Post-peak seroepidemiological studies of pandemic influenza (H1N1-2009) among a general population.

†<p>Subjects, sample size and sampling period refer to those after observing the peak incidence of H1N1-2009. For several studies examining pre-existing immunity, the same or additional samples before the 2009 pandemic were investigated at different time periods, but are not included in this Table.</p>‡<p>Estimated proportions seropositive before and after observing an epidemic peak. When age-standardized estimate was given in the original study, we used it as the population mean.</p><p>*Three studies did not estimate the proportion seropositive before the 2009 pandemic, and we assume that 7.5% of the population was initially immune based on a crude average among other studies.</p>§<p>After peak column represents if the sampling took place longer than 1 month after observing the highest incidence of cases.</p>¥<p>Vaccination column represents if a population-wide vaccination campaign of H1N1-2009 took place prior to the sampling.</p>¶<p>Laboratory methods to determine seropositivity; HI, hemagglutination inbibition assay and MN, microneutralization assay.</p

Distribution of the number of days between events in the USA Today 2014 to October 2017 mass killing sample, binned into integer days.

<p>Overlaid in red is the prediction of the Lankford & Tomek (2017) Exponential null hypothesis model, as fit to the 2006–2013 data and extrapolated to this sample. Overlaid in green is the extrapolated prediction of the Towers <i>et al</i> (2015) contagion model. The contagion model again does a significantly better job of describing the excess of events occurring a few days after a mass killing.</p

Distribution of the number of days between events in the USA Today 2006–2013 mass killing sample, binned into integer days.

<p>Overlaid in red is the prediction of the Lankford & Tomek (2017) Exponential null hypothesis model. Overlaid in green is the prediction of the Towers <i>et al</i> (2015) contagion model. While both models have similar predicted means and variances, and give similar predictions for the fraction of events occurring within 14 days, the contagion model does a significantly better job of describing the excess of events occurring a few days after a mass killing.</p

Illustrative example showing how two distributions can have exactly the same mean and variance, but can be quite different in shape.

<p>In red is the Exponential distribution with mean and variance equal to 14 days, and in blue is the Log-Normal distribution with exactly the same mean and variance. This underlines why simply testing the mean and variance of a distribution is not necessarily sensitive to differing distribution hypotheses. In addition, even though the shapes of these particular distributions are quite different, the fraction of events falling within 14 days happens to be quite similar; 65% for the Exponential, and 68% for the Log-Normal. This demonstrates how coarse binning of data reduces sensitivity to the shape of distributions.</p

The time difference between events, based on 1,000 Monte Carlo simulations of the eight year span of the USA Today mass killing data set.

<p>In red is the prediction of the Exponential null hypothesis model of Lankford & Tomek (2017), and in green is the self-excitation contagion model of Towers <i>et al</i> (2015). These simulated data highlight the difference in shapes of the distributions hypothesized in the two analyses. Despite the differences in shape, however, the means of the two distributions are the same, and the fraction of events occurring within 14 days of a previous event is only 2% different.</p

Minimum sample sizes required for post-epidemic seroepidemiological studies of final size as a function of the margin error, the reproduction number, and the coefficient of variation of the generation time.

<p>(A & B) Sample size with three different reproduction numbers as a function of the margin of error. (A) employs an estimation formula based Type I error alone (at <i>α</i> = 0.05), while (B) accounts for both Type I and II errors (at <i>α</i> = 0.05 and 1−<i>β</i> = 0.80). The margin of error represents random sampling error, around which the reported percentage would include the true percentage. Since (A) is a special case of (B) (with <i>β</i> = 0.50), <i>R</i> = 1.40 in (A) is also shown as dotted line in (B). The coefficient of variation (CV) of the generation time and the proportion of population with pre-existing immunity are fixed at 40.7% and 7.5%, respectively. (C & D) Sample size with three different coefficients of variation as a function of the margin of error. (C) accounts for Type I error alone (<i>α</i> = 0.05), while (D) accounts for both Type I and II errors (<i>α</i> = 0.05 and 1−<i>β</i> = 0.80). The reproduction number and the proportion of population with pre-existing immunity are fixed at 1.40 and 7.5%, respectively. CV = 0 corresponds to a constant generation time, whereas CV = 1 represents an exponentially distributed generation time. In (B) and (D), several lines are truncated, due to impossibility to account for larger margins of error in the estimation formula.</p

Sensitivity of minimum sample size for post-epidemic seroepidemiological studies to the reproduction number and the proportion of population with pre-existing immunity.

<p>(A). The minimum sample size with three different coefficients of variation (CVs) as a function of the reproduction number. (B). The minimum sample size with three CVs as a function of the proportion of population with pre-existing immunity. In (A), the proportion of population with pre-existing immunity is fixed at 0, and the estimates correspond to the margin of error of 10% and Type I and II errors at <i>α</i> = 0.05 and 1−<i>β</i> = 0.50, respectively. In (B), the reproduction number is fixed at 1.40, and the estimates correspond to the margin of error of 10% and Type I and II errors at <i>α</i> = 0.05 and 1−<i>β</i> = 0.50, respectively.</p

Relationship of state prevalence of firearm ownership, mental illness, and state rankings of strength of firearm legislation, to the state incidence of mass killings, school shootings, and mass shootings.

<p>Correlations with ∣<i>ρ</i>∣ ≥ 0.28 are significant to <i>p</i> < 0.05.</p

Results of fits of the self-excitation contagion model in Eq 4 to the various data samples considered in these studies, using a running mean calculation of <i>N</i>₀(<i>t</i>).

<p><i>N</i>_secondary is the average number of new incidents. The p-value is obtained from the likelihood ratio test comparing the likelihood of the full contagion model to the likelihood of the null hypothesis model of no contagion. The numbers in the square brackets indicate the 95% confidence interval on the parameter.</p

Contagion in Mass Killings and School Shootings

<div><p>Background</p><p>Several past studies have found that media reports of suicides and homicides appear to subsequently increase the incidence of similar events in the community, apparently due to the coverage planting the seeds of ideation in at-risk individuals to commit similar acts.</p><p>Methods</p><p>Here we explore whether or not contagion is evident in more high-profile incidents, such as school shootings and mass killings (incidents with four or more people killed). We fit a contagion model to recent data sets related to such incidents in the US, with terms that take into account the fact that a school shooting or mass murder may temporarily increase the probability of a similar event in the immediate future, by assuming an exponential decay in contagiousness after an event.</p><p>Conclusions</p><p>We find significant evidence that mass killings involving firearms are incented by similar events in the immediate past. On average, this temporary increase in probability lasts 13 days, and each incident incites at least 0.30 new incidents (<i>p</i> = 0.0015). We also find significant evidence of contagion in school shootings, for which an incident is contagious for an average of 13 days, and incites an average of at least 0.22 new incidents (<i>p</i> = 0.0001). All <i>p</i>-values are assessed based on a likelihood ratio test comparing the likelihood of a contagion model to that of a null model with no contagion. On average, mass killings involving firearms occur approximately every two weeks in the US, while school shootings occur on average monthly. We find that state prevalence of firearm ownership is significantly associated with the state incidence of mass killings with firearms, school shootings, and mass shootings.</p></div