On the dynamic behavior of the network SIR epidemic model

TL;DR

This paper investigates the complex dynamic behavior of network-based SIR epidemic models, revealing multimodal infection curves and stability conditions, especially in rank-1 interaction scenarios, with numerical insights into higher-rank cases.

Contribution

It introduces analysis of multimodal infection curves in network SIR models and provides explicit stability conditions and invariants for rank-1 interaction matrices.

Findings

Multimodal infection curves can occur in network SIR models with multiple subpopulations.

Explicit equilibrium expressions and stability conditions are derived for rank-1 interaction matrices.

Numerical results suggest multiple peaks in infection curves for higher-rank interaction matrices.

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.