ANALYSES OF AN ANTIVIRAL IMMUNE RESPONSE MODEL WITH TIME DELAYS

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1 CANADIAN APPLIED MATHEMATICS QUARTERLY Volume 6, Number 4, Winter 1998 ANALYSES OF AN ANTIVIRAL IMMUNE RESPONSE MODEL WITH TIME DELAYS KENNETH L. COOKE, YANG KUANG AND BINGTUAN LI ABSTRACT. This paper deals with the study of a model of the response of the immune system within a human to an invading virus. The model consists of three highly nonlinear differential equations, some of which involve discrete delays. The model was formulated in Tang et al. (211,where some preliminary simulations were reported for a special case without time delays. Here, among other things, we shall show that the model is uniformly persistent in the sense that all positive solutions eventually enter and stay in a compact set in the interior of the positive cone. We also provide some numerical simulation and bifurcation analyses. 1. Introduction. This paper presents an analysis and simulations of a model of the response of the immune system within a human to an invading virus. The model was formulated in Tang et al. [21], where some preliminary simulations were reported. Before stating some of the results in the present analysis, it will be useful to review some of the background for this problem. G.I. Marchuk in his book [14] and in papers with his co-workers, [15, 161, formulated a mathematical model to describe the interactions between the immune system and viruses and viral particles. The immune system is complicated, with many components, and despite some simplification, the Marchuk model consisted of ten differential equations for ten principal variables: viral particles and antigens, macrophages, helper T cells (precursors of CTL killer T cells), helper T cells (precursors of B cells), CTL killer T cells, B cells, plasma cells, antibodies, and in the affected organ (blood or liver) both infected cells and destroyed cells. Much effort was devoted to obtaining, from data, values for the many parameters in these equations. Because of the high dimensionality of the system, not much analysis could be reported. Received by the editors on October 23, 1996, and in revised form on January 26, Research of the first author partially supported b NSF rant DMS Research of the second author partially supportexby N S $rant ~ DMS , Key words and phrases. Globa stab~l~ty, immune system, tlme delay, Liapunov functional, saddle, uniformly persistent. AMS Mathematics Subject Classification. 34K15, 34K20, 92A15. Copyright Rocky Mountain Mathematics Consortium

2 322 K.L. COOKE. Y. KUANG AND B. LI Subsequently, several authors have looked at reduced and simplified models. Among these we may cite Barradas [I], Payne et al. [18], Srinivas and Pattabhi Ramacharyulu [20], and Forys [6]. For instance, these may contain only three or four main variables such as antigen concentration, plasma cell concentration (representing the immune system), and antibody concentration. In the model of Tang et al. [21], a different approach was adopted. This model contains three variables: healthy cells in the organ, infected cells in the organ and viral particle concentration. It does not explicitly include concentrations of T cells, B cells, or antibodies. Instead, it regards the immune response as a self-activated negative feedback control. The controlling influences due to antibodies, phagocytes, and CTL killer cells are built into the equations. If we let H(t) be the concentration of healthy cells in the organ, P(t) the concentration of virally infected cells in the organ, and V(t) the concentration of free viral particles in the plasma compartment, the rates of change of these variables are described as: H1(t) = - rate of healthy cells becoming infected + rate of regeneration; P1(t) = + rate of virus infecting healthy cells - rate of destruction by CTL killer T cells - rate of cell death caused by viruses; V1(t) = - rate of viruses infecting healthy cells - rate of neutralization by antibodies produced by B cells to induce phagocytosis - rate of absorption by phagocytes (not induced by antibodies) + rate of viral release from living and dead infected cells. Based on the above description, we arrive at a mathematical model of

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