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We study seasonal epidemic spreading in asusceptible-infected-removed-susceptible (SIRS) model on smallworld graphs. Wederive a mean-field description that accurately captures the salient featuresof the model, most notably a phase transition between annual and biennialoutbreaks. A numerical scaling analysis exhibits a diverging autocorrelationtime in the thermodynamic limit, which confirms the presence of a classicaldiscrete time crystalline phase. We derive the phase diagram of the model bothfrom mean-field theory and from numerics. Our work offers new perspectives bydemonstrating that small-worldness and non-Markovianity can stabilize aclassical discrete time crystal, and by linking recent efforts to understandsuch dynamical phases of matter to the century-old problem of biennialepidemics.

Seasonal epidemic spreading on small-world networks: Biennial outbreaks and classical discrete time crystals