# Mutually cooperative epidemics on power-law networks

## TL;DR

This paper investigates how cooperative interactions between two infectious diseases influence epidemic transitions on power-law networks, revealing conditions for continuous or discontinuous outbreaks based on network heterogeneity and disease cooperativity.

## Contribution

It introduces a mean-field theoretical framework for co-infection dynamics on uncorrelated power-law networks and clarifies how network heterogeneity affects epidemic transition types.

## Key findings

- Discontinuous transitions occur at high cooperativity when the second moment is finite.
- Always continuous transitions on scale-free networks with diverging second moment.
- Heterogeneity and system size significantly influence the nature of epidemic transitions.

## Abstract

The spread of an infectious disease can, in some cases, promote the propagation of other pathogens favouring violent outbreaks, which cause a discontinuous transition to an endemic state. The topology of the contact network plays a crucial role in these cooperative dynamics. We consider a susceptible--infected--removed (SIR) type model with two mutually cooperative pathogens: an individual already infected with one disease has an increased probability of getting infected by the other. We present an heterogeneous mean-field theoretical approach to the co--infection dynamics on generic uncorrelated power-law degree-distributed networks and validate its results by means of numerical simulations. We show that, when the second moment of the degree distribution is finite, the epidemic transition is continuous for low cooperativity, while it is discontinuous when cooperativity is sufficiently high. For scale-free networks, i.e. topologies with diverging second moment, the transition is instead always continuous. In this way we clarify the effect of heterogeneity and system size on the nature of the transition and we validate the physical interpretation about the origin of the discontinuity.