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Stability analysis of pathogen-immune interaction dynamics. (English) Zbl 1086.92029

Summary: The paper considers models of dynamics of infectious diseases in vivo from the standpoint of the mathematical analysis of stability. The models describe the interaction of the target cells, the pathogens, and the humoral immune response. The paper mainly focuses on the interior equilibrium, whose components are all positive, If the model ignores the absorption of the pathogens due to infection, the interior equilibrium is always asymptotically stable. On the other hand, if the model does consider it, the interior equilibrium can be unstable and a simple Hopf bifurcation can occur.

A sufficient condition that the interior equilibrium is asymptotically stable is obtained. The condition explains that the interior equilibrium is asymptotically stable when experimental parameter values are used for the model. Moreover, the paper considers the models in which uninfected cells are involved in the immune response to pathogens, and are removed by the immune complexes. The effect of the involvement strongly affects the stability of the interior equilibria. The results are shown with the aid of symbolic calculation software.

##### MSC:
 92C50 Medical applications of mathematical biology 92C60 Medical epidemiology 34D05 Asymptotic stability of ODE
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