Household transmission of influenza A and B within a prospective cohort during the 2013‐2014 and 2014‐2015 seasons

Abstract

People living within the same household as someone ill with influenza are at increased risk of infection. Here, we use Markov chain Monte Carlo methods to partition the hazard of influenza illness within a cohort into the hazard from the community and the hazard from the household. During the 2013‐2014 influenza season, 49 (4.7%) of the 1044 people enrolled in a community surveillance cohort had an acute respiratory illness (ARI) attributable to influenza. During the 2014‐2015 influenza season, 50 (4.7%) of the 1063 people in the cohort had an ARI attributable to influenza. The secondary attack rate from a household member was 2.3% for influenza A (H1) during 2013‐2014, 5.3% for influenza B during 2013‐2014, and 7.6% for influenza A (H3) during 2014‐2015. Living in a household with a person ill with influenza increased the risk of an ARI attributable to influenza up to 350%, depending on the season and the influenza virus circulating within the household.

1. INTRODUCTION

In accepting the family as a semiclosed group one must believe that infections in successive family members within a short period of time are more likely to be related to each other than to have been separately acquired from outside sources. This is essentially an act of faith and nearly impossible to establish in terms of exact probability. –Carol Buck, 1956 ¹

For almost a century, researchers have prospectively followed households to elucidate many of the intricacies of the natural history of influenza and other respiratory illnesses: the range of severity, the collection of symptoms, the diversity of etiologies, the spatiotemporal trends, the incidence within subgroups, and the transmission within the household. ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ , ⁹ , ¹⁰ , ¹¹ , ¹² , ¹³ , ¹⁴ , ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ , ²⁰ , ²¹ , ²² , ²³ , ²⁴ , ²⁵ , ²⁶ , ²⁷ , ²⁸ , ²⁹ , ³⁰ , ³¹ , ³² , ³³ , ³⁴ , ³⁵ , ³⁶ , ³⁷ , ³⁸ , ³⁹ , ⁴⁰ , ⁴¹ , ⁴² , ⁴³ , ⁴⁴ , ⁴⁵ , ⁴⁶ , ⁴⁷ , ⁴⁸ , ⁴⁹ , ⁵⁰ , ⁵¹ , ⁵² , ⁵³ , ⁵⁴ , ⁵⁵ , ⁵⁶ , ⁵⁷ , ⁵⁸ , ⁵⁹ , ⁶⁰ , ⁶¹ , ⁶² , ⁶³ , ⁶⁴ Additionally, case‐ascertained household studies have sought to also answer similar questions about influenza by following the household contacts of a case series. ⁶⁵ , ⁶⁶ , ⁶⁷ , ⁶⁸ , ⁶⁹ , ⁷⁰ , ⁷¹ , ⁷² , ⁷³ , ⁷⁴ , ⁷⁵ , ⁷⁶ , ⁷⁷ , ⁷⁸ , ⁷⁹ , ⁸⁰ , ⁸¹ , ⁸² , ⁸³ , ⁸⁴ , ⁸⁵ , ⁸⁶ , ⁸⁷ , ⁸⁸ , ⁸⁹ , ⁹⁰ , ⁹¹ , ⁹² , ⁹³ , ⁹⁴ , ⁹⁵ , ⁹⁶ , ⁹⁷ , ⁹⁸ , ⁹⁹ , ¹⁰⁰ , ¹⁰¹ , ¹⁰² , ¹⁰³ , ¹⁰⁴ , ¹⁰⁵ , ¹⁰⁶ , ¹⁰⁷ , ¹⁰⁸ , ¹⁰⁹ , ¹¹⁰ , ¹¹¹ , ¹¹² , ¹¹³ , ¹¹⁴ , ¹¹⁵ , ¹¹⁶ , ¹¹⁷ , ¹¹⁸ , ¹¹⁹ , ¹²⁰ , ¹²¹ , ¹²² , ¹²³ , ¹²⁴ , ¹²⁵ , ¹²⁶ , ¹²⁷ , ¹²⁸ , ¹²⁹ , ¹³⁰ , ¹³¹ , ¹³² , ¹³³ To summarize the transmissibility of influenza, studies of both types often report the secondary attack rate (SAR), a descriptive statistic that quantifies the risk of an exposed person becoming ill from an infectious person living within the same household. ¹ , ² , ³ , ⁴ , ⁵ , ⁶ , ⁷ , ⁸ , ⁹ , ¹⁰ , ¹¹ , ¹² , ¹³ , ¹⁴ , ¹⁵ , ¹⁶ , ¹⁷ , ¹⁸ , ¹⁹ , ²⁰ , ²¹ , ²² , ²³ , ²⁴ , ⁶⁵ , ⁶⁶ , ⁶⁷ , ⁶⁸ , ⁶⁹ , ⁷⁰ , ⁷¹ , ⁷² , ⁷³ , ⁷⁴ , ⁷⁵ , ⁷⁶ , ⁷⁷ , ⁷⁸ , ⁷⁹ , ⁸⁰ , ⁸¹ , ⁸² , ⁸³ , ⁸⁴ , ⁸⁵ , ⁸⁶ , ⁸⁷ , ⁸⁸ , ⁸⁹ , ⁹⁰ , ⁹¹ , ⁹² , ⁹³ , ⁹⁴ , ⁹⁵ , ⁹⁶ , ⁹⁷ , ⁹⁸ , ⁹⁹ , ¹⁰⁰ , ¹⁰¹ , ¹⁰² , ¹⁰³ , ¹⁰⁴ , ¹⁰⁵ , ¹⁰⁶ , ¹⁰⁷ , ¹⁰⁸ , ¹⁰⁹ , ¹¹⁰ , ¹¹¹ , ¹¹² , ¹¹³ , ¹¹⁴ , ¹¹⁵ , ¹¹⁶ , ¹¹⁷ , ¹¹⁸ , ¹¹⁹ , ¹²⁰ , ¹²¹ , ¹²² , ¹²³ , ¹²⁴ The SAR is the de facto measure of transmissibility of influenza, and trials of antivirals have used SAR to quantify changes in the transmissibility of influenza attributable to a therapeutic regimen given to infectious household members or a prophylactic regimen given to susceptible household members. ¹³ , ¹⁴ , ²¹ , ⁶⁶ , ⁷¹ , ⁷² , ¹¹¹ Here, we estimate the SAR of influenza A and B virus using data from a prospective cohort in New York City during the 2013‐2014 and 2014‐2015 seasons. ⁵⁵ Counting the number of cases within a household following the index case may overestimate the SAR of seasonal influenza, as household members may be infected outside the home, for example, at work or at school. ³⁷ , ⁷⁹ , ⁸¹ , ⁹³ , ⁹⁶ , ¹³⁴ , ¹³⁵ , ¹³⁶ , ¹³⁷ , ¹³⁸ , ¹³⁹ To avoid overcounting, we fit a compartmental model that quantifies both community transmission and household transmission of influenza.

2. METHODS

The Mobile Surveillance for Acute Respiratory and Influenza like Illness in the Community (MoSAIC) study was a prospective cohort study enrolling households in New York City to investigate the incidence, etiology, and risk factors for acute respiratory infections. ⁵⁵ Inclusion in the study required a household with at least three people, of whom at least 1 was a child under 18 years old. Investigators defined an episode of an acute respiratory illness (ARI) as the presence of at least two of the following symptoms: rhinorrhea, congestion, sore throat, cough, and fever. Investigators also included rhinorrhea among children under 1 year of age as an episode of ARI. Enrollees reported possible ARI within their household via text message. After confirming an episode of ARI over the telephone, researchers collected mid‐turbinate nasal swabs from enrollees with an ARI and tested swabs for a panel of 20 viral and bacterial respiratory pathogens using PCR. ¹⁴⁰ The institutional review boards of the Columbia University Medical Center (New York, NY) and the Centers for Disease Control and Prevention (Atlanta, GA) approved this study.

Here, we investigated a secondary objective of the study: to characterize the transmission of influenza within households enrolled in the study. We defined a case as an enrollee

1. who had a confirmed episode of ARI with onset of symptoms between October 5, 2013 and October 5, 2015 and

2. whose nasal swab was positive for influenza A, influenza A (H1), influenza A (H3), or influenza B by RT‐PCR.

We divided the study period into two influenza seasons. The 2013‐2014 influenza season started October 5, 2013 and ended September 27, 2014, and the 2014‐2015 influenza season started September 28, 2014 and ended October 5, 2015. For the 2013‐2014 and 2014‐2015 seasons, we tabulated the number of households with 0, 1, 2, 3, 4, or 5 cases for

• influenza A (H1);

• influenza A (H3);

• influenza A (H1) or A (H3);

• influenza B; and,

• influenza A (H1), A (H3), or B.

For each of these season‐virus combination, we considered enrollees at risk of influenza if they (1) were enrolled on or before the onset date of the first case for that season and (2) remained enrolled by the onset date of the last case of the season. Because investigators enrolled households into the cohort on a rolling basis, this time frame ensured all cases were included while also including as many households as possible. As this time frame was different for each season‐virus combination, a different number of households was at risk for each analysis. The attack rate of influenza was the proportion of cases among those at risk. We used Wilson's interval to compute 95% confidence intervals for the attack rate. ¹⁴¹

For the primary analysis of household transmission, we only considered pairs of cases both PCR‐positive for the same type or subtype of influenza. As a secondary analysis of household transmission, we included additional episodes of ARI up to 10 days after the onset date of a PCR‐positive case within the household, regardless of the test results for these additional episodes of ARI. The results from this secondary analysis simplify comparison of our results to previous work which used a syndromic case definition for secondary cases. For both analyses, we separately analyzed cases of influenza A (H1), influenza A (H3), and influenza B to reduce spurious household transmission in the data. We computed 3‐, 5‐, and 7‐day SARs by counting the number of apparent secondary cases among those at risk, as follows. For each household, we defined the index case (or index cases) as the enrollee(s) who met the case definition with the earliest onset date within that household. The enrollees at risk of secondary transmission were those enrollees who were not index cases but lived in a household with at least 1 case. We denoted the length of time at risk in days as r and consider $r=3,5,\text{or}7$. For our primary analysis, the r‐day secondary cases were those cases with an onset date at least 1 day and at most r days following the date of onset of the index case within their household. For our secondary analysis, we also counted the episodes of ARI regardless of test result within 10 days of the onset of a case when counting secondary cases. The r‐day SAR was the proportion of r‐day secondary cases among those enrollees at risk of secondary transmission. We used Wilson's interval to compute 95% confidence intervals for the r‐day SAR. ¹⁴¹

We continued our analysis of SAR using Markov chain Monte Carlo methods. We modeled the transmission of influenza within our cohort using a transmission model—built on a susceptible‐infectious‐recovered (SIR) compartmental model—modified from the work of Cauchemez and colleagues. ¹²⁵ Details follow in the remainder of this section.

2.1. Household transmission model

We built a compartmental SIR model for the transmission of influenza into and within a cohort of households. We used the framework that Cauchemez and colleagues developed to analyze data from a case‐ascertained study. ¹²⁵ They assumed the length of the infectious period followed a gamma distribution and imputed the start and end of each case's infectious period. Additionally, they allowed for different susceptibility and infectiousness for children and adults. However, we needed to reduce the number of parameters in our model because cohort data have substantially fewer cases, and we wanted to avoid technical issues when imputing sparse data which leads to implausible results. ¹⁴² So, we assumed the shape parameter of the gamma distribution is 1; the infectious period began with symptom onset; and, we did not estimate separate parameters for children and adults. These simplifications allowed estimation with sparse data from a prospective cohort. In addition to information available from case‐ascertained studies, the prospective cohort has richer information on the hazard from the community over time. So, we also wanted to take advantage of this information in the model.

Now, we make the following assumptions to begin building our model.

• The time to infection follows an exponential distribution with a time‐dependent rate parameter.

• The time from infection to recovery follows an exponential distribution.

• The hazard from the community is proportional to the smoothed hazard within the cohort.

• The hazard within the household is proportional to the percent of household members who are infectious.

• The overall hazard is the sum of the community hazard and the household hazard.

• The observed onset dates are conditionally independent given the first case of the season, the overall hazard within the cohort, the onset dates of other household members, and the model parameters.

• The prior probability distribution of our parameters follows a uniform distribution on $\left[1\times 10^{-2},1\times 10^3\right]$.

In the remainder of this subsection, we will implement these assumptions to build a posterior, and we will detail our Metropolis algorithm.

2.1.1. Hazard within the cohort

Here, we construct smoothed estimates of the hazard within the cohort for use in our model. In the next subsection, we will use these smoothed estimates when estimating the hazard within each household.

We take the earliest onset date of the season among laboratory confirmed cases as $t=1$ and the last onset date of the season as day $t=T$. Then, we compute the Kaplan‐Meier estimates of survival $Ŝ(t)$ for $t=0,1,\dots ,T$: the probability of not meeting the case definition before time t. ¹⁴³ We estimate the cummulative hazard for $t=0,1,\dots ,T$ with

$$Ĥ(t)=-\log (Ŝ(t)).$$

We estimate the instantaneous hazard for $t=1,2,\dots ,T$ with

$$ĥ(t)=Ĥ(t)-Ĥ(t-1).$$

We use LOESS to smooth our estimates of $ĥ$. ¹⁴⁴ We name these locally smoothed estimates $\tilde{h}$.

2.1.2. Hazard within the household

The time‐dependent hazard within each household underlays the results from the model. Suppose we follow a cohort of households for T days and document the days on which household members first meet the case definition. For an individual, we take the sample space of the first day when a household member meets the case definition to be

$$\mathrm{\Omega }=\{1,2,3,\dots ,T-1,T,\infty \},$$

where individuals who do not meet the case definition during followup are assigned an onset date of $\infty$. Suppose a household has n members. Then, we may write the set of observed onset dates for the household in increasing order as

$$A=\{a_1\le a_2\le \cdots \le a_n\}\in {\mathrm{\Omega }}^n.$$

We write the onset dates in order to simplify our indices. We fit three parameters to our data: $\alpha$, $\beta$, and $\gamma$.

• If h is the instantaneous hazard within the cohort, then $\alpha h$ is the hazard attributable to community transmission.

• If $\pi$ is the proportion of household members who are infectious, then $\beta \pi$ is the hazard attributable to household transmission.

• The rate of recovery among infectious individuals is $\gamma$.

Next, we need an estimate of the proportion of infectious household members at day t. For individual $i=1,\dots ,n$, we define characteristic functions

$${\chi }_i(t)=\left\{\begin{array}{ll}1& \text{if}t>a_i\\ 0& \text{if}t\le a_i\end{array}\right.$$

The characteristic function reflects the fact that an individual is not infectious before their onset date. Using these characteristic functions, we define the expected proportion of infectious people in the household at time t as

$$\pi (t)=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{\chi }_i(t)e^{-\gamma (t-a_i)},$$

where $e^{-\gamma (t-a_i)}$ is the probability that the ith case in the household is still infectious at time t for $t\ge a_i$. Finally, we may define the time‐dependent hazard within the household:

$$\lambda (t)=\alpha \tilde{h}(t)+\beta \pi (t).$$

We summarize the model with Figure 1.

FIGURE 1.

Compartmental SIR model. Susceptible individuals become infectious with a time‐dependent, household‐specific hazard $\lambda (t)$, and recover at a constant rate of $\gamma$

2.1.3. Conditional probability of onset dates

We do not want the first case of the season to contribute to our likelihood. Instead, we want to condition our probability on the first case of the season. We use the symbol $\mathbf{1}$ to denote the fact that the season's index case is $1\in \mathrm{\Omega }$. Recall that $A=\{a_1\le a_2\le \cdots \le a_n\}\in {\mathrm{\Omega }}^n$ are ordered onset dates of a household with n people. Suppose $a\in A$ is the onset date of a household member. Because we want to condition our analysis on the season's index case, if $a=1$, then the household member is an index case for the season. So, if $a=1$, we take $p(a=1|\mathbf{1})=1$. If $2\le a\le T$, we take the conditional probability of onset on day a as

$$p(a|A\setminus \{a\},\alpha \tilde{h},\beta ,\gamma ,\mathbf{1})={\int }_{a-1}^a\lambda (t)e^{-{\int }_0^t\lambda (u)du}dt.$$ (1)

If $a=\infty$, we take the conditional probability of not meeting the case definition during followup as

$$p(a|A\setminus \{a\},\alpha \tilde{h},\beta ,\gamma ,\mathbf{1})={\int }_T^{\infty }\lambda (t)e^{-{\int }_0^t\lambda (u)du}dt.$$ (2)

The integrals in (1) and (2) are integrals of an exponential distribution with a time‐dependent rate parameter $\lambda$. Because $\chi$ is not continuous for cases, $\lambda$ is not continuous for households with at least one case. However, $\lambda$ is continuous between onset dates. So, we want to rewrite the integrals in (1) and (2) so that we may compute them easily. For a household with d unique onset dates, we order the unique onset dates in increasing order $a_{i_1}<a_{i_2}<\cdots <a_{i_d}$. Then, we define a non‐decreasing sequence of times $\{s_{2d+3}\}$ as $\left\{0,a_{i_1}-1,a_{i_1},\dots ,a_{i_d}-1,a_{i_d},T,\infty \right\}$. Now, for every $t\in \{s_{2d+3}\}$ we may compute the integral

$${\int }_0^{t=s_j}\lambda (u)du={\displaystyle \sum _{k=1}^{j-1}}{\int }_{s_k}^{s_{k+1}}\lambda (u)du.$$ (3)

Using (3) allows us to compute the integral in (1) for each onset date after day 1 and the integral in (2) for household members who do not meet the case definition during followup.

We assume conditional independence of onset dates within a household to estimate the conditional probability of onset dates A

$$p(A|\alpha \tilde{h},\beta ,\gamma ,\mathbf{1})={\displaystyle \prod _{l=1}^n}p(a_l|A\setminus \{a_l\},\alpha \tilde{h},\beta ,\gamma ,\mathbf{1}).$$

Similarly, we assume conditional independence of onset dates between households in order to compute the conditional probability of the onset dates within the cohort. We index the households with $1,\dots ,m$. Then, we label the onset dates within the hth household as $A_h=(a_{h1},a_{h2},\dots ,a_{h{n}_h})\in {\mathrm{\Omega }}^{n_h}$. We define the onset dates for the cohort as $𝒜=\{A_1,\dots ,A_m\}$. The overall conditional probability of $𝒜$ is

$$p(𝒜|\alpha \tilde{h},\beta ,\gamma ,\mathbf{1})={\displaystyle \prod _{h=1}^m}{\displaystyle \prod _{l=1}^{n_h}}p(a_{hl}|A_h\setminus \{a_{hl}\},\alpha \tilde{h},\beta ,\gamma ,\mathbf{1}).$$

In summary, we have a conditional probability of the observed onset dates. This is the likelihood we use when estimating our model parameters.

2.1.4. Metropolis algorithm

We use Markov chain Monte Carlo methods to estimate $\alpha ,\beta ,\text{and}\gamma$. Specifically, we use a stepwise Metropolis algorithm. ¹⁴⁵ We use a uniform prior with a large support for our parameters

$$p(\alpha ,\beta ,\gamma )\propto \left\{\begin{array}{ll}1& \text{if}\alpha ,\beta ,\gamma \in [10^{-2},10^3]\\ 0& \text{otherwise}\end{array}\right.$$

Before the algorithm begins, we tune mixing parameters using small chains such that the acceptance of the Metropolis algorithm is about $1/2$. Before the first iteration of the algorithm, we initialize the parameters using random draws from the uniform distribution on $[0.1,1]$. Then, we compute the conditional probability of the onset dates within the cohort $𝒜$ given these parameters and given the date of the first observed case of the season, $p(𝒜|\alpha \tilde{h},\beta ,\gamma ,\mathbf{1})$.

We begin the first iteration of our stepwise algorithm. To begin the first step, we take a random draw from a standard normal distribution and call this value u. Using u and the tuning parameter ${\delta }_{\alpha }$, we propose a new value for $\alpha$ as

$$\bar{\alpha }=\alpha e^{u{\delta }_{\alpha }}.$$

As $p(u)=p(-u)$, we have $p(\alpha |\bar{\alpha })=p(-u)=p(u)=p(\bar{\alpha }|\alpha )$. So, our proposal distribution is symmetric. Next, we decide whether to accept or reject our proposal $\bar{\alpha }$.

• If $p(\bar{\alpha },\beta ,\gamma )=0$, then we reject $\bar{\alpha }$.

• Otherwise, we compute $p(𝒜|\bar{\alpha }\tilde{h},\beta ,\gamma ,\mathbf{1})$.

  • –If $p(𝒜|\bar{\alpha }\tilde{h},\beta ,\gamma ,\mathbf{1})\ge p(𝒜|\alpha \tilde{h},\beta ,\gamma ,\mathbf{1})$, then we accept $\bar{\alpha }$ as the new value of $\alpha$.
  • –
    Otherwise, we accept $\bar{\alpha }$ as the new value of $\alpha$ with probability

    $$\frac{p(𝒜|\bar{\alpha }\tilde{h},\beta ,\gamma ,\mathbf{1})}{p(𝒜|\alpha \tilde{h},\beta ,\gamma ,\mathbf{1})}.$$

This completes the first step. For the second step, we propose $\bar{\beta }$ and either accept or reject using the same method as the first step. For the third step, we propose $\bar{\gamma }$ and either accept or reject using the same method as the first step. This completes the first iteration of the algorithm. The algorithm continues for $10^6$ iterations.

2.1.5. Estimates

We assume the parameter values from the Metropolis algorithm are a sample from the posterior probability distribution with density $p(\alpha ,\beta ,\gamma |𝒜,\mathbf{1})$. For each iterate, we compute the community attack rate (CAR) within the cohort by setting $\beta =0$, and we have

$$1-e^{-{\int }_0^T\alpha \tilde{h}(t)dt}=1-e^{-{\sum }_{t=1}^T\alpha \tilde{h}(t)}.$$

So, the CAR is the probability of influenza from the community during followup when no household members are infectious. We compute the SAR within a household of size n by setting $\alpha =0$, and we have

$$1-e^{-{\int }_0^{\infty }\frac{\beta }{n}{e}^{-\gamma t}dt}=1-e^{-\frac{\beta }{n\gamma }}.$$

So, the SAR is the probability of influenza from a single infectious household member when no community members are infectious.

For each iterate, we compute the relative attack rate of influenza within a household

• with a single infectious member and

• $n-1$ susceptible members

vs. within a household

• without any infectious people and

• of size n

using the formula

$$\text{RR}=\frac{1-e^{-{\int }_0^{\infty }\frac{\beta }{n}{e}^{-\gamma t}dt-{\int }_0^T\alpha \tilde{h}(t)dt}}{1-e^{-{\int }_0^T\alpha \tilde{h}(t)dt}}=\frac{1-e^{-\frac{\beta }{n\gamma }-{\sum }_{t=1}^T\alpha \tilde{h}(t)}}{1-e^{-{\sum }_{t=1}^T\alpha \tilde{h}(t)}}.$$

So, the denominator of our RR is simply the CAR. The numerator of the RR is the risk of influenza: the competing risk from the community and from a single infectious household member.

We use the median value as our point estimate. We create 95% credible intervals with the 2.5% and 97.5% quantiles.

2.2. Computer software

We used R version 3.5.1“Feather Spray” (Copyright 2018 The R Foundation for Statistical Computing) and GCC version 4.9.3 (Copyright 2015 Free Software Foundation, Inc.) for all computations. We used the R package Rcpp to execute our C++ functions within R. ¹⁴⁶

3. RESULTS

The number of households enrolled for our analyses ranged from a low of 230 for influenza A and B in 2013‐2014 to a high of 242 for influenza A (H3) in 2014‐2015 (Table 1). During the 2014‐2015 season, a single enrollee met the case definition twice: once for influenza A (H3) and once for influenza B. Otherwise, enrollees met the case definition at most once per season. The attack rate of influenza A and B was 4.7% during 2013‐2014 and 4.7% during 2014‐2015 seasons (Table 1). Most cases were attributable to community transmission (Tables 1 and 2). Two households had at least 2 cases of influenza A (H1) in 2013‐2014; 8 households had at least 2 cases of influenza A (H3) in 2014‐2015; and 3 household had at least 2 cases of influenza B in 2013‐2014 (Table 1).

TABLE 1.

Number of households with 0, 1, 2, 3, 4, or 5 cases; number of cases; number of people enrolled; and attack rate by virus and season

			Cases in household
Influenza virus	Season	Households enrolled	0	1	2	3	4	5	Cases	People enrolled	Attack rate % (95% CIa)
A and B	2013‐2014	230	189	34	6	1			49	1044	4.7 (3.6, 6.2)
A and B	2014‐2015	238	206	19	10	2	0	1b	50	1063	4.7 (3.6, 6.1)
B	2013‐2014	234	217	14	2	1			21	1062	2.0 (1.3, 3.0)
B	2014‐2015	238	234	4					4	1064	0.4 (0.1, 1.0)
A	2013‐2014	233	207	23	3				29	1067	2.7 (1.9, 3.9)
A	2014‐2015	242	212	18	10	1	1		45	1088	4.1 (3.1, 5.5)
A (H1)	2013‐2014	241	223	16	2				20	1144	1.7 (1.1, 2.7)
A (H1)	2014‐2015	242c	242						0	1088c	0 (0, 0.4)c
A (H3)	2013‐2014	233	225	8					8	1067	0.7 (0.4, 1.5)
A (H3)	2014‐2015	242	213	21	6	1	1		40	1088	3.7 (2.7, 5.0)

${}^{\mathrm{a}}$ Confidence interval.

${}^{\mathrm{b}}$ One enrollee met the case definition twice: once for influenza A (H3) and once for influenza B.

${}^{\mathrm{c}}$ Based on the number of people enrolled in the cohort for the influenza A (H3) analysis during 2014‐2015.

TABLE 2.

Attack rate, community attack rate, and secondary attack rate for seasons and influenza virus with at least one case

Influenza virus	Season	Attack rate % (95% CIc)	Community attack rate % (95% CIb)	Secondary attack rate${}^{\mathrm{a}}$ % (95% CIb)	Relative attack rate${}^{\mathrm{a}}$ RR (95% CIb)
A (H1)	2013‐2014	1.7 (1.1, 2.7)	1.5 (0.9, 2.3)	2.3 (0.3, 7.7)	2.6 (1.2, 6.8)
A (H3)	2013‐2014	0.7 (0.4, 1.5)	0.6 (0.3, 1.2)	0.2 (0.0, 6.8)	1.3 (1.0, 13.3)
A (H3)	2014‐2015	3.7 (2.7, 5.0)	2.7 (1.8, 3.8)	7.6 (3.7 13.3)	3.8 (2.2, 6.5)
B	2013‐2014	2.0 (1.3, 3.0)	1.5 (0.9, 2.4)	5.3 (1.5, 12.7)	4.5 (1.9, 10.9)
B	2014‐2015	0.4 (1.3, 3.0)	0.3 (0.1, 0.7)	0.3 (0.0, 14.3)	2.3 (1.0, 78.9)

Note: The relative attack rate is the attack rate of influenza in households with a single case relative to the attack rate of influenza in households with no cases.

${}^{\mathrm{a}}$ Computed for the average household size.

${}^{\mathrm{b}}$ Credible interval.

${}^{\mathrm{c}}$ Confidence interval.

Secondary attack rates varied depending on the season and the circulating virus type (Figure 2 and Table 2). Similarly, the CAR varied by season and circulating virus type (Figure 3 and Table 2). For a household of average size (4.7 people), the SAR of influenza A (H1) was 2.3% during 2013‐2014 (Table 2). Similarly, the SAR of influenza A (H3) was 7.6% during 2014‐2015 for a household of average size (4.5 people). The SAR of influenza B was 5.3% during 2013‐2014 for a household of average size (4.5 people).

FIGURE 2.

The estimated secondary attack rate by household size for influenza by selected type and subtype during 2013‐2014 and 2014‐2015. Vertical bars represent 95% credible intervals

FIGURE 3.

The estimated community attack rate over the followup period of (A) influenza A (H1) during 2013‐2014, (B) influenza A (H3) during 2014‐2015, and (C) influenza B during 2013‐2014. The black lines represent point estimates, and the shaded areas represent 95% credible regions

For an average‐sized household during 2013‐2014, if a household member became infectious with influenza A (H1), then the attack rate of influenza for the other household members increased 160% (Table 2). Similarly, the attack rate increased by 280% during 2014‐2015 if a household member became infectious with influenza A (H3). The attack rate increased by 350% during 2013‐2014 if a household member became infectious with influenza B.

The estimates of the SAR from the transmission model differed from the 3‐day, 5‐day, and 7‐day r‐day SAR (Table 3). As expected, estimates of SAR from the analysis, which also included episodes of ARI regardless of test result, were higher than when limited to only laboratory‐confirmed cases (Table 4).

TABLE 3.

Comparing secondary attack rate of PCR confirmed influenza from the transmission model with 3‐, 5‐, and 7‐day r‐day secondary attack rate

			r‐day secondary attack rate
Influenza virus	Season	Transmission model secondary attack rate ${}^{\mathrm{a}}$ % (95% CIb)	3‐day % (95% CIc)	5‐day % (95% CIc)	7‐day % (95% CIc)
A (H1)	2013‐2014	2.3 (0.3, 7.7)	2.5 (0.7, 8.8)	2.5 (0.7, 8.8)	2.5 (0.7, 8.8)
A (H3)	2013‐2014	0.2 (0.0, 6.8)	0.0 (0.0, 12.9)	0.0 (0.0, 12.9)	0.0 (0.0, 12.9)
A (H3)	2014‐2015	7.6 (3.7 13.3	7.4 (3.8, 13.9)	9.2 (5.1, 16.2)	9.2 (5.1, 16.2)
B	2013‐2014	5.3 (1.5, 12.7)	3.4 (1.0, 11.7)	5.2 (1.8, 14.1)	6.9 (2.7, 16.4)
B	2014‐2015	0.3 (0.0, 14.3)	0.0 (0.0, 24.2)	0.0 (0.0, 24.2)	0.0 (0.0, 24.2)

${}^{\mathrm{a}}$Computed for the average household size.

${}^{\mathrm{b}}$Credible interval

${}^{\mathrm{c}}$Confidence interval

TABLE 4.

Comparing secondary attack rate of episodes of ARI regardless of test results from the transmission model with 3‐, 5‐, 7‐, and 10‐day r‐day secondary attack rate

			r‐day secondary attack rate
Influenza virus	Season	Transmission model secondary attack rate${}^{\mathrm{a}}$ % (95% CIb)	3‐day % (95% CIc)	5‐day % (95% CIc)	7‐day % (95% CIc)	10‐day % (95% CIc)
A (H1)	2013‐2014	5.7 (2.0, 12.2)	6.3 (2.7, 14.0)	6.3 (2.7, 14.0)	6.3 (2.7, 14.0)	6.3 (2.7, 14.0)
A (H3)	2013‐2014	0.2 (0.0, 6.8)	0.0 (0.0, 12.9)	0.0 (0.0, 12.9)	0.0 (0.0, 12.9)	0 (0.0, 12.9)
A (H3)	2014‐2015	13.5 (8.6, 19.7)	12.0 (7.2, 19.5)	16.7 (10.8, 24.8)	17.6 (11.6, 25.8)	19.4 (13.1, 27.9)
B	2013‐2014	6.5 (2.1, 14.0)	5.2 (1.8, 14.1)	6.9 (2.7, 16.4)	8.6 (3.7, 18.6)	8.6 (3.7, 18.6)
B	2014‐2015	5.6 (0.2, 26.0)	8.3 (1.5, 35.4)	8.3 (1.5, 35.4)	8.3 (1.5, 35.4)	8.3 (1.5, 35.4)

${}^{\mathrm{a}}$Computed for the average household size.

${}^{\mathrm{b}}$Credible interval.

${}^{\mathrm{c}}$Confidence interval.

Estimates of model parameters are in Table 5. The 95% credible intervals of $\beta$ and $\gamma$ were wide, especially for those analyses with no apparent secondary cases. The point estimates and 95% credible intervals for $\alpha$ scale the overall hazard within the cohort, which we present for those analyses with at least one apparent secondary case (Figure 3).

TABLE 5.

Summary of parameter estimates from the household transmission models

Season	Virus	Case definition	$\alpha$ (95% CI)	$\beta$ (95% CI)	$\gamma$ (95% CI)
2013‐2014	A (H1)	PCR confirmed	0.82 (0.48, 1.29)	3.3 (0.04, 139)	33 (0.7, 845)
2013‐2014	A (H3)	PCR confirmed	0.80 (0.33, 1.58)	0.16 (0.01, 39)	59 (0.1, 886)
2014‐2015	A (H1)	PCR confirmed	0.73 (0.49, 1.03)	0.23 (0.08, 0.56)	0.65 (0.29, 1.29)
2013‐2014	B	PCR confirmed	0.73 (0.42, 1.16)	0.06 (0.01, 0.22)	0.24 (0.06, 0.63)
2014‐2015	B	PCR confirmed	0.60 (0.13, 1.63)	0.20 (0.01, 78)	46.9 (0.09, 878))
2013‐2014	A (H1)	ARI episode	0.72 (0.43, 1.11) 11.0	(0.3, 282)	40.5 (1.4, 845)
2013‐2014	A (H3)	ARI episode	0.80 (0.33, 1.58) 0.16	(0.01, 41)	59.3 (0.1, 886)
2014‐2015	A (H1)	ARI episode	0.60 (0.41, 0.83) 0.29	(0.14, 0.54)	0.44 (0.25, 0.74)
2013‐2014	B	ARI episode	0.70 (0.41, 1.11)	0.09 (0.01, 0.29)	0.30 (0.09, 0.71)
2014‐2015	B	ARI episode	0.47 (0.10, 1.29)	4.9 (0.03, 386)	24.3 (0.2, 828)

4. DISCUSSION

We found an attack rate of 4.7% for influenza illness each season and a SAR of 2.3% for influenza A (H1) in 2013‐2014, 5.3% for influenza B in 2013‐2014, and 7.6% for influenza A (H3) 2014‐2015. When the first person in an average sized household was infected with influenza, the attack rate increased up to 350% for the rest of the household, depending on the season and virus circulating in the home. Our results are qualitatively similar to published estimates of the attack rate and SAR of PCR‐confirmed influenza infections from household cohort studies in the United States. The Flu Watch cohort study reported the attack rate was on average 4% per season for the 2006‐2007 through 2008‐2009 seasons. ⁵³ The Household Influenza Vaccine Effectiveness (HIVE) study reported SARs of 2.9% for influenza A (H1), 7.7% for influenza B, and 15.3% for influenza A (H3) during the 2010‐2011 season and reported an attack rate of 8.7%. ¹⁸ For the 2014‐2015 season, the HIVE study reported a SAR of 17% for influenza A (H3), a SAR of 6% for influenza B, and an attack rate of 12%. ²³

When we included episodes of ARI regardless of test results, our estimates of SAR often increased substantially. For example, the estimate of SAR for influenza A (H3) during 2014‐2015 increased from 7.6% to 13.5% by including additional episodes of ARI regardless of test result. While we know that including all episodes of ARI likely overestimates the SAR, the opposite is true when we restricted analysis to laboratory‐confirmed secondary cases. The more inclusive definition of a secondary case allows comparison with previously published work from case ascertained household studies. The earliest reports from the United States on the household transmission of pandemic H1N1 2009 influenza reported a SAR of 13% and 27.3%. ⁷⁶ , ¹⁴⁷ Subsequent reports of estimates of the SAR ranged from 3.5% to 22.8%; and, our analogous estimate of SAR of influenza A (H1) during 2013‐2014 was 5.7%, which is in the middle of this range. ⁸¹ , ⁸² , ⁸⁵ , ⁹³ , ⁹⁴ , ¹⁴⁸

As accurate counts of the number of secondary cases are rarely available, researchers rely on statistical methods to estimate SARs. If we relied only on counting apparent secondary cases, then neither the 3‐, nor 5‐, nor 7‐day SAR could capture quantitatively similar estimates as our method. In effect, shortening the time at risk for these SARs corrects for over counting, but specifying the optimal time interval is difficult. Specifically, the number of uncounted cases attributable to household transmission must equal the number of over counted cases attributable to community transmission. While the 3‐, 5‐, or 7‐day SAR was similar to that from the transmission model for influenza A (H1) during 2013‐2014, neither the 3‐, 5‐, nor 7‐ day SAR was similar to the SAR from the transmission model for both influenza A (H3) during 2014‐2015 and influenza B during 2013‐2014 (Table 3). As risk from the community relative to the risk from the household increases, this bias increases. Using the model‐based approach obviates choosing the appropriate length of time at risk for household transmission, in exchange for a more cumbersome process of estimation. Additionally, the model‐based approach provides credible intervals for the CAR and the SAR, while simultaneously adjusting for uncertainty in both.

Our results may not be generalizable. In order to participate in this study, households must have at least one child. The dynamics of influenza transmission in homes without children may be qualitatively different than what we present here. ⁷⁶ In order to participate in this study, households must have at least 3 people. Smaller households account for about 60% of households in the United States (U.S. Census Bureau, 2012‐2016 American Community Survey 5‐Year Estimates). Our case definition relied on molecular evidence of influenza virus in a nasal swab as detected by RT‐PCR using a benchtop system. ¹⁴⁰ While highly specific, our methods may not have detected every infection with influenza virus among enrollees with an ARI. Our study did not capture asymptomatic infections with influenza, which would violate our assumptions that enrollees who never met the case definition remained susceptible for the entire influenza season. Although we analyzed the data separately for cases of influenza A (H1), influenza A (H3), and influenza B, more than one influenza virus may have circulated in some households. For example, most of the circulating influenza A (H3) viruses belonged to the 3C.2a genetic group during the 2014‐2015 season in New York, but influenza A (H3) viruses from other genetic groups also circulated within the state. ¹⁴⁸

Estimates of the transmissibility of influenza, measured by SAR within a household, ranges widely over recent history. The estimates we present here reinforce the variation of the expected transmissibility of influenza viruses circulating in the population from historical data (Table S3). Future data about this cohort may enable analyses which account for differential susceptibility within the cohort, for example, by extending the compartmental model to allow for immunity. The changing susceptibility of the population to the circulating influenza viruses could explain part of the observed dynamics of the transmissibility of influenza. Quantifying this relationship could strengthen public health messaging about preventing influenza.

CONFLICT OF INTEREST

The authors declare no conflicts of interest.

Supporting information

Data S1: Supporting information

Dahlgren FS, Foppa IM, Stockwell MS, Vargas CY, LaRussa P, Reed C. Household transmission of influenza A and B within a prospective cohort during the 2013‐2014 and 2014‐2015 seasons. Statistics in Medicine. 2021;40(28):6260–6276. 10.1002/sim.9181

DATA AVAILABILITY STATEMENT

The data are not available due to privacy or ethical restrictions.