Periodic Solutions For Two Classes Of Periodic HIV Dynamical Models With Random Perturbations

Human Immunodeficiency Virus(HIV) is a lentivirus that causes Acquired Immune Deficiency Syndrome(AIDS),and its mechanism of infection and reproduction is very complicated.There is no way to totally eliminate it so far.It makes a difference in infectors' health and quality of life.At the same time it also bring infinite harm to the society.Mathematical models play a key part in the research process of interaction existing between virus and immune system.We reveal the infection mechanism by analysing the dynamic behaviors of virus model and provide theoretical basis for clinical.Since the infection process are affected by many complicated biological phenomena,systems' dynamic behaviors can be more truly and accurately described by stochastic virus model.Nowadays there have been many results of stochastic virus model,but the research on stochastic virus model with periodic change is scarce.In view of this,the main content of this article is as follows:1.The dynamics of periodic HIV model affected by High Active Anti-Retroviral Therapy(HAART) and random perturbations is considered.Firstly,we show that the solution with any positive initial value is global and positive by It(?)'s formula.Secondly,the sufficient conditions for extinction of infected cells and virus particles are obtained by the stochastic comparison theorem.And furthermore,the effect of random perturbations on the dynamics of HIV model are discussed.Then,by using Has' minskii theory,the sufficient conditions for periodic change of uninfected cells,infected cells and virus particles are given.Lastly,numerical examples are given to illustrate the theoretical results.2.We investigate the dynamical behaviors of periodic HIV model with CTL immune response and infection delay in random perturbations.First,by It(?)'s formula,the existence and uniqueness of positive solution is proved.Then,by the stochastic comparison theorem,the sufficient conditions for extinction of virus-producing cells and CTLs are shown,and the influence of random perturbations on the dynamics of HIV model are analysed.Next,by Has' minskii theory,the sufficient conditions to ensure that uninfected cells,virus-producing cells and CTLs have a periodic change are established.Finally,numerical simulations illustrate all the results stated above.