Global dynamics for a class of discrete SEIRS epidemic models with general nonlinear incidence.

Zhidong Teng,Xiaolin Fan,Lei Wang

doi:10.1186/s13662-016-0846-y

Abstract

In this paper, a class of discrete SEIRS epidemic models with general nonlinear incidence is investigated. Particularly, a discrete SEIRS epidemic model with standard incidence is also considered. The positivity and boundedness of solutions with positive initial conditions are obtained. It is shown that if the basic reproduction number mathcal{R}_{0}leq1, then disease-free equilibrium is globally attractive, and if mathcal{R}_{0}> 1, then the disease is permanent. When the model degenerates into SEIR model, it is proved that if mathcal{R}_{0}> 1, then the model has a unique endemic equilibrium, which is globally attractive. Furthermore, the numerical examples verify an important open problem that when mathcal{R}_{0}>1, the endemic equilibrium of general SEIRS models is also globally attractive.

Highlights

As is well known, many infectious diseases possess a latent period, such as Hepatitis, HIV, SARS, Ebola, MERS, etc
7 Discussion In this paper, we proposed a discrete SEIRS epidemic model ( ) with general nonlinear incidence, which is described by the backward difference scheme
By our discussions presented in this paper, necessary and sufficient conditions for the global attractivity of the for SEIRS model ( ), when the basic reproduction number is greater than one, we do not obtain the local asymptotic stability and global attractivity of the endemic equilibrium

Summary

Introduction

Many infectious diseases possess a latent period, such as Hepatitis, HIV, SARS, Ebola, MERS, etc. In [ ]), we obtain that the disease-free equilibrium P of model ( ) is globally attractive Using the theorems of stability of difference equations, we obtain that the endemic equilibrium P∗ of model ( ) is globally attractive. The parameters μ , α, and σ will be chosen later All these examples of numerical simulations show that when R > , no matter sufficiently greater than one or closer to one but still greater than one, we always obtain that the endemic equilibrium P∗ is globally attractive, which may offer an affirmative conjecture to the open problem given in Remark . In our future work, we expect to obtain the corresponding theoretical results for this open problem

Discussion

Methods