# Mathematical analysis of complex SIR model with coinfection and density dependence

## TL;DR

This paper analyzes a complex SIR model with coinfection and density dependence, revealing how environmental carrying capacity influences disease persistence and stability, with implications for resource management and infection control.

## Contribution

It provides a mathematical analysis of an SIR model with coinfection and environmental carrying capacity, deriving threshold conditions and stability results.

## Key findings

- Small carrying capacity leads to a globally stable disease-free state.
- Increasing carrying capacity can promote disease persistence.
- The transition of equilibrium points is continuous as carrying capacity varies.

## Abstract

An SIR model with the coinfection of the two infectious agents in a single host population is considered. The model includes the environmental carry capacity in each class of population. A special case of this model is analyzed and several threshold conditions are obtained which describes the establishment of disease in the population. We prove that for small carrying capacity $K$ there exist a globally stable disease free equilibrium point. Furthermore, we establish the continuity of the transition dynamics of the stable equilibrium point, i.e. we prove that (1) for small values of $K$ there exists a unique globally stable equilibrium point, and (b) it moves continuously as $K$ is growing (while its face type may change). This indicate that carrying capacity is the crucial parameter and increase in resources in terms of carrying capacity promotes the risk of infection.