Optimal control the prevalence of horizontally transmitted infectious diseases in the community

‎In this paper, we propose a three-component mathematical model, including Suspected-Infected-Recovered individuals (SIR), under the control of maple vaccination, for infectious diseases. In such, infectious disease can be transmitted through contact with an infected person (horizontal transmission). Vaccination of suspected population will reduce the horizontal transmission of patients in the community. The mathematical model has two disease-free and endemic equilibrium points. The basic reproduction rate of the model, the existence and local asymptotic stability of these two equilibrium points are investigated. By using Pontriagin's minimum principle, we have investigated the conditions of reducing the suspected and infected population and increasing the recovered population due to the use of vaccination in the community. Numerical simulations to the optimal control problem show that control measures can lead to a decrease in the number of suspected population and an increase in recovered population ‎a‎nd it prevents the spread of the disease and becoming into an epidemic.