The goal of this paper is to explore the impact of non-linearity of functional responses on the optimal control of infectious diseases. In order to address this issue, we consider a problem of minimization of the level of infection at the terminal time for a controlled SIR model, where the incidence rate is given by a non-linear unspecified function f(S,I). In this model we consider four distinctive control policies: the vaccination of the newborn and the susceptible individuals, isolation of the infected individuals, and an indirect policy aimed at reduction of the transmission. The Pontryagin maximum principle is used for the problem analysis. In this problem we prove that the optimal controls are bang-bang functions. Then, the maximum possible number of switchings of these controls is found. Based on this, we describe the possible behavior of the optimal controls.

Psoriasis is an autoimmune disorder, characterized by hyper-proli-feration of Keratinocytes for the abnormal activation of T Cells, Dendritic Cells (DCs) and cytokine signaling. Interaction of DCs and T Cells enable T Cell to differentiate into Type 1 (Th₁), Type 2 (Th₂) helper T Cell depending on cytokine release. Hyper-proliferation of Keratinocytes may occur due to over expression of pro-inflammatory cytokines secreted by Th₁-Cells viz. Interferon gamma ( $\mbox{IFN}-{γ}$ ), Transforming growth factor beta ( $\mbox{TGF}-β$ ) and Tumor necrosis factor alpha ( $\mbox{TNF}-α$ ) etc. Deregulation of epidermal happens due to signaling of anti-inflammatory cytokines like Interleukin 10 ( $\mbox{IL}-{10}$ ), Interleukin 4 ( $\mbox{IL}-{4}$ ) etc., released by Th₂-Cells. In this article, we have constructed a set of nonlinear differential equations involving the above cell population for better understanding the impact of cytokines on Psoriasis. System is analyzed introducing therapeutic agent (Biologic / $\mbox{IL}-{10}$ ) for reducing the hyper-proliferation of Keratinocytes. Effect of Biologic is used as a surrogate of control parameter to reduce the psoriatic lesions. We also studied its effect both in continuous and impulsive dosing method. Our study reveals that impulsive dosing is more applicable compare with continuous dosing to prevent Psoriasis.

The Susceptible-Infected-Recovered (SIR) model for the spread of an infectious disease in a population of constant size is considered. In order to control the spread of infection, we propose the model with four bounded controls which describe vaccination of newborns, vaccination of the susceptible, treatment of infected, and indirect strategies aimed at a reduction of the incidence rate (e. g. quarantine). The optimal control problem of minimizing the total number of the infected individuals on a given time interval is stated and solved. The optimal solutions are obtained with the use of the Pontryagin Maximum Principle and investigated analytically. Numerical results are presented to illustrate the optimal solutions.

A SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the problem of minimizing the weighted sum of total fractions of infected and exposed individuals and total costs of intervention control constraints at a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. According to it, these controls are bang-bang, and are determined using the same switching function. A linear non-autonomous system of differential equations, to which this function satisfies together with its corresponding auxiliary functions, is found. In order to estimate the number of zeroes of the switching function, the matrix of the linear non-autonomous system is transformed to an upper triangular form on the entire time interval and the generalized Rolle’s theorem is applied to the converted system of differential equations. It is found that the optimal controls of the original problem have at most two switchings. This fact allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of two variables. Results of the numerical solution to this problem and their detailed analysis are provided.