Fig. 2: The pandemic phase-diagram.

a Pandemic. For R₀ > 1 we observe the classic pandemic phase. Prevalence η(t) vs. t (top) exhibits several waves, as more transmissible mutants emerge, and the average fitness \(\overline{\psi }(t)\) (bottom) increases. b We observe two waves, first the wild-type (light-red) and then the fitter mutant variants (dark-red). c Mutation-driven. Setting R₀ < 1, but increasing σ to 2, \(\overline{\psi }(t)\) (bottom) reaches critical fitness within a short time (inset), and consequently η(t) (top) and r_∞ (middle) turn pandemic. d Sporadic instances of high fitness strains (dark-red nodes), take over the majority of the population. e Infection-free. Under slower mutations (σ = 0.02), R₀ remains sub-pandemic, η(t) vanishes (top), r_∞ ≪ 1 (middle) and \(\overline{\psi }(t)\) remains almost constant (bottom). Hence, the pathogen fails to reach critical fitness ψ_c (grey dashed-line). f The unfit pathogen (light red) tapers off without significant fitness gains. g Volatile. At σ = 7, \(\overline{\psi }(t)\) fluctuates erratically (bottom), failing to lock-in the fitter strains. Consequently, η(t) and r_∞ fail to reach measurable levels (top, middle). h High-fitness strains appear (dark red nodes), but are lost to the volatile mutations. i σ, R₀ phase-diagram. In addition to the classic pandemic-phase (R₀ ≥ 1, red), we observe our three predicted phases: infection-free (blue) for R₀ < 1, small σ; volatile (blue) under large σ; and in between - mutation-driven (green), in which breakthrough mutations are almost guaranteed (P → 1). The gaps between phases (grey) feature sharp transitions from P → 0 to P → 1, which are well-approximated by our theoretical predictions (solid black lines). The examples in panels (a–h) are shown as red, blue, green and purple dots. j r_∞ vs. R₀ under σ = 0.1 (yellow dashed-line in panel (i), left). We observe a first-order pandemic transition, in which r_∞ abruptly jumps from ~0 to ~1. The theoretically predicted transition is also shown (grey dashed-line). k The transition from volatile to mutation-driven (yellow dashed-line in panel (i), right) is continuous (see Methods Section for technical details).