (categorized in the I class), and the complementary portion (1 − π)b(I + A1 + A2) is considered susceptible (and moves to the susceptible class S). The removal rate of infected HIV individuals who enter the AIDS class is represented by α; the portion of HIV-infected individuals is δ. This model also assumes that at rate δα, some of the HIV-infected cases transition to the AIDS group, whereas the remaining HIV-infected cases move to the class of individuals who do not develop AIDS (pre-AIDS) after an HIV infection rate of (1 − δ)α, where 0 ≤ δ ≤ 1. The model also assumes the natural death rate μ of individual deaths from all four compartments. β is the contact rate between susceptible individuals and exposed or HIV-infected individuals. AIDS patients are given an additional disease-induced mortality rate: σ > 0, ε > 0 and ρ > 0 for I(t), A1(t) and A2(t), respectively. This form of a susceptible–infected–pre-AIDS–AIDS (SIA1A2) model can be used to model HIV disease based upon the assumption that once an individual becomes infected, that individual remains infectious for life, as shown in Fig 1.

This section discusses the various results obtained using Malaysian data to analyze the accuracy of our model. Ten parameters were used to enable us to determine the inaccuracies of the model and to obtain better graphical representations.

This study demonstrates how to model the spread of AIDS after HIV infection. As with any modeling study of such a complex system as HIV and AIDS, several assumptions were necessary to make the analysis tractable. We assumed that there were sexual interactions between the susceptible and HIV-infected populations, that infected newborn babies moved directly to the HIV class and that a fraction of the remaining population also moved to the susceptible class to increase the growth of the total population. The model also assumed that γ is the rate at which an individual will fully move from A1(t) class to A2(t) class; this rate was not considered by the model reported in a previous study [14]. HIV/AIDS continues to infect the susceptible population if no control measures are swiftly enacted, and the endemic point, if it existed, could have been stable if all the eigenvalues were negative. The reproduction number R0 is a threshold value or number that determines the stability of the disease-free equilibrium. If R0 > 1, then an epidemic of AIDS occurs, and if R0 < 1, then the disease-free equilibrium is locally asymptotically stable and disease becomes endemic. Our results show that the disease-free steady state is unstable because the basic reproduction number R0 was 13.22657. These results show that the number of HIV cases and AIDS cases is still epidemic within the Malaysian population, and this will have policy implications for the most at-risk groups of populations, especially the HIV-infected population (Fig 2 and Fig 3). The public health implication of this instability is that HIV will continue to infect the susceptible population because in the rate of newborn babies b(I + A1 + A2), b is the parameter with the highest value compared with the other parameters. Thus, there must be effective intervention measures that will continue to minimize the spread of the HIV epidemic within the unaffected population. Furthermore, there must be effective ways to minimize the spread of pre-AIDS A1(t) cases that progress to AIDS after HIV infection, especially the rate of newborn babies b(I + A1 + A2), because this had the highest impact on disease spread and indicated that more infected HIV/AIDS individuals are born at these stages than at the other stages. Our results further suggest that without the intervention of antiretroviral medication (drug treatment), the rate γ at which an individual will fully move from the A1(t) class to the A2(t) class is 0.99/year. This information will assist policymakers in deciding at which stage to introduce intervention measures. The analysis presented herein with MCMC can be applied to a large class of HIV/AIDS epidemic models by taking into account both the uncertainty in the model parameters and other characteristics of the target posteriors by generating chains of samples. Contrasts were found as the posterior standard deviations exceeded the standard errors, as shown in Table 2 and Table 3. The graphical descriptions further demonstrate and support the empirical results and the long-term model prediction. We conclude that the predictive distributions generated predicted the model to a large degree of accuracy, as shown in Fig 6. Finally, there were some significant differences in the estimated parameters that will be useful to public health, potentially representing a practical and more effective way to epidemiologically model AIDS disease after HIV infection.   Source: