Stability of a reaction–diffusion alcohol model with the impact of tax policy

Abstract

To understand the impact of tax policy on the persistence of a drinking behavior, a reaction–diffusion alcohol model with the impact of tax policy is studied, with the focus on the positivity and boundedness of solutions and particularly the asymptotic profile of the equilibria. The basic reproduction number $R_0$ is calculated, and existence of a drinking-free equilibrium and a unique drinking-present equilibrium is established for $R_0$ above the threshold value 1. For both of ODE system and reaction–diffusion model, it is shown that the drinking-free equilibrium is globally asymptotically stable if $R_0\le 1$, and if $R_0>1$, we obtain a sufficient condition for global asymptotically stability of the drinking-present equilibrium. Numerical simulations are also provided to illustrate our analytical results.