Study Of A Discrete-time SIRS Epidemic Dynamical Model

Epidemic dynamical model has an essential part of mathematical biology modelstheory. Under normal conditions, we use continuous-time diferential equation model todescribe epidemic dynamical model. It is appropriate when the population density is verylarge and the infectious diseases spread very fast. But the transfrom in the number ofpopulation is discrete,so,we can use discrete-time dynamical model to research infectiousdisease. It can mirror the law of the spread of infectious diseases more exactly. Thediscretization of continuous model is an method to study epidemic models. Based on theidea,we study the dynamics behaviors of corresponding discrete-time epidemic models inthis paper.In the frst part of this paper, we investigate the dynamical behaviors of a class of dis-crete SIRS epidemic models with nonlinear incidence rate βSg(I). By using the inductivemethod, the positivity and the boundedness of all solutions are obtained. Furthermore,by constructing Lyapunov functions, only under the assumption that function g(I) andIg(I)are monotone increasing for all I∈(0,∞), the sufcient and necessary conditions onthe global asymptotic stability of the disease-free equilibrium and endemic equilibriumare established.In the second part of this paper, we investigate the dynamical behaviors of a classof discrete SIRS epidemic models with general nonlinear incidence rate βf(S)g(I) andvaccination in susceptible. By using the inductive method, the linearization method andthe theory of persistences in dynamical systems, the positivity and the boundedness ofsolutions, the existence and local stability of equilibria and the permanence of the modelare obtained. Furthermore, by constructing Lyapunov functions, under the conditionswhich functions f(S) and g(I) satisfy assumptions (H1)(H3), the global stability ofthe equilibria for the models is obtained. That is, the disease-free equilibrium is globallyasymptotically stable if basic reproduction number R0≤1, and the endemic equilibrium is globally asymptotically stable if R0>1.In the third part of this paper, we investigate the dynamical behaviors of a class ofdiscrete multi-group SIRS epidemic models with varying population sizes and the modelhas cross path infection between diferent groups. By using the inductive method, thepositivity and the boundedness of all solutions are obtained. Furthermore, by construct-ing Lyapunov functions, the sufcient conditions for the global asymptotic stability ofan endemic equilibrium of a discrete multi-group SIRS epidemic models with varyingpopulation sizes and we obtain that the model is permanent.