Stability Analysis Of Virus Infection Model With Reaction-diffusion

The outbreak of infectious diseases is a serious threat to human health and brings huge economic burden to society.Mathematical models play an important role in the prevention and control of hepatitis B and AIDS.Based on existing studies on HBV and HIV,two reaction-diffusion mathematical models with general incidences are proposed in this thesis.We focus on the effects of time delay and immunity on the dynamics of infectious diseases.The thesis is organized as follows:Chapter 1 provides the background and research status of HBV and HIV.We also cite theoretical knowledge of qualitative and stability analysis of differential equations needed in our discussion.In Chapter 2,we develop a delayed diffusive HBV model with double infection transmissions.The model includes both virus-to-cell and cell-to-cell transmission.We first consider the existence,positivity and boundedness of solutions.Two threshold parameters R0 and R1,called respectively the basic reproduction number and the immune response reproduction number,are calculated.The main results are established by the method of Lyapunov functionals.Precisely,the infection-free steady state is globally asymptotically stable when R0?1;the HBV infection is uniformly persistent and the immune-free steady state is globally asymptotically stable when R1?1<the CTL immune response is uniformly persistent and the chronic infection steady state is globally asymptotically stable when R1>1.Finally,numerical simulations are performed to confirm the theoretical results.In Chapter 3,taking into account of latent infection on HIV,we establish a reaction-diffusion dynamic model with double infection mechanisms and latent infection.Firstly,we prove the well-posedness of the model,the existence and uniqueness of the steady states.We also derive the expressions of the basic reproduction number and the immune response reproduction number.Secondly,we prove the local stability of the system by using the method of contradiction.Then we study global stability of the steady states with appropriate Lyapunov functionals.Finally,the global stability of the steady states are verified by numerical simulations.The thesis ends with a conclusion and a brief discussion on future work.