Journal of Mathematical Biology

, Volume 78, Issue 5, pp 1425–1438 | Cite as

Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations

  • Getachew K. BefekaduEmail author
  • Quanyan Zhu


In this paper, we consider the problem of controlling a diffusion process pertaining to an opioid epidemic dynamical model with random perturbation so as to prevent it from leaving a given bounded open domain. In particular, we assume that the random perturbation enters only through the dynamics of the susceptible group in the compartmental model of the opioid epidemic dynamics and, as a result of this, the corresponding diffusion is degenerate, for which we further assume that the associated diffusion operator is hypoelliptic, i.e., such a hypoellipticity assumption also implies that the corresponding diffusion process has a transition probability density function with strong Feller property. Here, we minimize the asymptotic exit rate of such a controlled-diffusion process from the given bounded open domain and we derive the Hamilton–Jacobi–Bellman equation for the corresponding optimal control problem, which is closely related to a nonlinear eigenvalue problem. Finally, we also prove a verification theorem that provides a sufficient condition for optimal control.


Diffusion processes Exit probability Epidemiology SIR compartmental model Prescription drug addiction Markov controls Minimum exit rates Principal eigenvalues Optimal control problem 

Mathematics Subject Classification

35J70 37C20 60J60 93E20 92D25 90C40 


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Electrical and Computer EngineeringMorgan State UniversityBaltimoreUSA
  2. 2.Department of Electrical and Computer Engineering, Tandon School of EngineeringNew York UniversityBrooklynUSA

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