# Critical behavior of a two-step contagion model with multiple seeds

## TL;DR

This paper analyzes how multiple infection seeds influence the critical behavior of contagion models, revealing a transition from hybrid to continuous epidemic spread as seed density increases, with distinct critical exponents.

## Contribution

It provides analytical and numerical insights into the phase transition nature of multi-seed contagion models, extending understanding beyond single-seed scenarios.

## Key findings

- Hybrid epidemic transition at low seed density
- Transition becomes continuous at high seed density
- Distinct critical exponents from single-seed models

## Abstract

A two-step contagion model with a single seed serves as a cornerstone for understanding the critical behaviors and underlying mechanism of discontinuous percolation transitions induced by cascade dynamics. When the contagion spreads from a single seed, a cluster of infected and recovered nodes grows without any cluster merging process. However, when the contagion starts from multiple seeds of $O(N)$ where $N$ is the system size, a node weakened by a seed can be infected more easily when it is in contact with another node infected by a different pathogen seed. This contagion process can be viewed as a cluster merging process in a percolation model. Here, we show analytically and numerically that when the density of infectious seeds is relatively small but $O(1)$, the epidemic transition is hybrid, exhibiting both continuous and discontinuous behavior, whereas when it is sufficiently large and reaches a critical point, the transition becomes continuous. We determine the full set of critical exponents describing the hybrid and the continuous transitions. Their critical behaviors differ from those in the single-seed case.