Global Dynamics in an Alcoholism Epidemic Model with Saturation Incidence Rate and Two Distributed Delays - Zejun Wang; Haicheng Liu; Mingyang Li; Mei Yang (Mathematics, 2023)
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 <semantics>R0</semantics> is calculated, and the threshold property of <semantics>R0</semantics> is established. By analyzing the characteristic equation, we demonstrate the local asymptotic stability of the different equilibria under various conditions: when <semantics>R0<1</semantics>, the alcoholism-free equilibrium is locally asymptotically stable; when <semantics>R0>1</semantics>, 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 <semantics>R0<1</semantics>, the alcoholism-free equilibrium is globally asymptotically stable; when <semantics>R0>1</semantics>, the alcoholism equilibrium is globally asymptotically stable. The sensitivity analysis of <semantics>R0</semantics> shows that reducing exposure is more effective than treatment in controlling alcoholism. Interestingly, we found that extending the latency delay <semantics>h1</semantics> and relapse delay <semantics>h2</semantics> also effectively contribute to the control of the spread of alcoholism. Numerical simulations are also provided to support our theoretical results.