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On the dynamic behavior of the network SIR epidemic model

Martina Alutto, Leonardo Cianfanelli, Giacomo Como, Fabio Fagnani

Abstract

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 $n\ge2$ 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.

On the dynamic behavior of the network SIR epidemic model

Abstract

We study a susceptible-infected-recovered (SIR) epidemic model on a network of 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 subpopulations. We then focus on the special case of rank- 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 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 , the single nodes' infection curves may display multiple peaks.
Paper Structure (14 sections, 10 theorems, 78 equations, 3 figures)

This paper contains 14 sections, 10 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Network SIR epidemic model
  4. The network SIR epidemic model with rank-$1$ interaction matrices
  5. Unimodality of the weighted aggregate infection curve
  6. Invariants of motion, limit equilibria, and stability
  7. Dynamic behavior of the infection in the single subpopulations
  8. A classification of possible dynamic behaviors
  9. Sufficient conditions for multimodal infection curves
  10. Numerical Simulations
  11. Conclusion
  12. Proof of Lemma \ref{['lemma1']}
  13. Proof of Lemma \ref{['lemma:varphi']}
  14. Proof of Lemma \ref{['lemma2new']}

Key Result

Proposition 1

Consider the network SIR epidemic model eq:network-SIR-compact with irreducible interaction matrix $A$ in $\mathbb{R}_+^{n\times n}$ and recovery rate $\gamma>0$. Then, Moreover, for every initial state $(x(0),y(0))$ in $\mathcal{S}$:

Figures (3)

  • Figure 1: Numerical simulation of the network SIR epidemic model with $n=2$ nodes with interaction matrix $A = \boldsymbol{1}\boldsymbol{1}^T$, recovery rate $\gamma =1$, and initial state $y_{1}(0)=1-x_1(0)=\textcolor{black}{0.15}$ and $y_{2}(0)=1-x_{2}(0)=0$ satisfying \ref{['eq:conditions']}-\ref{['oveps']}.
  • Figure 2: Numerical simulations of the network SIR epidemic model with $n=5$ nodes and rank-$1$ interaction matrix.
  • Figure 3: Numerical simulation of the network SIR epidemic model with $n=4$ nodes and full-rank interaction matrix.

Theorems & Definitions (29)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Example 1
  • Remark 1
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • ...and 19 more