A linear optimal control model of immunotherapy for recurrent autoimmune disease

In this work, we improve a recent mathematical model for evaluating the effects of drug treatments in autoimmune diseases, incorporating the natural death of all cell populations due to interactions with cells in the host environment and taking into account a constant input of self-antigen presenting cells, due to external environmental factors that are believed to trigger autoimmunity in people with susceptibility to this disease. We derive macro-analogies of the kinetic model and demonstrate the positivity and well-posedness of the solution. We then examine the equilibrium of the corresponding dynamical system and its stability. We show that continuous oscillations occur due to the existence of a Hopf bifurcation. We formulate a linear optimal control problem relevant to the model such that the number of self-reactive T cells and the amount of interleukin-2 cytokines that is administrated are simultaneously minimized. Numerical simulations of the model show the effectiveness of the therapeutic strategies.