Social Distancing, Vaccination and the Paradoxical Optimality of an Endemic Equilibrium

We analyze the impact of public health policy on the spread of a disease using a version of the SIR model that includes vital statistics, waning immunity, and vaccination. This model is rich enough to accommodate endemic steady states and disease-free steady states. We choose social distancing and vaccines to maximize an objective function that penalizes lost output resulting from social distancing, deaths resulting from the disease, and the cost of vaccination. Surprisingly, even though a disease-free equilibrium is attainable, optimal policy leads to an endemic steady state, though with a small number of deaths and negligible loss of output.