@article {2022,
	title = {Mathematical modelling of oscillating patterns for chronic autoimmune diseases},
	journal = {Mathematical Methods in the Applied SciencesMathematical Methods in the Applied SciencesMath Meth Appl Sci},
	volume = {n/a},
	year = {2022},
	month = {2022/04/01},
	abstract = {<p>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 nonlinear 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.</p>},
	keywords = {autoimmune diseases, cellular interactions, Dynamical systems, Hopf bifurcation, kinetic theory, mathematical biology},
	isbn = {0170-4214},
	url = {https://doi.org/10.1002/mma.8229},
	author = {Rossella Della Marca and Maria da Piedade Machado Ramos and Carolina Ribeiro and Ana Jacinta Soares}
}
@article {Berti2013199,
	title = {Existence and stability of quasi-periodic solutions for derivative wave equations},
	journal = {Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni},
	volume = {24},
	number = {2},
	year = {2013},
	note = {cited By (since 1996)0},
	pages = {199-214},
	abstract = {In this note we present the new KAM result in [3] which proves the existence of Cantor families of small amplitude, analytic, quasi-periodic solutions of derivative wave equations, with zero Lyapunov exponents and whose linearized equation is reducible to constant coefficients. In turn, this result is derived by an abstract KAM theorem for infinite dimensional reversible dynamical systems*.},
	keywords = {Constant coefficients, Dynamical systems, Existence and stability, Infinite dimensional, KAM for PDEs, Linearized equations, Lyapunov exponent, Lyapunov methods, Quasi-periodic solution, Small divisors, Wave equations},
	issn = {11206330},
	doi = {10.4171/RLM/652},
	author = {Massimiliano Berti and Luca Biasco and Michela Procesi}
}
