Advertisement

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
Article
  • 391 Downloads

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.

Keywords

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 

References

  1. Amano K (1979) A necessary condition for hypoellipticity of degenerate elliptic–parabolic operators. Tokyo J Math 2:111–120MathSciNetCrossRefzbMATHGoogle Scholar
  2. Battista NA, Pearcy LB, Strickland WC (2018) Modeling the prescription opioid epidemic. Preprint, June 2018, arXiv:1711.03658 [q-bio.PE]
  3. Befekadu GK, Antsaklis PJ (2015) On the asymptotic estimates for exit probabilities and minimum exit rates of diffusion processes pertaining to a chain of distributed control systems. SIAM J Control Optim 53:2297–2318MathSciNetCrossRefzbMATHGoogle Scholar
  4. Befekadu GK, Zhu Q (2018) A further study on the opioid epidemic dynamical model with random perturbation. Preprint, June 2018, arXiv:1805.12534 [math.OC]
  5. Beneš V (1970) Existence of optimal strategies based on specified information for a class of stochastic decision problems. SIAM J Control 8:179–188MathSciNetCrossRefzbMATHGoogle Scholar
  6. Bicket MC, Long JJ, Pronovost PJ, Alexander GC, Wu CL (2017) Prescription opioid analgesics commonly unused after surgery: a systematic review. JAMA Surg 152:1066–1071CrossRefGoogle Scholar
  7. Borkar VS (1989) Optimal control of diffusion processes. Longman Scientific and Technical, HarlowzbMATHGoogle Scholar
  8. Boué M, Dupuis P (2001) Risk-sensitive and robust escape control for degenerate diffusion processes. Math Control Signals Syst 14:62–85MathSciNetCrossRefzbMATHGoogle Scholar
  9. Centers for Disease Control and Prevention (2018) Understanding the epidemic. https://www.cdc.gov/drugoverdose/epidemic/index.html. Accessed 12 April 2018
  10. Dowell D, Haegerich TM, Chou R (2016) CDC guideline for prescribing opioids for chronic pain—United States, 2016. MMWR Recomm Rep 65:1–49CrossRefGoogle Scholar
  11. Dupuis P, McEneaney WM (1997) Risk-sensitive and robust escape criteria. SIAM J Control Optim 35:2021–2049MathSciNetCrossRefzbMATHGoogle Scholar
  12. Elliott DL (1973) Diffusions on manifolds arising from controllable systems. In: Mayne DQ, Brockett RW (eds) Geometric methods in system theory. Reidel Publ. Co., Dordrecht, pp 285–294CrossRefGoogle Scholar
  13. Frieden TR, Houry D (2016) Reducing the risks of relief—the CDC opioid-prescribing guideline. N Engl J Med 374:1501–1504CrossRefGoogle Scholar
  14. Hörmander L (1967) Hypoelliptic second order differential operators. Acta Math 119:147–171MathSciNetCrossRefzbMATHGoogle Scholar
  15. Ichihara K, Kunita H (1974) A classification of the second order degenerate elliptic operators and its probabilistic characterization. Z Wahrscheinlichkeitstheor Verw Geb 30:253–254MathSciNetCrossRefzbMATHGoogle Scholar
  16. Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond A 115:700–721CrossRefzbMATHGoogle Scholar
  17. Kratz P, Pardoux E (2018) Large deviations for infectious diseases models. In: Donati-Martin C, Lejay A, Rouault A (eds) Séminaire de Probabilités XLIX, vol 2215. Lecture Notes in Mathematics. Springer, ChamCrossRefGoogle Scholar
  18. Krylov NV (1980) Controlled-diffusion processes. Springer, BerlinCrossRefzbMATHGoogle Scholar
  19. Ma M, Liu S, Li J (2017) Bifurcation of a heroin model with nonlinear incidence rate. Nonlinear Dyn 88:555–565CrossRefzbMATHGoogle Scholar
  20. Njagarah HJB, Nyabadza F (2013) Modeling the impact of rehabilitation, amelioration and relapse on the prevalence of drug epidemics. J Biol Syst.  https://doi.org/10.1142/S0218339013500010
  21. Office of the Assistant Secretary for Planning and Evaluation. Opioid abuse in the U.S. and HHS actions to address opioid-drug related overdoses and deaths. Published 26 March 2015. Accessed 20 June 2018Google Scholar
  22. Pardoux E, Samegni-Kepgnou B (2017) Large deviation principle for epidemic models. J Appl Prob 54:905–920MathSciNetCrossRefzbMATHGoogle Scholar
  23. Pardoux E, Samegni-Kepgnou B (2018) Large deviation principle for reflected Poisson driven SDEs in epidemic models. Preprint, August 2018, arXiv:1808.04621 [math.PR]
  24. Quaas A, Sirakov B (2008) Principal eigenvalue and the Dirichlet problem for fully nonlinear elliptic operators. Adv Math 218:105–135MathSciNetCrossRefzbMATHGoogle Scholar
  25. Ryan Haight Online Consumer Protection Act of 2008, Public Law 110-425 (H.R. 6353)Google Scholar
  26. Rudd RA, Seth P, David F, Scholl L (2016) Increases in drug and opioid-involved overdose deaths—United States, 2010–2015. MMWR Morb Mortal Wkly Rep 65:1445–1452CrossRefGoogle Scholar
  27. Samanta G (2011) Dynamic behaviour for a non autonomous heroin epidemic with time delay. J Appl Math Comput 35:161–178MathSciNetCrossRefzbMATHGoogle Scholar
  28. Stroock D, Varadhan SRS (1972) On degenerate elliptic–parabolic operators of second order and their associated diffusions. Commun Pure Appl Math 25:651–713MathSciNetCrossRefzbMATHGoogle Scholar
  29. Sussmann HJ, Jurdjevic V (1972) Controllability of nonlinear systems. J Differ Equ 12:95–116MathSciNetCrossRefzbMATHGoogle Scholar
  30. Volkow ND, McLellan AT (2016) Opioid abuse in chronic pain—misconceptions and mitigation strategies. N Engl J Med 374:1253–1263CrossRefGoogle Scholar
  31. White E, Comiskey C (2007) Heroin epidemics, treatment and ode modelling. Math Biosci 208:312–324MathSciNetCrossRefzbMATHGoogle Scholar

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

Personalised recommendations