Global Bifurcation And Stability Of Steady-States Of Reaction-Diffusion Epidemic Models With Standard Incidences

In this paper,global bifurcation and stability of steady states of reaction-diffusion epidemic models with standard incidences under homogeneous Neumann boundary condition are investigated.First,the stability of constant steady states to both ODE and PDE models and diffusion-driven Turing instability are analyzed by linearization method.Then,using the maximum principle,we derive some priori estimates of positive steady states.The nonexistence and existence of nonconstant positive steady states are established by using energy method,implicit function theorem and the Leray-Schauder degree theory.Finally,by regarding diffusion coef-ficient d2 of infected as the bifurcation parameter,using asymptotic analysis method,we obtain the specific expression of the local bifurcating solutions and conditions of the stability of the steady states.By applying the global bifurcation theory of Rabinowitz,we prove that the local bifurcation curves can be extended to the global branches.