Bifurcation Analysis And Pattern Information Of A SI Diffusive Epidemic Model

Epidemic dynamics is an important mathematical method for theoretical quantitative research of epidemic models.In recent years,the mathematical model of infectious disease has been widely concerned,and great progress has been made in the study of the disease transmission by using the method of epidemic dynamics.This paper is concerned qualitatively and quantitatively with a spatial SI epidemic model with the generalized nonlinear incidence rate subject to the homogeneous Neumann boundary condition.Firstly,the ordinary differential model corresponding to the spatial epidemic model is considered.The existence and stability of the equilibrium and the stability and direction of the Hopf bifurcation are given.By calculating the algebraic equations and analyzing the actual meanings of the parameters in the system,five parameter conditions are obtained.The existence conditions and the number of positive equilibria are given respectively.Then,the stability of disease-free equilibrium and endemic equilibria is determined by using the stability theory and the central manifold theorem.Dulac function is used to find the parameter conditions without closed orbit.When the eigenvalues corresponding to the linearized matrix of the system at the endemic equilibrium contains zero real part,the supercritical Hopf bifurcation and unstable periodic solution are obtained by using the Hopf bifurcation theory and the central manifold theorem.Then,the effect of the diffusion coefficient on the stability of the reaction diffusion model is analyzed,and Turing instability and Hopf bifurcation are discussed.By discussing the eigenvalue problem of the reaction diffusion model,the stability condition of the endemic equilibrium and the existence condition of Turing instability are obtained step by step.The existence of spatially homogeneous Hopf bifurcation and spatially inhomogeneous Hopf bifurcation is discussed.The bifurcation points and the existence conditions are given.By using the center manifold theorem and the normal form theory,we get the spatially homogeneous Hopf bifurcation is unstable and the direction of the bifurcation is supercritical.Finally,the existence of the local steady-state bifurcation is discussed for steady state model in one-dimension space.A priori estimation of the nonconstant positive solutions is given by using the maximum principle.We establish the local structure of steady state bifurcation from simple eigenvalues by using Crandall-Rabinowitz local bifurcation theory.On the other hand,when the eigenvalues appear multiple roots,the local structure of the steady-state bifurcation from the double eigenvalues is studied by using the spatial decomposition theorem and the implicit function theory under certain conditions.This paper mainly focuses on a diffusive epidemic model with nonlinear incidence rate.We carry out the conclusions of the stability of equilibria,Turing instability,Hopf bifurcation and local steady state bifurcation.On the basis of these qualitative analysis,we realize the numerical simulation.The dynamic behavior of ordinary differential model is shown by the time diagram and the phase portrait.The diagram of nonconstant steady state from simple eigenvalues and double eigenvalues is obtained.The diagram of spatially homogeneous periodic solutions and spatially inhomogeneous periodic solutions is gained.These numerical simulations verify the theoretical analysis results.