Fig. 3: The transition to the mutation-driven phase.

For mutation-driven contagion a critical mutation must arise before the pathogen is eliminated, namely before η(t) crosses 1/N (horizontal grey dashed lines). This represents the unit-line, a state in which a single infected individual remains among the N node population. a η(t) vs. t (grey solid line) as obtained from Eq. (12) in the infection-free phase (R₀ = 0.25, σ = 0.1). The critical mutation occurs at the minimum point (t_c), which is below the unit line. Therefore the pathogen is eliminated prior to the appearance of the critical mutation. Indeed, the stochastic simulation (blue solid line) approaches zero prevalence, never reaching the exponentially growing branch of η(t), which arises at t > t_c. b Setting σ = 0.16 the system is at criticality. η(t_c) is adjacent to the unit line, and hence we observe critical behavior: some realizations decay (blue), whereas others successfully mutate (green). At criticality, as the number of infected individuals reaches ~1, the dynamics become stochastic, and hence the outcome differs across realizations. The long term behavior of the mutation-driven branch (green) diverges from the theoretically predicted grey line, as it reaches saturation. Indeed, Eq. (12) is only designed to capture the initial stages of the spread, and disregards the exhaustion of the susceptible population that occurs at large t. c Under σ = 0.5, the system is in the mutation-driven phase, η(t_c) is sufficiently above the unit line and the critical mutation is reached with probability P → 1. d The phase boundary in Eq. (15) depends on the initial size of the infected population \({{{{{{{\mathcal{{I}}}}}}}_{0}}}\). Here we show this boundary for \({{{{{{{\mathcal{{I}}}}}}}_{0}}}=1{0}^{2},\ldots,1{0}^{8}\) (grey solid lines), finding that the larger is the infected population (\({{{{{{{\mathcal{{I}}}}}}}_{0}}}\)), the broader is the coverage of the mutation-driven phase.