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A Novel Aggregated SIR Model for Spatial Epidemic Propagation

M. Soledad Aronna, Mariana Bergonzi, Ernesto Kofman

TL;DR

An extension of the classical susceptible infectious recovered (SIR) model that incorporates the effects of spatial propagation of an epidemic through a small number of additional compartments is proposed, offering a parsimonious alternative for studying spatially structured epidemic dynamics when only aggregated data are available or when model simplicity is essential.

Abstract

We propose an extension of the classical susceptible infectious recovered (SIR) model that incorporates the effects of spatial propagation of an epidemic through a small number of additional compartments. The model is designed to capture the dynamics of disease spread across multiple interconnected cities or populated regions, while avoiding the high dimensionality and large parameter sets typical of network based or agent-based approaches. Instead of explicitly modeling individual locations or mobility networks, we introduce aggregate variables that describe whether the epidemic has not yet reached, is currently active in, or has already passed through different regions of the spatial domain. This formulation allows the model to reproduce key qualitative features observed in aggregated incidence data, such as prolonged plateaus and multiple infection waves arising from asynchronous local outbreaks. The resulting system consists of ordinary differential equations with a relatively small number of interpretable parameters, providing a tractable framework for analytical investigation and numerical simulation. Our approach offers a parsimonious alternative for studying spatially structured epidemic dynamics when only aggregated data are available or when model simplicity is essential.

A Novel Aggregated SIR Model for Spatial Epidemic Propagation

TL;DR

An extension of the classical susceptible infectious recovered (SIR) model that incorporates the effects of spatial propagation of an epidemic through a small number of additional compartments is proposed, offering a parsimonious alternative for studying spatially structured epidemic dynamics when only aggregated data are available or when model simplicity is essential.

Abstract

We propose an extension of the classical susceptible infectious recovered (SIR) model that incorporates the effects of spatial propagation of an epidemic through a small number of additional compartments. The model is designed to capture the dynamics of disease spread across multiple interconnected cities or populated regions, while avoiding the high dimensionality and large parameter sets typical of network based or agent-based approaches. Instead of explicitly modeling individual locations or mobility networks, we introduce aggregate variables that describe whether the epidemic has not yet reached, is currently active in, or has already passed through different regions of the spatial domain. This formulation allows the model to reproduce key qualitative features observed in aggregated incidence data, such as prolonged plateaus and multiple infection waves arising from asynchronous local outbreaks. The resulting system consists of ordinary differential equations with a relatively small number of interpretable parameters, providing a tractable framework for analytical investigation and numerical simulation. Our approach offers a parsimonious alternative for studying spatially structured epidemic dynamics when only aggregated data are available or when model simplicity is essential.
Paper Structure (12 sections, 2 theorems, 24 equations, 4 figures)

This paper contains 12 sections, 2 theorems, 24 equations, 4 figures.

Table of Contents

  1. Keywords
  2. Introduction
  3. Model Formulation
  4. Model Dynamics
  5. About the basic reproduction number $\mathcal{R}_0$
  6. About the model parameters
  7. Numerical Examples
  8. Synthetic metapopulation model
  9. Fitting real data
  10. Conclusions
  11. Computing $\mathcal{R}_0$ for the aggregated model
  12. Computing $\mathcal{R}_0$ for the network model

Key Result

Theorem 1

The basic reproduction number for model eq:aggmodel is given by which coincides with the basic reproduction number of the standard SIR model (see, for example, brauer2012mathematical).

Figures (4)

  • Figure 1: Aggregated model scheme.
  • Figure 2: Infected population trajectories in the Multi-Population model.
  • Figure 3: Infected population trajectories in the Aggregated and Multi-Population model.
  • Figure 4: Daily cases of COVID-19 in Argentina (with 7-day moving average) and simulation results with the aggregated model.

Theorems & Definitions (3)

  • Theorem 1
  • Proposition B.1
  • proof