Diversify or focus? Spending to combat infectious diseases when budgets are tight.

We consider a health authority seeking to allocate annual budgets optimally over time to minimize the discounted social cost of infection(s) evolving in a finite set of groups. This optimization problem is challenging since the standard SIS epidemiological model describing the spread of the disease contains a nonconvexity. Neither optimal control nor standard discrete-time dynamic programming can be used to identify the optimal policy. We modify the standard dynamic programming algorithm and show how familiar, elementary arguments can be used to reach conclusions about the optimal policy. We show that under certain conditions it is optimal to focus the entire annual budget on one group at a time rather than divide it among several groups, as is often done in practice. We also show that under certain conditions it remains optimal to focus on one group when faced with a wealth constraint instead of an annual budget.