Analysis Of Virus Dynamics Model Including Delay And Epidemic Model With Backward Bifurcation

In this paper, the virus dynamics models including delay and epidemic models with backward bifurcation are studied. In the first part of this paper, the immune response of HIV-I with delay is studied. We give completely qualitative analysis to this model. By using a discrete delay to model the intracellular delay, it is shown that the sufficient condition to ensure the stability of the infected equilibrium does not change would be enlarged by Sturm sequence. Numerical simulations are presented to illustrate the results.In the second part, the model of cell-to-cell spread of HIV-I with two distributed delays are studied. We consider the lag between the time a new cell is reproduced and when it becomes mature and the lag between the time a cell is infected and when it begins to infect other cells, and model the two phases by gamma distributions. If the susceptible cells proliferate logistically, we can obtain conditions in which the model undergoes the Hopf bifurcation, and conditions in which the positive equilibrium is always unstable, and conditions in which the positive equilibrium is always asymptotically stable, and conditions in which stability changes from stable to unstable to stable , by using two average time delays as bifurcation parameters. Then, numerical simulations are presented to illustrate these results.In the third part, a simple SIS model with treatmente is studied. We use the treatment rate which is proportional to the number of the infectives when the capacity of treatment is not reached, and otherwise, takes the maximal capacity. When we do not consider the disease-induced death, the completely qualitative analysis are given by means of the theory of asymptotically autonomous system. It is found that backward bifurcation and bistable endemic equilibria occur. When we consider the disease-induced death. It is shown that the backward bifurcation and bistable endemic equilibria also occur. Moreover, there is no limit cycle under some conditions, and the subcritical Hopf bifurcation will occurs under another conditions. Lastly, numerical simulations are presented to illustrate these results...