Dynamic Properties For Several Types Of Pathogen-Immune Models

To study the dynamic behavior of pathogens and immune cells is helpful to explore the transmission mechanism of infectious diseases.This paper mainly studies dynamic properties of several types of pathogen immune models,the main contents are as follows:Firstly,according to the diffusion of pathogens and immune cells in vivo,a reaction-diffusion model of pathogen-immune with homogeneous Neumann boundary condition is constructed.To discuss the effect of diffusion factors on dynamics between pathogens and host immune cells,taking the ratio of the diffusion coefficient between pathogens and immune cells as the parameter,the critical condition of Turing instability is obtained by analyzing the root distribution of the characteristic equation of the linearized system at the positive steady-state solution.Matlab numerical simulations are performed to show the dynamic phenomenon of Turing instability in the pathogen-immune model.Furthermore,the change of pathogens and immune cells density in the dynamic phenomenon of Turing instability is discussed.Secondly,based on the diffusion model,according to a phenomenon of immune delay in the process of pathogen immunity,a delayed pathogen-immune reaction diffusion model with homogeneous Neumann boundary condition is established.To discuss effects of diffusion and time-delay factors on the dynamics between pathogens and immune cells,by using the diffusion ratio of pathogen-immune cells and immune delay as two parameters,the characteristic root distribution of the linearized system at the positive equilibrium solution is analyzed,and the necessary and sufficient conditions for the positive equilibrium solution to experiencing Turing instability and Hopf bifurcation are obtained by usingthe bifurcation theory of functional differential equations.In addition,the dynamic behavior closed to the critical value of Turing instability and Hopf bifurcation is corresponding dynamic behaviors are discussed.The results provide certain theoretical support for controlling the growth of pathogen.Finally,according to the randomness of the immune process,a stochastic pathogen-immune model with environmental white noise is established.The existence and the stability of equilibrium of the bilinear model are studied.The existence of globally unique positive solutions of the stochastic model and the sufficient conditions for extinction and strong persistence in the mean of the pathogen are derived by applying the Ito formula and the Lyapunov function.In addition,theoretical results are verified by numerical simulations.