How generation intervals shape the relationship between growth rates and reproductive numbers

J. Wallinga () 1 M. Lipsitch 0 0 Department of Epidemiology and Department of Immunology and Infectious Diseases, Harvard School of Public Health , 677 Huntington Avenue, Boston, MA 02115 , USA 1 Department of Infectious Diseases Epidemiology, National Institute of Public Health and the Environment , PO Box 1, 3720 BA Bilthoven , The Netherlands Mathematical models of transmission have become invaluable management tools in planning for the control of emerging infectious diseases. A key variable in such models is the reproductive number R. For new emerging infectious diseases, the value of the reproductive number can only be inferred indirectly from the observed exponential epidemic growth rate r. Such inference is ambiguous as several different equations exist that relate the reproductive number to the growth rate, and it is unclear which of these equations might apply to a new infection. Here, we show that these different equations differ only with respect to their assumed shape of the generation interval distribution. Therefore, the shape of the generation interval distribution determines which equation is appropriate for inferring the reproductive number from the observed growth rate. We show that by assuming all generation intervals to be equal to the mean, we obtain an upper bound to the range of possible values that the reproductive number may attain for a given growth rate. Furthermore, we show that by taking the generation interval distribution equal to the observed distribution, it is possible to obtain an empirical estimate of the reproductive number. 1. INTRODUCTION The past decade has seen a dramatic increase in the attention paid to infectious disease epidemics as a potential health threat. This is due in part to disease outbreaks in domestic livestock (Keeling et al. 2001), the fear of bioterrorist attacks with smallpox virus (Gani & Leach 2001), the emergence of severe acute respiratory syndrome (SARS) in 2003 (Lipsitch et al. 2003) and the risk of an influenza pandemic among human populations (Longini et al. 2004; Ferguson et al. 2005). Planning for the mitigation and control of such health threats relies increasingly on mathematical models of infection transmission. One of the key parameters in mathematical transmission models is the reproductive number R0, defined as the number of secondary infections that arise from a typical primary case in a completely susceptible population. When infection is spreading through a population that may be partially immune, it is often more convenient to work with an effective reproductive number R, which is defined as the number of secondary infections that arise from a typical primary case. The magnitude of R is a useful indicator of both the risk of an epidemic and the effort required to control an infection (Anderson & May 1991; Roberts & Heesterbeek 2003; Heffernan et al. 2005). Accurate estimation of the value of the reproductive number is crucial to planning for the control of an infection. For new emerging infections, such as SARS in 2003, the available information about the transmissibility of a new infectious disease epidemic is likely to be restricted to daily counts of new cases. It is well known that these counts increase exponentially in the initial phase of an epidemic. The rate of exponential growth, r, is defined as the per capita change in number of new cases per unit of time. The observed value of the growth rate r can be related to the value of reproductive number R through a linear equation: RZ1CrTc (Anderson & May 1991; Pybus et al. 2001; Ferguson et al. 2005). Here, Tc is the mean generation interval, defined as the mean duration between time of infection of a secondary infectee and the time of infection of its primary infector (sometimes this is called the serial interval or generation time). Demographers, ecologists and evolutionary biologists take a slightly different approach. They derive the growth rate from fecundity rates, survival rates and the reproductive number R according to the so-called LotkaEuler equation (Dublin & Lotka 1925; Feller 1941; Metz & Diekmann 1986; Keyfitz & Caswell 2005). Ecological textbooks suggest simplifying this equation by ignoring variability in generation time (Begon et al. 1996). The result is, after rearranging, an exponential equation: RZexp(rTc). Here, Tc is the cohort generation time, a demographic analogue of the epidemiological mean generation interval. Having two alternative equations for relating the desired value of reproductive number to the observed value for growth rate, we face the difficulty of choosing the most appropriate one. For example, the growth rate of the Hepatitis C epidemic is estimated to be rZ0.96 per year, Downloaded from http://rspb.royalsocietypublishing.org/ on November 14, 2014 600 J. Wallinga & M. Lipsitch Reproductive numbers from growth rates (b) A moment generating function expression for the reproductive number R While the LotkaEuler equation is a basic part of demography, in which one may be interested in deriving population growth rates from life tables, a related problem in epidemiology is to estimate the reproductive number, R, from growth rates of a disease. We have previously defined n(a) as the rate of production of female offspring by a mother at age a. It is readily seen that if we integrate n(a) over the whole lifespan, we obtain the total number of female offspring produced by a mother over her lifespan, known as R and the mean generation interval of this infection is of the order of TcZ20 years (Pybus et al. 2001). The value for the reproductive number by the linear equation is RZ1C0.096!20Z2.9, whereas the value that is obtained using the exponential equation is RZexp(0.096!20)Z6.8. Such large discrepancies do matter in planning for public health interventions. There exist several other expressions that relate the reproductive number to the growth rate (Dublin & Lotka 1925; Lipsitch et al. 2003; Wearing et al. 2005) and expressions for estimating the reproductive number from time-series of case counts (Wallinga & Teunis 2004). How should we choose the most appropriate equation for inferring the reproductive number from observed growth rates for a particular infection? We start by recapitulating the LotkaEuler equation in terms of human demography, and we rephrase this equation into more convenient terms for infectious disease epidemiology, following Levin et al. (1996). We use the rephrased LotkaEuler equation to examine the assumptions that underlie the alternative relationships between reproductive number and observed change in a number of cases. We will illustrate our findings by estimating the reproductive number R for influenza A infections. The key variables and their interpretation in ecological, demographical and epidemiological terms are presented in the electronic supplementary material. 2. INFERRING R FROM r (a) Deriving the LotkaEuler equation We introduce the LotkaEuler equation using (...truncated)