Mathematical Modeling, Analysis, and Simulation of Patient Addiction Journey

Authors

Adan Baca, University of ArizonaFollow
Diego Gonzalez, University of La VerneFollow
Alonso G. Ogueda, George Mason UniversityFollow
Holly C. Matto, George Mason UniversityFollow
Padmanabhan Seshaiyer, George Mason UniversityFollow

Publication Date

8-13-2024

Keywords

Compartmental Models, Neural Networks, Drug Addiction, Literate Programming

Disciplines

Data Science | Mathematics | Numerical Analysis and Computation | Ordinary Differential Equations and Applied Dynamics | Physical Sciences and Mathematics | Science and Mathematics Education

Abstract

This paper aims to develop a mathematical model to study the dynamics of addiction as individuals go through their detox journey. The motivation for this work is three fold. First, there has been a significant increase in drug overdose and drug addiction following the COVID-19 pandemic, and addiction may be interpreted as a infectious disease. Secondly, the dynamics of infectious disease could be modeled via compartmental models described by differential equations and one can therefore leverage the existing analytical and numerical methods to model addiction as a disease. Finally, the work helps to inform how mathematical models governed by differential equations, can help to provide insights into societal challenges such as addition. The proposed work describes a modified SIR differential equation system for studying the non-linear dynamics various sub-populations among the drug users. A detailed mathematical analysis along with the derivation of the basic reproduction number for the proposed model is also presented. Furthermore, we numerically simulate the dynamics and validate the model against synthetic data for COVID-19 cases to show the robustness and reliability of our model. We also introduce a Physics Informed Neural Network approach that helps to estimate the parameters in the model efficiently. Integrated with the research is an instructional exposition using Literate Programming that an help educators to enhance their own pedagogical practices to engage student interaction in the classroom.

Recommended Citation

Baca, Adan; Gonzalez, Diego; Ogueda, Alonso G.; Matto, Holly C.; and Seshaiyer, Padmanabhan (2024) "Mathematical Modeling, Analysis, and Simulation of Patient Addiction Journey," CODEE Journal: Vol. 18, Article 4.
Available at: https://scholarship.claremont.edu/codee/vol18/iss1/4