Optimal Control of an SIR Model with Noncompliance as a Social Contagion

We propose and study a compartmental model for epidemiology with human behavioral effects. Specifically, our model incorporates governmental prevention measures aimed at lowering the disease infection rate, but we split the population into those who comply with the measures and those who do not comply and therefore do not receive the reduction in infectivity. We then allow the attitude of noncompliance to spread as a social contagion parallel to the disease. We derive the reproductive ratio for our model and provide stability analysis for the disease-free equilibria. We then propose an optimal control scenario wherein a policy-maker with access to control variables representing disease prevention mandates, treatment efforts, and educational campaigns aimed at encouraging compliance minimizes a cost functional incorporating several cost concerns. Via careful analysis of the control-to-state map, we are able to prove existence of optimal controls. Our proof applies to dynamics which can be nonlinear in the control variables and general cost functionals including the case of $L^1$ control costs. We numerically resolve optimal strategies using the sequential quadratic Hamiltonian method, a relatively new numerical method for optimal control which is easy to implement and has good convergence theory, as we demonstrate. We test our model in several parameter regimes with specific interest in observing how the policy-maker's optimal strategies depend on their particular preferences which are expressed via design of different cost functionals.