A statistical inference in an epidemic model with combinational drug treatment: HIV as a case study

Stochastic models are the systems of stochastic differential equations (SDEs) that account the variability in cellular reproduction and death, the infection process, viral reproduction and the immune response against the infection. Generally, the stochastic models are derived based on the dynamics of deterministic models and they provide new insights, distinct from the basic deterministic models. In several years, few mathematical models have described the immunological survival against epidemiological diseases, like HIV. But complete cure to such diseases is still a problem to our human society. Here, we construct a three-compartmental system based on the HIV dynamics and apply the stochastic approach to the deterministic version of our mathematical model. The key feature of this research is to capture the probable time to extinction of infected cells by promoting antiretroviral treatment as well as in vitro monocyte-derived dendritic cell (moDC) vaccination in the population. We introduce the hypothesized transition rates for the stochastic model and the quasi stationary distribution of the infected immune cells. The aim of the analysis of this model is to identify the effect of such combinational drug treatments, to count the unsuccessful invasion of virus during the course of drug treatment and predict the time for complete extinction of the disease. Keywords: Stochastic model, HIV, Quasi-stationary distribution, Time to extinction, Combinational drug treatment, MSC: 93A30, 60H10, 60H35