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Modelling the Opioid Epidemic in New York City vs. New York State

This project is a collaboration between myself, Noah Igra, and Simon Zhao.

Motivation:

How is the opioid epidemic affecting different people of different demographics? That’s the question we started with when we began this project. We wanted to see if, given a set of initial conditions, there is a difference between the urban and the rural demographic when it comes to discussing the most impactful components of the opioid epidemic. Differences across regions would imply that different approaches in mitigating the epidemic should be applied to urban and rural areas of the same state. In our case, we chose to use New York State vs. New York City. We were successful in modelling both New York City and New York State. Results look promising, but we found that there are no considerable differences between the groups for the most part. Moving forward, we need to find better comparative measures to further analyze how the two groups differ.

Our strategy was thus: First, model the epidemic and evaluate its performance. Second, perform a sensitivity analysis to see how parameters (such as prescription rate and entry into treatment rate) affect each group differently. Third, compare the groups sensitivities for difference.

Background:

In 1927, Kermack and McKendrick introduced the original SIR (Susceptible, Infected, Recovered) model to evaluate the factors in contagious epidemics. This model has since been modified and repurposed to evaluate different diseases using population dynamical methods. In 2007, White and Comiskey used this model to address the heroin epidemic in Ireland, one of the first applications of these methods on a substance dependent epidemic. In 2019, Battista, Strickland and Pearcy took the next step by redesigning this model to include the potential for both addiction and rehabilitation. Focusing on prescription opioids, they used this model to evaluate the possibility for an addiction free equilibrium given current prescription rates on a national scale. For legitimacy, they validated their model by comparing their model outputs to real values for given years. They also performed a sensitivity analysis on the model’s parameters to confirm sufficiency (for example, using a linear parameter as opposed to a non-linear one to define the relationship between different classes in the population).

Data Sourcing and Model Validation:

The parameters for the model were sourced from data reflecting rates of change on different levels of the population; national, state, sub-state, etc. Furthermore, the data was manipulated to reflect yearly rates of change from one class of the population to another. However, some parameters were harder to find, such as ε (end prescription without addiction rate), and Ϛ (rate of rehab entry). Some of these parameters have a wider variation across years, such as α (prescription rate). To evaluate the model accurately, these coefficients are sampled from realistic ranges.

Model:

Instead of a traditional SIR [Susceptible, Infected, Recovered] model, Battista et al. used a modified 4 category system, namely SPAR [Susceptible, Prescribed, Addicted, Rehabilitation/treatment]

The parameters are presented in the chart below. There are 10 parameters that are present within Battista’s mathematical model.

For our own work, we sourced data to reflect the epidemic in New York City and in New York State excluding NYC. Additionally, the Addicted class in our case refers to people addicted to all opioids as opposed to only prescription opioids (which is what Battista’s model used). This accounts for more cases that are reflected in the data we sourced, and as such was a better fit.

The equations that stem from these coefficients and its interactions is given below:

Our contribution to this project involved dimensional analysis in an effort to reduce the parameter space as follows:

Notice that we have assumed δ + σ + μ = 1. This means that we assume that after each year those that join rehab are treated, pass away, or relapse. Also, by changing the equations to a ratio-based system, we are focusing on relative values as opposed to actual ones, simplifying sourcing. In this way, we are able to remove δ (successful treatment rate) from the system of equations. We find that our newly crafted system equations generate simulated overdoses consistent with real data:

This graph displays the difference in simulated overdoses between the original system of equations and the modified system of equation (only at most 2 deaths difference per year!). The different colors correspond to different values of end prescription without addiction rate (ε).

Model Results:

NYC

NYS excluding NYC

These graphs are important because the graph on the left shows that the values we have chosen correspond to real values of opioid related death. That is to say that our modelling is consistent with real data! The graph on the left shows us that there is a large range of output for the three difficult parameters, and the graph on the right shows that there is a range of combos resulting in the correct fit.

Sensitivity Analysis:

A sensitivity analysis allows us to see how each parameter contributes to the overall yearly rate of change for each class (again, S, P, A, R). In essence, we are finding the effect of each parameter on the overall system. In doing so, we are able to evaluate any differences in effect of each parameter between New York City and New York State. Not only can we find the effect of each parameter, but also its interactions with other parameters as well. For the sensitivity analysis, we followed the procedure in Saltelli’s 2010 paper. We used a variance-based sensitivity analysis, using a Monte Carlo method in order to solve a decomposed scalar of multiple parameters. This involves calculating the contribution of each parameter on the total variance, namely how much each parameter changes the variance. This tells us the effect that each parameter has on the output of our model. In our work, we looked for both first order sensitivity, which is the isolated effect of each parameter on the variance. We also looked for total order sensitivity, or the total effect each parameter had, including interactions, on the variance. In our project, we ran everything through a python program containing SALib, Saltelli’s own Sensitivity Analysis Library.

Results:

After finding the data for parameters, creating the program for modelling the epidemic, and running the two different groups through sensitivity analysis, we produced the following:

New York City

New York State excluding NYC

Prescription rate is (obviously) very important. There is a huge jump between first-order and total-order sensitivity for treatment entry rate. This implies a non-linear effect of treatment entry would be more relevant, similar to the one needed to model the effect of addiction via illicit drugs. Considering that an increase in the proportion in the population in treatment can reflect an increase in investment in overall treatment, it is reasonable to propose that future versions of these models should include a treatment entry rate dependent on both the addicted class and the treatment class. To test for difference between regions, we used a z statistics test for paired sensitivity ratios across sub-classes (S,P,A and R) using Holms-Sidak correction for multiple comparisons. There weren’t major differences between New York City and New York State, but relapse rate on the Addicted subgroup of the population had a statistically significant difference (p-value < 0.05). The results were not mind-blowing or earth shattering, but are encouraging. Comparing proportions is difficult in such a large parameter space, and there is needs to be more work on the methodology of these comparisons.

Future of this study:

There is still much to learn about the opioid epidemic by focusing on smaller demographics. This allows for a more granular approach that can make solving this national issue more specified for each area, and also gives a clearer sense of the issue in each location. In the future, we hope to gain more data about the epidemic for better data prediction.

Additionally, the increased presence of synthetic opioids such as fentanyl in this epidemic will be highly influential to this area of research. The benefit of using data accounting for Substance Use Disorder for all opioids, as opposed to prescription opioids alone, is that we are able to account for fentanyl as well as any other illicit opioids involved in the epidemic. Data on fentanyl and illicit opioids have not been thoroughly collected yet, but their contribution to the ongoing crisis in the United States requires us to include them in future studies.

References:

Saltelli, Andrea, et al. “Variance Based Sensitivity Analysis of Model Output. Design and Estimator for the Total Sensitivity Index.” Computer Physics Communications, vol. 181, no. 2, Feb. 2010, pp. 259–270., doi:10.1016/j.cpc.2009.09.018.

Battista, Nicholas A., et al. “Modeling the Prescription Opioid Epidemic.” Bulletin of Mathematical Biology, vol. 81, no. 7, 2019, pp. 2258–2289., doi:10.1007/s11538–019–00605–0.

Sobol, I.M. “Global sensitivity indices for nonlinear mathematical models and their Monte Carlo estimates.” Mathematics and Computers in Simulation, Volume 55, Issues 1–3, (2001), Pages 271–280,

Sobol, I.M. “Sensitivity estimates for nonlinear mathematical models.” Matem. Modelirovanie 2 (1) (1990) 112–118 (in Russian), MMCE, 1(4) (1993) 407–414 (in English),

Parsells Kelly J, Cook SF, Kaufman DW, et al. “Prevalence and characteristics of opioid use in the US adult population.” Pain. (2008), 138:507–513.

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