Characterizing the reproduction number of epidemics with early subexponential growth dynamics

Abstract

Early estimates of the transmission potential of emerging and re-emerging infections are increasingly used to inform public health authorities on the level of risk posed by outbreaks. Existing methods to estimate the reproduction number generally assume exponential growth in case incidence in the first few disease generations, before susceptible depletion sets in. In reality, outbreaks can display subexponential (i.e. polynomial) growth in the first few disease generations, owing to clustering in contact patterns, spatial effects, inhomogeneous mixing, reactive behaviour changes or other mechanisms. Here, we introduce the generalized growth model to characterize the early growth profile of outbreaks and estimate the effective reproduction number, with no need for explicit assumptions about the shape of epidemic growth. We demonstrate this phenomenological approach using analytical results and simulations from mechanistic models, and provide validation against a range of empirical disease datasets. Our results suggest that subexponential growth in the early phase of an epidemic is the rule rather the exception. Mechanistic simulations show that slight modifications to the classical susceptible-infectious-removed model result in subexponential growth, and in turn a rapid decline in the reproduction number within three to five disease generations. For empirical outbreaks, the generalized-growth model consistently outperforms the exponential model for a variety of directly and indirectly transmitted diseases datasets (pandemic influenza, measles, smallpox, bubonic plague, cholera, foot-and-mouth disease, HIV/AIDS and Ebola) with model estimates supporting subexponential growth dynamics. The rapid decline in effective reproduction number predicted by analytical results and observed in real and synthetic datasets within three to five disease generations contrasts with the expectation of invariant reproduction number in epidemics obeying exponential growth. The generalized-growth concept also provides us a compelling argument for the unexpected extinction of certain emerging disease outbreaks during the early ascending phase. Overall, our approach promotes a more reliable and data-driven characterization of the early epidemic phase, which is important for accurate estimation of the reproduction number and prediction of disease impact.

Keywords: effective reproduction number; epidemics; exponential growth; generalized-growth model; phenomenological model; subexponential growth.

Figure 1.

Simulated profiles of epidemic growth (a) and corresponding effective reproduction number (b) based on the generalized-growth model during the first five disease generation intervals. The parameter p is varied from 0.2 to 1.0, whereas the growth rate parameter r is fixed at 0.65 per day and C₀ = 1. The length of the generation interval is assumed to be fixed at T_g = 3 days. When p = 1 (exponential growth), the reproduction number is constant at . When p < 1 (subexponential growth) the reproduction number gradually declines over time and approaches 1.0 asymptotically. The black circles correspond to calculated using the analytical formula in equation (2.6), whereas the solid lines correspond to the estimates from numerical solutions based on renewal equation (2.7). (Online version in colour.)

Figure 2.

Simulated profiles of the reproduction number during the first five disease generation intervals under the generalized-growth model, for different values of p. The parameter r is fixed at 0.65 per day and C₀ = 1. Estimates of the reproduction number are numerically obtained using equation (2.7) assuming four different distributions for the generation interval: (a) uniform distribution in the range 2–4 days, (b) exponential distribution with the mean of 3 days, (c) gamma distribution with the mean of 3 days and variance of 1 day and (d) fixed generation interval at 3 days. When p = 1 (exponential growth), the reproduction number is invariant irrespective of the shape of the generation interval distribution and is given by when the generation interval is exponentially distributed [20], when the generation interval is fixed [20], and when the generation interval follows a gamma distribution with the shape and scale parameters given by a = 9 and b = 1/3 [20], respectively. For p < 1 (subexponential growth), the effective reproduction number approaches 1.0 asymptotically over disease generations. The reproduction number at fixed generation intervals can be explicitly computed using equation (2.6) (black circles). (Online version in colour.)

Figure 3.

Simulations of case incidence (a,c,e) and the effective reproduction number R_g (b,d,f) for the first few consecutive disease generations, using a network-based transmission model with household–community structure [12,41] for different values of the community mixing parameter (, 45 and 65 households) while keeping the household size H fixed at 5. The household reproduction number R_0H was set at 2.0 and the community reproduction number R_0c was set at 0.7 based on a previous Ebola mathematical modelling study [12]. Each simulation started with one infectious individual. Early subexponential growth dynamics are observed across all scenarios, which is consistent with a declining trend in the effective reproduction number R_g. For comparison with exponential growth dynamics, the dark solid line illustrates exponential growth in case incidence over disease generations with a fixed reproduction number of 2.7. The horizontal dashed line at R_g = 1.0 is shown for reference. (Online version in colour.)

Figure 4.

Simulations of case incidence (a) and the effective reproduction number R_g (b) for the first few consecutive disease generations, using model (2.9) for different values of the decline rate parameter q with R₀ = 2 and a large population size (N = 10⁸). Each simulation started with one infectious individual. When q = 1, exponential growth during the early epidemic phase is evident as a straight line fits well for several consecutive disease generations of the incidence curve in semi-logarithmic scale, and the effective reproduction number, R_g, remains invariant at 2. A concave down curve is indicative of subexponential growth with q > 0. The horizontal dashed line at R_g = 1.0 is shown for reference. (Online version in colour.)

Figure 5.

Simulations of case incidence (a) and the effective reproduction number R_g (b) for the first few consecutive disease generations, using model (2.11) for different values of scaling parameter α, with R₀ = 2, γ = 1/5, and a large population size (N = 10⁸). Each simulation started with one infectious individual. When α = 1, exponential growth during the early epidemic phase is evident as a straight line fits well for several consecutive disease generations of the incidence curve, and the effective reproduction number, R_g, remains invariant at 2. The horizontal dashed line at R_g = 1.0 is shown for reference. (Online version in colour.)

Figure 6.

Estimates of the deceleration of growth parameter and the corresponding 95% CIs derived from various infectious disease outbreak datasets of case incidence series by fitting the generalized-growth model to the initial epidemic phase comprising of approximately the first three (green), four (blue) and five (red) disease generation intervals. (Online version in colour.)

Figure 7.

Estimates of the effective reproduction number and the corresponding 95% CIs derived from various infectious disease outbreak datasets of case incidence series by fitting the generalized-growth model to the initial epidemic phase comprising of approximately the first three (green), four (blue) and five (red) disease generation intervals. The generation interval is assumed to follow a gamma distribution with the corresponding mean and variance provided in table 1. (Online version in colour.)