On stability analysis and optimal control of an sir epidemic model

In this paper, we present an SIR (susceptible, infected, recovered) epidemic model with controls describing the interactions between susceptible, infected and recovered populations. Both vaccination and education as controls are applied to the model in order to reduce the spread of the disease. SIR model is fitted to the diphtheria data from East Java province. From these estimated parameters, we found that the basic reproductive number is greater than one which implies that the outbreaks will occur in the population. The aim of this optimal control is to minimize both the disease outbreaks and the intervention cost. Pontryagin’s Maximum Principle is used to characterize the optimal control. Furthermore, we derive the optimality system and solve it numerically using the forward-backward sweep method. The results of numerical simulations show that control over both vaccination and education can be effective in reducing the infected population.