SIAM News Blog

Mathematical Model Exposes Contagion Factor in Mass Shootings

By Lina Sorg

Every 17 minutes, someone in the United States is killed by a gun. The incidence and prominence of mass shootings in America has increased over recent years, dominating televisions and news outlets and fueling the continual debate over gun control. The 1999 Columbine High School massacre, 2007 Virginia Tech shooting, and 2012 shooting at Sandy Hook Elementary School in Newton, Conn. drew international attention. The 2012 movie theater shooting in Aurora, Colo., the 2015 attack in San Bernardino, Calif., and the 2016 shooting at Pulse nightclub in Orlando, Fla., are among the deadliest mass shootings in U.S. history. The frequency and severity of these events have intensified societal focus on and criticism of U.S. gun culture. 

While it is understood that mental health complications and firearm access often expedite such events, at-risk individuals are in some cases inspired to act upon media exposure to details of similar episodes. During a minisymposium presentation at the 2017 SIAM Conference on Applications of Dynamical Systems, held in Snowbird, Utah, in May, Sherry Towers (Arizona State University) presented a mathematical model measuring the contagion level of mass killings and school shootings in the U.S. “The U.S. leads the world by far in gun-related homicides,” Towers said. “Over the years, estimated gun sales per month have skyrocketed.” People frequently purchase guns in response to large political events (such as the 2008 presidential election) or mass killings (like the Sandy Hook massacre and San Bernardino shooting). As a result, there are now more guns than people in the U.S., and a mass shooting—during which three or more people are shot, but not necessarily killed—occurs more than once a week on average. “What we have is a large majority of the population that owns no guns, and a large minority that owns a lot of guns,” Towers said. “If you own a gun in America, you own an average of 10 guns.”

On January 21, 2014, Towers was preparing for a scheduled meeting at Purdue University that never happened. A Purdue student shot and killed a fellow student, then waited for the police to arrest him before committing suicide in prison. Upon learning of the incident, Towers realized that it was the third school shooting she had heard about in 10 days. “Even for a country that has a problem with school shootings, that seemed like an unusually large cluster,” she said. She recognized that youth suicide can be dangerously contagious, and wondered if the same was true of high-profile shootings and mass killings.

A mass shooting—during which three or more people are shot, but not necessarily killed—occurs more than once a week on average in the United States. Image credit: Wikimedia Commons.
Towers began to investigate gun-related contagion. She started with USA Today’s extensive downloadable database of all mass killings (most of which are shootings) in the U.S. from 2006 onward; this yielded 232 gun-related instances. She also contacted the Brady Campaign, an advocacy agency in favor of gun control, knowing that the organization would have statistics on mass killings and school shootings. Brady offered her their data, which Towers had to format herself; this work was both tedious and emotionally taxing. 

Using her collected data and the Hawkes model, Towers created a point-process contagion model. She found that each prior event happening at some instance in time will on average excite another event. “If a mass killing occurs, it increases the probability that a similar event will occur in the near future,” she said. “Then it decreases exponentially.” This trend manifests in the data as an unusual bunching in time, much different from what one would expect of basic probability. Periodicity is of course present in the model, as school shootings do not occur on weekends or in the summer because school is not in session. Additionally, mass shootings in bars are much more likely to occur on Saturday nights due to the high traffic. Upon optimizing her model, Towers realized that she couldn’t use the least-squares statistic because it assumes that data is normally distributed; since most days see no school shootings and thus no data, this is untrue. Instead, she employed a negative binomial likelihood to account for over-dispersion, then fit the null model and the full model with self-excitation to the observed number of shootings per day. 

Towers stressed the importance of model validation. “We bootstrapped 50 percent of our data, fit the model to that, then took that model and used it to predict the other 50 percent,” she said. “If your model has good predictive power, you should find that a large percent of the time you get a better description of your testing data.” Testing revealed that Towers’ model has a high level of predictive power.

Based on the full data set from USA Today, Towers found that the average length of contagion in mass shootings is 13 days; this is followed by an exponential decay. School shooting data from the Brady Campaign also indicated a 13-day contagion period. However, Towers did not find any contagion in mass shootings where less than four people were shot, a factor she attributes to media coverage. “School shootings and mass killings usually receive national—or even international—media attention,” she said. “In contrast, mass shootings with low causality counts happen so often that they rarely make it past the local news.” While school shootings nearly always make national news regardless of casualty count, small domestic disputes or gang-related shootings often go unreported. Towers hypothesizes that the media is unknowingly playing an active role in shooting contagion. “Widespread media attention is perhaps acting as a vector reaching those people who are vulnerable in society,” she said.

Towers described the general contagious process, during which a susceptible individual must be exposed to an infectious individual. In most cases, quarantine or isolation can limit infection during the crucial contagion period, a task that is far easier when preventing the spread of communicable disease than limiting copycat shootings. Although one’s health status can affect susceptibility, Towers’ analysis indicates that mental illness is not directly related to the prevalence of gun-related tragedies. Firearm ownership, however, has a significant correlation. Towers also introduced the Federal Bureau of Investigation’s five stages of an active shooter: fantasy, planning, preparation, approach, and implementation. Awareness of these stages could impact mental healthcare access, gun laws, and media coverage, all of which could lessen the incidence of subsequent shootings. 

Towers concluded by emphasizing that more research is absolutely essential to better understand contagion and act accordingly. Unfortunately, she is one of only few researchers modeling gun violence; no federal funding is available due to the topic’s sensitive nature. “Since 1996, there has been a congressional moratorium on federal funding for most gun violence research, under the premise that gun violence research a priori advocates for and promotes gun control,” she said. Ultimately, while not all shootings are contagious, Towers presented significant evidence that some mass killings involving firearms are inspired by similar happenings in the immediate past. Simply knowing that there is in fact a media-related contagion factor of subconscious ideation can help researchers better understand the complex dynamics and aftermath of violent events.

Lina Sorg is the associate editor of SIAM News
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