Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Early warning signals of non-critical transitions from linearised time-varying dynamics with applications to epidemic systems

Joshua Looker, Kat S. Rock, Louise Dyson

TL;DR

The paper tackles predicting early warning signals in non-critical transitions of time-varying epidemic systems where the Jacobian is non-normal and trajectories do not cross an equilibrium bifurcation. It develops a time-dependent covariance framework using the linear-noise approximation, solving the non-autonomous Lyapunov equation in both the J-eigenbasis and the symmetric H-basis to decompose variance contributions from modal growth and transient amplification. Applying this framework to the SIR model with demography, it shows that variance in the infectious class, Var(I), can peak due to transient geometry even when $R_t$ does not cross 1 and the equilibrium remains stable. Simulations across population sizes validate the theory for large populations and highlight data-related caveats, supporting the use of time-varying covariance signals as early warnings of epidemic waves or extinction in non-bifurcating regimes.

Abstract

In the wake of the SARS-CoV-2 pandemic, there has been heightened interest from applied mathematicians in infectious disease modelling. Modelling efforts often focus on predicting whether diseases are likely to be eliminated or, instead, (re-)emerge, especially as a result of control measures.This tipping point between elimination and infection waves has been successfully anticipated in the literature through the use of early warning signals and such signals often rely on the theory of critical slowing down. Recent developments have shown that these signals (increases in fluctuation variance and return time) can emerge from the system geometry in the case of non-normal dynamics rather than a change in asymptotic stability. We show how such dynamical behaviour occurs in the fluctuations from the mean-field in general stochastic systems. Using the susceptible-infectious-recovered model as an example application, we analyse how critical-like behaviour can be exploited to anticipate infection waves in the absence of an equilibrium bifurcation.

Early warning signals of non-critical transitions from linearised time-varying dynamics with applications to epidemic systems

TL;DR

The paper tackles predicting early warning signals in non-critical transitions of time-varying epidemic systems where the Jacobian is non-normal and trajectories do not cross an equilibrium bifurcation. It develops a time-dependent covariance framework using the linear-noise approximation, solving the non-autonomous Lyapunov equation in both the J-eigenbasis and the symmetric H-basis to decompose variance contributions from modal growth and transient amplification. Applying this framework to the SIR model with demography, it shows that variance in the infectious class, Var(I), can peak due to transient geometry even when does not cross 1 and the equilibrium remains stable. Simulations across population sizes validate the theory for large populations and highlight data-related caveats, supporting the use of time-varying covariance signals as early warnings of epidemic waves or extinction in non-bifurcating regimes.

Abstract

In the wake of the SARS-CoV-2 pandemic, there has been heightened interest from applied mathematicians in infectious disease modelling. Modelling efforts often focus on predicting whether diseases are likely to be eliminated or, instead, (re-)emerge, especially as a result of control measures.This tipping point between elimination and infection waves has been successfully anticipated in the literature through the use of early warning signals and such signals often rely on the theory of critical slowing down. Recent developments have shown that these signals (increases in fluctuation variance and return time) can emerge from the system geometry in the case of non-normal dynamics rather than a change in asymptotic stability. We show how such dynamical behaviour occurs in the fluctuations from the mean-field in general stochastic systems. Using the susceptible-infectious-recovered model as an example application, we analyse how critical-like behaviour can be exploited to anticipate infection waves in the absence of an equilibrium bifurcation.
Paper Structure (10 sections, 2 theorems, 40 equations, 20 figures, 1 table)

This paper contains 10 sections, 2 theorems, 40 equations, 20 figures, 1 table.

Table of Contents

  1. Introduction
  2. Time-dependent critical slowing down behaviour
  3. Variance-based indicators in practice
  4. SIR model: theoretical results
  5. Jacobian and its symmetric part
  6. Theoretical covariance evolution
  7. Simulation results
  8. Discussion
  9. Deriving the fluctuation bounds
  10. Further simulation results

Key Result

Proposition 2.1

Under the above definitions and assumptions, the following matrix equality holds where $(\cdot)^*$ denotes the Hermitian transpose, and, for the diagonal modal variances $p_i(t):=(\Sigma_y)_{ii}$, the exact scalar ordinary differential equation (ODE) is for each $i=1,\dots,n$. Remark. In eq:modal_scalar, the first term $2\Re(\lambda_i)p_i$ is the pure modal growth/decay; $(B_y)_{ii}$ is the moda

Figures (20)

  • Figure 1: Phase diagram for the 'wedge' where the black dot indicates the EE and arrows indicate the direction of the flow.
  • Figure 2: Phase plane of the SIR model with $R_0=4,1/\mu=80\text{ years}$ and heat map denoting the discriminant of the Jacobian. The curved dashed black line shows the $D=0$ manifold, the vertical dashed black line corresponds to $R_t=1$ and the near collinear vertical solid red line indicates the $S$ coordinate of the endemic equilibrium. The solid yellow dot indicates the endemic equilibrium.
  • Figure 3: Phase plane of the SIR model with $R_0=4,1/\mu=1/73\text{ years}$ and heat map denoting the discriminant of the Jacobian. The curved dotted black line shows the $D=0$ manifold, the vertical dotted black line corresponds to $R_t=1$ and the vertical solid red line indicates the $S$ coordinate of the endemic equilibrium. The solid yellow dot indicates the endemic equilibrium.
  • Figure 4: Phase plane of the SIR model with $R_0=0.840,1/\mu=80\text{ years}$ and heat map denoting the discriminant of the Jacobian. The curved dotted black line shows the $D=0$ manifold and there is no endemic equilibrium.
  • Figure 5: Phase plane of the SIR model with $R_0=4,1/\mu=80\text{ years}$ and heat map denoting the determinant of $H$. The curved dotted black line shows the $H_D=0$ manifold, the vertical dotted black line corresponds to $R_t=1$ and the vertical solid red line indicates the $S$ coordinate of the endemic equilibrium. The solid yellow dot indicates the endemic equilibrium.
  • ...and 15 more figures

Theorems & Definitions (4)

  • Proposition 2.1: Modal ( $J$-eigenbasis) decomposition
  • proof
  • Proposition 2.2: Symmetric ($H$) eigenbasis decomposition
  • proof