Could society itself spiral into a Lorenz-like chaos when facing an epidemic threat?

TL;DR

The paper addresses how social feedbacks—specifically the coupling of transmission, risk perception, and collective memory—can generate complex epidemic-related dynamics. It develops a Lorenz-type model with three interacting variables and performs a qualitative bifurcation analysis, revealing a pitchfork bifurcation at $r_0=1$ and a Hopf bifurcation at $r_0=r_H$, along with dissipativity ensuring global existence. Numerical results indicate the possibility of a Lorenz-like strange attractor for certain parameter regimes, offering a social interpretation of chaotic cycles as waves of vigilance and fatigue, mediated by an analogue to $\mathcal{R}_0$. The study highlights the importance of incorporating behavioural feedbacks into epidemic models and outlines limitations and avenues for extending the framework to heterogeneous populations and stochastic shocks.

Abstract

Understanding how societies react to epidemic threats requires more than tracking infection curves. Public perception, collective memory and behavioural adaptation interact through feedback loops that can amplify or suppress the spread of fear, vigilance and precaution. In this work we reinterpret the classical Lorenz system in a socioepidemic context, governed by nonlinear interactions between perceived infection, social transmission behaviour and memory of past risk. We provide a qualitative analysis of the model and show that small fluctuations in perception or behaviour can trigger transitions between stable, oscillatory and chaotic collective responses. These results suggest that social reactions to epidemics may evolve according to intrinsic dynamical rules, generating complex patterns of vigilance, fatigue and renewed concern that mirror the irregular rhythms observed in real outbreaks. Our findings highlight the importance of incorporating behavioural feedbacks into epidemic modeling and reveal how chaotic dynamics may arise not only from pathogens but from society itself.

Figures (4)

• Figure 1: Schematic representation of the socioepidemiological model \ref{['modeloSIR']}. The variables $T$, $I$ and $M$ denote social transmission, perceived infection and social memory, respectively. Transmission increases perceived infection through exposure $(+)$, while perceived infection reduces transmission through behavioural response $(-)$. Both $T$ and $I$ contribute positively to the construction of social memory $(+)$, which later acts on transmission, promoting caution and adherence to norms $(-)$.
• Figure 2: Phase portraits of system \ref{['modeloSIR']} under two parameter regimes. Left: Several initial conditions with $r_0 < 1$ (Parameters: $\sigma = 2$, $r_0 = 0.5$, $\beta = 1$). Right: Initial conditions near the origin with $1 < r_0 < r_H$ (Parameters: $\sigma = 10$, $r_0 = 20$, $\beta = 2.7$). Black dots denote the trivial equilibrium point $P_0$ (a sink on the left and a saddle on the right), while red dots denote the non-trivial equilibria $P_e^{\pm}$ (stable foci).
• Figure 3: Schematic bifurcations and stability of the equilibria for model \ref{['modeloSIR']}. The trivial equilibrium point $P_0$ undergoes a supercritical pitchfork bifurcation at $r_0 = 1$ (black dot), changing its stability from a sink to a saddle and giving rise to two stable non-trivial equilibria $P_e^{\pm}$. For $r_0 = r_H$ (white dot), the equilibria $P_e^{\pm}$ undergo a subcritical Hopf bifurcation and lose their stability. For $r_0 > r_H$, the sketch of a Lorenz-like strange attractor indicates, in a purely schematic way, that for certain parameter values beyond this threshold the dynamics may become chaotic (as suggested by the numerical simulation in Figure \ref{['str_atr']} of Section \ref{['str_str_sec']}).
• Figure 4: Strange attractor generated by system \ref{['modeloSIR']} for $\sigma = 10$, $\beta = 8/3$ and $r_0 = 28$. The geometry of the attractor reflects the chaotic interplay among social transmission $T$, perceived infection $I$ and social memory $M$, illustrating how their nonlinear feedbacks can give rise to irregular and unstable dynamics. The black dot denotes the trivial equilibrium point $P_0$, while the red dots denote the non-trivial equilibria $P_e^{\pm}$. All these equilibria are unstable.

Theorems & Definitions (7)

• Lemma 1: Dissipativity and global existence
• proof
• Theorem 1
• Lemma 2
• proof
• Lemma 3
• proof