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

Abstract

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 R0 is calculated, and the threshold property of R0 is established. By analyzing the characteristic equation, we demonstrate the local asymptotic stability of the different equilibria under various conditions: when R0<1, the alcoholism-free equilibrium is locally asymptotically stable; when R0>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 R0<1, the alcoholism-free equilibrium is globally asymptotically stable; when R0>1, the alcoholism equilibrium is globally asymptotically stable. The sensitivity analysis of R0 shows that reducing exposure is more effective than treatment in controlling alcoholism. Interestingly, we found that extending the latency delay h1 and relapse delay h2 also effectively contribute to the control of the spread of alcoholism. Numerical simulations are also provided to support our theoretical results.