Compartmental modelling of social dynamics with generalised peer incidence

Abstract

A generalised compartmental method for investigating the spread of socially determined behaviour is introduced, and cast in the specific context of societal smoking dynamics with multiple peer influence. We consider how new peer influence terms, acting in both the rate at which smokers abandon their habit, and the rate at which former smokers relapse, can affect the spread of smoking in populations of constant size. In particular, we develop a three-population model (comprising classes of potential, current, and former smokers) governed by multiple incidence transfer rates with linear frequency dependence. Both a deterministic system and its stochastic analogue are discussed: in the first we demonstrate that multiple peer influence not only modifies the number of steady-states and nature of their asymptotic stability, but also introduces a new kind of non-linear "tipping-point" dynamic; while in the second we use recently compiled smoking statistics from the Northeast of England to investigate the impact of systemic uncertainty on the potential for societal "tipping". The generality of our assumptions mean that the results presented here are likely to be relevant to other compartmental models, especially those concerned with the transmission of socially determined behaviours.