No-verdose: Using SEIRP and Optimal Control Theory to Model the Opioid Epidemic and Mitigate Opioid-Induced Mortality

Abstract

The opioid epidemic was declared a public health emergency in 2017, and continues to worsen the wellbeing of communities and account for hundreds of thousands of deaths annually. Drug epidemics can be modeled using Susceptible-Infected-Recovered (SIR) epidemiological models, which simulate the dynamics of addiction and inform intervention optimization strategies. In this project, we model opioid addiction using a variation of the SIR model called the SEIRP model that includes Exposed (E) and Prevented (P) compartments. We then introduce a control variable representing a digital education and prevention campaign as an epidemic intervention. We implement optimal control theory using Pontryagin’s Maximum Principle and the Forward-Backward Sweep Algorithm to maximize the impact of the education campaign. We use literate programming to enhance the communication of our research process. The uniqueness of our SEIRP model is that it accounts for individuals who avoid opioids for a lifetime due to prevention efforts. We found that for both quadratic and linear cost functions, the optimal intervention intensity remained at its maximum value at the beginning of the time frame. For quadratic cost, the optimal intervention intensity decreased continuously from the middle to the end of the time frame. For linear cost, the optimal intervention intensity dropped discontinuously to its minimum value in the middle and remained at its minimum through the end of the time frame. This project informs future opioid epidemic mitigation policies, in particular the effectiveness of digital education campaigns in relation to the costs of these efforts. Our research contributes to the United Nations’ Sustainable Development Goal #3: Good Health and Well-Being.