Epidemic Dynamics And Stability Analysis

In this paper, we firstly take into consideration two ways of apoptosis which is stimulated by internal signals within the cells and by external signals at the infected cell surface. On the basis of the simplified two-dimensional virus infection dynamics model, we propose two extended models respecting to two ways of apoptosis which directly affects the population of uninfected cells in viral infection and study the influence of apoptosis in viral infection by mathematical theoretical analysis and numerical simulation. The results show that increasing apoptosis plays a positive role in control of virus infection.Then, we incorporate the third population of cytotoxic T lymphocytes (CTLs) immune response into the above model which describe the apoptosis by external signals in HIV infected patients and study its dynamical properties. Both theoretical results and numerical simulations show that depending on intensity of the apoptosis of healthy cells, the apoptosis can either promote or comfort the long term evolution of HIV infection.Further, we introduce the discrete-time delay of apoptosis into the pervious HIV infection model with apoptosis. The bifurcation theoretical study shows that stability switching occurs as the time delay in apoptosis increases and numerical simulations are performed to illustrate the theoretical results and display the different impacts of a delay in apoptosis.Finally, we introduce the fractional-order derivatives into a HIV infection model with nonlinear incidence and show that the established model possesses non-negative solution, as desired in any population dynamics. We also deal with the stability of the infection-free equilibrium, the immune-absence equilibrium and the immune-presence equilibrium. Numerical simulations are carried out to illustrate the results.