Generation interval contraction and epidemic data analysis
Introduction
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 from i to j to be a contact that is sufficient to infect j if i 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], [4] and pandemic influenza [5], [6]. More recently, generation interval distributions have been used to calculate the incubation period distribution of SARS [7] and to estimate 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 . For simplicity, assume that infectious contact from i to j occurs with probability one, . Let be a continuous nonnegative random variable denoting the first time at which i makes infectious contact with j. Person j is infected at time . Since all infectious persons were infected at time zero, is the generation interval. If we repeat the experiment with larger and larger m, the expected value of 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 be the hazard of infectious contact from any potential infector to j at time t and let be the expected infection time of j given m potential infectors. Thenso 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.
Section snippets
General stochastic SIR model
We start with a very general stochastic “Susceptible-Infectious-Removed” (SIR) epidemic model. This model includes fully-mixed and network-based models as special cases, and it has been used previously to define a mapping from the final outcomes of stochastic SIR models to the components of semi-directed random networks [10], [11].
Each person i is infected at his or her infection time , with if i is never infected. Person i recovers from infectiousness or dies at time , where the
Generation interval contraction
In this section, we show that the mean infectious contact interval given that i infects j is shorter than the mean infectious contact interval given that i makes infectious contact with j. In the notation from the previous section,(note that implies but not vice versa). In general, this inequality is strict when j is at risk of infectious contact from any source other than i. This inequality implies the contraction of generation and serial intervals during an
Simulations
We refer to the “race” to infect a susceptible person as competition among potential infectors. In this section, we illustrate two types of competition among potential infectors: Global competition among potential infectors results from a high global prevalence of infection. Local competition among potential infectors results from rapid transmission within clusters of contacts, which causes susceptibles to be at risk of infectious contact from multiple sources within their clusters even if the
Consequences for estimation
The effect of generation interval contraction on parameter estimates obtained from models that assume a constant generation or serial interval distribution is difficult to assess. The assumption of a constant serial or generation interval distribution may be reasonable in the early stages of an epidemic with little clustering of contacts, in an epidemic with near one, or in an endemic situation. However, this ignores the more fundamental issue that estimates of these distributions are
Discussion
Generation and serial interval distributions are not stable characteristics of an infectious disease. When multiple infectious persons compete to infect a given susceptible person, infection is caused by the first person to make infectious contact. In Section 3, we showed that the mean infectious contact interval given that i actually infected j is less than or equal to the mean given i made infectious contact with j. That is,with strict inequality when is
Acknowledgments
This work was supported by the US National Institutes of Health cooperative agreement 5U01GM076497 “Models of Infectious Disease Agent Study” (E.K. and M.L.) and Ruth L. Kirchstein National Research Service Award 5T32AI007535 “Epidemiology of Infectious Diseases and Biodefense” (E.K.). We also wish to thank Jacco Wallinga and the anonymous reviewers of Mathematical Biosciences for useful comments and suggestions.
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