Estimating individual and household reproduction numbers in an emerging epidemic - PubMed (original) (raw)

Estimating individual and household reproduction numbers in an emerging epidemic

Christophe Fraser. PLoS One. 2007.

Abstract

Reproduction numbers, defined as averages of the number of people infected by a typical case, play a central role in tracking infectious disease outbreaks. The aim of this paper is to develop methods for estimating reproduction numbers which are simple enough that they could be applied with limited data or in real time during an outbreak. I present a new estimator for the individual reproduction number, which describes the state of the epidemic at a point in time rather than tracking individuals over time, and discuss some potential benefits. Then, to capture more of the detail that micro-simulations have shown is important in outbreak dynamics, I analyse a model of transmission within and between households, and develop a method to estimate the household reproduction number, defined as the number of households infected by each infected household. This method is validated by numerical simulations of the spread of influenza and measles using historical data, and estimates are obtained for would-be emerging epidemics of these viruses. I argue that the household reproduction number is useful in assessing the impact of measures that target the household for isolation, quarantine, vaccination or prophylactic treatment, and measures such as social distancing and school or workplace closures which limit between-household transmission, all of which play a key role in current thinking on future infectious disease mitigation.

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Conflict of interest statement

Competing Interests: The author has declared that no competing interests exist.

Figures

Figure 1

Figure 1. Concept of approach.

As a starting point, consider a household of size 4. A, illustrates an infection being transmitted in this household, with the index case (i) being infected outside the house at time t0. They then infect exactly one person (ii) at time t1, and (ii) in turn infects person (iii) at time t2. Person (iv) escapes infection altogether. The infectiousness of each individual over time is also shown, with infection events highlighted. B, illustrates how these events can be reinterpreted by taking the unit of infection as the household, infected at time t0 and with total infectiousness of the household defined as the sum of the four individual curves shown in A. The aim of the method is then to average this process over all possible chains of infection in the household, and all household sizes, to obtain the characteristic infectiousness profile of a household, as shown in C, where the household is no longer decomposed into its constituent units at all.

Figure 2

Figure 2. The infectiousness of households.

The average infectiousness of a fully susceptible household infected with influenza (A) or measles (B). The infectiousness of individuals (denoted β (τ)) is shown, as is the infectiousness of the typical infected household (denoted β* (τ*)). This latter curve is obtained by simulating over 10,000 epidemics of transmission within households starting from one infected case. The two analytical approximations described in the text are also shown. “Approx 1” is the main approximation described, while “Approx 2” is the one obtained by assuming that all infections occur in the second generation of infection within the household. Parameters are as described in the main text, and the curves are arbitrarily scaled such that each individual infects on average one person outside of the household (i.e. RG = 1).

Figure 3

Figure 3. Reproduction numbers for an exponentially growing epidemic.

Estimates of the individual reproduction number R (r ) (the average number of people infected by each individual) and the household reproduction number R* (r ) (the average number of people infected by each household) are shown as a function of the epidemic growth rate r for influenza (A) and measles (B). The two analytical approximations which bracket the household reproduction number R* (denoted upper and lower) are shown along with numerically estimated values (‘+’ symbols). The approximation to the individual reproduction number given by equation (12) is shown (dashed line) along with numerical estimates (‘Δ’ symbols). The estimate for the quantity RG, the average number of people each person infects outside his or her home, is also shown.

Figure 4

Figure 4. Simulated epidemics.

To check the method for consistency, I simulate ten epidemics of influenza (A) and measles (B) within a fully susceptible community of 2,000 households. I use parameters estimated for an epidemic growth rate r = 0.20 per day, and condition on non-extinction of the epidemic. In C and D, the natural logarithm of the incidence is compared to the fixed slope curve r = 0.20 predicted by the model (thick line). Linear regression through these data yields the estimate r̂ = 0.19 in both cases.

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