Study On Traveling Wave Solutions Of Several Kinds Of Reaction-Diffusion Epidemic Models

It is an urgent and important task to study the transmission of infectious disease through population dynamics and disease transmission mechanism.In this thesis,we will establish the reaction-diffusion infectious disease models and the virus infection models,the traveling wave solutions of these models are studied by means of the basic theory of Ordinary Differ-ential Equation,Reaction-Diffusion Equation,Functional Differential Equation,Upper and Lower Solution,Fixed-point Theorem,etc..The main research contents are as follows:In the first chapter,the significant impact of infectious diseases on the development of human society are introduced,as well as significance of using mathematical models to study the transmission mechanism of infectious diseases.The research status of travelling wave solutions of nonlocal delay and nonlocal dispersal SIR(Susceptible-Infected-Removed)epidemic model with nonlinear incidence,the periodic environment reaction diffusion model and the virus infection model are given.In the second chapter,a nonlocal dispersal SIR epidemic model with nonlinear incidence and distributed delay is proposed and the existence of traveling wave solutions of the models is investigated.First,the basic regeneration number(?)₀>1 and the critical wave velocity(8^*of the model are defined.Then,by using the upper and lower solution,auxiliary system,solution-map,Fixed-point theorem,diagonal extraction of subsequences and Lyapunov func-tion,we obtain the existence of traveling wave solutions satisfying the boundary conditions.The nonexistence of the traveling wave solution is proved by using the asymptotic propaga-tion theory when(?)₀and 0<(8<(8^*.Finally,the theoretical results are verified by numerical simulations.In the third chapter,considering the interaction between spatial diffusion and time delay,an infectious disease model with nonlocal delay and nonlinear incidence is proposed,and the existence of the traveling wave solution is discussed.By the existence of traveling wave solutions is proved by constructing an auxiliary system,and using the method of upper and lower solution,Schauder Fixed-point theorem and the diagonal extraction subsequence when(?)₀>1 and(8>(8^*.When(?)₀>1 and 0<(8<(8^*,the nonexistence of traveling wave solutions is proved by the asymptotic propagation theory.In the fourth chapter,the traveling wave solutions of a non-autonomous reaction-diffusion SIR epidemic model with general nonlinear incidence are investigated.It is found that the existence of traveling wave solutions directly depends on the basic reproduction number(?)₀and the critical wave speed(8^*.We obtained the existence of periodic traveling waves for each wave speed(8>(8^*using the Schauder's fixed points theorem when(?)₀>1.The nonexis-tence of periodic traveling waves for two cases(i)(?)₀>1 and 0<(8<(8^*,(ii)(?)₀ 1 and(8 0 are also obtained.Finally,the numerical experiments verify the theoretical results.In the fifth chapter,a periodic reaction-diffusion SIR epidemic model with demography is studied.Since the boundedness of the infected part is difficult to obtain,an auxiliary system is introduced.For every wave speed(8>(8^*,there exist traveling wave solutions when(?)₀>1,and there does not exists such periodic traveling waves for any(8>0 when(?)₀<1.Finally,the theories are verified by numerical examples.In the sixth chapter,the traveling waves for a virus infection model with humoral immu-nity,cell-to-cell transmission and general nonlinear incidence is discussed.Our results show that the existence of traveling wave solutions are not only determined by the basic reproduc-tion number(?)₀of virus infection and the antibody response reproduction number(?)₁,but al-so by the critical wave speed(8^*.We adopt the Schauder's fixed-point theorem and Lyapunov function methods to obtain the existence of traveling wave solution connecting infection-free equilibrium₀and antibody-free infection equilibrium₁for(?)₀>1,(?)₁<1 and(8>(8^*,and the existence of traveling wave solution connecting infection-free equilibrium₀and antibody-present infection equilibrium^*for(?)₀>1,(?)₁>1 and(8>(8^*.The existence of traveling wave solution connecting₁and^*is discussed.Some numerical simulations are carried out to illustrate our analytical results.