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

Getachew K. Befekadu, Quanyan Zhu

Research output: Contribution to journalArticle

Abstract

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.

Original languageEnglish (US)
JournalJournal of Mathematical Biology
DOIs
StateAccepted/In press - Jan 1 2018

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Opioids
Random Perturbation
narcotics
Epidemic Model
Dynamical Model
Diffusion Process
Opioid Analgesics
Optimal Control
Strong Feller Property
Hypoellipticity
Compartmental Model
Controlled Diffusions
Transition Density
Nonlinear Eigenvalue Problem
Transition Probability
Probability density function
fellers
Optimal Control Problem
Minimise
Imply

Keywords

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

ASJC Scopus subject areas

  • Modeling and Simulation
  • Agricultural and Biological Sciences (miscellaneous)
  • Applied Mathematics

Cite this

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abstract = "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.",
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KW - Epidemiology

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