Dynamics Analysis Of Two Infectious Disease Models With Reaction-diffusion

In history,large-scale outbreaks of infectious diseases have caused great pain to human society.They have not only taken hundreds of millions of lives,but also caused a huge impact on society and economy in a short period of time.More and more scholars began to devote themselves to the study of infectious diseases.Through qualitative and quantitative research on the transmission laws of infectious diseases,the establishment of transmission models of infectious diseases can provide reliable help for understanding the development scale,transmission speed and prevention and control of infectious diseases.Next,we will consider two different models of infectious diseases are studied under the environment of spatial heterogeneity:1.The SEIVR model with reaction-diffusion and indirect transmission was established by considering different diffusion rates and droplet infection,in which S represented suscep-tible patients,E represented asymptomatic infected patients,I represented infected patients,V represented virus,and R represented recovered patients.Firstly,the global existence of the model solution and the existence of the global attractor are proved Then the basic regen-eration number is defined and its threshold effect is proved:when(?)₀?1,the disease-free equilibrium is globally asymptotically stable;When(?)₀>0,the solution of the model is uniformly persistent and there is a positive steady state.Finally,in the case of spatial homo-geneity,by constructing Lyapunov function,it is proved that when(?)₀>0,the only positive steady state stability of the model.2.By considering spatial heterogeneity and reaction-diffusion,a diffusion virus kinetic model with different diffusion rates and latent infection was established.First,the existence,uniqueness,boundedness and positivity of the model solutions are proved.Then the basic regeneration number is defined,and it is proved that the infection-free steady state is globally asymptotically stable when(?)₀<1,and the model is uniformly persistent when(?)₀>1.In particular,we show that the infection-free homeostasis is globally asymptotically stable when(?)₀=1.Finally,we discuss that the infective homeostasis is globally asymptotically stable when(?)₀>1.