Study On Stability Of Epidemic Models With Relapse And Delays

Infectious diseases have always threatened human health since its inception.In order to better understand the transmission law of infectious diseases,many scholars establish mathematical models to research them basing on the transmission mechanism of infectious diseases.This paper mainly consists of two parts: an SEIR epidemic model with recurrence and a delayed HIV-1 model with cell-to-cell and virus-to-cell transmissions.In the first part,we study an SEIR model with relapse.Here,the recurrence rate is a function of time,and we consider the cases in which infectious patients in the epidemic model are directly converted to recoveres by drugs or autoimmunity.Firstly,the threshold and equilibrium of infectious model are given.Non-negativity and boundedness of the model solution are proved by Comparison lemma,Fluctuation lemma,Lebesgue-Fatou lemma and so on.Using the Hurwitz criterion,we obtain the global asymptotic stability of the infection-free equilibrium when R₀＜1.Secondly,we take two specific functions for the recurrence rate.When R₀＜1,the stability of the positive equilibrium under two kinds of recurrence rates was obtained by Hurwitz criterion and other methods.Finally,numerical simulations agree with our theoretical results.In the second part,we study a delayed HIV-1 model with cell-to-cell and virus-to-cell transmissions.The threshold and equilibrium of the model are given.Firstly,non-negativity and boundedness of solutions are proved by Comparison lemma and so on.Secondly,using the Hurwitz criterion,we obtain the global asymptotic stability of the disease-free equilibrium when R₀＜1.In addition,the global stability of the positive equilibrium are obtained by constructing Liapunov functionals.Finally,numerical simulations agree with our theoretical results.