arxiv: 2604.10995 · v1 · submitted 2026-04-13 · 🧬 q-bio.PE

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How complex behavioural contagion can prevent infectious diseases from becoming endemic

Michael J. Plank , Matt Ryan , Lloyd Chapman , Roslyn I. Hickson , Thomas House , Emma McBryde , James M. McCaw

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Pith reviewed 2026-05-10 16:15 UTC · model grok-4.3

classification 🧬 q-bio.PE

keywords behavioural contagionepidemic modelsnonlinear dynamicsdisease eliminationbistabilitysocial contagioninfectious disease

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The pith

Nonlinear behavioural contagion can cause an epidemic to trigger sustained protective behaviour that eliminates the disease.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper builds an epidemic model that couples disease transmission with nonlinear social contagion of protective behaviours. In some parameter ranges this produces multiple stable disease-free equilibria, so an outbreak that begins in a low-behaviour population can push the system across a threshold into a high-behaviour state where the pathogen dies out. The same nonlinearity can make higher values of the basic reproduction number R0 lead to elimination rather than endemicity. A reader would care because conventional models that assume linear behaviour change miss this feedback route by which an epidemic can resolve itself.

Core claim

In an SIR-type model extended with complex behavioural contagion, the behaviour uptake rate is a nonlinear function of the number of behaving contacts. This functional form creates regions of parameter space containing multiple stable disease-free equilibria. An epidemic starting from low behaviour levels can therefore induce a transition to the high-behaviour equilibrium, after which the disease is eliminated. The outcome occurs for certain combinations of R0, the strength of the behavioural effect on transmission, and the relative speed of behaviour uptake and abandonment, and it is robust to the presence or absence of temporary post-infection immunity.

What carries the argument

The nonlinear function that sets the rate of behaviour uptake from the number of contacts already practicing the behaviour, which generates multiple stable disease-free equilibria.

If this is right

Moderate R0 values produce endemic disease while higher R0 values can produce behaviour-driven elimination.
An epidemic can serve as the perturbation that switches a population from low to high protective behaviour.
The elimination effect holds whether or not the model includes temporary post-infection immunity.
The existence of multiple disease-free equilibria requires both the nonlinear uptake function and suitable values of behavioural strength and speed parameters.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Public-health planning could explore whether brief, controlled exposure to a pathogen or information campaign might be used to cross the behaviour threshold in low-adoption groups.
The same bistable structure might appear in models of other socially contagious actions such as vaccine uptake or climate-related behaviour.
Field studies that simultaneously track incidence and behaviour adoption during outbreaks could directly test for the predicted switch between equilibria.
Standard policy models that omit nonlinear contagion may systematically underestimate the chance that some outbreaks will self-limit through behaviour change.

Load-bearing premise

The rate at which people adopt protective behaviour must increase nonlinearly with the number of others already doing so, in a way that produces multiple stable states without the disease.

What would settle it

Longitudinal data from a real population showing that, after an outbreak with high R0 and initially low protective behaviour, behaviour remains permanently elevated and disease prevalence reaches zero, while moderate-R0 outbreaks in the same setting become endemic without a sustained behaviour increase.

Figures

Figures reproduced from arXiv: 2604.10995 by Emma McBryde, James M. McCaw, Lloyd Chapman, Matt Ryan, Michael J. Plank, Roslyn I. Hickson, Thomas House.

**Figure 1.** Figure 1: Two-parameter bifurcation diagram (R0, qc) for the transmission-modulated model: (a) when τ < 4 (i.e., no BDFEs exist); (b) when τ > 4 (i.e., two BDFEs exist). χ = 0.2 in both plots. The black vertical line is the standard disease invasion threshold at R0 = 1, which is a transcritical bifurcation (TC0) of the NDFE and an EE. Solid yellow and red curves are transcritical bifurcations (TC− and TC+ respective… view at source ↗

**Figure 2.** Figure 2: Dynamics in parameter regimes 2 (a,b,c), 3 (d,e,f) and 4 (g,h,i) for: the transmission-modulated model with qc = 0.3 shown as phase plots in the (S, B) plane (left column) and time series solutions (centre column); and for the susceptibilitymodulated model with qs = 0.3 shown as time series solutions (right column). All plots have τ = 3.8 (so there are no BDFEs), χ = 0.2, and α = 0.25. Phase plots show th… view at source ↗

**Figure 3.** Figure 3: Dynamics in parameter regimes 6 (a,b,c), 7 (d,e,f) and 8 (g,h,i) for: the transmission-modulated model with qc = 0.7 shown as phase plots in the (S, B) plane (left column) and time series solutions (centre column); and for the susceptibilitymodulated model with qs = 0.7 shown as time series solutions (right column). All plots have τ = 5 (so there are two BDFEs), χ = 0.2, and α = 0.25. Phase plots show the… view at source ↗

**Figure 4.** Figure 4: Dynamics in parameter regimes 9 (a,b,c) and 10 (d,e,f) for: the transmission-modulated model with qc = 0.3 shown as phase plots in the (S, B) plane (left column) and time series solutions (centre column); and for the susceptibility-modulated model with qs = 0.3 shown as time series solutions (right column). All plots have τ = 5 (so there are two BDFEs), χ = 0.2, and α = 0.25. Phase plots show the vector fi… view at source ↗

**Figure 5.** Figure 5: Solutions of the SIRS model showing: (a) prevalence of infection [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗

read the original abstract

Infectious disease transmission in human populations has a complex two-way interaction with changes in host behaviour. It is increasingly recognised that incorporating adaptive behavioural change into epidemic models is important for improving understanding of infectious disease dynamics and developing policy-relevant modelling tools. An important aspect of behavioural dynamics is social contagion, where people tend to adopt behaviours exhibited by others around them. In a simple behavioural contagion model, the behaviour uptake rate increases linearly with the number of contacts who have adopted a given behaviour. Here, we explore an epidemic model with complex behavioural contagion, where the behaviour uptake rate is a nonlinear function of the number of behaving contacts. We identify key bifurcation parameters of the model, which include the basic reproduction number $R_0$, the strength of the behavioural effect on disease transmission, and the speed of behaviour uptake relative to behaviour abandonment. We show that, in some regions of parameter space, the model has multiple disease-free equilibria. In this situation, the occurrence of an epidemic in a population with an initially low level of behaviour practice can trigger a self-sustaining increase in behaviour, which then causes the disease to be eliminated. In some cases, while moderate values of $R_0$ lead to the disease becoming endemic, higher values of $R_0$ may lead to behaviour-driven disease elimination. We demonstrate that this mechanism of epidemic-triggered uptake of behaviour leading to disease elimination can occur in the presence and absence of temporary post-infection immunity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Nonlinear behavioral contagion creates multiple stable disease-free equilibria that an epidemic can flip into a protective state, sometimes more readily at higher R0.

read the letter

The main thing to know is that the model replaces linear behavior uptake with a nonlinear function of behaving contacts. This produces multiple disease-free equilibria in some parameter regions. An epidemic that starts in the low-behavior state can trigger a self-reinforcing increase in protective behavior, driving effective transmission below one and eliminating the disease. The paper also reports that moderate R0 values can lead to endemicity while higher R0 values sometimes produce elimination instead.

Referee Report

2 major / 2 minor

Summary. The paper develops a compartmental epidemic model incorporating complex behavioural contagion, in which the rate of adopting a protective behaviour is a nonlinear function of the number of contacts already exhibiting the behaviour. The authors perform bifurcation analysis with respect to the basic reproduction number R0, the strength of the behavioural effect on transmission, and the ratio of behaviour uptake to abandonment rates. They report the existence of multiple disease-free equilibria in certain parameter regimes, such that an epidemic starting from low behaviour levels can drive the system across a separatrix to a high-behaviour disease-free equilibrium that eliminates the pathogen. They further claim that, for some parameter values, increasing R0 can shift the long-term outcome from endemicity to elimination via this behavioural mechanism, and that the phenomenon persists both with and without temporary post-infection immunity.

Significance. If the results hold under reasonable variations in model assumptions, the work identifies a concrete mechanism by which nonlinear social contagion can produce counter-intuitive epidemic outcomes, including the possibility that higher transmissibility leads to disease elimination rather than endemicity. This underscores the potential for behavioural dynamics to alter effective reproduction numbers and equilibrium structure in ways not captured by standard models, with implications for both theoretical epidemiology and the design of interventions that leverage social norms.

major comments (2)

[Model formulation] Model formulation section: The headline result of multiple stable disease-free equilibria whose basins permit an epidemic to trigger self-sustaining behaviour uptake and subsequent elimination depends on the specific nonlinear functional form chosen for the behaviour uptake rate. The manuscript demonstrates the phenomenon for its chosen nonlinearity but does not test whether the bistable region and the R0-dependent elimination regime survive under alternative plausible complex-contagion maps (e.g., Hill functions with different exponents or different saturation thresholds) while keeping the same R0 range and behavioural effect strength. This functional-form sensitivity is load-bearing for the central claim.
[Bifurcation and equilibrium analysis] Bifurcation and equilibrium analysis: The abstract states that key bifurcation parameters are identified and that multiple DFEs exist in some regions, yet the explicit equilibrium conditions, Jacobian or next-generation matrix derivations, and numerical continuation details required to verify the claimed multiple equilibria and the non-monotonic dependence on R0 are not supplied at a level that permits independent reproduction of the reported regimes.

minor comments (2)

The distinction between simple (linear) and complex (nonlinear) behavioural contagion could be drawn more sharply in the introduction to clarify the incremental contribution.
Bifurcation diagrams would benefit from explicit indication of the basins of attraction for each disease-free equilibrium and from clearer annotation of the parameter values at which the qualitative transitions occur.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive comments, which help clarify the robustness and reproducibility of our findings on nonlinear behavioural contagion in epidemic models. We address each major comment point by point below, indicating planned revisions to the manuscript.

read point-by-point responses

Referee: [Model formulation] Model formulation section: The headline result of multiple stable disease-free equilibria whose basins permit an epidemic to trigger self-sustaining behaviour uptake and subsequent elimination depends on the specific nonlinear functional form chosen for the behaviour uptake rate. The manuscript demonstrates the phenomenon for its chosen nonlinearity but does not test whether the bistable region and the R0-dependent elimination regime survive under alternative plausible complex-contagion maps (e.g., Hill functions with different exponents or different saturation thresholds) while keeping the same R0 range and behavioural effect strength. This functional-form sensitivity is load-bearing for the central claim.

Authors: We chose the nonlinear uptake function to capture the accelerating, threshold-like nature of complex contagion as described in the social sciences literature. We agree that demonstrating robustness across functional forms strengthens the central claim. In the revised manuscript we will add a new subsection (or supplementary analysis) that numerically explores alternative maps, including Hill functions with exponents 2 and 3 and varied saturation thresholds, while holding R0 and behavioural effect strength fixed. We will report the parameter regimes in which multiple disease-free equilibria and the non-monotonic R0 dependence persist or disappear, thereby addressing the functional-form sensitivity directly. revision: yes
Referee: [Bifurcation and equilibrium analysis] Bifurcation and equilibrium analysis: The abstract states that key bifurcation parameters are identified and that multiple DFEs exist in some regions, yet the explicit equilibrium conditions, Jacobian or next-generation matrix derivations, and numerical continuation details required to verify the claimed multiple equilibria and the non-monotonic dependence on R0 are not supplied at a level that permits independent reproduction of the reported regimes.

Authors: We acknowledge that additional explicit derivations will improve independent verification. The revised manuscript will include the algebraic conditions for the disease-free equilibria obtained by setting all time derivatives to zero. We will also present the Jacobian matrix at these equilibria, outline the next-generation matrix construction used to obtain R0, and describe the numerical continuation procedure (including software or algorithm details) employed to trace the bifurcations. Supplementary material will contain the corresponding code or pseudocode so that the reported regimes and the counter-intuitive R0 dependence can be reproduced. revision: yes

Circularity Check

0 steps flagged

Standard bifurcation analysis of an assumed nonlinear contagion model with no definitional or fitted circularity

full rationale

The paper introduces an explicit compartmental ODE model with a chosen nonlinear function for behavior uptake rate (a modeling assumption, not derived from the target outcome). It then performs standard equilibrium and bifurcation analysis to identify parameter regions with multiple disease-free equilibria. No quantity is defined in terms of itself, no parameter is fitted to the epidemic outcome and then called a prediction, and no load-bearing step reduces to a self-citation chain or imported uniqueness theorem. The central claim (epidemic-triggered behavior shift eliminating disease) is a direct consequence of solving the model equations under the stated assumptions, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The model rests on a nonlinear uptake function whose exact form is not given in the abstract, plus standard assumptions of compartmental epidemic modeling and the existence of bifurcation parameters that produce multiple equilibria.

free parameters (3)

R0
Basic reproduction number varied across regimes to demonstrate endemic versus elimination outcomes
behavioral effect strength
Parameter controlling reduction in transmission due to adopted behavior
uptake versus abandonment speed ratio
Relative rate of behavior adoption versus abandonment

axioms (2)

domain assumption Behavior uptake rate is a nonlinear function of the number of contacts who have adopted the behavior
Central modeling choice that generates the complex contagion dynamics and multiple equilibria
standard math Standard mass-action or frequency-dependent transmission in a compartmental epidemic framework
Background structure assumed for coupling disease and behavior states

pith-pipeline@v0.9.0 · 5580 in / 1489 out tokens · 71256 ms · 2026-05-10T16:15:45.474818+00:00 · methodology

discussion (0)

Reference graph

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