A mathematical model with nonlinear relapse: conditions for a forward-backward bifurcation

TL;DR

This paper develops a nonlinear relapse model for addiction dynamics, analyzing conditions for bifurcations and stability, and includes stochastic effects and numerical simulations to understand the impact of reformed individuals.

Contribution

It introduces a novel Susceptible-Addicted-Reformed model with nonlinear relapse, deriving conditions for forward-backward bifurcation and analyzing stochastic effects.

Findings

Conditions for forward-backward bifurcation are established.

The stability of the addicted-free equilibrium depends on R_0.

Reformed individuals' influence is highly sensitive to initial addiction levels.

Abstract

We constructed a Susceptible-Addicted-Reformed model and explored the dynamics of nonlinear relapse in the Reformed population. The transition from susceptible considered {\it at-risk} is modeled using a strictly decreasing general function, mimicking an influential factor that reduces the flow into the addicted class. The {\it basic reproductive number} is computed. Furthermore, $R_0$ determines the local asymptotically stability of the addicted-free equilibrium. Conditions for a forward-backward bifurcation were established using $R_0$ and other threshold quantities. A stochastic version of the model is presented, and some numerical examples are shown. Results showed that the influence of the temporarily reformed individuals is highly sensitive to the initial addicted population.