Estimating reproduction numbers using a Kermack McKendrick transmission model

Though less well known than their compartmental counterparts (SIR, SIS, etc…), time-since-infection models offer a more intuitive starting point for modelling infectious disease transmission, and importantly for this application, they provide two other major advantages. First, it is typically easier to identify their key parameters, and second they more readily adapt to describe multi-level transmission (by multi-level, I mean here within-household and between household). A disadvantage is that it can be harder to include heterogeneities. Nomenclature is confusing, since both types of model have their origin in the same classic paper of Kermack and McKendrick [21], and both the SIR model and the simplest time-since-infection model are known as “the Kermack-McKendrick model”.

The model, in the formalism chosen here, predicts the changing incidence rate I (t) as a function of calendar time t in terms of the transmissibility, denoted β (t, τ ), an arbitrary function of calendar time t and time since infection τ. β (t, τ) typically reflects pathogen load, or perhaps more precisely pathogen shedding. It is commonly a single peaked function reflecting pathogen growth followed by immune suppression, or host death, but can be more exotic such as the double peaked profile associated with early and late transmission of HIV [22], or the repeated peaks of malaria [23]. β (t, τ) also reflects the effective contact rate between infectious and susceptible individuals, which can change during the course of a single infection, increasing for example if a person coughs or sneezes due to respiratory disease, or decreasing if a person takes to bed with illness, and during the course of the epidemic as public health measures are implemented. More discussions of the components (infectiousness and contact) of β (t, τ) can be found in [24]. Because I am interested in outbreaks of emerging infections, I will not describe explicitly reductions in the susceptible population caused by the epidemic. Formally this corresponds to working in the infinite population limit. This assumption is not essential for this section however, since β (t, τ) could also be thought of as incorporating the proportion of cases that are susceptible; the assumption becomes more important in the later sections on household models.

Mathematically, transmission is defined by a Poisson infection process such that the probability that, between time t and t+δ, someone infected a time t ago successfully infects someone else is β (t, τ)δ, where δ is a very small time interval.

This assumption then results in a prediction that the mean incidence I (t ) at time t follows the so-called renewal equation(1)

This equation states that the number of newly infected individuals is proportional to the number of prevalent cases multiplied by their infectiousness. It may often be convenient (and realistic) to truncate the function β (t, τ) at a time τ_m such that β (t, τ) = 0 for all τ>τ_m.

The asymptotic behaviour of incidence I (t ) is determined by reproduction numbers [21], [25]. Two intuitively defined reproduction numbers are the case reproduction number, which I denote R_c (t ), and the instantaneous reproduction number, which I denote R (t). The case reproduction number R_c (t ) is a property of individuals infected at time t, and is the average number of people someone infected at time t can expect to infect. For a person infected at time t it is the total infection hazard from time t onwards, i.e.(2)

While the case reproduction number has been widely used, it may also be worth considering a quantity which I call the instantaneous reproduction number R (t ), a property of the epidemic at time t. It is the average number of people someone infected at time t could expect to infect should conditions remain unchanged. It is given by(3)

To illustrate the distinction between R_c (t ) and R (t ), consider a situation where the transmission rate is abruptly reduced at a time t = t_I. The instantaneous reproduction number R (t ), which estimates how many people one case would infect if circumstances were to remain fixed, would abruptly switch from a high to a low value at time t_I. The case reproduction number R_c (t ), on the other hand, estimates how many people each case actually infects. It will thus account for the fact that someone infected at time t<t_I may spend part of their infectious period before and after the reduction in transmission which occurs at time t_I and thus R_c (t ) will smoothly transition from higher to lower values.

To derive simple estimating equations for R (t ), I consider the case where this function is separable, which corresponds to saying that the relative progression of infectiousness as a function of time since infection is independent of calendar time. In this case β (t, τ) can be written as the product of two functions Φ₁ (t ) and Φ₂ (τ), i.e.(4)

A counter-example might be when reactive patient isolation is introduced and acts to reduce infectiousness in late stage infection, in which case β (t, τ) can't be decomposed in this way. For this type of situation, it may be reasonable to assume the β (t, τ) can be decomposed separately in different stages of the epidemic, pre- and post- implementation of isolation measures, for example.

Since β (t, τ) is a product, I can arbitrary normalise one or other of the functions Φ₁ (t) and Φ₂ (τ), so without loss of generality, I choose Φ₂ (τ) to have total integral 1, i.e. . Substituting equation (4) into equation (3) I find that(5)

The function Φ₁ (t) is equal to the instantaneous reproduction number R (t). The function Φ₂ (τ) is then the distribution of how these infection events are distributed as a function of time since infection τ. This is an idealised definition of the generation time distribution, which I denote w (t). Thus, infectiousness can be decomposed as the product of the instantaneous reproduction number and the generation time distribution, i.e.(6)

The relationship between the idealised generation time distribution w (τ) and the distribution of observed generation times can be rather complex for a number of reasons. First, infections are rarely observed, and thus must be either back-calculated or the generation times must be based on a surrogate such as the appearance of symptoms [1], [12]. Second, right censoring can cause the observed generation times to be shorter or longer than expected for a growing or declining epidemic, respectively [26]. Third, as apparent here, if the reproduction number R (t) changes due to depletion of susceptibles, changes in contact rates or public health measures, then this will also change the observed generation times for infectious individuals during that period of change. Thus the distribution w (τ) is really intended as a measure of infectiousness which will correspond to generation times for an index case in an ideal large closed setting where contact rates are constant. It can be inferred from data on the timing of cases, as in [10], [13].

Inserting (6) into (1) yields a novel estimator for instantaneous reproduction number(7)

By substituting the decomposition (6) into equation (2), a relation between the instantaneous and case reproduction number is obtained:(8)i.e. the case reproduction number is a smoothed function of the instantaneous reproduction number.

Usually, incidence is reported as a discrete time series of the form I_i incident cases reported between time t_i and time t_i+1, in which case the generation time distribution should be appropriately discretised into a form w_i such that . The estimators for the reproduction numbers become(9)and(10)

Equation (10) was proposed by [12], [27] as a real time estimator of the reproduction number, while equation (9) was first used for analysing polio transmission in India [28] (based on the work presented in this manuscript).

While the case reproduction number is an intuitively appealing quantity, the instantaneous reproduction number estimated by equation (9) should also be considered for practical applications as it may suffer fewer problems of right censoring in an incompletely observed epidemic. Right censoring is a real problem in using the case reproduction number to track an epidemic in real time, since the estimator for R_c (t) at time t is seen in equation (10) to rely on knowing the incidence at future time-points. An algorithm to deal with this issue was proposed by [29], but switching instead to the instantaneous reproduction number estimated by equation (9) may be a simpler solution. Right censoring is not however the only complication associated with estimating reproduction numbers in practice, and is not completely absent from (7) due to the delay in detecting infections. Left censoring may also arise due to not knowing the baseline number infected if an epidemic has been unfolding for some time before observations are recorded. Finally, estimating the generation time distribution may not be straightforward.

Several strategies are possible to deal with the fact that one never observes infections, but rather as a time series of cases of the form C_i, where case definitions could be based on symptoms, hospitalisation or seroconversion. One strategy, used in [12], is simply to ignore this and use cases as surrogates of infection for estimation of both the generation time and the reproduction numbers. Often though, it may be possible to characterise a distribution of the time from infection to becoming a case, say ξ_i where . If a case is defined by symptoms then this would be the incubation period distribution. One can then back-calculate incidence as follows(11)

A drawback of this approach is that the estimated incidence time series Î_i will tend to be over-smoothed relative to the original time series I_i. It also makes clear that there is still a problem of right censoring in an incompletely observed epidemic in the estimator of equation (9), though less than in equation (10).

Statistical properties of these estimators are straightforward [12], [28]. One previously noted point [12], [28] is that because these estimators are essentially ratios of incidences, they can be used in cases where only a fraction of cases are observed, such as for polio where only a tiny fraction of infections lead to disease (of the order of 1 in 200), though the confidence intervals will change.

A special case applicable to many cases where surveillance is poor is when only the epidemic growth/decline rate r is known. In this case the incidence takes the form I (t) = I (0)exp (rt) and both estimators (7) and (8) for the reproduction number become(12)where the reproduction numbers are now expressed as a function of the exponential rate of change r. This is likely a useful formula, presented and studied in detail in [30], where the links to earlier ecological and demographic modelling were also highlighted. Much of the subsequent analysis will concern itself with deriving an equation equivalent to (12) for the household reproduction number R^*(r).

Extending the model to heterogeneous natural histories of infection

The model defined above assumes that the function β (t, τ) describes the “natural history” of infection in each infected individual. Before specialising to the model of household transmission, it is first worth considering the case where different individuals experience different “natural histories”, defined here by the susceptibility to infection, and infectiousness after infection.

I denote a vector of random variables X = {X1, X2, …} to describe factors which influence susceptibility or infectiousness. For example for the standard SEIR model of infection the random variables would be the durations of the latent period (L) and the infectious period (D), i.e. X = {L,D}. Let f (X ) denote the probability distribution of these random variables amongst new infections (taking into account differences in susceptibility), defined such that(13)where the integral is taken over the domain of the random variables. In other words, f (X ) is the proportion of new infections that have state X.

Let β (X, t, τ) denote the infectiousness profile of an individual with state X.

Assuming that all individuals mix homogeneously, then the transmission model defined earlier by equation (1) is generalised to(14)where I (X , t) is the incidence of infections with state X. I define the function K(t) to denote the integral(15)which clearly depends only on time t and not state X. The total incidence at time t is defined by the integral(16)

By substituting equation (14), which can be rewritten as I (X, t ) = f (X ) K (t ), into equation (16), I obtain that K (t ) = I_tot (t ) and thus that(17)

I can now substitute (17) into (14) to obtain(18)

Dividing both sides of this equation by f(X)yields an equation for the total incidence(19)

If I define the average infectiousness as follows(20)then equation (19) can now be seen to be the standard Kermack-McKendrick model of equation (1), i.e.(21)

In other words, in this model of an emerging infectious disease epidemic with heterogeneities in susceptibility and infectiousness, the dynamics of mean total incidence of infection is exactly equivalent to the basic model where the infectiousness is appropriately averaged using equation (20). Once an expression is derived for the average infectiousness β̃(t, τ), the results such as equations (9) or (12) can be used without further consideration of the heterogeneities in infectiousness or susceptibility.

Heterogeneities which are transmitted or preserved from one infection to the next, for example due to non-random mixing between different risk groups, a situation not considered here, lead to a more complex result. Some public health interventions such as isolation and contact tracing can induce such heritability even if it is not a basic property of the transmission process [31], [32].

A useful exercise in applying this formalism (not elaborated here) is the derivation of standard formulae for the basic reproduction number as a function of the exponential growth rate r for the SEIR model [30].

Estimating reproduction numbers for a model of transmission within and between households

One approach to estimating household reproduction numbers is simply to switch perspective from individual to household, directly estimate the generation time distribution (times taken for one household to infect another) and incidence of infection of households, and apply the results of equations (9) or (12) to estimate reproduction number as a function of time, R^*(t), or exponential growth rate, R^*(r). Because, as I have shown, the linearised Kermack-McKendrick model is applicable even when susceptibility and infectiousness are heterogeneous, this method is acceptable despite the fact that households may be quite heterogeneous in size and in the number of people infected. One analogous situation where this approach has been used is in estimating farm-to-farm reproduction numbers in the 2001 UK foot-and-mouth virus epidemic [27]. However, unless specifically tailored to this task, it is unlikely the data will be collected in the requisite form for this approach to be used in the human household situation. Thus, in this section I explore the alternative approach of explicitly modelling transmission within and between households.

Homogeneous transmission models can be interpreted as two-level hierarchical models, where the processes which guide the natural history of infection within the host are considered separate from those which drive transmission between hosts. The link between the two can be thought of as the function β (t, τ) which translates the impact of changing processes within the host into changing infectiousness as a function of time since infection. The approach taken here to modelling household transmission is to study a three-level hierarchical model of transmission. The three levels are within-host, within-household, and between households. The natural history of infection is described by the individual infectiousness function β (t, τ). I assume in this section that individuals are homogenous in infectiousness and susceptibility. I then use this to predict the course of epidemics within households, and derive a function β ^* (t, τ ^*) which describes the average infectiousness of a household towards other households as a function of the time since the household was infected, τ ^* (from here-on, I use the starred symbols to denote properties of households, and un-starred symbols to denote properties of individuals). The basic idea behind this analysis is illustrated in Fig 1.

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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.

https://doi.org/10.1371/journal.pone.0000758.g001

To simplify the notation, and because the main aim of this section is to study the case of an epidemic growing exponentially, I consider the situation where infectiousness is independent of calendar time t. This could be relaxed, though only if variation in time is somewhat slower than the typical duration of infection within a household.

More specifically, the model assumptions are that:

• individuals are distributed into households, and mix randomly and homogeneously outside of their household;
• within a small time interval δ, an individual who has been infected a time τ ago infects a person at random in the population with probability β_G (τ )δ;
• within this same time interval he or she infects each susceptible individual in his or her household with probability β_L (τ, n)δ (this is allowed to depend on the household size n, since empirical evidence suggests such variation may occur [10]);
• the population is large, and the disease has low prevalence, so that the probability of a household being repeatedly infected is negligible;
• the functions β_G (τ ) and β_L (τ, n ) are proportional to each other as functions of the time since infection t.

As a result of the last assumption and of the discussion around equation (6), the infectiousness functions can be decomposed as β_G (τ) = R_G w (τ) and β_n (n, τ) = r_n w (τ), where R_G is the average number of people each infected individual infects through random (non-household) contacts, w (τ) is the generation time distribution for between household transmission, and r_n is a parameter describing infection within the household whose interpretation will be clarified below.

I start by analysing the process of transmission within a single infected household of size n in terms of the functions r_n and w (τ). Consider first a household of size 2, where one individual is infected at time τ ^* = 0. Given the Poisson process described by the assumptions listed above, the probability that the second individual remains uninfected at time τ ^* is . The probability that the second person is never infected is . The distribution of times of infection of the second individual, conditional on infection, is then (22)where −∂ q₂ (τ ^*)/∂t ^* is the rate of change of the cumulative probability of not being infected, i.e. the probability density of being infected at time τ ^*, and the normalising factor 1-Q₂ is the total probability of being infected. The difference between w₂ (τ ^*) and the standard generation time distribution w (τ) is a saturation effect, so that the second case tends to get infected earlier as the infectiousness of the index case (ρ₂) is increased.

The infectiousness of the second individual towards other non-household members of the population, conditional on his or her infection, and described as a function of the time τ ^* since the infection of the household is thus the convolution of w₂ (τ ^* ) and β_G (τ), so that the total infectiousness of the household is (23)

Generalising this exact result to larger households involves some complications. Consider for example a household of size 3, where one individual is infected at time τ ^* = 0. The probability that neither of the other two individuals is infected by the first individual at time τ ^* is directly analogous to the situation for households of size 2. However this is somewhat greater than the actual probability that they are not infected at all, since once one of these two is infected, they can also infect the other, and thus the probability that they each escape infection is somewhat less than .

To progress further with analysing this system, I propose to approximate the process by assuming that infections within a household can be approximately described by a discrete generation Reed-Frost model, i.e. where the probability of not being infected in each generation is (Q_n )^m where m individuals are infected in the previous generation and Q_n ° exp (−r_n ). Q_n is the escape probability of each infectious-susceptible pair of individuals considered in isolation. In the formalism proposed by Ludwig, this corresponds to using infectious rank as a surrogate for infectious generation [33]. Dynamics are recovered by assuming the times between generations are described by the standard generation time distribution w (τ).

The ordering of infection events has no influence on the final number of individuals infected [33], and therefore this approximation will produce exact results for the final number of people infected in each household. Because of the possibility of “later” generations preceding “earlier” ones, as noted in the case of households of size 3 above, and because of ignoring the saturation effect present in equation (22) in terms of the actual generation times within households, this approximation will overestimate the time taken for individuals to become infected in the household. Because of the general form of the relation between generation time and reproduction number seen in equation (12), this will result in over-estimates of the household reproduction number R^* (r). To provide a counter-balancing under-estimate of R^* (r), I also consider an alternative approximation obtained by assuming the same total number of cases as predicted by this Reed-Frost model, but where all cases are assumed to be infected by the first index case. This is not a formal lower bound, since in the limit of infinite infectiousness within the household, all members of the household will be infected simultaneously upon introduction of the infection into the household. I find however that even for the example of highly infectious measles virus (below), the under-approximation is sufficient to provide a practical lower bound.

The probability of different chains of infection within households can easily be computed from the assumed Reed-Frost model [2]. I denote pr( {m₁, m₂, …, m_n }|n) the probability of a chain of infection occurring in a household of size n where m₁ index cases infects m₂, who in turn infect m₃ tertiary cases and so on, up to a maximum of n generations of infection. It is an assumption of the model that the number of index cases is 1, so m₁ = 1. The full probabilities can be obtained as products of binomials linking each generation to the previous ones as follows(24)where(25)

The second approximation is that the time taken for one infected to infect the next is distributed according to the standard generation time distribution w (τ). The time at which someone in the (i+1)^th generation of infection is infected is as a result drawn from the i^th auto-convolution of this distribution, denoted here w_[i](τ ^*) and defined by the recursive convolution equation (26)which satisfies .

Consider now an individual in the i^th generation of infection in the household, and consider this household at a time τ ^* after the first index case was infected. This individual must have been infected at some earlier time s ⩽ τ ^* distributed according to the distribution w_[i-1](s). His or her infectiousness to others outside of the household will be given by β_G (τ^*-s). Thus, by averaging over all possible values of s, the average infectiousness of such an individual in the i^th generation is (27)

Thus having averaged over all possible times of infection in the chain of transmission events in the household, infectious households are stratified by their size and by the number of cases in each generation. Using the notation defined earlier, I define the state vector X = {n, m₁, …, m_n} of variables which define the infectiousness and susceptibility of the infected household, where n is the household size and m_i is the number of infected individuals in the i^th generation of infection in the household. The infectiousness of a household with this state X towards other households, mediated by random mixing of individuals between households, is the sum of the infectiousness of all the individuals each given by equation (27), i.e. (28)

Given that this infection process involves random mixing of individuals outside their household, the distribution of sizes of households which get infected is the so-called size-biased household distribution. This is the distribution of sizes one obtains by sampling individuals at random in the population and recording the size of their household, as opposed to the more commonly recorded household size distribution which is obtained by sampling households at random. If k_n denotes the household size distribution, then (29)is the size-biased household size distribution. Given a household of size n gets infected, the probability of a chain of infections is given by the Reed-Frost probabilities pr ({m₁, …, m_n}|n). The distribution of the random variables X = {n, m₁, …, m_n} at infection is thus(30)

The mean infectiousness of a household is (31)where μ_i is the expectation of m_i under the distribution f (X), i.e. the expected number of cases in the i^th generation of infection in a typical infected household given by (32)

Let μ = Σ_iμ_i be the average total number of cases in an infected household. The household reproduction number takes an intuitive and well known form derived in [3], [6], expressed in terms of the parameter R_G as follows: (33)i.e. the household reproduction number is the product of the expected number of infections in a household multiplied by the number of people each individual infects out of their household.

The mean household generation time distribution (time for one household infecting the next) is (34)

The mean generation time for households, T_g^*, can be expressed in terms of the individual generation time T_g as (35)

The generation time distribution w ^* (τ ^* ) can be used for the previously defined estimators of reproduction numbers (7)–(12) using household incidence data or just exponential growth rates. The exponential growth rate r for an exponentially growing epidemic is the same whether measured for individual or household incidence. For an exponentially growing or declining epidemic, one obtains the estimator (36)

Now consider the integration (37)where the first equality uses the definition of the auto-convolution, the second is a re-ordering of integrals, the third involves changing variables to u = τ ^*-s, the fourth is a factorisation and the fifth arises by induction. The sixth uses the definition of the individual reproduction number R (r ) one obtains ignoring household structure from equation (12). The household reproduction number can be expressed in terms of the individual reproduction number R (r ) as (38)

Examination of equation (33) immediately reveals that the estimate for the number of people each person infects out of the household is (39)

I have thus derived a simple analytic relation between the individual and household reproduction numbers. Both are approximations, ignoring the effects of local saturation on the generation time, which will tend to produce overestimates of the reproduction number. An alternative approximating to the household reproduction number, which provides an underestimate, is found when all secondary household cases are assumed to arise in the second generation, i.e. using equation (38) but substituting μ₁→μ′₁ = μ₁≡1, μ₂→μ′₂ = Σ^∞_i = 2μi and μ_i→μ′_i = 0 for i>2.

Application to influenza transmission

There are two reasons for considering household structure in analysing the pandemic influenza situation. First, influenza transmission is known to be concentrated within the household, and thus parameter estimates which ignore this heterogeneity are likely to be frail. Second, many public health policies for future pandemics are likely to be organised around the household. The net effect of social distance measures such as school and workplace closures and cancellation of social gatherings is effectively to reduce transmission out of households (and perhaps inadvertently to increase transmission within them). Furthermore, antiviral treatment and prophylaxis and quarantine measures are likely to be targeted at whole households rather than individuals (though restricting families with one suspect case to stay together without any other support is possibly undesirable) [16], [17], [20].

A number of studies have identified the parameters needed to estimate the household reproduction number for influenza [8], [10], [11], [17]. It is important to bear in mind that these parameters could be quite different in future pandemics, and thus that robust methodology may be more useful in responding to new outbreaks than numerical estimates obtained for past outbreaks. While it would be straightforward to use demographic data and exponential growth rates from earlier pandemics combined with inter-pandemic data on the transmissibility of influenza within households to obtain estimates of R ^* for historical pandemics, it has not been shown that the within household transmission parameters for inter-pandemic influenza adequately describe the pandemic situation, so I focus instead on providing illustrative examples using current demographic data (on the household size distribution from the UK) [34], and recent data on the transmissibility of influenza in modern households [10].

The household size data from 2001is truncated to size 6, and I assume that all households of size 6 or greater have size exactly 6. The data are k₁ = 29% (i.e. 29% of households are single person households), k₂ = 35%, k₃ = 16%, k₄ = 14%, k₅ = 5% and k₆ = 2%. The size of the mean household is thus 2.38 (average size of households where households are sampled at random), while the household of the mean individual has size 3.06 (average size of household to which individuals belong, where individuals are sampled at random).

From the French influenza study [10], I obtain maximum likelihood estimates of the within household transmission parameter of ρ_n = 1.35/n^1.0 (which is consistent with the best fit to the Tecumseh data [8] of ρ_n = 1.27/n^0.97). The former study followed seronegative households for a two week winter outbreak of seasonal influenza. The corresponding escape probabilities are Q₂ = 50.9% (i.e. the probability of not being infected by the other household member in a household of size two is 50.9%), Q₃ = 63.8%, Q₄ = 71.4%, Q₅ = 76.4% and Q₆ = 79.9%. On the scale of other infections, this places influenza as being approximately as infectious as mumps, but a lot less infectious than either varicella-zoster or measles [1]. By applying the Reed-Frost model to these data with this distribution of households, I obtain estimates of the average number of infections in each generation of infection of μ₁ ° 1, μ₂ = 0.64 (i.e. the first index case directly infects an average of 0.64 people in his or her household), μ₃ = 0.19, μ₄ = 0.036, μ₅ = 0.0037 and μ₆ = 0.00021, and thus the estimate for the total expected number of cases in an infected household is μ = Σ⁶_i = 1μ_i = 1.87, to be compared to the mean size of 3.06. These calculations are performed in Microsoft Excel 2007 using equation (25).

There is not yet a consensus on the generation time of influenza [13], [14], [16], [30], [35], with estimates ranging from 2.6 days in [13] to 5.3 days in [14]. I use a Gamma distribution with mean T_g = 2.85 days and standard deviation 0.93 days, as reported in [30].

Based on these data, I compare the predicted and simulated infectiousness of households in Fig. 2, which shows the average over all households sizes and compares this to the final analytical approximation given by equation (31) for β^* (τ ^*), and also the alternate approximation which considers all secondary infections to arise in the second generation of infection ; the simulations and the first approximation are clearly in good agreement. Individual based stochastic simulations were programmed using Berkeley Madonna, and are described in Appendix S1.

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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. R_G = 1).

https://doi.org/10.1371/journal.pone.0000758.g002

For the case of an exponentially growing epidemic, the estimates of the individual and household reproduction numbers, R and R ^* respectively, are shown in Figure 3, along with the estimate of the number of people one person infects outside their household, R_G. For R ^*, both the under- and over-estimating approximations are shown, along with estimates obtained from the simulated generation time distribution. As expected for this low-infectiousness scenario, the simulated values are closer to the over-estimating approximation. The range between these approximations which bracket the true value is rather narrow, indicating that the method is predictive.

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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 R_G, the average number of people each person infects outside his or her home, is also shown.

https://doi.org/10.1371/journal.pone.0000758.g003

For the 1918 “Spanish Flu” H1N1 pandemic, the median growth rate in large US cities was r = 0.20 per day [30], [35], with comparable estimates in the UK [17]. This value also serves as an upper estimate for the spread of the H2N2 pandemic virus in 1957 [17]. Based on this growth rate, the estimated individual reproduction number is R = 1.74, while the estimated household reproduction number is R ^* = 2.26, and thus the out-of-household reproduction number is R_G = 1.21. Of course, households were bigger in 1918 than now, so that the actual value of R^* was likely higher than this. These estimates would imply that a proportion 1−1/R ^* = 56% of between household transmission would need to be blocked to prevent epidemic spread. Figure 3 could provide a rough guide to the likely values of R^* and R_G for a new influenza pandemic where the rate of exponential growth can reliably be determined.

Consider someone who the index case in their household; they would be expected to infect R_G = 1.21 people out of their household and μ₂ = 0.64 within their household. This validates assumed proportions of transmission within and between households from earlier simulation studies [17], [20]. The sum of these is greater than R since the reproduction number R is an average over different generations of infection within the household. For this value, the estimate of R which takes into account local saturation effects was determined numerically to be R = 1.79. Fig 3 shows that for all values of r, numerically estimated values for R (r ) are close to the curve estimated from application of equation (12) which ignores local saturation effects.

As a final check of the method, epidemics within a community of 2,000 households were simulated using an individual based stochastic model (see Appendix S1). I choose R_G = 1.21 as inferred from an epidemic growth rate of r = 0.20 per day, and the other parameters as described above. The exponential rate of growth was then re-estimated directly from the simulated incidence time-series to be r = 0.19 (Figure 4), close to the predicted value of r = 0.20. This provides further support for the validity of this method, especially since no restrictions were placed on re-infection of households within this small simulated community.

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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.

https://doi.org/10.1371/journal.pone.0000758.g004

Application to measles transmission

As noted above, influenza is relatively uninfectious compared to other common viruses. For a contrasting application of the method, I now focus on measles which was the most infectious of the pathogens studied in [1]. Measles also has a more peaked generation time distribution, so that generations of infection are more distinct, and to make the contrast with the influenza estimates yet greater, I also use demographic data on household size chosen from the national census in 1961, when household sizes were greater than they are now. This analysis is perhaps a little artificial when applied to measles, since a large proportion of the population will have immunity either due to past infection or vaccination with the live MMR vaccine. The principal motivation is to further test and illustrate the methods in a case where good data on the transmission dynamics within households are available. Stratification by household of the recent outbreaks of measles caused by decreasing uptake of the MMR vaccine could reveal whether household heterogeneities should have be accounted for in estimating the changing reproduction number of measles [36].

The household size data from 1961 is truncated to size 6, and I assume that all households of size 6 or greater have size exactly 6. The data are k₁ = 14% (i.e. 14% of households are single person households), k₂ = 30%, k₃ = 23%, k₄ = 18%, k₅ = 9% and k₆ = 7%. The size of the mean household is thus 2.99 (average size of households where households are sampled at random), while the household of the mean individual has size 3.66 (average size of household to which individuals belong, where individuals are sampled at random).

Hope-Simpson reported susceptible-infectious escape probabilities of Q = 69.9% for mumps, Q = 39% for varicella, and Q = 24.4% for measles in under 15s [1]. The results were reported independent of household size, and were regarded as unreliable in over-15s. Based on applying the Reed-Frost model to the measles estimate with this distribution of households, I obtain estimates of the average number of infections in each generation of infection of μ₁ ° 1, μ₂ = 2.01 (i.e. the first index case directly infects an average of 2.01 people in his or her household), μ₃ = 0.50, μ₄ = 0.020, μ₅ = 0.00036 and μ₆ = 0.0000031, and thus the estimate for the total expected number of cases in an infected household is μ = Σ⁶_i = 1μ_i = 3.54, to be compared to the mean size of 3.66.

Hope-Simpson also reported the intervals between linked cases in households using different case definitions [1]; the intervals for what he regarded as the most reliable case definition, “maximum rash”. These data is well described by a Gamma distribution (not shown). The maximum likelihood estimate of the generation time is T_g = 10.5 days with standard deviation 2.4 days.

Based on these data, I repeat the simulations of the previous section on influenza but with parameters for measles in Figs 2, 3 and 4. Figure 2B shows that, as expected, the average infectiousness of a household is less well approximated by either approximation than for the much less infectious case of influenza. In this case, multiple peaks of infectiousness corresponding to generations of infection within the household can be clearly distinguished, and there are more cases in the second generation of infection than in the first.

In terms of the predicting of the household reproduction number R^*, the method is still found to be strongly predictive (as evidence by the small gap between upper and lower estimate) and reliable (compared to numerical estimates). While in influenza, the simulations were close to the upper approximation, here they are closer to the lower approximation, as expected for the more infectious situation of measles transmission. Simulations of transmission within a community of households were again found in Figure 4B to validate the approach. The difference in the shape of the epidemic curve with influenza reflects the different shape of the generation time distribution, though the exponential growth rate is the same.