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\begin{document}

\title{Mathematical modelling of oscillating patterns for chronic autoimmune
diseases}



\author[1]{Rossella Della Marca}%
\author[2]{M. Piedade Ramos}%
\author[2]{Carolina Ribeiro}%
\author[2]{Ana Soares}%
\affil[1]{University of Parma}%
\affil[2]{University of Minho}%


\vspace{-1em}



  
  \date{May 18, 2024}


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\begin{abstract}
Many autoimmune diseases are chronic in nature, so that in general
patients experience periods of recurrence and remission of the symptoms
characterizing their specific autoimmune ailment. In order to describe
this very important feature of autoimmunity, we construct a mathematical
model of kinetic type describing the immune system cellular interactions
in the context of autoimmunity exhibiting recurrent dynamics. The model
equations constitute a non-linear system of integro-differential
equations with quadratic terms that describe the interactions between
self-antigen presenting cells, self-reactive T cells and
immunosuppressive cells. We consider a constant input of self-antigen
presenting cells, due to external environmental factors that are
believed to trigger autoimmunity in people with predisposition for this
condition. We also consider the natural death of all cell populations
involved in our model, caused by their interaction with cells of the
host environment. We derive the macroscopic analogue and show positivity
and well-posedness of the solution, and then we study the equilibria of
the corresponding dynamical system and their stability properties. By
applying dynamical system theory, we prove that steady oscillations may
arise due to the occurrence of a Hopf bifurcation. We perform some
numerical simulations for our model, and we observe a recurrent pattern
in the solutions of both the kinetic description and its macroscopic
analogue, which leads us to conclude that this model is able to capture
the chronic behaviour of many autoimmune diseases.%
\end{abstract}%



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