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eAppendix for Quantifying the impact of school openings on the reproductive number of the 2009 A/H1N1 influenza epidemic in the United States
Karen E. Huang, Marc Lipsitch, Jeffrey Shaman, Edward Goldstein
S1. More on incidence, reproductive numbers, measurement errors and confidence intervals: We use weekly state-specific surveillance data on outpatient consultations for influenza-like illness, and virological testing of respiratory specimens collected by the US CDC EMBED Equation.DSMT4 . These data were provided to us by Dr. Lyn Finelli, Lead of the Surveillance and Outbreak Response Team, Influenza Division, National Center for Immunization and Respiratory Diseases.
Given that some of the virological data positive for influenza A is un-subtyped, the weekly share of un-subtyped influenza A specimens EMBED Equation.DSMT4 attributed to influenza A(H1N1)pdm is defined to be proportional to the share of A(H1N1)pdm among the sub-typed specimens:
EMBED Equation.DSMT4
The total weekly number of A(H1N1)pdm specimens is therefore
EMBED Equation.DSMT4
The weekly incidence proxy EMBED Equation.DSMT4 for A(H1N1)pdm on week EMBED Equation.DSMT4 is defined as the percent of outpatient consultations in the surveillance data which are for influenza-like illness times the number of weekly A(H1N1)pdm specimens (described above) divided by the total number of tested weekly specimens. This incidence proxy is related to the daily growth (decline) rate EMBED Equation.DSMT4 of incidence during week EMBED Equation.DSMT4 by
EMBED Equation.DSMT4 (S1)
This definition assumes that the daily growth rate during week EMBED Equation.DSMT4 is constant.
The latter assumption also allows one to express the reproductive number EMBED Equation.DSMT4 during week EMBED Equation.DSMT4 (defined as the average number of infections causes by an individual who got infected during week EMBED Equation.DSMT4 ) as follows: Suppose that incidence of influenza on day EMBED Equation.DSMT4 is EMBED Equation.DSMT4 . The assumption of a constant growth rate suggests that for days EMBED Equation.DSMT4 and EMBED Equation.DSMT4 , EMBED Equation.DSMT4 . Incidence EMBED Equation.DSMT4 is caused by individuals EMBED Equation.DSMT4 infected on days EMBED Equation.DSMT4 , with the incidence on day EMBED Equation.DSMT4 caused by EMBED Equation.DSMT4 being EMBED Equation.DSMT4 , where EMBED Equation.DSMT4 is the serial interval. Thus, assuming a serial interval supported over a 7-day period
EMBED Equation.DSMT4
and hence
EMBED Equation.DSMT4 (S2)
We note that that there is measurement error associated with the estimation of EMBED Equation.DSMT4 , and hence the outcomes for the regression model in equation (2), and those measurement errors are state-specific and cannot be readily incorporated into a standard regression model. Those measurement errors come from the weekly estimates of the state-wide percentages of outpatient consultations for influenza-like illness (measured from count EMBED Equation.DSMT4 among EMBED Equation.DSMT4 patients in the surveillance system), percentages of respiratory specimens positive for influenza A and percentages of influenza A specimens attributable to influenza A(H1N1)pdm.
We incorporate the above measurement error into inference by generating a random sample from the distribution of the effect EMBED Equation.DSMT4 of school opening on the ln(reproductive number) via the following principle: for two random variables EMBED Equation.DSMT4 and EMBED Equation.DSMT4 , to generate a random sample EMBED Equation.DSMT4 for EMBED Equation.DSMT4 , we can first generate a random sample EMBED Equation.DSMT4 for EMBED Equation.DSMT4 and then generate EMBED Equation.DSMT4 from the conditional distribution of EMBED Equation.DSMT4 . Thus we generate a random sample for EMBED Equation.DSMT4 in two steps:
1. We generate a posterior estimate of the true weekly state-specific proportions EMBED Equation.DSMT4 of medical consultations for influenza-like illness among all outpatient consultations in the population using the observed counts EMBED Equation.DSMT4 for influenza-like illness patients among EMBED Equation.DSMT4 visits to providers in the sentinel surveillance system for each state EMBED Equation.DSMT4 /week EMBED Equation.DSMT4 , assuming that influenza-like illness cases represent a binomial sample among the sentinel surveillance system visitors, with the posterior distribution (under a flat prior) given by
EMBED Equation.DSMT4
Similarly, we generate posterior samples for the weekly state-specific percentages EMBED Equation.DSMT4 of respiratory specimens positive for influenza A, and the percentages EMBED Equation.DSMT4 of influenza A specimens attributable to influenza A(H1N1)pdm. We combine those to form a sample EMBED Equation.DSMT4 ( EMBED Equation.DSMT4 ) of week/state specific collections of those quantities.
2. For each collection EMBED Equation.DSMT4 above, the effect EMBED Equation.DSMT4 of school opening on the ln(reproductive number) is estimated as an intercept in the corresponding linear regression model. Thus
EMBED Equation.DSMT4 (S3)
Here EMBED Equation.DSMT4 is the estimate of the intercept in the regression model, EMBED Equation.DSMT4 is the estimate of the standard deviation of the intercept, and EMBED Equation.DSMT4 is a t-distribution with the same number of degrees of freedom as the regression model ADDIN EN.CITE Weisberg200588(2)88886Weisberg, SanfordApplied linear regressionWiley series in probability and statisticsxvi, 310 p.3rdRegression analysis.2005Hoboken, N.J.Wiley-Interscience0471663794 (acid-free paper)13613080Jefferson or Adams Building Reading Rooms - STORED OFFSITE QA278.2; .W44 2005Table of contents only http://www.loc.gov/catdir/toc/wiley051/2004050920.htmlContributor biographical information http://www.loc.gov/catdir/enhancements/fy0618/2004050920-b.htmlPublisher description http://www.loc.gov/catdir/enhancements/fy0618/2004050920-d.html( HYPERLINK \l "_ENREF_2" \o "Weisberg, 2005 #88" 2). We generate a value EMBED Equation.DSMT4 from the distribution in eq. S3.
Combining these two steps, we obtain a random sample EMBED Equation.DSMT4 for the effect of school opening on the reproductive number. The mean and the 95% credible interval for that sample are reported for each choice of a model in Table 1 in the main text. The effect of the humidity covariate is estimated in a similar manner.
S2. Sensitivity with regard to the choice of the serial interval
Our results in the main body of the text utilize the serial interval estimate for the 2009 A(H1N1)pdm obtained in EMBED Equation.DSMT4 . We note that the results should be sensitive to the length of the serial interval, with longer serial intervals resulting in larger departures from 1 for the estimates of the reproductive numbers given by equation (2), and larger values for the difference in the (logarithms of) the reproductive numbers before and after school openings.
The serial interval estimate we utilize has a mean EMBED Equation.DSMT4 days (and standard deviation EMBED Equation.DSMT4 days). Several other estimates of the serial interval for 2009 A(H1N1)pdm are available EMBED Equation.DSMT4 with mean ranging between 2.5 and 3 days. To explore the sensitivity of our estimates to the choice of serial interval, we proceed as follows: Let EMBED Equation.DSMT4 be the serial interval in EMBED Equation.DSMT4 (whose distribution is given by EMBED Equation.DSMT4 ). We consider the following family of serial intervals
EMBED Equation.DSMT4 (S4)
whose mean/standard deviation is
EMBED Equation.DSMT4 , EMBED Equation.DSMT4
We consider the following means/standard deviations for the serial interval given by eq. S4: EMBED Equation.DSMT4 and EMBED Equation.DSMT4 . Table S1 documents the estimated effect of school opening on the reproductive number for the corresponding 25 choices of serial interval mean and standard deviation.
Mean\SD 0.95 1.15 1.35 1.55 1.752.45 17.6% (9.4%,26.5%) 17.53%
(9.3%,26.3%) 17.37%
(9.2%,26.1%) 17.19%
(9%25.6%) 16.91% (8.9%,25.5%)2.6 18.83%
(10.1%,28.1%) 18.65% (9.6%,28.1%) 18.44%
(9.6%,27.5%) 18.3% (9.6%,27.4%) 18.08% (9.6%,27.2%)2.75 19.96% (10.5%,30%) 19.88% (10.4%,30%) 19.79% (10.6%,29.5%) 19.62% (10.6%,29.4%) 19.4% (10.1%,30%)2.9 21.28% (11.2%,32.3%) 21.13% (11.1%,32%) 20.97% (11.1%,31.7%) 20.75% (10.9%,31.3%) 20.6% (10.8%,30.8%)3.05 22.43% (11.7%,34.2%) 22.32% (11.8%,33.5%) 22.11% (11.4%,33.9%) 22.02% (11.6%,33.3%) 21.8% (11.4%,32.6%)
Table S1: Sensitivity of the estimated impact of school openings on the reproductive number with respect to the serial interval mean and standard deviation. Serial intervals considered are described by equation (S4).
We see that, as expected, the estimates increase with increasing means of the serial interval. Moreover, there is less sensitivity with respect to the standard deviation of the serial interval, with slight decreases in estimates as the standard deviation increases.
S3. Analysis with a three week window around school openings: In our analysis in the main body of the text, we have considered the increments in the natural logarithm of the epidemics reproductive number for a pair of weeks prior and following school opening (Eq. (3)) under the premise that such increments are largely attributed to the effect of school opening if the weeks in the considered pair are sufficiently close to each other. We have attempted to restrict the weeks used for inference to be as close to the opening week as possible for at least two additional reasons:
To avoid going deeper into the summer when potential differences in mixing patterns exist.
To avoid temporary peaks in incidence proxy 1-3 weeks after school opening, as were observed in some states, possibly caused by local depletion of susceptibles in places where incidence was higher than average when school opened. Reproductive numbers estimated during those periods would not represent the reproductive number immediately following school opening.
Here we present the results of the corresponding analysis for a three-week window around school openings. More precisely, for each state appearing in the two-week analysis in the main body of the text, we consider the following quantity
EMBED Equation.DSMT4
Those quantities are then regressed either against an intercept, or against an intercept and a humidity covariate, whose value is given by
EMBED Equation.DSMT4
(compare with Eqs. (3) and (4)), with the estimation accounting for measurement error in the outcomes performed as described in section S1.
EMBED Equation.DSMT4 (cutoff) 15 20 25 30 EMBED Equation.DSMT4 increase 17.59 % (9.3%,26.6%) 16.91% (7.6%,26.9%) 17.97% (6%,30.8%) 17.97% (6%,30.8%) EMBED Equation.DSMT4 increase after adjusting for humidity 19.0% (9.0%,30%) 18.38% (7.8%,30%) 19.28% (5.9%,34.4%) 19.28% (5.9%,34.4%)Average humidity covariate contribution -1.27%
(-5.4%,3.1%) -1.27%
(-5.3%,3%) -1.1%
(-4.5%,2.5%) -1.1%
(-4.5%,2.5%)No. of states included 10 9 7 7
Table S2: Increase in the effective reproductive number associated with school opening for the 3-week analysis.
We see that the estimates for the effect of school opening on the reproductive number are similar though somewhat lower than the corresponding estimates in the main body of the text.
ADDIN EN.REFLIST 1. US CDC. Overview of Influenza Surveillance in the United States [Internet]. 2013. Available from: http://www.cdc.gov/flu/weekly/overview.htm.
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3. Cauchemez S, Donnelly CA, Reed C, Ghani AC, Fraser C, Kent CK, et al. Household transmission of 2009 pandemic influenza A (H1N1) virus in the United States. The New England journal of medicine. 2009 Dec 31;361(27):2619-27.
4. Yang Y, Sugimoto JD, Halloran ME, Basta NE, Chao DL, Matrajt L, et al. The transmissibility and control of pandemic influenza A (H1N1) virus. Science. 2009 Oct 30;326(5953):729-33.
5. Donnelly CA, Finelli L, Cauchemez S, Olsen SJ, Doshi S, Jackson ML, et al. Serial intervals and the temporal distribution of secondary infections within households of 2009 pandemic influenza A (H1N1): implications for influenza control recommendations. Clinical infectious diseases 2011;52 Suppl 1:S123-30.
6. White LF, Wallinga J, Finelli L, Reed C, Riley S, Lipsitch M, et al. Estimation of the reproductive number and the serial interval in early phase of the 2009 influenza A/H1N1 pandemic in the USA. Influenza and other respiratory viruses. 2009; 3(6):267-76.
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