| Summary: | Social distancing plays an essential role in controlling the spread of an epidemic, but changing the behavior of individuals regarding social distancing is costly. In order to make a rational decision, individuals must compare the cost of social distancing and the cost of infection. People are typically more likely to change their behavior if they are aware that the government is willing to incur additional cost to shorten the duration of an epidemic. I extend an optimal control problem of social distancing by integrating with the SIR model which describes the disease process. I present an optimal control problem to consider the behavior of susceptible individuals and the government in investment as control strategies and compute the equilibrium strategies under the potency of investment, using relative risk functions according to the investment that is made by susceptible individuals and the government. The equilibrium of this problem represents the optimal control strategies for minimizing the cost and duration of controlling an epidemic. Additionally, the model is evaluated using COVID-19 data from Egypt, Japan, Italy, Belgium, Nigeria, and Germany. The findings extracted from this model could be valuable in developing public health policy in the event of an epidemic.
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