Stability Analysis of a Mathematical Model of the Immune Response with Delays

The immune system is a complex network of cells and signals that has evolved to respond to the presence of pathogens (bacteria, virus, fungi). By pathogen we mean a microbial non-self recognized as a potential threat by the host. Some pathogens preferentially survive and proliferate better inside cells (intracellular pathogens) and others are extracellular (Medzhitov et al. 2002). The two basic types of immunity are innate and adaptive. The innate response is the first line of defense; this response targets any type of microbial non-self and is non-specific because the strategy is the same irrespective of the pathogen. Innate immunity can suffice to clear the pathogen in most cases, but sometimes it is insufficient. In fact, pathogens may possess ways to overcome the innate response and successfully colonize and infect the host. When innate immunity fails, a completely different cascade of events ensues leading to adaptive immunity. Unlike the innate response, the adaptive response is tailored to the type of pathogen. Immune responses that clear intracellular pathogens typically involve effector cells (such as cytotoxic T cells, or CTLs) while extracellular pathogens are cleared mostly by effector molecules (e. g. antibodies) involving a different cascade of cells (such as B cells) (Janeway 2001). There is a natural temporal kinetic that arises as part of these immune responses. The innate immune response develops first occurring on the order of minutes and hours. Adaptive immunity follows innate and occurs on the order of days or weeks. Each has an inherent delay in their development (see next section), and this timing may be crucial in determining success or failure in clearing the pathogen.

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