| Despite of the rapid development of science and technology,people from all over the world have been suffering from lots of diseases such as HIV,hepatitis B,malaria,measles every year,therefore it is important to investigate the dynamics of infectious diseases through mathematical models.Compared with the models in continuous media,epidemic models under the discrete media are more realistic in describing the spread of infectious diseases in many cases,the discrete media considered by the model are called patches.In reality,a patch may represent a city,a town or an island.As a result of highly differentiated zones,the natural environment is usually heterogeneous.As a typical representative,a heterogeneous medium with certain periodicity has attracted extensive attention of researchers.Meanwhile,the latent period of the disease is also an important factor to influence the spread of infectious diseases(mathematically,the equation has time delay),it means that newly infected individuals do not infect others immediately but do so after a period of time.Epidemic models with periodic medium or time delay not only lose the monotonicity,but also possess complicated structures,which makes it essentially difficult to analyze.Therefore,it is of significant importance both in theory and practice to study the propagation phenomena of the following epidemic models with spatial periodic structure and time-delay effects:In the second chapter,we study an epidemic system in a periodic patchy environment with bilinear incidence and recruitment.It is assumed that all the parameters share the same period N ∈ N.By the next generation matrix,we define the basic reproduction number R0.Then we construct a closed convex set,and use the Schauder’s fixed point theorem to get the existence of a nontrivial traveling wave solution when the basic reproduction number R0>1 and c>c*,where c*is the minimum speed.Since we adopt the bilinear incidence,it is more difficult to verify the boundedness of traveling waves.Then by novel analysis,we obtain the boundedness of the solution.And on this basis,we prove the existence of traveling wave solutions when c=c*.After that,we prove that there is no nontrivial traveling wave solution when R0<1,or R0>1 and c ∈(0,c*)by contradiction method.As a result of incorporating the periodicity,it is challenging to get the uniqueness of the endemic equilibrium and the convergence of the traveling waves as ξ→∞.At last,numerical simulations show the uniqueness of the endemic equilibrium and the convergence of traveling waves at +∞.In the third chapter,we study an epidemic model in a periodic patchy environment with standard incidence and recruitment.It is also assumed that all the parameters share the same period N ∈ N.First,we establish the spreading speed of auxiliary equations using the theory of the spreading speeds for monotone semiflows,and analyze the upper and lower bounds of the initial value solution.By comparison principle,we identify the spreading speed for the solutions of our system when R0>1.Then by the similar method to the Chapter 2,we prove the existence of traveling waves when c ≥ c*and R0>1,and the nonexistence of traveling wave solution when R0 ≤ 1 and c>0,or R0>1 and c ∈(0,c*).Comparing the asymptotic speeds of spread with the minimal wave speed of traveling waves,we get that these two speeds coincide with each other.In order to find the effects of spatial heterogeneity on the spatial spread of infectious diseases,we discuss the dependence of R0 and c*on the heterogeneity and amplitude of the parameters of our model by simulation.We find that the heterogeneity of the transmission and removed rates can increase both R0 and c*.However,the heterogeneity of diffusion coefficients do not influence Ro,but decrease c*.In the fourth chapter,we establish an epidemic model in a periodic patchy environment with bilinear incidence but without recruitment,that means,we doesn’t take into account the effects of popution dynamics in this model.Using a similar approach to Chapter 2,we prove the existence and the boundedness of traveling wave solutions when R0>1,c>c*,and the nonexistence of traveling wave solution when R0 ≤1 and c>0,or R0>1 and c ∈(0,c*).The main difficulty of the model is to prove the convergence of the traveling waves U(the susceptible term).To solve this problem,we get the Harnack type property for U,and find that U(ξ)is increasing when ξ goes to +∞.In view of the boundedness of U,we obtain the convergence of U(ξ).At the same time,we find that the traveling wave solution V(ξ)(the infected term)goes to zero as ξ→+∞,that is,V(ξ)is a pulse wave,which is different from the traveling waves with positive lower limits in Chapters 2-3.Furthermore,by theoretical analyses,we find that the heterogeneity of diffusion coefficients does not influence R0 but decrease c*,meanwhile the heterogeneity of transmission rates and removed rates can increase R0 and c*.In the fifth chapter,we consider a two-group epidemic model with bilinear incidence and time delay in a patchy environment.It is assumed that the infectious disease has a fixed latent period and spreads between two groups.Firstly,we establish our model,in which S-equation has cross infection term and I-equation has a sum of infinitely infection term.When the basic reproduction number R0>1 and speed c>c*,we prove that the system admits a nontrivial traveling wave solution with the help of upper and lower solutions and the Schauder’s fixed point theorem.When R0≤1 and c>0,or R0>1 and c ∈(0,c*),we also show that there is no positive traveling wave solution by contradiction method and two-sided Laplace transform,where k=1,2.Finally,we discuss and simulate the dependence of the minimum speed c*on the parameters,and find that c*increases when the cross infection rate βij(i≠j)increases,but decreases when the latent period τ increases. |
