Optimal strategy of vaccination and treatment in an SIRS model with Markovian switching

Abstract

Medical treatment and vaccination decisions are often sequential and uncertain. Markov decision process is an appropriate means to model and handle such stochastic dynamic decisions. This paper studies the near-optimality of a stochastic SIRS epidemic model that incorporates vaccination and saturated treatment with regime switching. The stochastic model takes white noises and color noise into account. We first prove some priori estimates of the susceptible, infected, and recovered populations. Moreover, we establish some sufficient and necessary conditions of the near-optimality by Pontryagin stochastic maximum principle. Our results show that the two kinds of environmental noises have great impacts on the infectious diseases. Finally, we illustrate our conclusions through numerical simulations. 767 768 MU AND ZHANG Strategies for prevention and control of infectious diseases include vaccination, treatment, quarantine, isolation, and prophylaxis. Vaccination and treatment control are two effective ways to control diseases. For most infectious diseases, treatments exist that can cure or lessen the effects of the diseases and improve the life of the patients. To better study these control measures, there are a few research results of control strategy. [17][18][19][20][21][22][23] For example, Kar and Batabyal 20 considered an SIR epidemic model using vaccination as control and analyzed stability. Lashari 22 used optimal control to study an SIR epidemic model with a saturated treatment. However, these systems are the deterministic epidemic model. Moreover, the exact solution of the state equation and the adjoint equation are difficult to be found and the precise control is difficult to be given. Hence, it is more practical to study the near-optimal control problem of the epidemic model. This paper is aimed at considering a stochastic SIRS epidemic model 6 with Markov chains, which includes vaccination and saturated treatment control. The objective function is formulated as an optimal control problem with two control variables to minimize the susceptible and infected individuals as well as the cost of implementing the two controls. Moreover, we will establish the sufficient and necessary conditions for the near-optimal control problems. The novelty of this study are as follows:(i) The Markov chains, vaccination control, and saturated treatment are taken into account the SIRS epidemic model simultaneously, which will obtain a novel model. (ii) The near-optimal control problem of a stochastic SIRS model is discussed.The layout of this paper is as follows: In Section 2, some basic concepts are presented and the SIRS model is formulated. The sufficient and necessary conditions of the near-optimal control for the stochastic SIRS model are obtained in Sections 3 and 4, respectively. In Section 6, we make simulations to confirm our results. Section 7 is given some conclusions to finish the paper. BASIC CONCEPTS AND THE MODEL FORMULATIONWe first introduce the following definitions and notations in Section 2.1 before we formu...

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