Societal self-regulation induces complex infection dynamics and chaos

TL;DR

This work shows that delaying and saturating human mitigation, when coupled with seasonal forcing, can destabilize classic endemic infection dynamics and generate complex, even chaotic, regimes in epidemiological models. By extending the SIRS framework with a hazard-based behavioral response $h(t)$ and a saturating mitigation function $m(h)$, the authors reveal Hopf bifurcations, Arnold tongues, and coexisting attractors in parameter regions that also minimize societal costs. They demonstrate that the cost-optimal mitigation can lie in or near these complex regimes, implying reduced predictability of infection dynamics under optimal control. Comparing model predictions to COVID-19 and influenza data suggests that COVID-19 mitigation may have favored these complex dynamics, highlighting the practical impact on forecasting and policy design in real-world epidemics.

Abstract

Classically, endemic infectious diseases are expected to display relatively stable, predictable infection dynamics. Accordingly, basic disease models such as the susceptible-infected-recovered-susceptible model display stable endemic states or recurrent seasonal waves. However, if the human population reacts to high infection numbers by mitigating the spread of the disease, then this delayed behavioral feedback loop can generate infection waves itself, driven by periodic mitigation and subsequent relaxation. We show that such behavioral reactions, together with a seasonal effect of comparable impact, can cause complex and unpredictable infection dynamics, including Arnold tongues, coexisting attractors, and chaos. Importantly, these arise in epidemiologically relevant parameter regions where the costs associated to infections and mitigation are jointly minimized. By comparing our model to data, we find signs that COVID-19 was mitigated in a way that favored complex infection dynamics. Our results challenge the intuition that endemic disease dynamics necessarily implies predictability and seasonal waves and show the emergence of complex infection dynamics when humans optimize their reaction to increasing infection numbers.

Figures (10)

• Figure 1: Model overview (a) The extended SIRS model including periodic seasonality ($\Gamma$, green) and delayed behavioral feedback for mitigation ($m$, blue). (b) For a novel disease outbreak, classic SIRS models (with or without seasonality $\Gamma$, resp. green and dotted gray) feature initial exponential growth of infection numbers, eventually stopped only by immunity. With behavioral feedback, mitigation keeps the infection number considerably lower (dashed blue, see inset). (c) Following a transient phase, infection numbers settle into their endemic state. In the SIRS model with neither seasonality nor feedback, this endemic state is stable (dotted gray). In contrast, both seasonality (green) and behavioral feedback (dashed blue) can induce oscillations on their own. Parameters: $\beta_0=0.5, \gamma=0.1$. Green: $a=0.25$, $m_{\rm max}=0$, $\nu=1/500$. Dashed blue: $a=0$, $m_{\rm max}=0.84$, $\tau=30$, $\nu=1/100$.
• Figure 2: The relative weighting between mitigation and seasonality gives rise to three qualitatively different dynamical regimes. Four different scenarios are considered, that differ in their relative weighting of mitigation and seasonality (increasing maximal mitigation from scenario 1 to 4). (a) Stability diagram of the endemic equilibrium. Scenario 1 lies in the stable region (green zone), while scenarios 2-4 lie in the region of oscillations that are caused by a Hopf bifurcation if maximal mitigation $m_{\rm max}$ and delay $\tau$ are large enough ($m_{\rm max}\geq 1-\frac{1}{R_0}=0.8$, blue zone). (b) Stability diagram of the endemic equilibrium for winter- and summer-adjusted $R_0$ (by fixing seasonality to its minimum and maximum values ($1\pm a$): $R_0^{\rm summer}=R_0(1-a)$, $R_0^{\rm winter}=R_0(1+a)$). The Hopf bifurcation curves separate three different regimes: Seasonality-dominated (green, stable endemic equilibrium), mitigation-dominated (blue, unstable endemic equilibrium) and interfering (cyan, stable endemic equilibrium for $R_0^{\rm winter}$ and unstable for $R_0^{\rm summer}$). (c) Without seasonality, scenario 1 has a stable endemic equilibrium and scenarios 2-4 display periodic oscillations. (d) With seasonality, scenarios 1-4 show four qualitatively different dynamics (from bottom to top): Yearly waves of infections (scenario 1), chaotic dynamics (scenario 2), two waves per year (scenario 3) and high (or quasi-) periodic dynamics (scenario 4). Scenario parameters: $R_0=5$, $a=0.25$, $\tau=30$, $m_{\rm max}=0.75, 0.835, 0.85, 0.9$.
• Figure 3: Chaos, phase-locking and coexisting attractors in the extended SIRS model. (a) A peak diagram, i.e., a scatter plot of peak infection numbers $I_k$ of the steady-state timeseries against the seasonal amplitude $a$. Parameters: $\tau=32.5$, $m_{\rm max}=0.85$. (b,c) Arnold tongues for the averaged number of waves per year $W$ emerge as a result of phase-locking. Parameters: $a=0.25$. $m_{\rm max}=0.86$ in (b), $m_{\rm max}=0.84$ in (c). (d) For $m_{\rm max}=0.84$, areas of different tongues overlap, giving rise to coexisting attractors. Parameters: $a=0.248$, $\tau=32.8$ (red cross), $S(0)=0.521$ and $I(0)=0.001$ vs. $I(0)=0.361$.
• Figure 4: Different measures reveal complex infection dynamics for various mitigation strategies. (a) Arnold tongues for the number of waves per year $W$ in the $\tau$-$m_{\rm max}$-plane. (b) Time averaged infections $\langle I \rangle$ drop steeply if $m_{\rm max}$ is large enough. The complexity of the dynamics in that region is characterized by different quantities visualized in two insets (gray rectangles, in panels (a),(b)): (c) The largest Lyapunov exponent $\lambda_1$ is as large as $\lambda_1 = 0.62years^{-1}$, which corresponds to a Lyapunov time of $1.6years$ (left). Parameter variations obtained by Gaussian-weighted samples of trajectories around each point (widths $\sigma_\tau=4$, $\sigma_{m_{\rm max}}=0.03$) change the number of waves per year $\delta W$ as large as $\delta W=0.53$ (absolute, mid) and the expected change of average infections $\delta \langle I \rangle$ as large as $\delta \langle I \rangle=0.27\%$ (absolute, right). (d) The number of waves per year $W$ and average infections $\langle I \rangle$ differ between coexisting attractors, approached by different initial conditions. This can lead to one additional peak every two years ($\Delta W=0.5$, left) and up to $50\%$ more infections (local difference between the attractors with minimal infections $\langle I\rangle_\mathrm{min}$ (mid) and maximal infections $\langle I\rangle_\mathrm{max}$ (right). [min,max] values of the color bar, from left to right: (c) [-3.02,0.62 yr$^{-1}$], [-1.04,0.53], [-0.34%,0.27%]; (d) [0,0.5], [0.14%,0.22%], [0.14%,0.22%].
• Figure 5: Minimizing the societal costs of infections $C_I$ and mitigation $C_m$ can cause complex infection dynamics. (a) Both infections and the implementation of mitigation strategies come at a cost to society. The costs increase in opposite direction for changes of maximal mitigation $m_{\rm max}$ (here $\tau=15$) (b) The cost-optimal region (red crosses on black lines) is quite invariant against changing the relative weighting parameter between $C_I$ and $C_m$ (dashed lines for factors of $5$ and $1/5$ relative to the solid line). (c) Costs in the $\tau$-$m_{\rm max}$-plane. (d) The cost-optimal region can coincide with the region of complex dynamics, where the Arnold tongues start to overlap. Costs are overlayed onto Fig. \ref{['fig:3-complexity']}a. Minimal costs are displayed as red crosses for various $\tau$. Second minima (orange crosses) emerge for large $\tau$ at lower $m_{\rm max}$. The figure displays time averaged costs $\langle C_I(t)\rangle$, $\langle C_m(t) \rangle$.