Traveling Wave Solutions Of A SIR Epidemic Diffusive Model With External Supplies

Epidemic waves play a crucial role in the study of disease invasion, since it means that the disease spreads at a constant speed if an epidemic wave exists. As is known, traveling waves may be changed even in a small perturbation. Thus it is important to study the qualitative property of traveling waves such as the exact asymptotic behavior, the stability and uniqueness and so on. However, it is much challenging to study the traveling waves of epidemic models since the systems are non-cooperative. To the best of our knowledge, most of the existing work in this topic focus on the existence of the traveling waves. In this thesis, we shall study the minimal speed and qualitative properties including existence, stability and uniqueness of traveling waves of a diffusive SIR epidemic model with external supplies. The main results are as follows:Firstly, we study the minimal speed of traveling wave of the reaction-diffusion SIR epidemic model. By showing the existence and nonexistence of traveling waves including the critical case, we get that the critical speed is equivalent to the minimal speed. Since the system is non-cooperative, the traveling waves lack of monotonicity and the critical wave can not be obtained simply by a limiting approach. The main di?cult is to verify the boundary conditions. We answer the question by constructing suitable Lyapunov functional and applying the theory of spreading speed. Moreover, we claim the existence of damped oscillating traveling waves, which implies that the external supplies can induce damped spatio-temporal oscillations.Next, we consider the stability and uniqueness of traveling waves of the SIR model. By using the weighted energy method, we ?rst prove the exponential stability of traveling waves under the so-called small initial perturbation(i.e. the initial perturbation around the traveling waves is suitable small in a weighted norm). Then we establish the exact asymptotic behavior of traveling waves at-∞ by using Ikehara’s theorem. Finally the uniqueness(up to translations) of traveling waves is proved by the stability result. Here we choose the weighted function ω(·) satis?esω(+∞) = 0, which are different from the previous works where the weighted functions are selected to be greater than 1. And hence the results are improved in the sense that the initial perturbation is allowed to be uniformly bounded at +∞ but may not be 0.Finally, we consider a nonlocal dispersal version of the SIR epidemic model. We are mainly concerned with the minimal speed, the local exponential stability and uniqueness of traveling waves of the nonlocal dispersal SIR model, and we discuss how the nonlocal dispersal affects the traveling waves. The conclusion shows that the nonlocal dispersal increases the minimal speed of traveling waves and slows down the convergence rate of the solution to the traveling waves.