From elementary processes to chaos: a discrete-time two-stage contagion model and its continuous limit
Abstract
Two-stage contagion, in which adoption requires two successive contacts with adopters, combined with population turnover produces bistability and relaxation oscillations in Katriel's differential-equation model, but no chaos. We derive from the elementary reactions of the process, without reference to the differential equations, a discrete-time model in which the population is updated in rounds of length τ, as in elections, meetings or turns of a game: each individual makes at most one transition per round, events compete as exponential risks, and adoptions take effect from the next round. The model is a Markov chain whose large-population limit is a map Fτ; Katriel's equations are recovered as τ→0. Taking the probability p of adoption after a single contact as bifurcation parameter, with rates and round length fixed, Fτ goes from extinction through a window of chaos to a stable equilibrium; for β/γ=46, μ/γ=0.002, γτ=5.5 the window is 0.023≤ p≤0.056, bounded by the invasion threshold and by a boundary crisis beyond a subcritical Neimark--Sacker point. One-stage contagion (p=1), p=0 and permanent adoption showed no chaos. The chaotic attractor is essentially unique, with one positive Lyapunov exponent and an outbreak--recharge geometry. In the same window Katriel's equations have a periodic relaxation cycle with troughs of 10−44--10−164: the round structure does not create the oscillation but turns it into chaos and shrinks the oscillatory range in p from $(0.022,0.41)$ to $(0.022,0.056)$. The chaos is therefore not an artefact of discretising a stable state, but a property of two-stage contagion with synchronous updating.
Related
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BibTeX
@misc{munoz2026from,
title = {{From elementary processes to chaos: a discrete-time two-stage contagion model and its continuous limit}},
author = {Francisco J. Muñoz and Juan Carlos Nuño and Luca Meacci},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08353},
eprint = {2611.08353},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08353},
}
RIS
TY - UNPB
TI - From elementary processes to chaos: a discrete-time two-stage contagion model and its continuous limit
AU - Francisco J. Muñoz
AU - Juan Carlos Nuño
AU - Luca Meacci
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08353
UR - https://arxiv.org/abs/2611.08353
AB - Two-stage contagion, in which adoption requires two successive contacts with adopters, combined with population turnover produces bistability and relaxation oscillations in Katriel's differential-equation model, but no chaos. We derive from the elementary reactions of the process, without reference to the differential equations, a discrete-time model in which the population is updated in rounds of length τ, as in elections, meetings or turns of a game: each individual makes at most one transition per round, events compete as exponential risks, and adoptions take effect from the next round. The model is a Markov chain whose large-population limit is a map F_τ; Katriel's equations are recovered as τ→0. Taking the probability p of adoption after a single contact as bifurcation parameter, with rates and round length fixed, F_τ goes from extinction through a window of chaos to a stable equilibrium; for β/γ=46, μ/γ=0.002, γτ=5.5 the window is 0.023≤ p≤0.056, bounded by the invasion threshold and by a boundary crisis beyond a subcritical Neimark--Sacker point. One-stage contagion (p=1), p=0 and permanent adoption showed no chaos. The chaotic attractor is essentially unique, with one positive Lyapunov exponent and an outbreak--recharge geometry. In the same window Katriel's equations have a periodic relaxation cycle with troughs of 10^−44--10^−164: the round structure does not create the oscillation but turns it into chaos and shrinks the oscillatory range in p from $(0.022,0.41)$ to $(0.022,0.056)$. The chaos is therefore not an artefact of discretising a stable state, but a property of two-stage contagion with synchronous updating.
ER -