Dynamics of an epidemiological model of drinking

Resum

In this paper, we review and carry out the bifurcation analysis of the drinking model proposed by Manthey et al. in 2008. We show that the model exhibits a drinking-free equilibrium point and a problem drinking-free equilibrium which exists when the reproductive number RY 0 > 1. Each of these equilibria is a stable node if the threshold parameters RY 0 < 1,RZ 0 < 1 and R1 < 1,RY 0 > 1 respectively. Moreover, the existence of multiple endemic equilibria reveals the complex dynamics of the model by the occurrence of two saddle-node bifurcations leading to the phenomenon of backward bifurcation. Taking the transmission rate of non-drinkers to problem drinkers κ as a bifurcation parameter, we show the two saddle-nodes SN and SN2 for the extinction and persistence of the drinking problem. Before the disappearance of E1 and E2 after SN in the interval (0,RZc1 0 ) the system is bistable with E0 and E2 being the stable equilibrium points. The system is rendered mono stable after the SN before the SN2 emerged in (RZc1 0 ,RZc2 0 ) with E0 as the only stable equilibrium point. Then after the emergence of SN2 in the interval (RZc2 0 ,κ∗), the system becomes bistable again with E0 and E4. But for κ > κ∗, there is endemic persistence of the drinking problem with E4 as the only stable attractor. When such a scenario does not exist a forward bifurcation occurs for which the drinking-free equilibrium exchanges stability at κ = κc with drinking-present equilibrium point. Numerical simulations of the model support these extinction and persistence regions of the drinking problem.