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J R Soc Interface. 2016 Oct;13(123). pii: 20160659.

Characterizing the reproduction number of epidemics with early subexponential growth dynamics.

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School of Public Health, Georgia State University, Atlanta, GA, USA Division of International Epidemiology and Population Studies, Fogarty International Center, National Institutes of Health, Bethesda, MD, USA
Division of International Epidemiology and Population Studies, Fogarty International Center, National Institutes of Health, Bethesda, MD, USA.
Department of Public Health, University of Copenhagen, Copenhagen, Denmark Department of Global Health, George Washington University, Washington, DC, USA.
Agent Based Modelling Laboratory, York University, Toronto, Canada.


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.


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

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