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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Notes
Including the Ryan Haight Online Pharmacy Consumer Protection Act of 2008 which prohibited the Internet distribution of controlled substances without a valid prescription (Ryan Haight Online Consumer Protection Act 2008); see also Dowell et al. (2016) for CDC guideline for prescribing opioids for chronic pain—United States, 2016.
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Befekadu, G.K., Zhu, Q. Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations. J. Math. Biol. 78, 1425–1438 (2019). https://doi.org/10.1007/s00285-018-1314-y
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DOI: https://doi.org/10.1007/s00285-018-1314-y
Keywords
- Diffusion processes
- Exit probability
- Epidemiology
- SIR compartmental model
- Prescription drug addiction
- Markov controls
- Minimum exit rates
- Principal eigenvalues
- Optimal control problem