What a contagion model found when fitted to US mass-shooting data

The idea that publicity around a violent event can raise the chance of another one is old, and for suicide it is reasonably well established. Several studies have found that media reports of suicides and homicides appear to increase the incidence of similar events afterwards, on the reasoning that coverage plants ideation in individuals already at risk.

In 2015 a team led by Sherry Towers, a physicist at Arizona State University, asked whether the same statistical signature shows up in higher-profile incidents: school shootings and mass killings. The paper, Contagion in Mass Killings and School Shootings, was published in PLOS ONE.

What the model does

The method is borrowed directly from infectious-disease epidemiology, and it is worth being precise about what it assumes, because the word “contagion” invites a stronger reading than the mathematics supports.

The team fitted a contagion model to US incident data. The model includes terms allowing that a school shooting or mass murder may temporarily raise the probability of a similar event in the immediate future, and it assumes that this raised contagiousness decays exponentially after an event. Mass killings were defined as incidents with four or more people killed.

What the model tests, then, is a specific alternative to independence. If incidents were independent of one another, a contagion term would add nothing to the fit. If they cluster in time beyond what chance produces, it will.

The numbers

Both parts of the analysis returned a significant contagion term, and both returned the same window.

• Mass killings involving firearms: the temporary rise in probability lasts an average of 13 days, and each incident is estimated to incite at least 0.30 further incidents (p = 0.0015).
• School shootings: contagious for an average of 13 days, inciting at least 0.22 further incidents (p = 0.0001).

The p-values come from a likelihood ratio test comparing the contagion model against a null model with no contagion. That is the right comparison for the question asked, and it is a comparison between two models rather than a demonstration of a mechanism.

For base rates, the paper notes that mass killings involving firearms occur in the United States roughly every two weeks on average, and school shootings roughly monthly.

The gun-ownership finding

Separately from the timing analysis, the study found that state prevalence of firearm ownership was significantly associated with state incidence of mass killings with firearms and with school shootings.

This is a cross-sectional association between states, not a causal estimate, and it is subject to every confounder that distinguishes one US state from another. It is reported here because it is part of the paper’s findings, not because it settles anything.

What “contagious for 13 days” does and does not mean

Read carefully, the result is narrow. It says that in the fortnight following an incident, the modelled probability of another is elevated, and that the elevation is large enough to beat a no-contagion null. It does not identify a route. It does not distinguish media coverage from other channels, and it cannot, because the data are incident dates rather than exposure measurements.

The reproduction figures are also easy to misread. “At least 0.30 new incidents” is a number below one, which in epidemic terms means the process is not self-sustaining: clusters die out rather than growing without limit. A figure of that size describes a tendency to bunch, not an outbreak.

There is a further limitation the authors are explicit about in their framing: the definitional threshold of four or more killed determines which events enter the dataset at all, and any timing result inherits whatever bias that boundary introduces.

Why it reached practitioners

The paper’s clearest audience was never the general reader. It circulated among the organisations that train the response to these events, including the Advanced Law Enforcement Rapid Response Training centre at Texas State University, which lists it in its news collection.

For that audience the useful content is not the mechanism debate but the window. If risk is measurably raised for around a fortnight after a high-profile incident, that is an operational planning input, independent of whether anyone ever pins down why.

The literature it was built on

The contagion framing did not originate with this paper, and the earlier work is what made it a reasonable hypothesis rather than a provocation.

Several previous studies had found that media reports of suicides and homicides appear to increase the incidence of similar events afterwards. The proposed mechanism there is specific: coverage plants ideation in individuals already at elevated risk, rather than creating risk where none existed. That distinction matters, because it predicts a short-lived effect concentrated in a small susceptible population, which is exactly the shape a contagion model with exponential decay describes.

Extending the same method to mass killings was therefore a test of whether a known effect scales to higher-profile incidents. The answer the paper reports is that it does, with an estimated window of around a fortnight and a reproduction figure well below one.

This is also the strongest argument for reading the result narrowly. The suicide literature supports recommendations about how events are reported — restraint about method, about the perpetrator, about prominence — rather than conclusions about the population as a whole. A timing result inherits the same limitation.