A SIRS Model With Delay And Moving Boundary

Abstract Mathematic has been applied to various fields of scientific research, and has impacted more on our lives. Epidemic dynamics is one of mathematical fields used in the actual. As the incidence of disease and its spreading character within the populations, we can establish mathematical models to reflect the development of disease, and analyze the reason of the prevalence of disease by using mathematical methods and ideas. Epidemic dynamics also could provide quantitative and theoretical bases for people to make decision to prevent and cure disease. Epidemic dynamics, which contacts facts closely and plays an important part in human health, attracts much attention, and now becomes one of the hot spots in applied mathematics.With the development of the epidemic dynamics, many important biological factors have been introduced into the epidemic model to reflect the fact. In this paper, based on the models established by ordinary differential equations, we set up the corresponding partial differential equations, analyze a SIRS epidemiological model with distributed delay and free boundary. The distributed delay is introduced to reflect the incubation time which influences the infection force of the disease and the free boundary is introduced to reflect spreading or vanishing process of the disease.In Section1, we first present the importance of the epidemical research since epidemics poses serious threats to human survival and development, and then describe the recent development of epidemic dynamics. We then introduce the classical SIRS epidemic model and its development process. On the basis of previous research, we focus on the SIRS ordinary differential system with a bilinear incidence rate and distributed delays. Further, considering of spreading of the disease in space, diffusion is introduced and improved model of partial differential equations with a bilinear incidence and distributed delays is established, which will be discussed in this paper.In Section2, we will prove that the solution to the system is bounded. The lower bound of solution is given first and the upper bound is established by using Gaghardo-Nirenberg inequality and the Moser iteration. Section3deals with the basic reproduction number R0, we will show that if R0>1the disease-free equilibrium is unstable, while if R0<1the disease-free equilibrium is locally asymptotically stable. If R0>1the endemic equilibrium is locally asymptotically stable, and if R0<1the disease-free equilibrium does not exist. In particular, we construct Lyapunov functions to show the global stability for both cases of R0<1and R0>1.Section4is devoted to the corresponding free boundary problem. The existence and uniqueness of the solution are first given, and then the properties of the free boundary are discussed.Section5deals with the spreading or vanishing of the disease. Firstly, we give the definitions of the spreading or vanishing of the disease, and then sufficient conditions for spreading or extinction are given. Finally a brief conclusions are given.