| Abstract: |
| This work proposes a novel approach for examining a SATR model using epidemiological modeling. Using a convex incidence function, which is a measure of the frequency of interaction between an addict and a non-addict, we want to explore the dynamics of addiction transmission within a community. We establish the well-posedness of the model, which ensures the existence, uniqueness, and positivity of the solution. We also determine the next-generation operator, which allows us to calculate $\mathcal{R_0}$, the fundamental reproduction number. We examine in detail how $\mathcal{R_0}$ serves as a threshold quantity to regulate the behavior of the addiction. Specifically, we show that when $\mathcal{R_0} < 1$, the addiction-free steady state is globally asymptotically stable for all values of $q$. Conversely, we establish the uniform persistence of the associated equilibrium when $\mathcal{R_0} > 1$ and construct a steady state of addiction that is globally asymptotically stable for all values of $q$. We also analyze the asymptotic profile of the addiction stable state, considering the effects of small and high addiction rates on the state dynamics. To validate our theoretical findings, we provide numerical simulations that both confirm our findings and provide additional insights into the dynamics of the addiction. |
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