The Stability Analysis Of SIRS Epidemic Models With Nonlinear Incidence Rate

This paper frstly discusess a class of single-group SIRS epidemic model-s with nonlinear incidence rate by using the method of theory analytical, thepositivity and boundedness of solutions, the existence and global stability ofequilibria are obtained. Besides, we also discuss a class of single-group SIRSepidemic models with general nonlinear incidence rate and vaccination in sus-ceptible and a class of multi-group SIRS epidemic models with general nonlinearincidence rate, respectively. In the same way, we can obtain the positivity andboundedness of solutions, the existence, local stability and global stability ofequilibria as well as the permanence of disease for models, respectively.There are fve sections in all. The frst section is introduction, the back-ground, purposes and signifcance of epidemic model will be introduced, mainlydescribing the present investigation of epidemic models with nonlinear incidencerate. Finally, we give the structure of our paper.In the second section, we will give some results about a class of single-group SIRS epidemic models with nonlinear incidence rate by using the theoryof stability of diferential equation and the method of Lyapunove function. Thatis, the positivity and boundedness of solutions, the existence and global stabilityof equilibria, at the same time, we will give the sufcient and necessary conditionsof equilibria are global asymptotically stable. We give a numerical simulationcorrespond to our results in this section at last, and we obtain the correctness ofour results.In the third section, we will give some results about a class of single-groupSIRS epidemic models with general nonlinear incidence rate and vaccination in susceptible by using the linearization method, theory of persistence in dynam-ical systems and the method of Lyapunov function. That is the positivity andboundedness of solutions, the existence, local stability and global stability of e-quilibria as well as the permanence of disease for models, at the same time, wewill give the sufcient conditions of equilibria are global asymptotically stable.We give a numerical simulation correspond to our results in this section at last,and we obtain the correctness of our results.In the fourth section, we will give some results about a class of multi-groupSIRS epidemic models with general nonlinear incidence rate by using the methodof Lyapunov function. That is the positivity and boundedness of solutions, theexistence, local stability and global stability of equilibria as well as the perma-nence of disease for models, at the same time, we will give the sufcient condi-tions of endemic equilibrium is global asymptotically stable. We give a numericalsimulation correspond to our results in this section at last, and we obtain thecorrectness of our results.The ffth section is a conclusion, we will summarize the main results of thethree classes of epidemic models in this paper, and give several open problemsabout them.