Research On Reaction-diffusion Epidemic Models Based On Environmental Pathogen Infection

In recent years,due to the deterioration of the environment,human beings are faced many infectious diseases caused by environmental pollution,such as SARS,MERS,Norovirus,Dengue fever,Malaria,and COVID-19.These infectious dis-eases have the characteristics of strong infectivity,fast transmission speed,and wide epidemic range,etc..At present,scientific research on the transmission of these dis-eases has become a hot topic at the forefront of the theoretical research on infectious diseases.Inspired by these studies,this paper studies the dynamic behavior of a reaction-diffusion epidemic model based on environmental pathogen infection.The main contents can be divided into two parts:1.In the first part(section 2),a reaction-diffusion model for nested within-host and between-host dynamics in an environmentally-driven infectious disease is proposed.For the within-host virus infectious model,the predecessors have con-ducted in-depth research,and only the main conclusions are summarized here.For the between-host isolated slow model,the well-posedness of solutions and the basic reproduction numberR_bare obtained by using the theory of reaction-diffusion e-quation.According to the theory of the Lyapunov function,the threshold condition for the global asymptotic stability of equilibrium is established.For the between-host nested slow model,the positivity and ultimate boundedness of the solution,and the existence of equilibrium are obtained by using reaction-diffusion equation theory.Particularly,the existence of equilibrium shows that the nested slow model has more complex dynamical behavior than the corresponding isolated slow model.The threshold condition of local asymptotic stability of equilibrium is established by using the linearization method and Routh-Hurwitz criterion.Lastly,numerical examples are given to explain the main conclusions.2.In the second part(section 3),a reaction-diffusion SIR epidemic model via environmental driven infection in heterogeneous space is proposed.To reflect the prevention and control measures of disease in allusion to the susceptible in the model,the nonlinear incidence functionEf(S)is applied to describe the protective measures of susceptible.In the general spatially heterogeneous case of the model,the theory of reaction-diffusion equation is used to obtain the well-posedness of the solution and basic reproduction numberR₀.By using the asymptotically autonomous semiflow theory,the theory of Lyapunov function,and the theory of persistence in dynamical systems,the threshold condition for the global asymptotic stability of disease-free equilibrium and the uniform persistence of the model is established.Besides,in the spatially homogeneous case of the model,the threshold condition of the global asymptotic stability of endemic equilibrium is established by using the theory of Lyapunov function.Lastly,the numerical examples are enrolled to explain the open problems.