An Optimal Policy for Lockdown Periods in a Pandemic Based on the Stochastic SIR Model with Time Delay

In this paper, we combine the optimal stopping problem of timing (beginning and ending) a lockdown with a stochastic control model based on the stochastic SIR model to analyze how the progression of a pandemic can be affected by controlling the rate of disease transmission (contact rate) from infected to susceptible individuals in the population. A number of important lessons emerge from model results: (1) the sequence of lockdown decision making in even a simple stochastic SIR model can provide useful quantitative information to policy makers in charge of the timing of a lockdown, (2) policy makers are likely to start a lockdown late and end the lockdown late when there is greater volatility in the number of infected patients, (3) the analysis shows that an optimal lockdown policy can be characterized by two threshold values that can be computed for four parameters if the payoff functions are linear, (4) the optimal level of restrictions on human contact depends not only on the contact rate and the recovery rate but also on the number of susceptible individuals remaining at decision time, and (5) anticipating the effect of a time delay, the policy maker may start the lockdown earlier but belittle the time delay regarding when to end it. As a result, the lockdown periods become longer when there is a time delay.