Nature of equilibria and effects of drug treatments in some simple viral population dynamical models

Abstract

We examine some simple mathematical models which have been recently employed to predict the evolution of population dynamical systems involving virus particles. They include: (1) A general two-component antibody-viral system; (2) A simplified two-component model for HIV-1 dynamics (3) An HIV-1 three-component model including virions and (4) A four-component HIV-1 dynamical model which includes both latently and actively infected cells. For each system we find equilibrium points and analyse their local stability properties in order to obtain a global phase portrait. Analytical methods are complemented with numerical solutions. In all four models there are at most two equilibrium points for physically meaningful values of the variables. As the viral growth rate parameter increases through a critical value, a transcritical bifurcation occurs. One critical point (P1) is always found at zero viral or infected cell levels and non-zero antibody or uninfected cell levels. For parameter values in their usual ranges, P1 is either an asymptotically stable node or a saddle point. When the critical point P2 occurs at biologically meaningful values, it is either an asymptotically stable node or an asymptotically stable spiral point. For all three HIV-1 models, the values of the parameters at which P2 makes a transition to physically meaningful values are precisely those at which P1 changes from an asymptotically stable node to an unstable saddle point. The global pictures for all four models are similar and examples are represented graphically. No limit-cycle solutions were found in any of the models for parameter values in their usual ranges. In the four-component HIV-1 model, the effects of varying each parameter are found and conditions under which P2 changes from spiral point to node are investigated numerically. The effects of reverse transcriptase inhibitors and protease inhibitors, two classes of drugs used to treat HIV-1 infection, are examined in the three-component model foor early HIV-1 dynamics.