A nonconservative kinetic model under the action of an external force field for modeling the medical treatment of autoimmune response

TL;DR

This paper introduces a nonconservative kinetic model with an external force field to analyze immune system dysregulation and autoimmune treatments, providing analytical results and numerical simulations for different treatment scenarios.

Contribution

It develops a novel kinetic framework for autoimmune response modeling, including stability analysis and treatment strategy simulations.

Findings

Existence, uniqueness, positivity, and boundedness of solutions established.

Stability results obtained for autonomous (drug-free) case.

Numerical simulations demonstrate model behavior under treatment strategies.

Abstract

In this paper, we develop a nonconservative kinetic framework to be applied to the study of immune system dysregulation. From the modeling viewpoint, the model regards a system composed of stochastically interacting agents, under the action of an eternal force field. According to the application perspectives of this paper, the external force field has a specific analytical shape. In this case, some analytical results are proved, i.e. existence, uniqueness, positivity, and boundedness of solution of the related Cauchy problem, at least locally in time. Then, the model is refined to be implemented for the study of treatment strategies in case of autoimmune response. Specifically, we distinguish the autonomous case from the nonautonomous one, representing the absence or delivery of drugs, respectively. The former allows us to gain some stability results. Whereas, the latter is qualitatively studied. Numerical simulations are provided for both schemes.