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.