A Backward Branching Problem In An Infectious Disease Model

In this paper,we formulate a SIVS epidemic model with special recovery rate to study the impact of limited medical resource on the transmission dynamics of diseases with vaccination.It is shown that limited medical resource leads to vital dynamics,such as bistabili?ty.The backward bifurcation has been proved precis'ely.Backward bifurcation implies that even if the basic reproduction number is s-maller than unity,there may be a stable endemic equilibrium and the basic reproductive number itself is not enough to describe whether a disease will prevail or not and we should pay more attention to the initial conditions.It is also shown that sufficient medical services and medicines are very important for the disease control and eradication.Besides,the impact of vaccination has been explored too.This paper is composed of five chapters.In chapter 1,we introduce the current research status,history and prevalence of infectious disease.In chapter 2,we mainly introduce the SIVS model based on standard SIS model with the incidence of mass action.In chapter 3,we analyze the existence of equilibria.Disease-free equilibrium E0 always exists.Besides,there exist only one endemic equilibrium if R0>1 and at most two equilibria if R0<1.By analyz-ing the existence of equilibria,we can present the system undergoes backward bifurcation.In chapter 4,we consider the stability of equilibria.It con-eludes that E0 is locally asymptotically stable if R0<1 and unstable if R0>1.When R0 = 1,E0 is a saddle-node.In addition,en-demic equilibria E(I,V)have different stability under different con-ditions.Further,we present the condition of backward bifurcation.When R0=1,R0=1,(?),the model will un-dergo backward bifurcation.Finally,we give some simulations to illustrate the main result.