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. 2007 Feb 22;274(1609):599-604.
doi: 10.1098/rspb.2006.3754.

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

J Wallinga  1 , M Lipsitch
Affiliations

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

J Wallinga et al. Proc Biol Sci. .

Abstract

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.

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Figures

Figure 1
Figure 1
Generation interval distributions uniquely characterize the relationship between the reproductive number R and the growth rate r: (a) generation interval distributions g(a) with identical mean Tc and increasing coefficient of variation σ/Tc from 0.25 (long dashes), 0.5 (long and short dashes), 0.75 (short dashes) to 1 (dots) and (b) the corresponding relationships between growth rate r and reproductive number R, the drawn curve indicates the upper bound obtained for the delta distribution.
Figure 2
Figure 2
Observed generation intervals for influenza A virus infections and the empirical relationship between reproductive number and growth rate: (a) histogram of observed generation intervals between two subsequent influenza cases in a household (after Hirotsu et al. 2004), the lighter shading indicates observations that could be due to co-primary or tertiary infections and (b) the corresponding empirical relationship between R and r, according to equation (3.6).
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