In infectious disease epidemiology, the serial interval
is the difference between the symptom onset time of an infected person and the symptom onset time of his or her infector [1
]. This is sometimes called the “generation interval.” However, we find it more useful to adopt the terminology of Svensson [2
] and define the generation interval
as the difference between the infection time of an infected person and the infection time of his or her infector. By these definitions, the serial interval is observable while the generation interval usually is not. We define infectious contact
to be a contact that is sufficient to infect j
is infectious and j
is susceptible, and we define a potential infector
of person i
to be an infectious person who has positive probability of making infectious contact with i
. Finally, we use the term hazard
rather than force of infection
to highlight the similarities between epidemic data analysis and survival analysis.
The generation interval has been an important input for epidemic models used to investigate the transmission and control of SARS [3
] and pandemic influenza [5
]. More recently, generation interval distributions have been used to calculate the incubation period distribution of SARS [7
] and to estimate R0
from the exponential growth rate at the beginning of an epidemic [8
]. It is generally assumed that the generation interval distribution is characteristic of an infectious disease. In this paper, we show that this is not true. Instead, the expected generation interval decreases as the number of potential infectors of susceptibles increases. During an epidemic, generation intervals tend to contract as the prevalence of infection increases. This effect was described by Svensson [2
] for an SIR model with homogeneous mixing. In this paper, we extend this result to all time-homogeneous stochastic SIR models.
A simple thought experiment illustrates the intuition behind our main result. Imagine a susceptible person j in a room. Place m other persons in the room and infect them all at time t = 0. For simplicity, assume that infectious contact from i to j occurs with probability one, i = 1, ..., m. Let tij be a continuous nonnegative random variable denoting the first time at which i makes infectious contact with j. Person j is infected at time tj = min(t1j, ..., tmj). Since all infectious persons were infected at time zero, tj is the generation interval. If we repeat the experiment with larger and larger m, the expected value of min(t1j, ..., tmj) will decrease.
When a susceptible person is at risk of infectious contact from multiple sources, there is a “race” to infect him or her in which only the first infectious contact leads to infection. Generation interval contraction is an example of a well-known phenomenon in epidemiology: The expected time to an outcome, given that the outcome occurs, decreases in the presence of competing risks. In our thought experiment, the outcome is the infection of j by a given i and the competing risks are infectious contacts from all sources other than i.
Adapting our thought experiment slightly, we see that the contraction of the generation interval is a consequence of the fact that the hazard of infection for j
increases as the number of potential infectors increases. Let λ(t
) be the hazard of infectious contact from any potential infector to j
at time t
and let E
] be the expected infection time of j
potential infectors. Then
so the expected generation interval decreases as the number of potential infectors increases. A hazard of infection that increases with the number of potential infectors is a defining feature of most epidemic models, so generation interval contraction is a very general phenomenon. We note that a very similar phenomenon occurs in endemic diseases, where increased force of infection results in a decreased average age at first infection [9
The rest of the paper is organized as follows: In Section 2, we describe a general stochastic SIR epidemic model. In Section 3, we use this model to show that the mean generation interval decreases as the number of potential infectors increases. As a corollary, we find that the mean serial interval also decreases. In Section 4, we consider the role of the population contact structure in generation interval contraction and illustrate the effects of global and local competition among potential infectors with simulations. In Section 5, we argue that hazards of infectious contact should be used instead of generation or serial interval distributions in the analysis of epidemic data. Section 6 summarizes our main results and conclusions.