Traveling Wave Solutions Of Two Types Of Age-structured Epidemic Diffusion Models

As we all know,traveling waves are used to describe the spatial spread of disease.The age structure of individual is an important factor in the study of epidemic.In contrast to the epidemic models without age structure,the wave profile system of the age-structured epidemic model is an elliptic-parabolic system,which brings essential difficulty to the study of traveling waves.In this paper,we study two types of age-structured epidemic diffusion models and it includes the existence,nonexistence and asymptotic behavior at plus or minus infinity of traveling waves.We study the traveling waves for a diffusive age-structured SIR epidemic model.Firstly,we construct a pair sub-and super-solutions of the model and consider a truncation problem.Then,using Schauder's fixed point theorem,we prove the existence of solution of this truncation problem.Further,we establish some priori estimates for the solutions of the truncation problem.Next,by taking limit,we prove the existence of the traveling wave solution when the wave velocity is greater than the minimum wave velocity and the basic reproduction number is greater than 1.Finally,using the comparison principle,we obtain the nonexistence of traveling wave solution when the basic reproduction number is less than 1.Through the theory of operator semigroup,we prove the nonexistence of traveling wave solution when the basic reproduction number is greater than 1 and the wave velocity is greater than 0 and less than the minimum wave velocity.We study the qualitative properties of the traveling waves for a two-group infection-age structured epidemic model with external supplies.First,we find the minimal speed of propagation of the disease by the linear approximation and Perron-Frobenius theorem.Next,according to the similar method as above,we obtain the existence of traveling wave solutions when the wave velocity is greater than the minimum wave velocity and the basic reproduction number is greater than 1.Due to the introduction of external supplies,it is difficult to obtain the asymptotic behavior of the traveling wave solution at positive infinity,which is solved by constructing an appropriate Lyapunov function.Finally,using the comparison principle,we obtain the nonexistence of traveling wave solution when the basic reproduction number is less than or equal to 1.Through the theory of operator semigroup,we prove the nonexistence of traveling wave solution when the wave velocity is greater than 0 and less than the minimum wave velocity.