Researches On An Eco-epidemiological Model And Stochastic SIR Epidemic Model With A Generalized Incidence Rate

Firstly,a generalized eco-epidemiological model with disease in prey is proposed and the dynamical behaviors are studied,including the positivity and boundedness of solutions of the proposed model,and the stability properties of the equilibrium points of the proposed model.The results show that if the predator's coefficient k=k0 is constant(independent of delay (?),gestation period),then the positive equilibrium point of the proposed model is locally asymptotically stable when the time delay (?) is smaller than the threshold value (?)0.A Hopf bifurcation can occur as the delay increases to (?)0.If k=k0e-d?(d is the death rate ofpredator),the results show that the time delay has both destabilizing and stabilizing effects,that is to say,the positive equilibrium,if it exists,will become stable again for relatively time delay.Secondly,a SIR epidemic model with stochastic perturbation is presented and the dynamical behaviors are investigated.The stability of the equilibria of the corre-sponding deterministic model is considered and the corresponding conditions for the extinction and the permanence of the epidemic model are obtained.Meanwhile,the threshold dynamics of a stochastic SIR model is considered,including the extinction and the permanence in mean of epidemic disease.The results show that relatively stochastic disturbance can cause infections diseases to be extinct,in other words,the persistent infections disease in a deterministic model can become extinct due to the white noise stochastic disturbance.This implies that the stochastic disturbance is con-ductive to the epidemic diseases control.However,the disease may become the endemic disease while the white noise stochastic disturbance is small enough.Finally,selecting the suitable parameters and initial conditions,some numerical simulations are conducted in order to test the theoretical results,and the reasonability and reliability are elaborated.