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A mathematical model with nonlinear relapse: conditions for a forward-backward bifurcation
Fabio Sanchez1, Jorge Arroyo-Esquivel2, Juan G Calvo1
1Centro de Investigación en Matemática Pura y Aplicada-Escuela de Matemática, Universidad de Costa Rica, San José, Costa Rica.
Journal of Biological Dynamics
|March 21, 2023
Summary
This study introduces a mathematical model for addiction, revealing that the number of temporarily reformed individuals significantly impacts relapse rates, especially with larger initial addicted populations. Understanding these dynamics is crucial for addiction intervention strategies.
Area of Science:
- Mathematical Biology
- Epidemiology
- Addiction Studies
Background:
- Addiction is a complex issue with significant public health implications.
- Relapse dynamics in reformed populations require robust mathematical modeling.
- Understanding factors influencing the transition from at-risk to addicted states is critical.
Purpose of the Study:
- To develop and analyze a novel mathematical model (Susceptible-Addicted-Reformed) for addiction dynamics.
- To investigate the nonlinear dynamics of relapse within the reformed population.
- To determine conditions for bifurcations and stability of equilibria.
Main Methods:
- Construction of a deterministic Susceptible-Addicted-Reformed (SAR) model.
- Analysis of the basic reproductive number (R0) for equilibrium stability.
- Investigation of forward-backward bifurcation conditions.
- Development and analysis of a stochastic version of the SAR model.
Main Results:
- The basic reproductive number determines the stability of the addiction-free equilibrium.
- Conditions for forward-backward bifurcations were established.
- Model simulations indicate high sensitivity of relapse dynamics to the initial addicted population size.
- The influence of temporarily reformed individuals on addiction spread is significant.
Conclusions:
- The SAR model provides insights into addiction and relapse nonlinear dynamics.
- Temporarily reformed individuals play a crucial role in modulating addiction prevalence.
- Initial addicted population size is a key factor in relapse dynamics.
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