Optimal Treatment Strategies for Control Model of Psoriasis

Abstract

Psoriasis is an autoimmune disorder with the symptom of chronic inflammation of the skin. Psoriasis makes skin cells build up quickly on the skin surface which causes the skin to appear red, dry, and with scaly patches. Adequate treatment for psoriasis is very challenging and a total cure of the disease still does not exit. Mathematical models have long been effective means to predict cellular behaviors of skin regulation under normal or pathological circumstances. In this paper, a proposed control model of psoriasis is described on a given time interval by a nonlinear control system of three differential equations involving the concentrations of dendritic cells (tissues macrophages), T-lymphocytes, and keratinocytes with medication intake as a control function. An optimal control problem of minimizing the release of keratinocytes at the end of the time interval is stated and studied using the Pontryagin maximum principle. Different types of optimal control dependent on model parameters are obtained analytically. Possible applications to an optimal drug therapy are discussed.

Chattering and its approximation in control of psoriasis treatment

“…is the switching function that using formula (14) describes the behavior of the control u * ξ (t); • the Hamiltonian…”

“…Corollary 3.2. Formula (14) and Lemma 3.1 yield the following relationship for the optimal control u * 0 (t) :…”

“…Therefore, the optimal control u * 0 (t) does not have a singular arc. By formula (14), we see that it is bangbang control taking the values {u min ; 1} on the interval [0, T ]. Here, we can consider the question of estimating the number of zeros of the function L 0 (t), and hence the estimate of the number of switchings of the control u * 0 (t).…”

“…Here, we can consider the question of estimating the number of zeros of the function L 0 (t), and hence the estimate of the number of switchings of the control u * 0 (t). This question was studied in more detail in [14].…”

“…Psoriasis is not a disease that directly leads to death, and, as it was noted in [14], its treatment implies the usage of a skin cream and an oral medication. Both prescribed medications have regular daily dosage and are not as harmful for patients as the drugs used in chemotherapy for cancer treatment ( [38]).…”