Fig. 4: Timing our mitigation.

a The prevalence η(t) vs. t without mitigation (red), under early mitigation (blue, t_R = 20) and under late mitigation (green, t_R = 75). Starting from R₀ = 1.2, σ = 0.03, the mitigation reduces R₀ to R_R = 0.6, a factor of one half. While early mitigation ensures the elimination of the pathogen (blue), under late mitigation (green), the initial decline is eventually reversed due to the appearance of a critical mutation. Note that for evolving pathogens the late mitigation does not simply prolong the pandemic, but rather, as the green curve clearly shows, causes it to enter the mutation-driven phase. Consequently, we risk a second wave that may occur much later than t_R, but whose seeds were sown during our initial delayed response. b The probability P to reach critical fitness (ψ_c, Eq. (11)) vs. t_R, as obtained for different values of σ. The later we instigate our mitigation, the higher the risk for a breakthrough mutation. c Setting t_R = 20, we have P → 0, and hence a successful mitigation. d At t_R = 75 we predict P ≈ 0.5, and hence in some realizations the pathogen is eliminated (light green), while in others, it reemerges in the form of a second pandemic wave (dark green). e Further delaying our response to t_R = 100 we enter P ≲ 1, predicting an almost inevitable mitigation failure.