Numerical Approximations And Controls For Stochastic Age-Structured HIV Models

Human immunodeficiency virus(HIV)can cause Acquired Immune Deficiency Syn-drome(AIDS).It targets the body's CD4+T cells,causes a deficiency in the body's immune system,so that the body gradually loses the ability to fight off many opportunistic infections,leading to a variety of clinical symptoms.HIV has strong infectivity and high pathogenici-ty,which seriously endangers people's health and life.Since there is no effective treatment method for HIV,it is of great practical significance to study the mechanism of HIV transmis-sion and control strategy by using mathematical model.In this dissertation,according to the characteristics of HIV transmission,four class of stochastic age-structured HIV models are established.On the basis of discussing existence and uniqueness of global positive solutions,the convergence of the numerical solution for the system and the numerical approximation of the threshold are studied.The stability distribution of the time-delay system is analysed,and the finite time stability and control for the system are further investigated.The details are as follows:1.The numerical approximation of the basic reproduction number for an age-structured epidemic model is studied.For a deterministic age-structured epidemic system and its s-tochastic version,by using the ? method to discretize the linear operator produced by the infective population in a finite horizon,then the basic reproduction number defined by the spectral radius of the next generation matrix is solved,which obtaining the corresponding numerical solution.The convergence of the corresponding threshold is given by using the spectral approximation theory.This algorithm can be used to calculate the basic reproduction number for the HIV model.2.The numerical approximation of an impulsive stochastic age-structured HIV model with Markov switching is studied.The model involves the virus-to-cell infection and cell-to-cell transmission.And the randomness is introduced by a mean-reverting process(Ornstein-Uhlenbeck process).Since the coefficients of the system do not satisfy the global Lipschitz condition,the classical Euler-Maruyama(EM)method may cause explosion.In order to fill this gap,the explicit numerical approximation for the system is investigated by applying the truncated EM method.At the same time,the p-th moment boundedness of the numerical solution and the strong convergence of such algorithm are given.3.The stationary distribution for stochastic age-structured HIV model with delay is investigated.A stochastic age-structured HIV model with time delay is established to in-vestigate the effect of delay existed in cell-to-virus infection and cell-to-cell transmission on viral spread.Based on the discussion of the existence and uniqueness of the global positive solution for the system,the p-th moment boundedness of the solution is studied by utilizing Lyapunov function,and the existence and uniqueness of the stationary distribution for the solution is further studied.4.The finite-time stability and optimal impulsive control for stochastic age-structured HIV model with time-varying delay and driven by Levy process are studied.By employing the stochastic comparison theorem and the bounded impulsive interval method,the sufficient conditions of finite-time stability for the stochastic HIV system are given by constructing Lyapunov function.Furthermore,by studying the optimal impulse control of the system,the optimal solution to control the transmission of HIV is obtained.5.The finite time contraction stability of a stochastic age-structured HIV model with time-varying delay and Markov switching under control strategy is investigated.By using Lyapunov function and stochastic comparison theorem,the sufficient conditions for the finite time contraction stability are obtained.The effects of control strategy,noise intensity,and time delay on the finite time contraction stability are analyzed.