Modeling the immune system response: an application to leishmaniasis

In this paper, we present a mathematical model of the immune response to parasites. The model is a type of predator-prey system in which the parasite serves as the prey and the immune response as the predator. The model idealizes the entire immune response as a single entity although it is comprised of several aspects. Parasite density is captured using logistic growth while the immune response is modeled as a combination of two components, activation by parasite density and an autocatalytic reinforcement process. Analysis of the equilibria of the model demonstrate bifurcations between parasites and immune response arising from the autocatalytic response component. The analysis also points to the steady states associated with disease resolution or persistence in leishmaniasis. Numerical predictions of the model when applied to different cases of Leishmania mexicana are in very close agreement with experimental observations.