# A generalized model of social and biological contagion

## TL;DR

This paper introduces a comprehensive contagion model that unifies social and biological spread mechanisms, accounting for individual memory, exposure dose, and susceptibility heterogeneity, with implications for understanding epidemic thresholds and persistence.

## Contribution

It presents a novel generalized model of contagion that incorporates memory, dose effects, and heterogeneity, classifies contagion dynamics, and links parameters to measurable real-world data.

## Key findings

- Identifies three classes of contagion models: epidemic threshold, vanishing critical mass, and critical mass.
- Shows conditions for class membership depend on memory length and infection probabilities.
- Finds that epidemics with permanent immunity tend to die out but can persist with memory effects.

## Abstract

We present a model of contagion that unifies and generalizes existing models of the spread of social influences and micro-organismal infections. Our model incorporates individual memory of exposure to a contagious entity (e.g., a rumor or disease), variable magnitudes of exposure (dose sizes), and heterogeneity in the susceptibility of individuals. Through analysis and simulation, we examine in detail the case where individuals may recover from an infection and then immediately become susceptible again (analogous to the so-called SIS model). We identify three basic classes of contagion models which we call \textit{epidemic threshold}, \textit{vanishing critical mass}, and \textit{critical mass} classes, where each class of models corresponds to different strategies for prevention or facilitation. We find that the conditions for a particular contagion model to belong to one of the these three classes depend only on memory length and the probabilities of being infected by one and two exposures respectively. These parameters are in principle measurable for real contagious influences or entities, thus yielding empirical implications for our model. We also study the case where individuals attain permanent immunity once recovered, finding that epidemics inevitably die out but may be surprisingly persistent when individuals possess memory.