Stability Analysis Of Two Kinds Of Epidemic Models In Patchy Environment

In this paper,based on the classical SIR model,by considering the factors of transport- related infection between patchies and nonlinear infection rate ?SiIi/1+aIi,we establish two kinds of epidemic models with saturation incidence in patchy environment.For the first kind of model which based on SIS structure,we proved the positivity and boundedness of the solution of the system and obtained invariant sets,basic reproduction numbers,disease-free equilibrium and endemic equilibrium.The results show the basic reproduc-tion number R0 is a threshold to distinguish whether the disease is epidemic.With the help of constructing Lyapunov function and characteristic root method,we get the sta-bility analysis of disease-free equilibrium and endemic equilibrium.When R0>1 the disease-free equilibrium is unstable,the endemic equilibrium is globally and asymptoti-cally stable.When R0<1,the disease-free equilibrium is globally and asymptotically stable.For the second kind of model,we added recovered(R)to SIS model and con-structed SIRS model.By using the same methods as before,we proved the positivity and boundedness of the solution of the system,and obtained the invariant sets,basic re-production numbers,disease-free equilibrium and endemic equilibrium.The results show the basic reproduction number R0 is a threshold to distinguish whether the disease is epi-demic.When R0>1 the disease-free equilibrium is unstable,the endemic equilibrium is locally and asymptotically stable.When R0<1,the disease-free equilibrium is locally and asymptotically stable.Finally,in order to prove the results of the theoretical analysis,we did some numer-ical simulations to verify them as the patchy number set to 2.