Singular Regimes in Controlled Models of Disease Treatment

This talk consists of two parts. Firstly, a model of psoriasis dynamics is considered, consisting of three differential equations. These equations describe the interaction between cell populations that determine the onset, progression, and treatment of this disease. This model is then transformed into three controlled models by introducing two constrained controls reflecting the doses of medications. The first and second models each contain one control, while the third model has two controls. For each model, a corresponding minimization problem is formulated. In the first minimization problem, the application of Pontryagin's maximum principle reveals a potential form of optimal control. It can be either a relay function throughout the specified interval or, in addition to relay segments, it can include a second-order special regime. The report presents the results of studying this special regime. As the optimal treatment protocol with such a special regime does not have medical significance, a method for approximating optimal control with a sequence of relay controls that are optimal in certain perturbed minimization problems is proposed. It is demonstrated how such a perturbation occurs and the convergence of the family of perturbed optimal solutions to the optimal solution of the unperturbed minimization problem is justified in the corresponding metrics. Furthermore, the possibility of a third-order special regime arising in the optimal control for the considered minimization problem is shown. To simplify the corresponding study, a special system of differential equations for the switching function and its associated auxiliary functions is introduced, which allows for sequential calculation of the required derivatives of this function. The connection between such a special regime and an adjacent relay segment of the optimal control is also studied. In the second minimization problem, the use of Pontryagin's maximum principle also reveals a potential form of optimal control. It can be either a relay function throughout the specified interval or, in addition to relay segments, it can include a first-order special regime. The report presents the results of studying this special regime. In the third minimization problem, the application of Pontryagin's maximum principle also yields potential forms of optimal controls. The possibility of simultaneous occurrence of special regimes in these controls is studied. After showing that this is not possible, the potential forms of each control separately are demonstrated.
Next, we will consider a Lotka-Volterra competition model consisting of two differential equations. These equations describe the interaction between cell populations in blood cancer diseases. This model is then transformed into a controlled model by introducing bounded control, reflecting the dose of medication or intensity of therapy. For such a model, an appropriate minimization problem is formulated. The use of the Pontryagin maximum principle reveals a possible form of optimal control. It can be either a piecewise constant function with no more than one switch, or a relay function, or, in addition to relay sections, it may contain a special first-order regime. The presentation will include results of studying this special regime. We will also consider the situation where therapy in the model is administered indirectly, through the use of a so-called therapy function and an additional linear differential equation containing bounded control. Cases of monotone and non-monotone therapy functions will be examined. The application of the Pontryagin maximum principle to this resulting minimization problem also yields a possible form of optimal control. It turns out that in the case of a monotone therapy function, the mentioned special first-order regime becomes a special second-order regime. In the case of a non-monotone therapy function, the optimal control may contain both special first and second-order regimes simultaneously. The presentation will include results of studying these special regimes.