Mathematical analysis of HIV/AIDS stochastic dynamic models

In this paper, we present a group of stochastic dynamic models of the infection of the HIV/AIDS in a host population of four classes of epidemiological classes: susceptible individuals, HIV infected, AIDS infected individuals and treatment class (i.e. infected individuals who are under ARV treatment). The mathematical analysis of the deterministic model has been previously consider by Emvudu and Bongor (2012). The mathematical analysis of the group of the stochastic models associated to the deterministic model is performed to determine the conditions for the stability of the random equilibriums. The results of this mathematical analysis are the next one: the global and asymptotic stability of the disease free equilibrium of the mean system when the effective reproduction number is less than 1; the global, asymptotical, and p-exponential stability of the disease free equilibrium of the stochastic model occurs when the basic reproduction number is less than 1; the stability in probability of the trivial equilibrium of the stochastic perturbations model occurs if the basic reproduction number is bigger than 1.