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The authors have declared that no competing interests exist.

Conceived and designed the experiments: DW NK. Performed the experiments: DW NK. Analyzed the data: DW NK. Contributed reagents/materials/analysis tools: DW NK. Wrote the paper: DW NK.

In the USA, the relationship between the legal availability of guns and the firearm-related homicide rate has been debated. It has been argued that unrestricted gun availability promotes the occurrence of firearm-induced homicides. It has also been pointed out that gun possession can protect potential victims when attacked. This paper provides a first mathematical analysis of this tradeoff, with the goal to steer the debate towards arguing about assumptions, statistics, and scientific methods. The model is based on a set of clearly defined assumptions, which are supported by available statistical data, and is formulated axiomatically such that results do not depend on arbitrary mathematical expressions. According to this framework, two alternative scenarios can minimize the gun-related homicide rate: a ban of private firearms possession, or a policy allowing the general population to carry guns. Importantly, the model identifies the crucial parameters that determine which policy minimizes the death rate, and thus serves as a guide for the design of future epidemiological studies. The parameters that need to be measured include the fraction of offenders that illegally possess a gun, the degree of protection provided by gun ownership, and the fraction of the population who take up their right to own a gun and carry it when attacked. Limited data available in the literature were used to demonstrate how the model can be parameterized, and this preliminary analysis suggests that a ban of private firearm possession, or possibly a partial reduction in gun availability, might lower the rate of firearm-induced homicides. This, however, should not be seen as a policy recommendation, due to the limited data available to inform and parameterize the model. However, the model clearly defines what needs to be measured, and provides a basis for a scientific discussion about assumptions and data.

Gun violence has been an ongoing problem in the United States of America

This debate cannot be settled satisfactorily by verbal arguments alone, since these are often driven by opinion, and lack a solid scientific backing. What is under debate is essentially an epidemiological problem: how do different gun control strategies affect the rate at which people become killed by attackers, and how can this rate be minimized? This question can be addressed with mathematical models that describe the interaction between a criminal shooter and one or more people that are the target of the shooter. The gun policy is defined as the fraction of the population that can legally and readily obtain firearms. We aim to analyze the above-described tradeoff and to examine which type of gun policy minimizes firearm-related deaths under different assumptions. Calculations are performed for two scenarios: the assault by a shooter of a single potentially armed victim (what we call a one-on-one attack), or the assault of a crowd of people that can be potentially armed (a one-against-many attack). Note that the former scenario has been documented to be the most prevalent cause of gun-related homicides

The models are based on assumptions that are grounded in and supported by epidemiological data. Yet, it is important to note that a limited amount of such data currently exist, and that future studies will need to confirm these. Because model predictions are a direct consequence of the underlying assumptions, this is necessary to keep in mind. The models give rise to the suggestion that two alternative strategies can minimize the rate of firearm-induced homicides: either a ban of private firearm possession, or a strategy allowing the general population to carry guns. Which strategy minimizes gun-induced homicides depends on the parameters, and here lies the most important contribution of the model: it identifies what needs to be measured statistically in order to improve our understanding about the relationship between legal gun availability and the rate of gun-induced homicides. The important parameters include the fraction of offenders that illegally possess a gun, the degree of protection provided by gun ownership, and the fraction of the population who take up their right to own a gun and carry it when attacked. Limited data exist in the literature that allows a first and preliminary estimate of these parameters, which was performed for illustrative purposes. In the context of these estimates, the model suggests that a ban of private firearm possession, and possibly a certain reduction in the degree of gun availability, might reduce the rate at which firearm-induced homicides occur. However, because these parameter estimates are based on data that were not collected with model parameterization in mind, and because only very limited studies that are relevant currently exist, this model suggestion cannot be interpreted as a solid result that can recommend specific policies. The models represent a first mathematical formulation that examines the relationship between legal gun availability and the rate at which firearm-induced deaths occur. The models give rise to specific predictions, and highlight what needs to be measured to improve understanding. This can serve as a guide for future statistical, epidemiological, and modeling studies.

Because this analysis is very technical in nature and at the same time concerned with a topic that is of interest to a lay audience, it is important to clearly spell out its scope, the inherent limitations, and how results should be interpreted. Any mathematical model of a biological or behavioral process represents by definition a simplification and abstraction of a complex system. A number of assumptions are made about what drives these processes, which are rooted in data that are available in the literature. The model formulates these assumptions in terms of equations, which allow us to take those assumptions to their precise logical conclusions. The results then clearly depend on the assumptions underlying the model, and this is very important to keep in mind when reading this paper, or any paper that deals with mathematical models in biological and behavioral sciences.

In the field of gun violence, it is especially important to be aware of this notion because compared to other fields of biology or epidemiology, only a limited amount of data is available to inform the construction of mathematical models. Therefore, the results reported here should not be viewed as final policy recommendations, but as a first approach to scientifically and logically formulate the issues involved in the gun control debate. One of the aims of this paper is to stimulate a debate about assumptions, statistics, and techniques. By identifying the quantities and parameters that need to be measured, our paper paves the way to further studies which will refine this approach and eventually provide a detailed understanding of how gun availability influences the amount of gun-related and other violence in human populations.

In our opinion, the most valuable yield from our study is a better understanding of what needs to be measured statistically and epidemiologically in order to improve understanding. The models highlight crucial parameters that can determine how gun availability affects the level of firearm-induced homicides, and they suggest what types of statistical data will need to be collected in future research to drive progress. The work performed here can therefore help as a guide for designing statistical studies that are crucially important for a better understanding of these issues, and this would not be possible without designing a mathematical framework. This is a much more important contribution of our work than the actual results that are dependent on assumptions, the nature of which are uncertain to some extent due to the limited amount of data currently available.

To summarize, this paper presents a first mathematical framework to analyze the debate about gun control in the United States, with the aim to steer the debate towards arguing about assumptions and statistics. Equations are used to capture key processes and assumptions that are based on a limited amount of statistical data that are available in the literature. Based on these assumptions, the model gives rise to certain predictions, which are preliminary in nature due to the scarcity of solid statistical studies. The most important contribution, however, is that the model identifies what types of future statistical studies need to be performed in order to improve our understanding of this complex issue.

To calculate the effect of different gun control policies on the gun-induced death rate of people, we turn to a mathematical framework that is constructed in this section. This is a new approach, but should be viewed in the context of the larger area of mathematical models that examine aspects of crime, as well as specific shooting scenarios. In the context of crime, a variety of mathematical studies have been performed

We consider the correlates of the total rate (per year, per capita) at which people are killed as a result of shootings. We introduce the variable

An important aspect of this model is the form of the dependency of these two quantities on the gun control policy,

Here we consider the situation where an attacker faces a single individual who can be potentially armed. The fraction of people owning guns in the population is defined by the legal possibility and ease with which guns can be acquired (

The first important result is that the killing rate can only be minimized for the extreme strategies

Further, we can provide simple conditions on which of the two extreme strategies minimizes death. Let us first assume that

(a) The fraction of people who possess the gun and have it with them when attacked is relatively low,

Next, let us suppose that condition (4) is violated, that is,

In the opposite case, the “gun availability to all” strategy minimizes death. Condition (5) defines a threshold value for

The case where inequality (4) is violated is illustrated in

Note that the key condition (4) relates two quantities, which in some sense are the opposites of each other: The first quantity is

An important question is as follows. Let us suppose that the total gun ban is impossible due to e.g. constitutional or cultural constraints. Would a partial restriction of gun ownership help reduce the firearm-related homicide rate? It follows that if

It is, however, possible that the population of victims is similar to the population of the attackers in the context of gun ownership, especially if we model the situation in different socio-economic conditions, such as inner cities. The following model is more appropriate for such situations:

It states that a victim who is entitled to a legal weapon will have the gun available at the time of the attack with probability

The

Note that if

Furthermore, partial measures to reduce

As before, with

Here, we consider a situation where a shooter attacks a crowd of people, such as in a movie theater or mall shooting. The difference compared to the previous scenario is that multiple people can potentially be armed and contribute to stopping the attacker. In this context, it is worth to point out that the protective effect of gun ownership assumes that the people in the crowd who are being attacked are sufficiently trained in the use of the weapons such that accidental collateral damage does not occur during attempted defense. If this is not the case, this might lower or void the benefits afforded by gun ownership.

We suppose that there are

At each time-step, the attacker shoots at one person in the crowd (with the probability to kill

This model is considered in detail in

The optimal strategy that minimizes the gun-induced death rate of people again depends on the degree of law enforcement (i.e. the probability for offenders to obtain firearms illegally). More precisely, we have to evaluate the inequality

There are two cases:

If inequality (10) holds (tight law inforcement and/or gun protection ineffective), then the “ban of private firearm possession” policy (

If

An important aspect of our model is the exact form of the dependency of the quantities

The probability for a person to die in an attack (once he or she is at an attack spot) is described by the function

The number of armed attackers (and thus the frequency of attacks) is a growing function of

What information can we extract from these general assumptions about the functions

To see how the overall risk of being killed depends on the gun-policy

It is easy to see that if both functions

More generally, if

In other words, as before, the most effective gun policy corresponds to either

An intermediate optimum is possible if

Can we derive a condition, in this very general setting, which would inform us which one of the two possible optima,

The next question we address axiomatically is whether a small change in a gun policy starting from

We analyzed mathematical models in order to calculate the gun-induced homicide rate of people depending on different gun control strategies. In particular, we examined the tradeoff that legal gun availability could either increase the firearm-induced death rates by increasing the number of gun-mediated attack, or reduce the death rate due to protection offered by gun ownership. Such a mathematical framework has so far not been constructed and analyzed, although our work falls into the larger area of shooting and crime modeling, which has been briefly reviewed above.

The gun control strategies in our model were expressed by a parameter that describes the fraction of the population that can legally own firearms. The strategies can range from a ban of private firearm possession to a “gun availability to all” strategy. We first investigated a situation in which one shooter is faced by only a single person that could potentially own a gun and that could fight back against the shooter. This can correspond to a one-on-one attack, such as a robbery, or a school shooting where the only person in the classroom that could carry a gun is the teacher. Subsequently, we examined a different scenario where a shooter faces a crowd of people, all of which could potentially own a gun and fight back against the attacker. This corresponds to shootings in public places such as movie theaters and malls.

In order to understand the implications of these modeling approaches, two aspects need to be considered. First, we discuss to what degree the model formulation is rooted in epidemiological data and how the validity of the model can be tested. Subsequently, we discuss how available statistical data can be used to parameterize the model and to derive specific predictions, based on the model’s assumptions.

In order to validate the model, data need to be available that inform the functions

For the one-against-one scenario, the function

The formulation of the function

One aspect we have no data on is how exactly the function

The shape of the function

So far, we have discussed how robust the model assumptions are with respect to available data. With these assumptions in mind, the model gives rise to a set of predictions about the effect of gun availability,

One of our results was that either a complete ban of private firearm possession or a policy allowing the general population to carry guns can minimize the rate of firearm-induced homicides, depending on the parameter values. Here, we discuss how the model can be parameterized based on published statistical data, and what such a parameterized model suggests. In this context we point out that any of the published data used to parameterize the model were certainly not designed with our mathematical study in mind and are therefore suboptimal for this purpose. While it is educational and important to attempt model parameterization in this context, the predictions that come out of this exercise are preliminary in nature and await further, more directed statistical studies to refine this work. They should certainly not be viewed as policy recommendations. The power of the mathematical modeling framework, however, is to identify what needs to be measured in future statistical studies to take this research further. It thus serves as a guide that would not exist without the analysis of mathematical models.

An important parameter is the degree of law enforcement relative to the amount of protection that gun ownership offers. If the law is enforced strictly enough, a ban of private firearm possession minimizes the gun-induced death rate of people according to this model.

The question arises how strict the law has to be enforced for the a ban of private firearm possession to minimize the gun-induced death of people. According to our results, this depends on the degree to which gun ownership protects potential victims during an attack and on the fraction of people who take up their legal right of gun ownership and carry the gun with them when attacked. These parameters in turn are likely to vary depending on the scenario of the attack and are discussed as follows.

The most prevalent use of guns is a one-against-one scenario and largely involves handguns

In order to examine the fraction of offenders that cannot legally obtain a gun but own one illegally,

The fraction of people who legally own a firearm and have it in possession when attacked,

For the sake of the argument, let us consider the extreme scenario where all people who can legally own a gun do so and carry it with them at the time of an attack. This would require an effort by the government to persuade people to purchase firearms and carry them around at all times. As mentioned above, gun-related death is now minimized if

Rather than considering a ban of private firearm possession, it can be more practical to consider the option of partially restricting firearm access. In general, the model suggests that this might either decrease of increase the gun-induced death rate, depending whether the general condition (12) (or its variant for the one-against-one scenario, condition (13)) is fulfilled. Thus, an important message of the calculation is that while a ban of private firearm possession might minimize the death rate, a partial reduction may or may not have this effect, depending on the exact circumstances. Hence, these circumstances need to be studied in detail when designing policies, and our model provides a guide for this. The crucial mathematical condition depends on measurable parameter values, and in particular, on the relative rate of change of the function

We can see that in general, if

Next, we discuss the one-against-many scenario. Here, two different gun control policies can again potentially minimize firearm-induced deaths of people: either a ban of private firearm possession, or arming the general population. However, in the latter case, not necessarily the entire population should carry firearms, but a certain fraction of the population, which is defined by model parameters. As in the one-against-one scenario, which policy minimizes gun-induced fatalities depends on the fraction of offenders that cannot legally obtain a gun but carry one illegally, the degree of gun-induced protection of victims during an attack, and the fraction of people who take up their right of gun ownership and carry the gun with them when attacked. In contrast to the one-against-one scenario, however, this dependence is more complicated here.

In this model, the degree of gun-mediated protection against an attack is given by the parameter

Presented are the contour-plots of the threshold value

The meaning of these numbers further depends on the weapon carried by the attacker. Strictly speaking, the model considered here was designed for attackers and victims with similar weapons. The victims would typically possess hand guns. If the attacker also uses a hand gun, it can be questioned whether the attacker is 30 times more efficient at fatally shooting someone than a victim, even if the attacker is better trained and has more experience. In other situations, more advanced weapons, such as semi-automatic guns can be used to assault crowds, where tens to hundreds of rounds per minute are fired. The victims typically will not possess such powerful weapons. Therefore, their ability to shoot is significantly lower than that of the attacker. We can interpret the results of the model for a situation where the attacker fires a semi-automatic or automatic weapons and the victims respond with hand guns. In this case we must assume that the probability of victims to fire and shoot the killer is significantly (perhaps 2 orders of magnitude) lower than that of the attacker. This shifts the outcome towards a strategy that bans private firearm possession, which would now minimize firearm-related death if a relatively large crowd of 40 people is attacked. For the opposite strategy to minimize death, the attacked crowd would now need to be significantly larger to make up for the increased shooting efficiency displayed by the attacker.

Now, for the sake of the argument assume that

Having discussed the one-against-many scenario in some detail, it has to be pointed out that while assaults on crowds generate the most dramatic outcomes (many people shot at once), the great majority of gun-related deaths occur in a one-against-one setting

We have discussed the model in the context of what we called the suburban model. That is, we separated attacker and victim populations. We have shown, however, that in the context of the inner city model, the condition for a ban of private firearm possession becomes easier to fulfill. This model does not separate the attacker and victim populations but instead describes a scenario where a large fraction of the population can have criminal tendencies, and where a person may either be an attacker or a victim depending on the situation, e.g. two armed people getting into a fight, drug related crimes, etc.

It also has to be kept in mind that parameter estimates could be different depending on the setting, although there is currently no information available about this in the literature. Related to this issue, it is clear that crime is not uniform with respect to spatial locations. There are areas with adverse socio-economic conditions which are characterized by high homicide rates, and there are areas of relative safety with very few gun crimes. While our model does not take space into account explicitly, it takes into account different scenarios (such as the suburban model or the inner-city model). The optimization problem solved here does not explicitly depend on the spatial distribution of different crime conditions. Further questions about crime management can however be asked if one utilized a spatial extension of this model.

In the present work, two basic scenarios (one-against-one and one-against-many) were discussed. We would like to note that these two scenarios do not include all possible settings in the context of gun violence. There can also be many-on-one or many-on-many cases, and gun violence can also include other important aspects such as accidental deaths, self-injuries, and suicides. Related to this, a blend of the one-against-one, one-against-many, and other scenarios could be considered, although this would introduce further uncertainty due to the necessity to weigh the importance of different scenarios. At present, we preferred to limit our focus to only two scenarios, because the problem is already characterized by a significant amount of complexity.

An issue that we have ignored in our discussion so far are possible deterring effects of gun ownership, i.e. the notion that non-homicide crimes, for example burglaries, could occur less often if those offenders are deterred by the presence of guns in households. Our analysis was concerned with minimizing gun-related homicides, and not crime in general, which is a different topic and should be the subject of future work. If a gun is present in households, and the burglar would consequently carry a gun during the offense, however, the number of gun-related deaths is likely to increase, even if perhaps the total number of burglaries might decrease. This applies especially if guns in the household are unlikely to protect against injury or death, as indicated in the literature

Another link to behavioral decisions which has implications for the influence of gun availability on homicides comes from the punishment literature. It has been shown that if people have the option to punish each other at a cost to themselves, then they use it often illogically as “pre-emptive” strikes

Finally, it is important to note that this paper only takes account of factors related to the gun control policy, and assumes a constant socio-economic background. Of course in the real world a reduction in gun-related (and other) homicides would require improvement of the living and work conditions and education of underprivileged populations. Here we do not consider these issues. It is important to emphasize that cultural and socio-economic differences exist between different countries, between cities or regions within the same country, and even between different time periods within the same location, making it difficult to draw direct comparisons. A lower or higher rate of gun-related deaths is not only a function of gun control policies, but also of those other factors. Comparisons in the context of this model are only possible within the same cultural and socio-economic space. We single out the direct effects of gun-control policies and investigate those under fixed cultural and socio-economic circumstances. Therefore, at present, there is no straightforward way to relate our results to many statistical studies that compare gun violence in different cities, countries, or over time following changes in gun policies

Related to this, it has been reported that homicide growth rates show the same form for different city sizes in a study quantifying aspects of murders in Brazilian cities for a defined period of time

This paper provides the first mathematical formulation to analyze the tradeoff in the relationship between legal gun availability and the rate of firearm-induced death: while more wide-spread legal gun availability can increase the number of gun-mediated attacks and thus the firearm-induced death rate, gun ownership might also protect potential victims when attacked by an armed offender, and thus reduce the firearm-induced death rate. The main contributions of this study are as follows. (1) We created a mathematical model which takes account of several factors that are often discussed in the context of gun-induced homicide. The model is based on a set of assumptions that are supported by previously published empirical data, as was discussed in detail. For assumptions where no epidemiological data were available for model grounding, we employed axiomatic modeling approaches which showed that most model properties remained robust within epidemiologically reasonable constraints. The model suggests that the rate of firearm-induced homicides can be minimized either by a ban of private firearm possession, or by the legal availability of guns for everyone, depending on the parameter values. While there is strong indication that the model assumptions and hence the properties are consistent with data, it will be important to collect more data to back up the underlying assumptions more strongly. (2) We illustrated how model parameterization can be useful in deciding the correct strategy to minimize the rate of gun-induced homicide by attempting parameter estimations, based on data that are available in the literature. These data have not been collected with this purpose in mind and are certainly not extensive enough to allow conclusive parameter estimates. With this constrain in mind, the preliminary parameterization of the model suggests that the firearm-induced homicide rate might be minimized by a ban of private firearm possession, and possibly reduced if gun availability is restricted to a certain extent. Due to the preliminary nature of the data used for model parameterization, however, this should not be viewed as a policy recommendation, which will require detailed epidemiological studies that collect extensive data sets specifically geared towards parameterizing the model. (3) Possibly the most important contribution of our study is as follows. Our model is based on several variables/parameters, and it shows in what way these parameters may contribute to the delicate balance of factors responsible for the prevalence of gun-related homicides. To improve understanding, these crucial parameters need to be measured by epidemiological and statistical studies. The model identifies these parameters and can thus serve as a guide for the design of these studies. Our work will hopefully stimulate such empirical studies, and also steer the debate about gun control towards a scientific approach where assumptions, data, and methodologies are discussed.

Detailed analysis of the stochastic models used in this paper is provided in

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