SIAM News Blog

The Mathematics Underlying Gun Violence

By Shelby M. Scott and Louis J. Gross

Each year, gun violence is responsible for approximately 31,000 deaths and 78,000 non-fatal injuries in the U.S. [1]. These casualties cost roughly $229 billion, with rural communities and younger individuals experiencing a higher burden [11]. Exposure to gun violence is associated with a greater propensity toward both chronic health conditions and risky social behaviors [11].

The consequences of gun violence impact diverse populations across a wide variety of scales, and research therefore involves multidisciplinary perspectives. Unfortunately, work in this field is limited. In 1996, the Dickey Amendment prevented the Centers for Disease Control and Prevention from using federal funds to promote or advocate for gun control, effectively shutting down research [9].

Last spring, the National Institute for Mathematical and Biological Synthesis and the Center for the Dynamics of Social Complexity hosted 29 individuals at an investigative workshop entitled “Mathematics of Gun Violence.” The workshop aimed to review existing literature, identify areas that require further research, develop cross-disciplinary collaborations, and suggest data collection to assist evidence-based policy recommendations. In summarizing workshop findings, we present valuable ideas surrounding the mathematics of gun violence.

Existing Literature

While statistical methods have addressed aspects of gun crime and violence in the U.S., mathematical modeling approaches are limited (with the exception of a few existing models). To analyze the dynamics of crime hotspots, Martin B. Short and his collaborators produced a partial differential equation model that interprets supercritical or subcritical bifurcations in a crime context [8]. Other researchers have studied the efficacy of law enforcement deployment and found that the dynamics of policing have significant effects on crime distribution [7].

In the context of crime, networks may correlate with the dynamics of spread. Ben Green and his coauthors evaluated the extent to which modeling contagion on a social network can predict victimization [6]. Paul Brantingham and his group used ecological modeling to address the intergroup dynamics of gangs [3]. Sara Bastomski and her collaborators examined the way in which neighborhood-level criminal networks shape crime distribution and determined that embeddedness has a positive association with local homicide rate [2]. Violence may propagate through networks, but the impacts of intervention can spill over in similar ways [12]. Analysis of network structure could suggest improved interventions.

A summary of the influences on gun violence research. Figure courtesy of Shelby M. Scott.

In addition to networks, other methods have found success in analyzing the spread of gun violence and crime. Due to the complicated relationships between individuals, systems dynamic models can be useful when investigating incarcerations and interventions [4]. Researchers have also used game theory to determine the efficacy of gun control policies [10], and developed systems of ordinary differential equations to analyze the dynamics of gun crime as it spreads throughout a population [5].

Key Takeaways

Below we identify some promising ideas that emerged from the workshop.

Collaboration with Stakeholders: A variety of relevant actors exist in situations of gun crime and violence. Therefore, identification of key players and their interactions with the affected community is important, and analysis of stakeholder network structures and their generalizability may improve interventions. Unfortunately, obtaining data to parameterize such models is difficult.

Interventions: Observing how different interventions affect various types of firearm-related events can allow the implementation of multifaceted, evidence-based approaches. It is also imperative to develop a consensus for intervention evaluation. Considering the root causes of crime and violence—rather than focusing on the outcomes—may improve response. Mathematical and statistical modeling can address both individual and population-level intervention effects.

Quantifying the Impact of Rare Events: Rarity is context-dependent, both in terms of scale and field of study. Rare events also introduce uncertainty due to outliers. Mass shootings are rare relative to interpersonal violence, and analysis of each scenario requires different methods. Borrowing from other disciplines that study rare events can help forecast the occurrence of future violent incidents and suggest appropriate reduction strategies.

Epidemiological Criminology: Applying tenets from epidemiology to criminology is common, and not just in the formal paradigm of “epidemiological criminology.” When employing multiscale models that connect the individual scale to the population and community scales (as in disease ecology), one must consider the nuances and limitations of the analogy between disciplines.

Theoretical Models: Investigating the factors and interactions that push an individual from nonviolence to violence, even in the absence of data, can offer insight about crime interruption. Ideas from sociology and psychology—when combined with quantitative models—may provide information about generalizable concepts and their interactions for violence analysis.

Technology: The introduction of new technologies that are relevant to gun violence and crime requires that one determine the technology’s specific purpose, evaluation, and impact — not only on the social system, but also for the individuals who are subject to the new tools. Practitioners must consider ethical issues and concerns of bias when working with collected data and conclusions drawn from technological advances.

Data Collection and Use: There is a dearth of data quality and quantity pertaining to crime assessment. Tradeoffs also occur in the applicability of available datasets between precision, realism, and generalizability. In light of this, interdisciplinary collaborations are necessary to fill gaps in the data using theory and methods from various fields. As with technology, it is important to address ethical matters and the data’s overall purpose.

Spatiotemporal Characteristics: Many quantitative models have contemplated the spatial, temporal, and spatiotemporal aspects of gun violence, but it is important to account for the differing data scales and appropriateness of analysis methods based on the situation. One must note measurement error—common in spatiotemporal data—when drawing conclusions or making actionable recommendations.

Network Models: Networks of individuals often perpetuate gun crime and violence, but these networks can also be instrumental in interrupting it. Determining the structure and influence of violence that perpetuates and interrupts networks could lead to improved interventions.

Participants of the “Mathematics of Gun Violence” workshop, organized by the National Institute for Mathematical and Biological Synthesis and the Center for the Dynamics of Social Complexity. Photo courtesy of Catherine Crawley.

Concluding Thoughts

Common discussion topics at the workshop included the need for more cross-disciplinary and interdisciplinary work. Leveraging diverse expertise across a wide range of fields can improve methods to study gun violence. Model verification, intervention programs, and technologies also require better evaluation techniques. Finally, workshop participants emphasized the importance of data improvements. The quality of available data is unreliable and necessitates the collection of many additional datasets. Enhanced opportunities for broadening the pool of researchers depends on increased funding agency support for projects in this major area of concern.

[1] American Psychological Association. (2013). Gun violence: Prediction, prevention, and policy (Technical report). Washington, D.C.: American Psychological Association.
[2] Bastomski, S., Brazil, N., & Papachristos, A.V. (2017). Neighborhood co-offending networks, structural embeddedness, and violent crime in Chicago. Soc. Netw., 51, 23-39.
[3] Brantingham, P.J., Valasik, M., & Tita, G.E. (2019). Competitive dominance, gang size and the directionality of gang violence. Crim. Sci., 8(7). 
[4] Cirone, J., Bendix, P., & An, G. (2019). A system dynamics model of violent trauma and the role of violence intervention programs. J. Surg. Res., 245.
[5] Gonzalez-Parra, G., Chen-Charpentier, B., & Kojouharov, H.V. (2018). Mathematical modeling of crime as a social epidemic. J. Interdiscipl. Math., 21(3), 623-643.
[6] Green, B., Horel, T., & Papachristos, A.B. (2017). Modeling contagion through social networks to explain and predict gunshot violence in Chicago, 2006 to 2014. JAMA Intern. Med., 177(3), 326-333. 
[7] Mohler, G.O., Short, M.B., Malinowski, S., Johnson, M., Tita, G.E., Bertozzi, A.L., & Brantingham, P.J. (2015). Randomized controlled field trials of predictive policing. J. Am. Stat. Assoc., 110(512), 1399-1411.
[8] Short, M.B., Brantingham, P.J., Bertozzi, A.L., & Tita, G.E. (2010). Dissipation and displacement of hotspots in reaction-diffusion models of crime. PNAS, 107(9), 3961-3965.
[9] Stark, D.E., & Shah, N.H. (2017). Funding and publication of research on gun violence and other leading causes of death. JAMA, 317(1), 84-86.
[10] Taylor, R. (1995). A game theoretic model of gun control. Int. Rev. Law Econ., 15(3), 269-288.
[11] U.S. Congress Joint Economic Committee. (2019). A state-by-state examination of the economic costs of gun violence. U.S. Congress Joint Economic Committee.
[12] Wiley, S.A., Levy, M.Z., & Branas, C.C. (2016). The impact of violence interruption on the diffusion of violence: A mathematical modeling approach. In Advances in the Mathematical Sciences: Research from the 2015 Association for Women in Mathematics Symposium (pp. 225-249). Switzerland: Springer International Publishing.

Shelby M. Scott is a Ph.D. candidate and National Defense Science and Engineering Graduate Fellow in the Department of Ecology and Evolutionary Biology at the University of Tennessee, Knoxville, where she is also pursuing a master’s degree in statistics. Her research interests include applying mathematical and statistical modeling to topics in public policy, ecology, and healthcare, as well as quantitative biology education and outreach. Louis J. Gross is Chancellor’s Professor of Ecology and Evolutionary Biology and Mathematics at the University of Tennessee, Knoxville, where he also directs the National Institute for Mathematical and Biological Synthesis, a synthesis center supported by the National Science Foundation. His research focuses on applications of mathematics and computational methods in disease ecology, landscape ecology, spatial control for natural resource management, climate change, and the development of quantitative curricula for life sciences students.

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