Optimal control model of immunotherapy for autoimmune diseases

Abstract

In this work, we develop a new mathematical model to evaluate the impact of drug therapies on autoimmunity disease. We describe the immune system interactions at the cellular level, using the kinetic theory approach, by considering self‐antigen presenting cells, self‐reactive T cells, immunosuppressive cells, and Interleukin‐2 (IL‐2) cytokines. The drug therapy consists of an intake of Interleukin‐2 cytokines which boosts the effect of immunosuppressive cells on the autoimmune reaction. We also derive the macroscopic model relative to the kinetic system and study the wellposedness of the Cauchy problem for the corresponding system of equations. We formulate an optimal control problem relative to the model so that the quantity of both the self‐reactive T cells that are produced in the body and the Interleukin‐2 cytokines that are administrated is simultaneously minimized. Moreover, we perform some numerical tests in view of investigating optimal treatment strategies and the results reveal that the optimal control approach provides good‐quality approximate solutions and shows to be a valuable procedure in identifying optimal treatment strategies.

Kinetic Modelling of Autoimmune Diseases

“…We have extended our research work in view of introducing drug therapies on the dynamics and investigating optimal treatment strategies. The results have been submitted for publication, see [6,7].…”

“…In paper [5], we have developed a rather complete qualitative analysis of the model equations and investigated the existence of biologically realistic solutions. Then, in paper [6], a new model has been proposed by considering a further population of IL-2 cytokines and an artificial inlet of external drug therapy with the aim of studying optimal policies for the immunotherapeutic treatment of autoimmune diseases. Paper [6] focus on the macroscopic formulation of this new model, whereas paper [7] introduces the kinetic system approach and exploits the corresponding cellular dynamics.…”

“…Then, in paper [6], a new model has been proposed by considering a further population of IL-2 cytokines and an artificial inlet of external drug therapy with the aim of studying optimal policies for the immunotherapeutic treatment of autoimmune diseases. Paper [6] focus on the macroscopic formulation of this new model, whereas paper [7] introduces the kinetic system approach and exploits the corresponding cellular dynamics. We believe that the kinetic approach, where the model is developed at the cellular scale, can give some insights concerning the biological processes involved in autoimmunity.…”

A kinetic model with time‐dependent proliferative/destructive rates

“…The applications of the kinetic approach cover several sectors. Mathematical biology [3,5], medicine [6,7], vehicular traffic [8,9], and economy [10,11] are the best known examples, thanks to a long-time interest from several research groups. Recently, this interdisciplinarity of mathematics has also reached typical human sciences.…”