Stability Analysis Of An HIV Model With CTL-Response Delay And Cell-to-Cell Transmission

The dynamics of infectious diseases refers to the study of the dynamics of a system by establishing a mathematical model to deepen people's understanding of the disease mechanism,optimize the prevention and treatment strategy.So more and more scholars research the pathogenesis and the transmission ways of HIV infection by establishing models.They consider healthy target cells,infected cells and viruses in the models.In this paper,we study an HIV model with cell-to-cell transmission and CTL-response immune delay motivated by the previous works.We also consider that the proliferation of target cells is described by logistic growth.We mainly study the stability of the equilibria of the system.We introduce the establishment process of the system to be studied in this paper,as well as the positivity and boundedness of solutions of the system.Furthermore,the equilibria of the system are calculated which are the infection-free equilibrium,the CTL-inactivated infection equilibrium and the CTL-activated infection equilibrium.We get the basic reproductive numbers of the system through the next generation matrix,including the basic reproductive numbers for viral infection R₀ and for CTL-response R₁.And the existence of the CTL-inactivated infection equilibrium and the CTL-activated infection equilibrium are obtained.We analyze the stability of the equilibria of the system.We show that the infection-free equilibrium is globally asymptotically stable by constructing a Lyapunov function.Specifically,locally dynamical properties of the CTL-inactivated infection equilibrium and the CTL-activated infection equilibrium are further investigated by Routh-Hurwitz criterion.Using the CTL-response delay?as the bifurcation parameter,we find that the CTL-activated infection equilibrium is locally asymptotically stable when the delay?varying in a small range around zero.But when?crosses a critical value ?₀,it becomes unstable and a Hopf bifurcation occurs.Numerical simulations are carried out to support our theoretical results.The nu-merical solution of system tends to CTL-activated infection equilibrium as time goes to infinity when?<?₀.And the solution tends to a stable periodic solution when?>?₀.In comparison with the HIV infection system without CTL-response immune delay,there are more challenges in theoretical deduction and numerical computing to investigate the stability of equilibria of the proposed system with necessary and reasonable assumption.And it is also observed that this system exhibits more complicated dynamical behaviors and new conclusive findings.From the sensitivity analysis of R₀ and R₁ with respect to the parameters,we obtain that it is noteworthy to assess the influences of CTL-response im-mune delay,cell-to-cell transmission and healthy T-cells logistic growth on the dynamic behaviors of system.