[Paper Review] A further study on the opioid epidemic dynamical model with random perturbation

Getachew K. Befekadu, Quanyan Zhu|arXiv (Cornell University)|May 31, 2018

COVID-19 epidemiological studies1 references3 citations

TL;DR

This paper develops a stochastic dynamical model of the opioid epidemic with random perturbations entering only through the susceptible population. It establishes two-sided bounds on the transition density function of the associated Fokker-Planck equation using a stochastic control framework, and derives an estimate for the exit probability from a bounded domain, offering theoretical tools for designing effective, data-informed public health interventions.

ABSTRACT

In this paper, we consider an opioid epidemic dynamical model with random perturbation that typically describes the interplay between regular prescription use, addictive use, and the process of rehabilitation from addiction and vice-versa. In particular, we provide two-sided bounds on the solution of the transition density function for the Fokker-Planck equation that corresponds to the opioid epidemic dynamical model, when a random perturbation enters only through the dynamics of the susceptible group in the compartmental model. Here, the proof for such bounds basically relies on the interpretation of the solution for the transition density function as the value function of a certain optimal stochastic control problem. Finally, as a possible interesting development in this direction, we also provide an estimate for the attainable exit probability with which the solution for the randomly perturbed opioid epidemic dynamical model exits from a given bounded open domain during a certain time interval. Note that such qualitative information on the first exit-time as well as two-sided bounds on the transition density function are useful for developing effective and fact-informed intervention strategies that primarily aim at curbing opioid epidemics or assisting in interpreting outcome results from opioid-related policies.

Motivation & Objective

• To analyze the propagation of random noise through an opioid epidemic model when perturbations affect only the susceptible compartment.
• To derive rigorous two-sided bounds on the transition density function of the Fokker-Planck equation governing the perturbed model.
• To estimate the attainable exit probability of the diffusion process from a bounded domain within a finite time interval.
• To connect the transition density to a value function in stochastic control, enabling theoretical justification for intervention strategies.
• To support the development of evidence-based, fact-informed policies for curbing opioid epidemics through probabilistic modeling.

Proposed method

• The transition density function is interpreted as the value function of an optimal stochastic control problem via logarithmic transformation.
• Two-sided bounds on the density are derived using the dynamic programming principle and verification arguments in stochastic control.
• The Fokker-Planck equation is analyzed under hypoellipticity assumptions to ensure smoothness and existence of solutions.
• A sequence of approximating boundary functions is constructed to handle the Dirichlet problem for the exit probability.
• The exit probability is estimated using convergence arguments and the dominated convergence theorem under weak convergence of the diffusion process.
• Theoretical results are established under assumptions of bounded, continuous coefficients and smooth domain boundaries.

Experimental results

Research questions

• RQ1 How does random perturbation entering only through the susceptible group affect the long-term dynamics of an opioid epidemic model?
• RQ2 What are the two-sided bounds on the transition density function of the Fokker-Planck equation in this stochastic model?
• RQ3 Can the transition density be represented as a value function in a stochastic control framework, and how does this enable tighter bounds?
• RQ4 What is the attainable exit probability of the system from a bounded domain within a finite time horizon?
• RQ5 How can these probabilistic estimates inform the design of effective, fact-based opioid intervention strategies?

Key findings

• Two-sided bounds on the transition density function are established by interpreting it as a value function in a stochastic control problem, providing rigorous probabilistic estimates.
• The exit probability of the diffusion process from a bounded domain is estimated using convergence of approximating functions and the dominated convergence theorem.
• The solution to the Fokker-Planck equation is shown to be smooth almost everywhere in the domain and continuous on the boundary under hypoellipticity and regularity conditions.
• The convergence of the perturbed process to the deterministic limit is proven as the noise intensity vanishes, ensuring consistency with the base model.
• The theoretical framework enables selection of admissible control strategies that prolong the time the system remains within a safe domain, relevant for policy design.
• The results provide a foundation for interpreting outcomes of opioid-related policies and designing interventions that minimize epidemic spread through probabilistic risk assessment.