On Optimal Control at the Onset of a New Viral Outbreak

An optimal control problem for the early stage of an infectious disease outbreak is considered. At that stage, control is often limited to non-medical interventions like social distancing and other behavioral changes. We show that the running cost of control satisfying mild, problem-specific, conditions generates an optimal control strategy that stays inside its admissible set for the entire duration of the study period $[0 ,T]$. For the optimal control problem, restricted by SIR compartmental model of disease transmission, we prove that the optimal control strategy, $u(t)$, may be growing until some moment $\bar{t} \in [0 ,T)$. However, for any $t \in [\bar{t}, T]$, the function $u(t)$ will decline as $t$ approaches $T$, which may cause the number of newly infected people to increase. So, the window from $0$ to $\bar{t}$ is the time for public health officials to prepare alternative mitigation measures, such as vaccines, testing, antiviral medications, and others. Our theoretical findings are illustrated with numerical examples showing optimal control strategies for various cost functions and weights. Simulation results provide a comprehensive demonstration of the effects of control on the epidemic spread and mitigation expenses, which can serve as invaluable references for public health officials.