### 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 language | English (US) |
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Journal | Journal of Mathematical Biology |

DOIs | |

State | Accepted/In press - Jan 1 2018 |

### Fingerprint

### 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

**Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations.** / Befekadu, Getachew K.; Zhu, Quanyan.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Optimal control of diffusion processes pertaining to an opioid epidemic dynamical model with random perturbations

AU - Befekadu, Getachew K.

AU - Zhu, Quanyan

PY - 2018/1/1

Y1 - 2018/1/1

N2 - 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.

AB - 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.

KW - Diffusion processes

KW - Epidemiology

KW - Exit probability

KW - Markov controls

KW - Minimum exit rates

KW - Optimal control problem

KW - Prescription drug addiction

KW - Principal eigenvalues

KW - SIR compartmental model

UR - http://www.scopus.com/inward/record.url?scp=85057965969&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85057965969&partnerID=8YFLogxK

U2 - 10.1007/s00285-018-1314-y

DO - 10.1007/s00285-018-1314-y

M3 - Article

JO - Journal of Mathematical Biology

JF - Journal of Mathematical Biology

SN - 0303-6812

ER -