Separate roles of the latent and infectious periods in shaping the relation between the basic reproduction number and the intrinsic growth rate of infectious disease outbreaks.

Two approximations are commonly used to describe the spread of an infectious disease at its early phase: (i) the branching processes based on the generation concept and (ii) the exponential growth over calendar time. The former is characterized by a mean parameter: the reproduction number R(0). The latter is characterized by a growth rate rho, also known as the Malthusian number. It is common to use empirically observed rho to assess R(0) using formulae derived either when both the latent and infectious periods follow exponential distributions or assuming both are fixed non-random quantities. This paper first points out that most of these formulae are special cases when the latent and infectious periods are gamma distributed, given by a closed-form solution in Anderson and Watson [1980. On the spread of a disease with gamma distributed latent and infectious periods. Biometrika 67 (1), 191-198]. A more general result will be then established which takes the result in Anderson and Watson [1980. On the spread of a disease with gamma distributed latent and infectious periods. Biometrika 67 (1), 191-198] as its special case. Three aspects separately shape the relationship between rho and R(0). They are: (i) the intensity of infectious contacts as a counting process; (ii) the distribution of the latent period and (iii) the distribution of the infectious period. This article also distinguishes the generation time from the transmission interval. It shows that whereas the distribution of the generation time can be derived by the latent and infectious period distributions, the distribution of the transmission interval is also determined by the intensity of infectious contacts as a counting process and hence by R(0). Some syntheses among R(0), rho and the average transmission interval are discussed. Numerical examples and simulation results are supplied to support the theoretical arguments.