# Exact solution of generalized cooperative SIR dynamics

## TL;DR

This paper analytically solves a generalized cooperative SIR model for co-infection, revealing bifurcation behavior, stability conditions, and effects of asymmetry on epidemic thresholds, advancing understanding of disease ecology dynamics.

## Contribution

It provides the first analytical solutions for the generalized cooperative SIR model, including bifurcation analysis and stability conditions, extending previous models.

## Key findings

- Existence of saddle-node bifurcation in cooperative SIR dynamics
- Hybrid transition behavior identified in the model
- Asymmetry in infection rates affects epidemic thresholds and transition types

## Abstract

In this paper, we introduce a general framework for co-infection as cooperative SIR dynamics. We first solve analytically CGCG model [1] and then the generalized model in symmetric scenarios. We calculate transition points, order parameter, i.e. total number of infected hosts. Also we show analytically there is a saddle-node bifurcation for two cooperative SIR dynamics and the transition is hybrid. Moreover, we investigate where symmetric solution is stable for initial fluctuations. Then we study asymmetric cases of parameters. The more asymmetry, for the primary and secondary infection rates of one pathogen in comparison to the other pathogen, can lead to the less infected hosts, the higher epidemic threshold and continuous transitions. Our model and results for co-infection in combination with super-infection [2] can open a road to model disease ecology.