Fig 1.
Effects of population heterogeneity on the dynamics of SIR models.
Examples for the time course of fraction of susceptible S/N (green), fraction of infected I/N (orange) and fraction of the cumulative number of infected C/N (blue) in a SIR model with N total individuals. (a)-(c) Homogeneous SIR model with R0 = 2.5 and γ = 0.13 day−1. (d)-(f) Heterogeneous SIR model with same R0 and γ and with α = 0.1. (a) and (d) show time course as linear plot, (b) and (e) show semi logarithmic plots of the same variables. (c) and (f) show the normalized time dependent reproduction number R(t)/R0 (yellow) and the average susceptibility (purple) as a function of time. The dotted lines in (a),(b),(d) and (e) indicate the herd immunity level CI. Other parameters: N = 8107 individuals and I0 = 10 initially infected.
Fig 2.
Fraction of susceptible individuals at long times.
(a) Fraction S∞/N of susceptible individuals that remain at long times as a function of the basic reproduction number R0 for different degrees of population heterogeneity characterized by the values of α. The limit α → ∞ corresponds to the classic case with homogeneous populations (green). In the limit α → 0 populations are most heterogeneous (blue). (b) Fraction SI/N of susceptible individuals as a function of R0 at the peak where the number of infected is maximal for different α. (c) Ratio of infections after the peak Sm − S∞ and infections before the peak N − Sm as a function of R0.
Fig 3.
(a) Daily new SARS-CoV-2 infections reported in the early months of 2020 in Germany. The number of new reported infections per day (red symbols) is shown together with the number of reported infections per day for those cases with later fatal outcome
(blue symbols). (b) Semi logarithmic representation of the same data. The dashed and solid lines represent linear and cubic fits to the data in specific time intervals. They are used to estimate the initial and final growth rates λ0 and λ∞ as well as
and
at the maximum of the rate of new cases Jmax. We find λ0 ≃ 0.269 day−1 (0.336 day−1), λ∞ ≃ −0.068 day−1 (−0.038 day−1), A2 ≃ −10−2 day−2 (−0.9110−2 day−2) and A3 ≃ 6.810−4 day−3 (7.510−4 day−3) for the fatal cases (for all reported cases).
Fig 4.
Time course of the SARS-CoV-2 epidemic in Germany (symbols) compared to solutions of the heterogeneous SIR model (lines).
(a) and (b) Data on daily reported infections (red) and on reported infections with later fatal outcome (blue) as logarithmic and linear plots. (c) and (d) same data and model solutions as in (a) and (b) but for cumulative numbers of cases. The horizontal dashed lines indicate scaled herd immunity levels. Parameter values for the model solution are R0 = 2.67 (R0 = 3.91), γ = 0.146 (γ = 0.069), α = 0.05 and N = 8107 for the fatal cases (for all cases). The case fatality rate that corresponds to this solution is f = 0.13% (f = 0.11%). (e) Time courses of the fraction of infected I/N (blue), the new cases per day J/N (red) and the fraction of cumulative cases C/N (yellow) for R0 = 2.67 and γ = 0.146. (f) Time course of the average susceptibility (blue), where R is the time dependent reproduction number and of
(red) for the solution shown in (e). Inset: distributions of susceptibility in the population for different values of τ.
Fig 5.
Role of population heterogeneity for the behavior of the generalized SIR model.
Plots of various dimensionless ratios of parameters characterizing the shape of the infection wave for different values of α. Here the limit α → ∞ corresponds to the homogeneous SIR model, the limit α → 0 to the strongly heterogeneous case. Here λ0 and λ∞ denote the initial and final growth rate, and
describe the shape of the wave at the maximum of new cases per day J. The horizontal dashed lines correspond to estimates from fits shown in Fig 3, the shaded regions indicate uncertainty ranges, see Appendix D.
Fig 6.
(a)-(c) Early mitigation by strong reduction of β for a homogeneous population (large α limit). The new cases per day are shown in (a) as symbols. A fit of the mitigated model is shown as solid lines. The solution for same parameter values R0 = 2 and γ = 0.24 but without mitigation is shown as dotted lines. The corresponding cumulative numbers of cases are shown in (b). Herd immunity levels corresponding to these solutions are indicated as horizontal dashes lines. The time dependence of β(t) are shown in (c) as solid lines. The time courses of β inferred from the data is shown as symbols. Mobility data indicating social activities in Germany relative to baseline values are shown in orange for comparison. (d)-(f) same plots as in (a)-(c) but for a moderate mitigation and heterogeneous population with R0 = 2, γ = 0.24 and α = 0.1. (g)-(i) Heterogeneous population with mild mitigation and release leading to almost herd immunity. Red symbols and lines correspond to the case of all reported infections, blue data and lines correspond to reported infections of fatal cases.
Fig 7.
Normalized coefficient at the peak of new cases per day.
(a) The ratio as a function of R0 is shown for different values of α. (b) The ratio
as a function of R0 for the same values of α. The dashed lines represent the values inferred from the data shown in Fig 3 for all cases (red) and fatal cases (blue). The shaded colored regions correspond to the uncertainties of the fits to the data.
Fig 8.
Mobility changes for a representative set of commonly visited places in Germany up to July 1 2020 from [34].