Limitations and Interpretation of the Analysis

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.

The Modeling Approach

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 [11]–[18], although they concentrate on questions that are different from the ones considered here. In the context of shootings in particular, specific scenarios have been investigated, some in the context of war [19]–[22]. Though not directly related to our work, these studies can be viewed as the beginnings of probabilistic and game-theoretic analyses that involve firearms, and thus form an important background for our explorations. At the beginning of the 20th century, Lanchester and Osipov independently developed extensive models of military combat, describing the strength of opposing armies as a function of time, and calculating expected casualties in different situations and in the context of different weapon types [19], [23]. Based on these models, data from specific wars have been used to gain further insights, e.g. [20]. Non-military mathematical analysis of shootings include models of duels and truels, which are generalizations of a duel involving three rather than two participants [21], [22].

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 to describe the gun control policy. This quantity denotes the fraction of the population owning a gun. A ban of private firearm possession is described by , while a “gun availability to all” strategy is given by . We assume that a certain small fraction of the population is violent (this assumption is relaxed later on by assuming that the population of offenders is not different from the population of victims), and that an encounter with an armed attacker may result in death. The number of offenders that own firearms is a function of the gun control policy and is denoted by ; this quantity is proportional to the frequency of gun-related attacks (the proportionality constant does not depend on and thus will not alter the results presented here). The probability of a person to die during an attack is also a function of the gun control policy because this determines whether the person and any other people also present at the place of the attack are armed and can defend themselves. This probability is denoted by . The overall risk of being killed by a violent attacker as a result of shooting is thus proportional to(1)

An important aspect of this model is the form of the dependency of these two quantities on the gun control policy, . The number of armed attackers, , is a growing function of , i.e. . Note, however, that even if offenders are not allowed to legally obtain firearms, there is a probability to obtain them illegally. Hence, the value of is non-zero for . One example of such a function is given by the following linear dependence,(2)with . The probability for a regular person to die in an attack (once he or she is at an attack spot) is discussed in the next sections, and depends on the exact situation. A general analysis of expression (1) is presented in the section Axiomatic modeling.

One-against-one Attack

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 (), as well as the personal choice to acquire a gun. Moreover, people who own a gun might not necessarily carry the firearm when attacked. Let us denote by the propensity of potential victims to buy a gun (if legal), and by their probability to have it with them at the time of the attack. Then, we can assume that the fraction of people armed with a gun when attacked is given by , where the parameter describes the fraction of people who take up their legal right of gun ownership and have the firearm in possession when attacked (). We will model the probability to be shot dead in an attack as(3)a linear function of , where is the probability for an unarmed person to die in an attack, and is the probability for an armed person to die in an attack, with . The number of attacks is proportional to given by equation (2). The aim is to find the value of that minimizes the death rate, , given by equation (1).

Optimization results.

The first important result is that the killing rate can only be minimized for the extreme strategies and , and that intermediate strategies are always suboptimal (this is because for all ). In other words, either a complete ban of private firearm possession, or a “gun availability to all” strategy minimizes gun-induced deaths.

Further, we can provide simple conditions on which of the two extreme strategies minimizes death. Let us first assume that(4)this case is illustrated in figure 1(a) and interpreted below. Now, a ban of private firearm possession always minimizes gun-induced deaths. This can be seen in figure 1(a), where we plot the shooting death rate, , as a function of the gun policy, , for several values of the quantity . For all of these functions, the minimum is achieved at .

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Figure 1. The rate of death caused by shooting in an one-against-one attack, as a function of the gun control policy, , where corresponds to a ban of private firearm possession, and to the “gun availability to all” policy.

(a) The fraction of people who possess the gun and have it with them when attacked is relatively low, with . The different lines correspond to different values of . For all values of , the shooting death rate is minimal for . (b) The fraction of people who possess the gun and have it with them when attacked is relatively high, with . As long as condition (5) holds, the shooting death rate is minimal for (ban of private firearm possession, solid lines). If condition (5) is violated, then the shooting death rate is minimized for (“gun availability to all”, dashed lines).

https://doi.org/10.1371/journal.pone.0071606.g001

Next, let us suppose that condition (4) is violated, that is, , see figure 1(b). In this case, a ban of private firearm possession minimizes death if the following additional condition holds,(5)

In the opposite case, the “gun availability to all” strategy minimizes death. Condition (5) defines a threshold value for , the fraction of offenders that cannot legally obtain a gun but possess one illegally. The threshold value provided by inequality (5) depends on the degree to which gun ownership reduces the probability for the attacked person to die, (the smaller the quantity , the higher the gun-mediated protection), and on the fraction of people who take up their right of legal gun ownership and carry the gun with them when attacked. The right hand side of inequality (5) decays with . When (everybody who has a right to a gun, carries a gun), condition (5) takes a particularly simple form:(6)

The case where inequality (4) is violated is illustrated in figure 1(b). Larger values of satisfy condition (5), see the solid curves in figure 1(b). For those curves, (the total firearms ban) corresponds to the minimum of the shooting death rate. If condition (5) is not satisfied (the dashed curves in the figure), then the (“gun availability to all”) policy is the optimum.

Note that the key condition (4) relates two quantities, which in some sense are the opposites of each other: The first quantity is , the probability that a potential attacker who cannot legally possess a gun will obtain it illegally and have it at the time of the attack. This can be a measure of law enforcement, with lower values of corresponding to stricter law enforcement. The other quantity is , the probability that a person who can legally have a gun will not have it available when attacked. To make the ban of private firearm possession work (that is, to make sure that it is indeed the optimal strategy), one would have to make an effort to enforce the law and fight illegal firearm possession to decrease . To make the “gun availability to all” policy work, one would have to increase , for example by encouraging the general population to have firearms available at all times.

One-against-many Attack

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 people within the range of a gun shot of the attacker, and of them are armed. We envisage the following discrete time Markov process. At each time-step, the state of the system is characterized by an ordered triplet of numbers, , where tells us whether the attacker has been shot down () or is alive (), where is the number of armed people in the crowd, and is the number of unarmed people. The initial state is .

At each time-step, the attacker shoots at one person in the crowd (with the probability to kill ), and all the armed people in the crowd try to shoot the attacker, each with the probability to kill . The following transitions are possible from the state with , (below we use the convention that expressions of type take the value for ):

• : one armed person is shot, the attacker is not shot, with probability ;
• : an unarmed person is shot, the attacker is not shot, with probability ;
• : one armed person is shot, the attacker is shot, with probability ;
• : an unarmed person is shot, the attacker is shot, with probability ;
• : no potential victims are shot, the attacker is shot, with probability ;
• : no one is shot, with probability .

This model is considered in detail in Text S1, and a Mathematica code is provided in File S1, which allows to visualize the results. For , is a decaying function of , with for all . The overall risk of being shot dead in a one-against-many attack can be expressed as the following function:(9)where parameter is again the fraction of all the people who take up their legal right of gun ownership and carry the gun with them when attacked. The parameter measures the effectiveness of the protection received from the guns (the expression for is given in the Methods section), and the function satisfies , .

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(10)

There are two cases:

• If inequality (10) holds (tight law inforcement and/or gun protection ineffective), then the “ban of private firearm possession” policy () is optimal.
• If (lax law inforcement and/or gun protection highly effective), then depending on the value of function we may have different outcomes, including the the optimum (gun availability to all) or an intermediate optimal policy.

Axiomatic Modeling

An important aspect of our model is the exact form of the dependency of the quantities and on the gun control policy, . The behavior of the overall risk function, (equation (1)) depends on our assumptions on these two fuctions. Below we discuss some general considerations that apply in different scenarios, and show that most of the conclusions obtained above continue to hold under a more general set of assumptions. We begin by discussion the functions and .

Model Assumptions, Model Testing, and Data

In order to validate the model, data need to be available that inform the functions and in our model. These are functions that are proportional to the mean rate at which people die during a gun attack, , and the mean rate at which gun attacks occur, . As indicated in the notation, both are assumed to depend on the probability for the general population to own guns, , which is determined by the gun policy.

For the one-against-one scenario, the function is defined in a straightforward way by . The parameter is the probability to die during an attack if the victim does not carry a gun, and the fraction of such victims is by definition . The parameter describes the probability for a victim to die during an attack if the victim does carry a gun, and the fraction of such victims is by definition . In our arguments, we assume that , i.e. that gun ownership can offer protection during an attack, because this is one of the central arguments in the current debate. If guns cannot protect against an attack, there is no tradeoff and private firearm possession would confer no benefit to the population. The extent to which gun possession protects against death when attacked is an important parameter that determines the outcome of the model, and is discussed further below in the context of model parameterization. In summary, the formulation of the function in the one-against-one scenario is robust and not subject to uncertainty.

The formulation of the function is not straightforward and needs to be backed up by epidemiological data. The most important assumption for our calculations is that increased legal gun availability leads to more attacks by criminals, i.e. is an increasing function of . It could potentially be argued that this is not the case because individuals with criminal predispositions can obtain guns illegally, regardless of the gun control policy in place. However, there is some data available in the literature to support our assumption. One study considered two types of populations in California [28]. Among the first group, gun purchases were denied because of a previous felony conviction. In the second group, purchases were approved. Although people in this group had previous felony arrests, they were not convicted. Compared to the group of people who could not obtain a gun, the gun purchasing group was characterized by a significantly higher risk of subsequent offenses that involved a gun. Up to a four-fold difference was observed. This certainly supports the model assumption that is an increasing function of , i.e. that the availability of legal gun purchases influences the rate of gun-mediated crimes. Another, related study, found similar results [29], and interestingly further pointed out that while a difference between these two groups was observed with respect to crimes involving guns, no significant difference was observed with respect to crimes that did not involve guns. These results demonstrate that the predisposition to engage in crimes remains the same after gun ownership was denied, and that the prevalence of gun-mediated attack depends significantly on the ability of potentially violent people to obtain guns legally, again supporting the core aspect of our model. In addition to these studies, other papers examined how the incidence of intimate partner homicide depended on the legal availability of guns to potential offenders. When states passed a law that denied guns to abusers under restraining orders, intimate partner homicide rates declined by about 7–12% [30], [31] again supporting the notions described above. Hence, our assumption is quite solidly rooted in available epidemiological data.

One aspect we have no data on is how exactly the function rises with . Interestingly, as explored above, axiomatic modeling approaches suggest that our central result continues to hold under all reasonable shapes of this function. More precisely, the optimal policies can be shown to be either a total ban or a guns-for-all policy, as long as the function is concave. Moreover, condition (5) holds for any shape of the function . There are, however, aspects of the present model for which additional information about the shape of function is necessary. Determining the derivative of at will be very important in the context of partial gun control measures. Quantitative estimates of this derivative (that is, the relative rate of change of ) will be crucial for evaluating conditions (12) and (13), which determine whether partial gun control would reduce the rate of gun-related homicides.

The shape of the function , and its derivative in particular, depend on several factors. In the simplest case, one can assume that the fraction of offenders that are not entitled to have a legal gun but get it illegally, , does not depend on , leading to the linear model (2). The opposite assumption however might also be true: as the prevalence of firearms in the general population increases, the fraction of criminals acquiring an illegal gun might increase too. In this case, becomes a nonlinear (concave) function of . The function might also be influenced by other processes. The theft of guns by criminals from legal owners could change the effectiveness of gun policies that limit the sales of firearms. It could reduce the potential of the general population to defend itself, while inducing relatively little change in the rate of attack. To some extent, this was addressed in our axiomatic modeling approach, although more detailed studies will be necessary to fully capture the effect of such a phenomenon. It might also be interesting to consider the possibility that without access to assault rifles, attackers resort to the use of explosives. While this might be relevant to the one-against-many type of attack, it is unlikely to play a role in the one-against-one setting.

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, , on the rate of firearm-induced homicides, which are discussed further below. It is, however, important to note that the prediction about the dependency of the rate of firearm-induced homicides on the gun availability parameter cannot be translated into the expected number of homicides that occur over time in any straightforward way. To draw an analogy, let us consider a hypothetical example about airbag design. We can ask how big the inflated airbag should be in order to minimize the chance of death upon impact. If it becomes too big, it can injure or suffocate the driver. If it inflates too little, it will not protect the driver. Hence, there is some optimum. If we construct a model to calculate the rate at which people get killed during a car crash depending on airbag design, we cannot use this model to fit a time series observed for fatal car accidents. We would have to include other aspects into the model, such as the density of traffic as a function of time, the level of tiredness of drivers or the chance of being intoxicated as a function of time, etc. Similarly, in order to predict gun crime time-series, other aspects would need to be modeled, and it will be those aspects that determine how good the model can fit the data. These are a set of complex factors that go much beyond the scope of our calculations. The occurrence of gun crime as a function of time is complicated, and depends on the time-frame considered [25], [26]. If one looks at this on the time-scale of hours, the number of gun crimes rises during night-time, and falls during the day. If one looks at the time-scale of months, data typically document a clear rise of gun crimes during the summer months. If one looks at the time-scale of years, other, more macroscopic patterns are found, e.g. an overall rise or decline of gun crimes over the years, depending on the exact scenario. Our model was not designed to address these aspects.

Model Parameterization

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.

One-against-many scenario.

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 . This is a growing function of (the number of people involved in the attack) and (the probability for a victim to shoot and kill the attacker with one shot). Further, is a decaying function of , the probability for the attacker to kill a victim with one shot. For a ban of private firearm possession to minimize gun-related deaths, the fraction of violent people that cannot obtain a gun legally but obtain one illegally must lie below the threshold given by condition (10), i.e. . Let us again assume that of the general population owns a gun, and for simplicity that they all carry the firearm with them when attacked (). The dependence of the function on the parameters , , and is studied numerically in figure 2. Parameter , the probability for the attacker to kill a person with one shot, varies between and . Parameter , the probability for an armed person to kill the attacker with one shot, varies between and . The of the right hand side of inequality (10) is represented by the shading, the lighter colors corresponding to higher values. The dependence on parameters and is explored for different numbers of people in the crowd that is being attacked (). For each case, we ask what values of and fulfill the inequality , assuming the estimated value . In the parameter regions where this inequality holds, a ban of private firearm possession will minimize deaths, and outside those ranges, it is advisable that a fraction of the population is armed. For we find that a ban of private firearm possession requires . i.e. the attacker needs to be at least 30 times more efficient at killing a victim than a single victim is at killing the attacker. For it requires . For and , a firearms ban minimizes gun-related deaths for any value of and . Thus, for smaller crowds, condition (10) is easily satisfied and a ban of private firearm possession would minimize deaths. For larger crowds, it is more likely that arming a certain fraction of the population would be the better strategy because the alternative strategy would require that the attacker is unrealistically more efficient at shooting than the defenders.

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Figure 2. One-against-many attacks: when is a ban of private firearm possession the optimal policy?

Presented are the contour-plots of the threshold value with , as a function of and for four different values of . Darker colors indicate smaller values, and the contour values are marked. For each pair of probabilities and , the plots show the highest possible value of still compatible with the ban of private firearm possession being the optimal policy. The black dashed lines on the bottom two plots indicate the approximate location of the contour corresponding to ; above those lines the ban of private firearm poss ession is the optimal solution. These lines are drawn according to the following relationship between and : for , and for . For and , the inequality holds for any values of and , and the ban of private firearm possession is the optimal solution in the whole parameter space.

https://doi.org/10.1371/journal.pone.0071606.g002

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 , i.e. that everybody who can legally own a gun does so and has it in possession when attacked. We go back to the scenario where the crowd is attacked with a hand-gun and examine the effect. The calculations yield the following results. For , a firearms ban requires that , i.e. the attacker needs to be at least 125 more efficient at killing a victim than a single victim is at killing the attacker. For , , and , the conditions are , , and . As expected, the assumption that everybody who legally owns a gun carries it when attacked shifts the numbers towards favoring the strategy that involves arming the general population to a certain degree.

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 [8]–[10], which generates less attention. Therefore, it is likely that the results from the simpler one-against-one scenario are the ones that should dictate policies when the aim is to minimize the overall gun-related homicides across the country.

Future Directions of Research

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 [41].

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 [42]. This abuse of costly punishment could be triggered by easy access to firearms. Integrating such behavioral modeling with the framework presented here could be an interesting question to address in future research.

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 [43], [44].

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 [45]. Similarly, it has been pointed out that processes relating to economic development are quite general and shared across cities, nations and different times, and many diverse properties of cities have been shown to be power law functions of population size [46]. At the present stage, however, it is not clear how this can apply to the current model. For example, both the UK and Switzerland show drastically lower rates of firearm-induced homicides than the US. Private firearm possession is largely banned in the UK, while gun prevalence in Switzerland is wide-spread. Extensive safety training is required for individuals possessing guns in Switzerland and the prevalence of wealth and poverty is significantly different from the US or the UK, making comparisons across different socioeconomic backgrounds very challenging. Future research may shed light onto ways of making the current model more “universal”, enabling to draw comparisons among different populations.

Conclusions

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.