# The effects of heterogeneity on stochastic cycles in epidemics

## TL;DR

This paper analyzes how heterogeneity in susceptibility and infectivity influences stochastic epidemic cycles, deriving analytical spectra and examining synchronization effects in a modified SIR model.

## Contribution

It introduces a reduced analytical framework for understanding the impact of heterogeneity on epidemic quasi-cycles and their spectral properties.

## Key findings

- Cycle frequency depends on population averages of heterogeneity.
- Amplitude of cycles is influenced by higher-order moments of heterogeneity.
- Heterogeneity affects synchronization and phase lag between groups.

## Abstract

Models of biological processes are often subject to different sources of noise. Developing an understanding of the combined effects of different types of uncertainty is an open challenge. In this paper, we study a variant of the susceptible-infective-recovered model of epidemic spread, which combines both agent-to-agent heterogeneity and intrinsic noise. We focus on epidemic cycles, driven by the stochasticity of infection and recovery events, and study in detail how heterogeneity in susceptibilities and propensities to pass on the disease affects these quasi-cycles. While the system can only be described by a large hierarchical set of equations in the transient regime, we derive a reduced closed set of equations for population-level quantities in the stationary regime. We analytically obtain the spectra of quasi-cycles in the linear-noise approximation. We find that the characteristic frequency of these cycles is typically determined by population averages of susceptibilities and infectivities, but that their amplitude depends on higher-order moments of the heterogeneity. We also investigate the synchronisation properties and phase lag between different groups of susceptible and infected individuals.