Hierarchical metapopulation model of disease spreading. In our model, individuals (dots) belong to groups (solid circles) that in turn belong to groups of groups, and so on, giving rise to a hierarchy of scales. Alternatively, one can imagine the hierarchy as a nested set of subpopulations of increasing size, as indicated schematically by the dashed boundaries. In this example, there are n = 8 individuals in each group and the hierarchy has l = 3 levels and a branching ratio of b = 2. Individuals in the same group are considered to be a distance x = 0 apart. As an example, the distance between individuals in groups i and j is x_ij = 2.

Average epidemic size ψ as a function of model parameters. (A) ψ and associated 95% confidence interval delimited by the 2.5th and 97.5th percentiles (vertical bars) as a function of P₀ as approximated by P₀ ≈ pψ′n/γ. This approximation, which becomes exact as p → 0, can be understood as follows. The quantity ψ′ is the expected normalized epidemic size for a stochastic, homogenous mixing model of size n and reproduction number R₀. Because infectives typically recover after 1/γ time steps, there are roughly ψ′n/γ opportunities for infected individuals to leave the initially infected group. Multiplying by p then gives the expected number who do. As discussed in the text, when P₀ > 1, an epidemic is expected to leave the initial context and spread nonlocally (providing R₀ > 1). The model parameters for the data shown here are basic reproduction number R₀ = 3, transport parameter ξ = 0.6, branching ratio b = 3, hierarchy depth l = 4, and group size n = 100. The results are averaged over 1,000 simulations. (B) The average epidemic size ψ (again with 95% confidence intervals) as a function of ξ (all parameters are the same but now P₀ is fixed at 5). For sufficiently low ξ, nonlocal epidemics never arise, regardless of whether or not P₀ and R₀ exceed unity. When ξ = 1/ln b, individuals are equally likely to travel to all distances; the disease typically spreads globally; and the epidemic size distribution is bimodal. For both plots, intermediate values of P₀ and ξ yield epidemics of a wide range of sizes (see Fig. 4).

Multiscale, resurgent behavior of real epidemics. (A and B) Frequency of epidemic sizes for measles (A) and pertussis (B) for Iceland, 1888–1990 (9). Each epidemic is identified as a contiguous sequence of months with nonzero caseloads bounded by months with zero caseloads, and the corresponding epidemic size ψ is normalized by the population size of Iceland at the midpoint of the epidemic. Both examples show a single mode for small outbreak sizes along with a relatively flat distribution of larger epidemic sizes. (Insets) The complementary cumulative frequency distributions (i.e., the number of epidemics of size at least ψ) for the same diseases on double logarithmic axes. A linear fit with slope –θ corresponds to a power-law distribution P[Ψ> ψ]∝ψ^–θ over some interval ψ_min ≤ ψ ≤ ψ_max, usually called the “scaling region.” Here, the largest, and most likely, candidate for a scaling region is 10^–4 ≤ ψ ≤ 10^–2, over which the exponent θ is 0.13 for measles and 0.16 for pertussis; both values are well outside the usual range of 1 ≤ θ < 2. We also identify a second, smaller scaling region, 10^–5 ≤ ψ ≤ 10^–4, in which θ is 0.40 for measles and 0.39 for pertussis, where the slightly larger exponents can be attributed to the spike near zero. Our conclusion is that neither distribution is well fit by a power law (see text). (C) Estimated worldwide number of cases of SARS as a function of time, showing resurgent behavior. Source: Mark Lipsitch, Harvard University, Boston.

Evidence of multiscale, resurgent behavior in simulated data. (A–C) Histograms of epidemic size ψ for three configurations of the model. (A) The bimodal size distribution generated from 5,000 simulations of a homogeneously mixing population of size N = 102,400 and reproduction number R₀ = 3. (B and C) The same size population as in A is hierarchically structured with branching ratio b = 4, depth l = 5, and group size n = 100, and for both of these examples, P₀ = 0.35 and ξ = 0.35. The models for B and C differ in the reproductive number: R₀ = 3 and 12, respectively. Nevertheless, both B and C show similar size distributions with modes near ψ = 0 and a relatively flat distribution for ψ > 0, qualitatively similar in form to that of the Icelandic data. (D–G) Example time series of total new cases, where in all cases, simulation parameters are R₀ = 3 and, again, N = 102,400. For D, the population is homogeneously mixing, and for E–G, the population is structured according to our model with the same parameters as for B and C. For the random mixing case of D, a typical epidemic trajectory rises rapidly once and then declines to zero, infecting most of the population in the process (see gray sidebar). In the examples with population structure, by contrast, epidemic trajectories exhibit dramatic resurgence, endure for markedly different time intervals, and infect very different fractions of the population (see gray sidebars), depending solely on stochastic fluctuations (i.e., all parameters are held constant).