The dynamics of two-stage contagion

Abstract

We explore simple models aimed at the study of social contagion, in which contagion proceeds through two stages. When coupled with demographic turnover, we show that two-stage contagion leads to nonlinear phenomena which are not present in the basic 'classical' models of mathematical epidemiology. These include: bistability, critical transitions, endogenous oscillations, and excitability, suggesting that contagion models with stages could account for some aspects of the complex dynamics encountered in social life. These phenomena, and the bifurcations involved, are studied by a combination of analytical and numerical means.

Predicting the Size and Duration of the COVID-19 Pandemic

“…But this model has no size or duration because I(t) can go to infinity when c > 0! It is worth mentioning the two-stage model proposed by Katriel showing instabilities in social contagions that are reminiscent of bursts and instabilities in the spread of COVID-19 [20]. His novel solution introduces the idea of multiple stages of infection and offers an alternative to the explanations given here.…”

When ideas go viral -- complex bifurcations in a two-stage transmission model

“…A similar compartmental model for two-stage contagion was previously proposed by Guy Katriel [8]. System (2) differs from Katriel's work in two aspects.…”

“…From a mathematical point of view, modeling social contagion in large communities is very similar to modeling the transmission of an infectious disease in a population. It seems hence natural that methods from the field of mathematical epidemiology, such as compartmental models [2,3,4,5,6,7,8], are used to model social contagion phenomena. In certain cases social contagion and disease spread have to be considered coupled to one another, as when a group in a social network criticizes vaccination [9].…”

“…In the context of social contagion, the spreading of a specific behavior or opinion in a population could be described as the transition of individuals from the "naive" (susceptible) status to the "promoter" (infectious) one. This transition might require several steps and depend on repeated exposure to promoters [8,13]. Therefore, we classify individuals in • naive/susceptible (S), those who have not yet been exposed to the considered behavior/opinion,…”

“…Second, we do not consider population demography (births/deaths). Such differences lead to major analytical challenges with respect to Katriel's study [8]. In the rest of this work we study the qualitative properties of system (2) by means of analytical and numerical methods.…”