Analysis of an epidemic model with peer-pressure and information-dependent transmission with high-order distributed delay (Q2281527)

In this paper, the authors propose and analyze an information-dependent behavioral model for the spread of high-risk alcohol consumption, viewed as a ``social disease''. To this end, they tailor a three-dimensional epidemiological model along with certain epidemiological concepts to suit the purpose at hand. The result of their considerations is an integro-differential model with infinite delay in which the force of persuasion (corresponding to the force of infection in disease propagation models) is assumed to depend on a so-called information index, which accounts for past and current prevalence of alcohol abuse in the community. The information index is introduced by means of an integral term defined using a \(n\)-th order Erlangian kernel, corresponding to human behavioral changes driven by risk perception, the model of interest being then reformulated as a system of ODEs via the linear chain trick. The authors determine the basic reproduction number \(R_0\) of the model, which characterizes the behavior of the model near the alcohol-free equilibrium (corresponding to the disease-free equilibrium in disease propagation models), via the next generation method and prove that if \(R_0<1\), then the alcohol-free equilibrium is locally stable, sufficient conditions for its global stability being determined via a Lyapunov stability analysis. If \(R_0>1\), it is shown that there is at least an endemic equilibrium, a convexity condition being sufficient for uniqueness. It is also determined that the system undergoes a transcritical bifurcation at \(R_0=1\) and that high levels of peer pressure may induce backwards bifurcation, leading to multiple endemic equilibria coexisting in the presence of awareness, which makes the eradication of high-risk alcohol consumption in the community significantly more difficult.