An integro-differential equation model for the spread of alcohol abuse.

*(English)*Zbl 1268.45006Summary: The alcohol abuse level of individuals in a population as it depends on resilience and peer influence is considered in this paper. Several simple models are studied as well as an integro-differential equation model which is derived using coarse graining from a pre-existing discrete network system. The connection structure of the discrete system tends to be richer than that of the integro-differential equation model; however, the continuum problem can be studied analytically using traveling wave, perturbation and phase plane techniques.

The analysis presented in this paper suggests that, in both the discrete network and integro-differential models, nearly alcoholic or highly sober individuals are relatively unaffected by peer pressure, and this aspect of the models leads to an inertia in the spread of alcohol abuse or sobriety depending on the connectivity, initial conditions and resilience of the population. A related but different model is introduced that avoids this inertia. A treatment scheme had also been developed for the discrete network system. A continuum version for the integro-differential model is provided here.

The analysis presented in this paper suggests that, in both the discrete network and integro-differential models, nearly alcoholic or highly sober individuals are relatively unaffected by peer pressure, and this aspect of the models leads to an inertia in the spread of alcohol abuse or sobriety depending on the connectivity, initial conditions and resilience of the population. A related but different model is introduced that avoids this inertia. A treatment scheme had also been developed for the discrete network system. A continuum version for the integro-differential model is provided here.

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\textit{D. A. French} et al., J. Integral Equations Appl. 22, No. 3, 443--464 (2010; Zbl 1268.45006)

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