Stability Analysis And Bifurcation In A Diffusive Epidemic Model With Two Delays

In 1979,Cappasso et al.proposed a system of ordinary differential equations for the cholera epidemic as a mathematical model to describe digestive tract infectious diseases.This model describes the interaction between bacteria and infected people,in which the infectivity of bacteria to the people is represented by a function without specific form.Multiple infectious diseases can be described by the specific form of a given function,such as typhoid fever,schistosomiasis,infectious hepatitis,etc.Due to its applicability,the model has been widely studied and a lot of achievements have been made.However,bacteria and people are diffusive in real life,and some diseases have the incubation period.In order to be more realistic and consider the influence of the fluidity of bacteria and population,and the incubation period of disease,people constructed the reaction-diffusion equation with time delay on the basis of the above model as the infectious disease model.This paper mainly studied the dynamic properties of a class of infectious disease models with diffusion and two delays from the perspective of bifurcation.First,In the case where the nonlinear term is associated with only one state variable,the existence of the steady-state solution of the system is studied.Then,through the distribution analysis of the characteristic values,the local stability of the steady-state solution and the sufficient conditions for the Hopf bifurcation to be generated at the positive constant steady state is obtained.On this basis,the central manifold theory and the canonical method of partial functional differential equation are used to calculate the quantity which determining Hopf bifurcation direction and the stability of periodic solution of Hopf bifurcation,and the computer numerical simulation is carried out.Then,in the case where the nonlinear term is related to both state variables,by using the geometric method of the analysis of the distribution of roots in the transcendental equation,the bifurcation curves describing the distribution of eigenvalues are obtained on the parametric plane with two delays as parameter s.Gave the stable region and the unstable region of the positive constant steady state solution respectively,and obtained the existence of Hopf bifurcation and double Hopf bifurcation.Then,a canonical form on the central manifold which can describe the dynamic properties near the steady-state solution of the original system is derived,and the complexity of the dynamic behavior of the system is revealed.Finally,numerical simulation is given.