Diversity and virulence thresholds in AIDS.

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Abstract
We propose a model for the interaction between human immunodeficiency virus and the immune system. Two differential equations describe the interactions between one strain of virus and one clone of T lymphocytes. We use the model to generalize earlier results pertaining to the AIDS diversity threshold [Nowak, M. A., Anderson, R. M., McLean, A. R., Wolfs, T. F. W., Goudsmit, J. and May, R. M. (1991) Science 254, 963-969]. Our model has (i) a stable steady state corresponding to the "controlled" persistence of the virus and (ii) a region corresponding to AIDS. The separatrix between the two regimes is formed by the stable manifold of a saddle point. We define a dimensionless "virulence" parameter which combines the infectivity and antigenicity of a virus strain. We derive analytically two parameter conditions involving virulence. The first corresponds to a saddle-node bifurcation which causes AIDS due to the loss of the stable equilibrium. The second corresponds to a global bifurcation which causes AIDS due to a change in the basins of attraction. Incorporating diversity into the model, we derive a diversity threshold corresponding to the saddle-node bifurcation. In this threshold condition diversity and virulence have an equivalent effect. By studying the effect of diversity on the critical virulence that is required for a new mutant to cause AIDS, we again establish that diversity and virulence are equivalent parameters. Because in our model increasing diversity decreases the critical virulence, the strain that eventually causes AIDS need not be a virulent one.