A Stochastic Differential Equation SIA HIV/AIDS Model

HIV/AIDS is an infectious disease caused by human immunodeficiency virus?HIV?.Human immunodeficiency virus?HIV?is a retrovirus that infects cells of the human immune system,destroying or impairing their function,inducing other diseases and threatening human health.Considering the influence of random factors in the environment on disease trans-mission,this paper adds a white noise disturbance?ie random disturbance factor?based on the deterministic model,and establishes a kind of SIA infectious disease model affected by random disturbance.We first analyze the deterministic SIA model with the characteristics of HIV transmission and the global stability of the deter-ministic model at the equilibrium point was proved.When the basic reproduction number R₀<1,the disease goes extinct;and when R₀>1,the disease persists.For the corresponding stochastic SIA epidemic model,the existence of the unique positive solution of the stochastic SIA model is proved,by using Markov semigroup-s and their asymptotic properties and Fokker-Planck equation,the conditions for disease extinction and persistence in stochastic SIA models are discussed.When the random reproduction number R₀^s<1 and under some conditions,the solution of the stochastic SIA model converges to the disease-free steady-state distribution,then the disease tends to extinction;when the random reproduction number R₀^s>1and under some conditions,the solution of stochastic SIA model converges to the steady state distribution of the endemic diseases according to the probability 1,and the disease persists.Finally,the numerical simulation verifies the correctness of the conclusion.