### Mathematical Biosciences and Engineering

2020, Issue 2: 1253-1271. doi: 10.3934/mbe.2020064
Research article Special Issues

# Modeling the immune system response: an application to leishmaniasis

• Received: 30 July 2019 Accepted: 15 November 2019 Published: 20 November 2019
• 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.

Citation: Ephraim O. Agyingi, Tamas I. Wiandt, Laurence U. Buxbaum, Bolaji N. Thomas. Modeling the immune system response: an application to leishmaniasis[J]. Mathematical Biosciences and Engineering, 2020, 17(2): 1253-1271. doi: 10.3934/mbe.2020064

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• 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.

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• © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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