The Global Dynamic Behaviors Of Coupled Viral Models With Delay

Infectious disease can be considered as one of the enemies for all mankind. SARS, avian influenza and tuberculosis that emerged in recent years, are undoubtedly huge challenges for human health defense. Scientists in every country are trying to study the propagation law of infectious diseases accurately by constructing a mathematical model, and then find more effective ways to prevent and eliminate these diseases.Infectious disease models and project background have been introduced in chapter 1. The microscopic viral infection models become the new object of study of many scholars in recent years. Subsequently, we introduce a viral infection model with a general incidence rate function f（x, v） in the second chapter. We consider delay and age structure as two important factors in the model, and obtain the global stability conditions of the equilibrium solution by analyzing our model.In the third chapter, we study the viral infection model with delay when the network is strongly connected and prove the positivity as well as boundedness of the solutions for the model. Meanwhile, we get that when the basic reproductive number R₀ <1, the infection-free equilibrium is globally asymptotically stable; the model is uniformly persistent, the infection-free equilibrium is unstable and the endemic equilibrium is globally asymptotically stable when the basic reproductive number R₀ >1 by the method of combining the knowledge of graph theory and Lyapunov functional method on the basis of La Salle’s invariance principle.However, the viral cell has its own characteristics in real life. So, there always existing a sort of virus that cannot propagate among cells, directly and indirectly, which suggests that the strongly connected network also has some limitations. Hence, we study the global dynamic behavior of the viral infection model under the conditions of non-strong connectivity network in the chapter 4. Based on the theoretical knowledge which is not strongly connected, in this chapter we mainly prove the stability of maximum equilibrium point P*. That is all the solutions of the system converge to a unique maximal P* of evaluate function E, and*P is positive equilibrium if and only if the basic reproduction numberR_0,H>1 for all the minimal element H ∈V（H）. Finally, all conclusions can be verified by numerical modeling.