Qualitative analysis of a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity

In this paper,a reaction-diffusion SIRS epidemic model with nonlinear incidence rate and partial immunity in a spatially heterogeneous environment is proposed.The well-posedness of the solution is firstly established.Then the basic reproduction number R0 is defined and a threshold dynamics is obtained.That is,when R_(0)<1,the disease-free steady state is locally stable,which implies that the disease is extinct,when R_(0)>1,the disease is permanent,and there exists at least one positive steady state solution.Finally,the asymptotic profiles of the positive steady state solution as individuals disperse at small and large rates are investigated.Furthermore,as an application of theoretical analysis,a numerical example involving the spread of influenza is discussed.Based on the numerical simulations,we find that the increase of transmission rate and spatial heterogeneity can enhance the risk of influenza propagation,and the increase of diffusion rate,saturation incidence for susceptible and recovery rate can reduce the risk of influenza propagation.Therefore,we propose to reduce the flow of people to lower the effect of spatial hetero-geneity,increase the transfer of infected individuals to hospitals in surrounding areas to increase the diffusion rate,and increase the construction of public medical resources to increase the recovery rate for controlling influenza propagation.

Dynamical analysis of a reaction-diffusion SEI epidemic model with nonlinear incidence rate

In this paper,a reaction-diffusion SEI epidemic model with nonlinear incidence rate is proposed.The well-posedness of solutions is studied,including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions.The basic reproduction numbers are given in both heterogeneous and homogeneous environments.For spatially heterogeneous environment,by the comparison principle of the diffusion system,the infection-free steady state is proved to be globally asymptotically stable if R_(0)<1,if R_(0)>1,the system will be persistent and admit at least one positive steady state.For spatially homogenous environment,by constructing a Lyapunov function,the infect ion-free steady state is proved to be globally asymptotically stable if,R_(0)<1,and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if R_(0)>1.Finally,two examples are given via numerical simulations,and then some control strategies are also presented by the sensitive analysis.