Internal HIV Dynamics

The research on the HIV virus dynamics can not only make us obtain more insight intothe infection of AIDS, but also provide theoretical basis to control the spread of HIV in aninfected individual. In this paper, we just consider two models of this virus-host interaction――the basic internal HIV dynamics and the internal HIV dynamics under HAART.For deterministic dynamics, on the basis of the theory of linear differential equation,we prove that if the initial value is positive, the solution of the system is always nonnegativeand bounded. Then, according to the characteristic equation theory for the stability of thedynamics system, we got the local stability of the equilibriums. On this basis we examinethe global asymptotic stability of equilibriums by using LaSalle invariance principle andKrasovskiii-LaSalle stability theorem.Based on the analysis of the deterministic system, two types of stochastic models aregot by introducing randomness into the virus infection rate and the mortality. This tech-nique of parameter perturbation is common in stochastic population modeling. Then, weprove the local solutions will not explode to infinity in a finite time by the theory of stop-ping time. For stochastic model adding disturbing term to the virus infection rate, we obtainthe asymptotic stability of solution near the equilibriums through the asymptotic stability ofthe trivial solution of linear systems. For the other one, we consider the moment bounded-ness of the solution respect to the virus-free equilibrium of the corresponding deterministicmodel. Then we prove that the smaller the noise intensity, the smaller the oscillation of thesolution of stochastic models, especially when the parameter σ1=0, the solution will beasymptotically stable.In addition, numerical simulations are presented to illustrate our findings and the clini-cal effectivity of HAART. In particular, we present that there is a stationary distributionin stochastic model.