Extended Data Fig. 4: Evolution of resistance in presence of vaccination.

(A) Before MT takeover, the decline in susceptible individuals (x) can be approximated by a linear function with slope equal to the vaccination rate c. Since vaccination is fast, individuals recovered from WT and non-vaccinated individuals recovered from MT are few. The equation of line (a) is x⁽t⁾ = x(0) − ct for t < t^* where t^* is the time of takeover of the MT. (B) The reproductive ratio R_WT is maintained at around 1 by dynamic social distancing. After mutant takeover, R_WT is less than 1, since the degree of social distancing is now adjusted to the population susceptible to the MT strain. The number of active WT infections before takeover and of active MT infections after takeover, is fluctuating around L/a until herd immunity to the MT is reached. Before MT takeover, the reproductive ratio of the MT grows as (b) R_MT = β₂(x⁽t⁾ + z₁⁽t⁾ + w₁⁽t⁾)/a. After takeover, R_MT is maintained around 1. (E) The number of vaccinated individuals (w₁) first increases linearly with slope equal to the vaccination rate. After MT takeover, the number of individuals vaccinated to the WT and recovered from MT (w₂) increased linearly with slope L. The equations of the lines are given by (c) w₁⁽t⁾ = ct for t < t^* (d) w₁⁽t⁾ = w₁⁽t^*) − L(t − t^*) for t > t^* (e) w₂⁽t⁾ = L(t − t^*) for > t^*. (F) Before MT takeover, the dynamic social distancing is adjusted to the WT. As the number of individuals immune to WT grows, social activity increases. When the MT emerges, social distancing measures are reinstated. Subsequently, social activity increases as the population immune to the MT grows. The equations for the lines given by (f) ⁽t⁾ = a/β₁⁽t⁾ for t < t^* ; (g) s⁽t⁾ = a/β₂(x⁽t⁾ + z₁⁽t⁾ + w₁⁽t⁾) for > t^*. Parameters: N = 10⁶; a = 0.25; d = 0.01; μ = 10⁻⁶; s₀ = 0.1; β₁ = β₂ = 7.5 ∙ 10⁻⁷; c = 10,000; L = 1500; q = 1.