Abstract
Within a given time interval we consider a nonlinear system of differential equations describing psoriasis treatment. Its phase variables define the concentrations of T-lymphocytes, keratinocytes and dendritic cells. Two scalar bounded controls are introduced into this system to reflect medication dosages aimed at suppressing interactions between T-lymphocytes and keratinocytes, and between T-lymphocytes and dendritic cells. For such a controlled system, a minimization problem of the concentration of keratinocytes at the terminal time is considered. For its analysis, the Pontryagin maximum principle is applied. As a result of this analysis, the properties of the optimal controls and their possible types are established. It is shown that each of these controls is either a bang-bang type on the entire time interval or (in addition to bang-bang type) contains a singular arc. The obtained analytical results are confirmed by numerical calculations using the software “BOCOP-2.0.5”. Their detailed analysis and the corresponding conclusions are presented.
Highlights
Psoriasis is an immune-mediated inflammatory skin disease that affects 2–3% of the population around the world [1]
We find the formulas of the optimal control and the corresponding optimal solutions of System (1) on a singular interval corresponding to such a singular arc
With the increase of the value of λ, the actual optimal control u∗(t) responsible for taking a drug that suppresses the interaction between T-lymphocytes and keratinocytes during most of the entire treatment period (95 days) has an interval corresponding to a smooth increase in the dosage of the used medication
Summary
Psoriasis is an immune-mediated inflammatory skin disease that affects 2–3% of the population around the world [1]. For mathematical models of psoriasis treatment, optimal control problems were considered in [15,16,17], where the optimal treatment strategies minimizing the weighted sum of the total concentration of keratinocytes and the total cost of the treatment were found numerically. In the optimal control problems for mathematical models of diseases and the spread of epidemics, the total cost of treatment can be expressed by an integral of control [18,19,20,21] In this case, the models are still described by systems of differential equations that are linear in control. This paper shows that optimal controls can contain singular arcs in the minimization problem for the mathematical model of psoriasis treatment even in the absence of the controls in the integral terms of the functional to be minimized responsible for the cost of this treatment.
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