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 h1 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.