Global Dynamics in an Alcoholism Epidemic Model with Saturation Incidence Rate and Two Distributed Delays

In this study, considering the delays for a susceptible individual becoming an alcoholic and the relapse of a recovered individual back into being an alcoholic, we formulate an epidemic model for alcoholism with distributed delays and relapse. The basic reproduction number $R_0$ is calculated, and the threshold property of $R_0$ is established. By analyzing the characteristic equation, we demonstrate the local asymptotic stability of the different equilibria under various conditions: when $R_0<1$, the alcoholism-free equilibrium is locally asymptotically stable; when $R_0>1$, the alcoholism equilibrium exists and is locally asymptotically stable. Furthermore, we demonstrate the global asymptotic stability at each equilibrium using a suitable Lyapunov function. Specifically, when $R_0<1$, the alcoholism-free equilibrium is globally asymptotically stable; when $R_0>1$, the alcoholism equilibrium is globally asymptotically stable. The sensitivity analysis of $R_0$ shows that reducing exposure is more effective than treatment in controlling alcoholism. Interestingly, we found that extending the latency delay $h_1$ and relapse delay $h_2$ also effectively contribute to the control of the spread of alcoholism. Numerical simulations are also provided to support our theoretical results.