ρ/τ ∈ {50, 93, 200, ∞}, corresponding to curves from right to left (or see color code), and different values for I_h, namely 0.01 in the top graph (a) and 0.0025 in the bottom graph (b). Solid curves: S(t). Dashed curves: α(t). The fraction of acutely infected citizens is kept at I_h in phase II until herd immunity is reached (S = 1/R₀, horizontal dashed line). If this is successful (if I_h > Î_h, see Eq 30) phase III begins, i.e., mitigation measures are being released (α = 0). For finite immune response (ρ/τ < ∞), S(t) oscillates around its limiting value S_∞ = 1/R₀. In the limit of infinite immune response (ρ/τ = ∞), there are no oscillations and S(t) converges to S_∞ = 1-R_∞ = 1+W(exp(-1-R₀I_h)), with the Lambert W function (see also Fig 3). Other parameters: R₀ = 3, τ = 10 days.

https://doi.org/10.1371/journal.pone.0315749.g001