Modeling the immune system response: an application to leishmaniasis

Ephraim O. Agyingi; Tamas I. Wiandt; Laurence U. Buxbaum; Bolaji N. Thomas; Ephraim O. Agyingi; Tamas I. Wiandt; Laurence U. Buxbaum; Bolaji N. Thomas

doi:10.3934/mbe.2020064

Mathematical Biosciences and Engineering

2020, Volume 17, Issue 2: 1253-1271. doi: 10.3934/mbe.2020064

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Modeling the immune system response: an application to leishmaniasis

1.
School of Mathematical Sciences, Rochester Institute of Technology, Rochester, NY 14623, USA
2.
Department of Medicine, Division of Infectious Diseases, University of Pennsylvania School of Medicine, Philadelphia, PA 19104, USA
3.
Department of Medicine, Corporal Michael J. Crescenz Veterans Affairs Medical Center, Philadelphia, PA 19104, USA
4.
Department of Biomedical Science, Rochester Institute of Technology, Rochester, NY 14623, USA

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.
- immune response,
- mathematical model,
- cutaneous leishmaniasis,
- bifurcation analysis
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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Abstract

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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