Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

How many outbreaks before an epidemic?

Fabio Rapallo, Enrico Scalas, Pietro Terna

TL;DR

This paper analyzes the finite-population behavior of the Reed-Frost epidemic model, showing that the final-size distribution can be bimodal with distinct small- and large-outbreak regimes. Using the exact final-size formula where stable and Monte Carlo simulations otherwise, the authors quantify the probability of small outbreaks and identify a critical threshold where bimodality emerges. They corroborate the analytic findings with an agent-based SIsaR model of COVID-19 diffusion and show that Reed-Frost provides a good approximation in finite populations. The work also links the outbreak dynamics to bond-percolation and discusses the impact of containment on outbreak probabilities, with broader implications for diffusion phenomena beyond epidemiology.

Abstract

In this work, we study the finite-population behaviour of the Reed-Frost epidemic model. Our analysis relies on the exact expression for the final epidemic size, replaced by Monte Carlo simulations in cases where the exact formula becomes numerically unstable. When the initial reproduction number is greater than a critical threshold, the distribution of the final size becomes bimodal. We therefore define the probabilities of small and large outbreaks, providing an intuitive answer to the question posed in the title through simple arguments based on the geometric distribution. Finally, an agent-based simulation confirms that the Reed-Frost model offers a good approximation in the case of the COVID-19 outbreak.

How many outbreaks before an epidemic?

TL;DR

This paper analyzes the finite-population behavior of the Reed-Frost epidemic model, showing that the final-size distribution can be bimodal with distinct small- and large-outbreak regimes. Using the exact final-size formula where stable and Monte Carlo simulations otherwise, the authors quantify the probability of small outbreaks and identify a critical threshold where bimodality emerges. They corroborate the analytic findings with an agent-based SIsaR model of COVID-19 diffusion and show that Reed-Frost provides a good approximation in finite populations. The work also links the outbreak dynamics to bond-percolation and discusses the impact of containment on outbreak probabilities, with broader implications for diffusion phenomena beyond epidemiology.

Abstract

In this work, we study the finite-population behaviour of the Reed-Frost epidemic model. Our analysis relies on the exact expression for the final epidemic size, replaced by Monte Carlo simulations in cases where the exact formula becomes numerically unstable. When the initial reproduction number is greater than a critical threshold, the distribution of the final size becomes bimodal. We therefore define the probabilities of small and large outbreaks, providing an intuitive answer to the question posed in the title through simple arguments based on the geometric distribution. Finally, an agent-based simulation confirms that the Reed-Frost model offers a good approximation in the case of the COVID-19 outbreak.

Paper Structure

This paper contains 9 sections, 9 equations, 4 figures, 4 tables.

Table of Contents

  1. Motivation
  2. The Models
  3. Agent based model
  4. The model by Reed and Frost
  5. Results
  6. Agent-based model: Unrestricted epidemic
  7. Agent-based model: Epidemic with containment measures
  8. Simulations of the Reed-Frost model
  9. Final remarks and discussion

Figures (4)

  • Figure 1: Scatterplot of the final total number of infected individuals of all categories versus the duration of the epidemic without containment measures.
  • Figure 2: Scatterplot of the final total number of infected individuals of all categories versus the duration of the epidemic with containment measures.
  • Figure 3: Six distributions of the final epidemic size. The grey rectangles are the Monte Carlo estimates, the segments are the sd, while red dots represent the exact probabilities. When the exact formula is not stable, only the Monte Carlo results are shown.
  • Figure 4: Critical values of $\tilde{p}$ (left panel) and $R_0$ (right panel) with $i_0=1$.