On the Dynamic Behavior of the Network SIR Epidemic Model
(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
- Alutto, Martina ; Cianfanelli, Leonardo ; Como, Giacomo LU and Fagnani, Fabio
- organization
- publishing date
- 2025
- 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
- 2025-09-10 10:33:08
@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}},
}