Existence Of Traveling Wave Solutions For A Class Of Si Infectious Disease Model With Diffusion

In the study of the dynamics of infectious diseases,the traveling wave solution represents the source of infection spreading through the space at a constant wave speed.In this paper,we consider a kind of SI infectious disease model,where both the susceptible and the infected spread(?)We first analyze the equilibrium point of the system,and prove that there is a heteroclinic trajectory between the boundary equilibrium point(1,0)and the positive equilibrium point(S~*,I~*),which is the traveling wave solutions we called,and then give the minimum wave velocity of the system.We apply the Wazewski theorem to construct a Wazewski set,which is large enough to satisfy that when z??,the trajectory of the system can satisfy the boundary conditions given before.That is to say the trajectory of phase space must be located on the stable manifold at the equilibrium point of endemic diseases.Then,let's find a set?in a sufficiently small circle near the boundary equilibrium point(1,0)and prove that there is a point on?,the trajectory through this point will not leave a bounded region in the Wazewski set.Finally,we will construct a Lyapunov function,combining with the LaSalle invariant principle to prove that the trajectory of the system finally tends to the positive equilibrium point and get the existence of the traveling wave solution.In this proof process,we use shooting method,which is a combination of Wazewski theorem,LaSalle's invariant principle and the stable manifold theorem.