Dynamic analysis of a fractional-order drug addiction model with generalized nonlinear incidence rate

This paper analyzes a fractional-order drug addiction model with a generalized incidence rate of the form $ f(P,A) $. The analysis begins by establishing the well-posedness of the model, verifying the existence, uniqueness, non-negativity, and boundedness of solutions. The basic reproduction number $ R_0 $ is derived, and two equilibrium points are identified: The drug-free equilibrium and the drug-persistent equilibrium. The local asymptotic stability of each equilibrium is investigated using the Routh-Hurwitz criterion, while global stability is established by constructing appropriate Lyapunov functionals and applying LaSalle’s invariance principle. Unlike classical integer-order models that assume memoryless dynamics, the incorporation of the Caputo fractional derivative effectively accounts for the 'memory effects' and historical dependence inherent in the neurological and behavioral progression of substance abuse. The main theoretical findings demonstrate that, under appropriate conditions, drug addiction will eventually die out when $ R_0 < 1 $, but will persist when $ R_0 > 1 $. Numerical simulations are conducted to validate the theoretical results, offering insights for intervention strategies and policy-making. In summary, the proposed framework provides a robust tool for understanding addiction dynamics and formulating public health policies to mitigate the impact of substance abuse in communities.