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.

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.

The objective of this paper is to explore possible impacts of nonlinearity of functional responses and a number of compartments of an infection disease model on principal qualitative properties of the optimal controls. To address this issue, we consider optimal controls for an Susceptible-Exposed-Infectious-Removed (SEIR) model of an endemically persisting infectious disease. We assume that the incidence rate is given by an unspecified nonlinear function constrained by a few biologically motivated conditions. For this model, we consider five controls (which comprise all controls that are possible for this model) with a possibility of acting simultaneously, and establish principal qualitative properties of the controls. A comparison with a similar SIR model is provided.

We consider a three-dimensional nonlinear control model based on the Wodarz HIV model. The model phase variables are populations of the uninfected and infected target cells and the concentration of an antiretroviral drug. The drug intake rate is assumed to be a bounded control function. An optimal control problem of minimizing the cumulative infection level (the infected cells population) on a given time interval is stated and solved, and the types of the optimal control for different model parameters are found by analytical methods. We thereby reduce the two-point boundary value problem (TPBVP) for the Pontryagin maximum principle to a problem of the finite-dimensional optimization. Numerical results are presented to illustrate the optimal solution.

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