Researches On An Epidemic Model And An Eco-epidemic Model With A Generalized Incidence Rate

This paper proposes a new generalized incidence function based on the nonlinear in-cidence of diseases.A generalized delay-induced SIRS epidemic model with nonlinear in-cidence rate,latency and relapse,and an eco-epidemic model with a generalized incidence rate,and Holling ? type functional response and two delays are proposed respectively.The dynamic behaviors of this two models are discussed by applying the theories and methods of ordinary differential equation,functional differential equation and differential dynamic system.Firstly,the dynamical behaviors of the proposed SIRS model are studied and the corresponding conclusions are obtained.It is proved that the solutions satisfying the ini-tial conditions are positively bounded.The basic reproduction number R0 is computed and the conditions for the existence of equilibrium points are found.After the stability analysis,it is obtained that the disease-free equilibrium point Q0 is globally asymptot-ically stable when ?=0 and R0<1.If R0>1 and under some certain conditions,the endemic equilibrium point Q*is locally asymptotically stable,and the disease-free equilibrium point Q0 is unstable.If ?>0,it is proved that disease-free equilibrium point Q0 is still globally asymptotically stable by constructing appropriate Lyapunov function.The sufficient conditions which guarantee the locally asymptotical stability of endemic equilibrium Q*are also obtained.The numerical simulations are conducted to verify the theoretical results and the influence of the relapse rate and non-linear rate on the stability of the epidemic model is simulated.Secondly,the dynamical behaviors of the presented eco-epidemic model are inves-tigated,and the uniform persistence of the solution satisfying the initial conditions is proved.The existences of all feasible equilibrium points of the presented eco-epidemic model are considered and the corresponding conditions are obtained.In the absence of time delay,the local stabilities of the five equilibrium points of the presented eco-epidemic model are proved.The local stability of the coexistence equilibrium point and the condi-tions for the existence of Hopf bifurcation are studied when there exists time delay and the following two cases are considered(?)?1>0?T2=0 and(?)?1=0??2>0.Furthermore,the conclusions of the above analysis are verified by numerical simulation,and the time sequence diagram and phase diagram of the coexistence equilibrium point are obtained.