Dynamics of drug resistance: Optimal control of an infectious disease

Antimicrobial resistance is a significant public health threat. In the United States alone, two million people are infected, and 23,000 die each year from antibiotic-resistant bacterial infections. In many cases, infections are resistant to all but a few remaining drugs. We examine the case in which a single drug remains and solve for the optimal treatment policy for a susceptible–infected–susceptible infectious disease model, incorporating the effects of drug resistance. The problem is formulated as an optimal control problem with two continuous state variables: the disease prevalence and drug’s “quality” (the fraction of infections that are drug-susceptible). The decision maker’s objective is to minimize the discounted cost of the disease to society over an infinite horizon. We provide a new generalizable solution approach that allows us to thoroughly characterize the optimal treatment policy analytically. We prove that the optimal treatment policy is a bang-bang policy with a single switching time. The action/inaction regions can be described by a single boundary that is strictly increasing when viewed as a function of drug quality, indicating that, when the disease transmission rate is constant, the policy of withholding treatment to preserve the drug for a potentially more serious future outbreak is not optimal. We show that the optimal value function and/or its derivatives are neither C¹ nor Lipschitz continuous, suggesting that numerical approaches to this family of dynamic infectious disease models may not be computationally stable. Furthermore, we demonstrate that relaxing the standard assumption of a constant disease transmission rate can fundamentally change the shape of the action region, add a singular arc to the optimal control, and make preserving the drug for a serious outbreak optimal. In addition, we apply our framework to the case of antibiotic-resistant gonorrhea.