Epidemic models typically predict convergence: the disease either dies out (R₀ < 1) or reaches an endemic equilibrium (R₀ > 1). But real epidemics come in waves — periodic surges separated by lulls. Where do the waves come from?
The SQIR-V model (arXiv: 2603.24107) adds quarantine, vaccination, and history-dependent transition rates to the standard compartmental framework. The nonlinear dependence on infection history — the rate at which susceptibles become infected depends on the cumulative infection trajectory — enables Hopf bifurcation: a stable equilibrium loses stability and gives birth to a limit cycle.
The mechanism is delay. Isolation and vaccination delays create a lag between infection and response. When the delay exceeds a threshold, the system overshoots the equilibrium: too many infections before quarantine kicks in, then too few after the response. The oscillation is self-sustaining because the delay is structural, not transient.
The bifurcation analysis identifies the critical delay and the direction of crossing: whether the eigenvalues cross the imaginary axis moving right (destabilizing) or left (stabilizing). Vaccination rate and isolation delay are the two control parameters; the waves emerge from their interaction.
The through-claim: epidemic waves are bifurcations, not stochastic fluctuations. The periodicity is deterministic — it emerges from the feedback loop between infection and delayed response. The delay creates a phase lag; the phase lag creates oscillation; the oscillation creates waves. Stochasticity adds noise to the waves, but the waves exist without it.
2603.24107. Mathematical epidemiology / Hopf bifurcation / epidemic waves / delay models / vaccination dynamics.