Dynamics of drug resistance: Optimal control of an infectious disease

Naveed Chehrazi, Lauren E. Cipriano, Eva A. Enns

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9 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)619-650
Number of pages32
JournalOperations research
Issue number3
StatePublished - 2019

Bibliographical note

Funding Information:
Funding: L. E. Cipriano was supported by the David G. Burgoyne Faculty Fellowship and a Discovery Grant from the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2014-04089]. E. A. Enns was supported by the National Institute for Allergy and Infectious Diseases at the National Institutes of Health [Grant K25AI118476 (PI: Enns)].

Publisher Copyright:
© 2019 INFORMS.


  • Antimicrobial resistance
  • Dynamic health policy
  • Healthcare management
  • Infectious disease models
  • Optimal control


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