Numerical and bifurcation analyses for a population model of HIV chemotherapy

A competitive implicit finite-difference method will be developed and used for the solution of a non-linear mathematical model associated with the administration of highly-active chemotherapy to an HIV-infected population aimed at delaying progression to disease. The model, which assumes a non-constant transmission probability, exhibits two steady states; a trivial steady state (HIV-infection-free population) and a non-trivial steady state (population with HIV infection). Detailed stability and bifurcation analyses will reveal that whilst the trivial steady state only undergoes a static bifurcation (single zero singularity), the non-trivial steady state can not only exhibit static and dynamic (Hopf) bifurcations, but also a combination of two types of bifurcation (a double zero singularity). Although the Gauss-Seidel-type method to be developed in this paper is implicit by construction, it enables the various sub-populations of the model to be monitored explicitly as time t tends to infinity. Furthermore, the method will be seen to be more competitive (in terms of numerical stability) than some well-known methods in the literature. The method is used to determine the impact of the chemotherapy treatment by comparing the population sizes at equilibrium of the treated and untreated infecteds.