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On the Dynamic Behavior of the Network SIR Epidemic Model

Alutto, Martina ; Cianfanelli, Leonardo ; Como, Giacomo LU and Fagnani, Fabio (2025) In IEEE Transactions on Control of Network Systems 12(1). p.177-189
Abstract

In this article, we study a susceptible–infected–recovered (SIR) epidemic model on a network of n interacting subpopulations. We analyze the transient and asymptotic behavior of the infection dynamics in each node of the network. In contrast to the classical scalar epidemic SIR model, where the infection curve is known to be unimodal (either always decreasing over time, or initially increasing until reaching a peak and from then on monotonically decreasing and asymptotically vanishing), we show the possible occurrence of multimodal infection curves in the network SIR epidemic model with n ≥ 2 subpopulations. We then focus on the special case of rank-1 interaction matrices, modeling subpopulations of homogeneously mixing individuals with... (More)

In this article, we study a susceptible–infected–recovered (SIR) epidemic model on a network of n interacting subpopulations. We analyze the transient and asymptotic behavior of the infection dynamics in each node of the network. In contrast to the classical scalar epidemic SIR model, where the infection curve is known to be unimodal (either always decreasing over time, or initially increasing until reaching a peak and from then on monotonically decreasing and asymptotically vanishing), we show the possible occurrence of multimodal infection curves in the network SIR epidemic model with n ≥ 2 subpopulations. We then focus on the special case of rank-1 interaction matrices, modeling subpopulations of homogeneously mixing individuals with different activity rates, susceptibility to the disease, and infectivity levels. For this special case, we find n invariants of motion and provide an explicit expression for the limit equilibrium point. We also determine necessary and sufficient conditions for stability of the equilibrium points. We then establish an upper bound on the number of changes of monotonicity of the infection curve at the single node level and provide sufficient conditions for its multimodality. Finally, we present some numerical results revealing that in the case of interaction matrices with rank larger than 1, the single nodes' infection curves may display multiple peaks.

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author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Infection curves, invariants of motion, limit equilibrium points, network epidemic models, stability, susceptible–infected–recovered (SIR) model
in
IEEE Transactions on Control of Network Systems
volume
12
issue
1
pages
13 pages
publisher
IEEE - Institute of Electrical and Electronics Engineers Inc.
external identifiers
  • scopus:105001085376
ISSN
2325-5870
DOI
10.1109/TCNS.2024.3448136
language
English
LU publication?
yes
id
5314b741-c9ff-4090-a8fa-d0ad33c596e4
date added to LUP
2025-09-10 10:32:12
date last changed
2025-10-14 11:24:56
@article{5314b741-c9ff-4090-a8fa-d0ad33c596e4,
  abstract     = {{<p>In this article, we study a susceptible–infected–recovered (SIR) epidemic model on a network of n interacting subpopulations. We analyze the transient and asymptotic behavior of the infection dynamics in each node of the network. In contrast to the classical scalar epidemic SIR model, where the infection curve is known to be unimodal (either always decreasing over time, or initially increasing until reaching a peak and from then on monotonically decreasing and asymptotically vanishing), we show the possible occurrence of multimodal infection curves in the network SIR epidemic model with n ≥ 2 subpopulations. We then focus on the special case of rank-1 interaction matrices, modeling subpopulations of homogeneously mixing individuals with different activity rates, susceptibility to the disease, and infectivity levels. For this special case, we find n invariants of motion and provide an explicit expression for the limit equilibrium point. We also determine necessary and sufficient conditions for stability of the equilibrium points. We then establish an upper bound on the number of changes of monotonicity of the infection curve at the single node level and provide sufficient conditions for its multimodality. Finally, we present some numerical results revealing that in the case of interaction matrices with rank larger than 1, the single nodes' infection curves may display multiple peaks.</p>}},
  author       = {{Alutto, Martina and Cianfanelli, Leonardo and Como, Giacomo and Fagnani, Fabio}},
  issn         = {{2325-5870}},
  keywords     = {{Infection curves; invariants of motion; limit equilibrium points; network epidemic models; stability; susceptible–infected–recovered (SIR) model}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{177--189}},
  publisher    = {{IEEE - Institute of Electrical and Electronics Engineers Inc.}},
  series       = {{IEEE Transactions on Control of Network Systems}},
  title        = {{On the Dynamic Behavior of the Network SIR Epidemic Model}},
  url          = {{http://dx.doi.org/10.1109/TCNS.2024.3448136}},
  doi          = {{10.1109/TCNS.2024.3448136}},
  volume       = {{12}},
  year         = {{2025}},
}