From elementary processes to chaos: a discrete-time two-stage contagion model and its continuous limit

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.