PLoS One 10(7): e0131950. doi:10.1371/journal.pone.0131950

**A Markov chain Monte Carlo approach to estimate AIDS after HIV infection**

Ofosuhene O. Apenteng^{1} and Noor Azina Ismail^{1}

^{1}Department of Applied Statistics, Faculty of Economics & Administration, University of Malaya, Kuala Lumpur, Malaysia, oapenten@siswa.um.edu.my and nazina@um.edu.my

**Abstract**

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.

**Keywords: **HIV/AIDS incidence, mathematical transmission modeling, basic reproduction number, uncertainty analysis.

**Supplement:**

**Description of the model**

The model classiﬁes the active population into four compartments: susceptible, *S(t)*; acute HIV infection, *I(t)*; cases of chronic HIV infection that did not progress to AIDS, *I _{1}(t)*; and cases of AIDS after HIV infection at time

*t, A(t)*. We assume that infected individuals are capable of having children that either are infected with HIV or are not infected with HIV. The susceptible class has a recruitment rate equivalent to the birth rate,

*b*. This model assumes that infected newborns may be incorporated into the model at a rate of

*b(I+I*, assuming that

_{1}+A)*I, I*and

_{1}*A*are sexually active individuals, that a fraction (π) of the offspring of HIV/AIDS patients are infected at birth (π

*b(I+I*) and that the infected newborns enter the acute HIV infection compartment, whereas the complementary proportion of newborns ((1-π)

_{1}+A)*b(I+I*enter into the susceptible compartment. π >0 represents the fraction of newborns who are uninfected (for more details, see [1]). The natural death rate is assumed to be proportional to the number of individuals in each compartment, µ >0. Our model assumes that γ is the rate at which an individual will fully transition from the

_{1}+A))*I*class to the

_{1}(t)*A(t)*class, which is a significant indicator of when an intervention should be introduced. The rate at which individuals with acute HIV infection move to the AIDS class is represented by α; the portion of individuals with chronic HIV infection is represented by δ. This model also assumes that at a rate of δα, some of the acute HIV infection cases transition to the AIDS group, whereas the remaining acute HIV infection cases move to the chronic HIV infection compartment at a rate of (1-δ)α, where 0≤δ≤1. β is the contact rate between susceptible individuals and individuals with acute HIV infection. AIDS patients are given an additional disease-induced mortality rate: σ>0, ε>0 and ρ>0 for

*I(t), I*and

_{1}(t)*A(t)*, respectively. The flow of individuals between these populations is shown in Figure 1.

Figure 1: Model flow diagram.

Comparison of yearly reported HIV case simulations with fitted parameters during the 25-year (1986-2011) [2,3] calibration and validation period. The performances of the plotted graphs could be compared and evaluated in future studies. The red lines represent the estimated parameters, with black lines representing the initial parameters.

Figure 2: For comparison, the initial model output and the best-fit model are plotted against the data.

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**Summary Statement:**

This study demonstrates how to model the spread of AIDS after chronic HIV infection. As with any modeling study of such a complex system as HIV/AIDS epidemiology, several assumptions were necessary to make the analysis tractable. There were some significant differences in the estimated parameters that will be useful to public health, potentially representing a practical and effective way to epidemiologically model the occurrence of AIDS disease after HIV infection.

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**Acknowledgements**: This work was financially supported by the University Malaya Research Grant RP004J-13ICT: Demographic Network Modelling of the Spread of Infectious Diseases, under the Social Network Dynamics Programme, Research Cluster Computation and Informatics, University of Malaya.

**References**

- Apenteng OO, Ismail NA (2015) A Markov Chain Monte Carlo Approach to Estimate AIDS after HIV Infection. PloS one 10: e0131950.
- (2014) The Global AIDS Response Progress Report 2014, Malaysia.: Ministry of Health Malaysia.
- (2012) Department of Statistics: Population Projections Malaysia 2010-2040.