On the Optimal Control of Vaccination and treatments for an SIR-Epidemic model with Infected Immigrants

This paper is proposed to study rigorously the optimal control of vaccination and treatments for an SIR(Susceptible-Infected-Recovered) Epidemic model with infected immigrants. The impact of infected immigrants is studied in order to understand and assess its influence on the transmission dynamics for an SIR(Susceptible-Infected-Recovered) Epidemic model. The main goal of this work is to find the optimal treatment and vaccination strategies in combination that will minimize the cost of vaccination and treatment as well as the number of infective and to measure the influence of the flow of infected immigrants in the transmission of disease in a population.Qualitative and quantitative analysis of the model with respect to stability of the disease free equilibrium and endemic equilibrium under the influence of the two control strategies are carried out.It is established that in the absence of infected immigrants the disease free equilibrium exist and is locally asymptotically stable. Hence, the optimal interventions for the disease control using Pontryagin’s Maximum Principle( PMP) are illustrated and various numerical simulations are performed to discuss the solution.