Birth and death of a limit cycle through Hopf bifurcations in a model of HIV infection with RT treatment

We study the occurrence of Hopf bifurcations in a three-dimensional system of ordinary differential equations, depending on ten parameters, proposed in Wang [1] as a mathematical model describing the interaction of HIV infection and CD4+ T cells, in which is considered the application of a treatment. In that paper is carried out the study of local and global stability of equilibria and is shown, via numerical simulations, the existence of periodic solutions for certain parameter values. In this work we prove that the existence of such periodic solutions is due to occurrence of Hopf bifurcations, under the variation of one of the parameters of the model. We identified two critical values associated to Hopf bifurcations which lead to the “birth” and “death” of a stable limit cycle. We also present some numerical simulations to confirm the analytical results obtained. Behind the analytical and numerical study performed, there is an interesting and difficult mathematical problem, related to the continuation of the limit cycle. We have numerical evidences that, under the variation of the same parameter, one limit cycle arises from a first Hopf bifurcation, grows as the parameter is varied, reaches its biggest amplitude and then diminishes, shrinking into a singular point in a second Hopf bifurcation.