Time Delays And Antibody Immune Response Of The Stability Of The Virus Dynamics Model Analysis

Throughout history communicable diseases have been a serious threat to human health, moreover virus dynamics play an more and more role in studying these diseases. In this paper, when studying virus dynamics models, time delay and antibody immune response were introduced so as to describe the dynamic characteristic of virus transmission and immune response. As the reason mentioned above, we discussed the stable properties of virus dynamics models of immune response with time delay. At the same time we analysed the dynamics properties of these models completely by constructing Lyapunov functions and using Routh-Hurwitz criterion. Based on these, this paper mainly discussed the following content:In the first of the paper, backgrounds and research status reffered to virus dynamics and epidemic dynamics were introduced.Following, considering bilinear infection rate was a extreme situation, using the Lyapunov-LaSalle invariant set we studied the global stable properties of virus dynamics models with nonlinear Infection Rate and antibody immune response based on the reaserch of Wang Xia and we obtained that the infection-free equilibrium Eo was globally asymptotically stable and the virus would not successfully invade host cell or it would be cleared in vivo immediately when basic reproduction number R0≤1; On the contrary if R0>1and immune response reproductive number R1≤1, the immune-free equilibrium E1was globally asymptotically stable which meant the virus would be alive in vivo but immune system didn’t respond to it as strongly; However if R0>1and R1>1, the virus and antibody would survive together in the host, in that case, endemic equilibrium was globally asymptotically stable which meant disease would be always there so that endemic disease would take shape. According to the discuss, we could control the basic reproduction number range from zero to one to get efficient measures to restrain the dieases. Along with the research of Huang Gang， four-dimensional situation was introduced to the model in which there was no health cell in the period of infection rate.In the last of the paper, owing to that the increase of health cells was constant, based on the study of Wang Xia the Logistic model was chosed to describe the growth process of health cells and with the use of time delay the incubation health cells changed to infectibility cells would easily found. In this part, we obtained the sufficient conditions of local asymptotical stability of disease-free equilibrium, immune-free equilibrium and endemic equilibrium together with the global asymptotical stability disease-free equilibrium. Based on the date got by calculations, we found that the sufficient conditions of Hopf bifurcation existed around the immune-free equilibrium and endemic equilibrium when time delay played on the role of bifurcation parameter. The existence of periodic solution meant virus period fluctuated around the threshold, which is very important to the formation of scheme related to medicine.