Assessing the Risk of Criminal Behavior Using a Mathematical Model

Introduction.

At present, mathematical models become widespread not only in technical sciences, but also in economics, medicine, biology, as well as in the social sciences. However, fewer publi‑ cations based on mathematical models can be found in legal sciences [19; 6]. They offer var‑ ious methods of assessing the risks of recidi‑ vism which can help judges in sentencing [8; 9]. Predictive models are being built [1]. Math‑ ematical methods for processing statistical information [5] and correlation analysis are used. There are also probabilistic prognostic approaches to assessing the risk of recurrence

• [1 8]. Still, predicting the individual criminal behavior is an extremely difficult task [17].

The authors propose a description of an individual’s behavior before and during a crime by numerical methods. Using the dis‑ crete probability distribution law, they develop a mathematical model of criminal behavior for persons who have committed a crime. This quantitative information on the person‑ al criminal behavior seems to be useful for predicting such criminal behavior taking into account external and internal factors which affect an individual’s behavior and can be used to prevent recidivism.

The proposed model is based on the grounds that an individual’s behavior is deter‑ mined by a multitude of negative factors that lead to criminal behavior. In the paper, these factors are formalized in a system of sets. In‑ dividual behavior can be formalized based on simple rules by summing up independent components that have a negative impact on a person. The paper introduces a hypothesis that the probability of committing a crime by an individual can be described by a discrete probability distribution over a known time pe‑ riod. This hypothesis allows us to formulate an equation where the unknown variable is the individual’s risk of committing a crime. Time discretization of a certain period of observa‑ tion of a person allows the resulting integral equation to be reduced to linear algebraic equations that can be easily solved.

In our time of universal digitalization, penetrating into all spheres of life (high‑per‑ formance computers, artificial intelligence, etc.), it seems possible to start creating math‑ ematical models describing the negative be‑ havior of a person prone to offence. Such models can allow us to predict the behavior of an individual prone to offence and assess the danger of this person in relation to other persons committing an offence. The authors try to describe the negative behavior of a per‑ son quantitatively before and during a crime.

The general interdisciplinary task of creat‑ ing a mathematical model of criminal behavior can be represented as:

• 1)    the task of mathematical formalization of the legislative and moral principles existing in society;

• 2)    the task of developing a mathematical model in the form of an integral equation that describes the function of a person’s danger, taking into account the severity of the offence committed,

• 3)    the task of predicting the individual negative behavior on the basis of extrapola‑ tion of social determinants affecting the per‑ sonality.

Material and methods.

To model individual criminal behavior [2; 3], a number of functions that take a numer‑ ical value have been used in the article. This is function C_C ( Characteristic of the Criminal ), a negative characteristic of the person‑ al qualities, which affects the probability of committing a crime. This function depends on time t and is determined by the set of values obtained by observing the person under study. The numerical value of this function can vary within 0 <  C C ( t ) <  1.

Function S d ( t ) ( Social Determinants ) is determined by negative processes in society and increases the probability of committing a crime. This function also has a numerical value within 0 <  S d ( t ) <  1.

From the mechanics of deformation of sol‑ ids it is known that the process of destruction of materials is identified with the process of accumulation of damage in them. Since the beginning of the last century starting with A. Palmgren’s research [13], numerous phe‑ nomenological theories of damage accumu‑ lation have been developed. At present, there is a large number of equations describing the process of damage accumulation, in which a certain number (parameter) reflecting the level of damage in the material is taken as a measure of damage.

The introduced C_C ( t ) and S d ( t ) functions which vary from zero to one are taken by analogy with the damage parameter in the mechanics of materials [10; 16]. The influence of various types of damage on the character‑ istics of the deformation process is known to be carried out with the scalar function ω — a measure of damage. The value of ω varies from ω = 0 for undamaged material to ω = 1 for completely destroyed material.

Function D ( Personality Dangerousness ) is introduced to model individual criminal behavior. This function determines the dan‑ gerousness of a person as the propensity to commit a crime.

Since the criminal’s personality can differ in its suggestibility or susceptibility to various social processes, coefficient s (Susceptibility) is used. This coefficient is determined by the psychophysiological characteristics of the personality and also has a quantitative value within 0 < s < 1. At the same time, it is believed that at s = 0 the person is not suscepti‑ ble, while at s = 1 they are strongly susceptible to various processes in society.

Thus, it is assumed that any person at each moment of time is under the influence of formed negative personal qualities and a number of social negative processes in soci‑ ety. Taking into account different levels of sus‑ ceptibility of a person, the negative “burden” on a person can be represented as a sum of influences C c ( t ) + sS d ( t ) that act on a person at each moment of time. Consequently, this function depends on the surrounding impact, personal qualities and the degree of percep‑ tion of the individual.

G ( t ) = ( C c ( t ) + sS d ( t ))/2. (1)

The mathematical formalization of the problem of determining the individual crim‑ inal behavior function D can be reduced to Fredholm’s integral equation of the first kind [14]. b

J K (x, О D (U d ^ = f (x), (2) a where f(x) and K (x, ^) are known functions; D (^) is an unknown function; variable x, as well as ^ are located in the interval (a, b). Function K(x, ^) is called the core of the integral equation [14]. For many problems of mathematical physics, function K (x, U has a visual physical interpretation. For example, in the theory of shells, it is Green’s function, which determines the deflection of the shell at point x from the force acting at point ^ [4; 7; 11].

Resolving Equation and Accepted Hypotheses.

In criminological studies, there is a concept of the risk of committing a crime [12] (Risk Asses sment). The risk is known to be a quan‑ titative measure of danger or the frequency of danger realization, the probability of one event when another occurs. Therefore the risk similar to the probability of committing a crime can be estimated as a measure of less value 0 < R < 1. It is obvious that the risk of committing a crime R will be proportional to the multiplication of the individual criminal behavior function and the function of negative “burden” on a person.

Since the risk of a crime is a probabilistic value, we define it as the probability of com‑ mitting a crime P under the prevailing condi‑ tions, and, consequently,

G x D = kR , (3)

where k is the coefficient of proportionality.

The meaning of equation (3) is that the risk of committing a crime is greater for a per‑ son with greater function D ( Personality Dangerousness ) and with more pronounced individual negative characteristic.

If a crime has occurred, the probability of its commission is equal to one.

P = 1. (4)

The set of personal characteristics and the range of social negative determinants have a maximum possible numerical value of 2, therefore, we get that the proportionality coefficient k = 1, while the function of the in‑ dividual danger D at the crime time under the most unfavorable external conditions and the maximum set of negative personal character‑ istics is equal to one.

D = 1. (5)

In case of more favorable external condi‑ tions, we have G < 1, therefore, the function of personality dangerousness when committing a crime can be greater than one D > 1.

The process of preparing and committing a crime is known to be a dynamic process which depends on a variety of events. There‑ fore, the probability of committing a crime will depend on the accumulated negative “burden” on a person over a period of time T L . Let us assume that the risk of committing a crime during a lifetime is equal to the probability density p of committing a crime. Thus, the mathematical model of committing a crime can be presented as an accumulation of neg‑ ative characteristics of a person leading to an offence

T L

J G ( t , t ’) D ( t *) dt * = p ( t ). (6) 0

Equation (6) is an integral equation that contains the desired function of the person‑ ality dangerousness D under the sign of the integral [14]. By analogy with equation (2), let us introduce a fixed time t^* in which informa‑ tion about the person under study is collected.

If the law of distribution of the probability density of a crime is known, then equation (6) is analogous to Fredholm’s integral equation of the first kind (2).

A System of Resolving Linear Equations Taking into Account the Discrete Law of the Distribution of Probabilities of Committing a Crime.

Since it is not possible to determine the core of integral equation (6) analytically from any social regularities, this equation can only be solved by numerical methods.

It is known that the integral equation can be reduced to a system of linear algebra‑ ic equations [4; 14]. Let us divide the period of a person’s life TL into L intervals in time (day, month, year, …). We study a period of life 10 < t < tend before the crime in which there is quantitative information on various negative social processes in society, and a quantitative negative status of the person’s characteristics. Therefore tend

J G ( t , t ’) D ( t *) dt * = p ( t ).         (7)

t 0

This observation period consist of k time intervals, and their distance is tend - t0 k

= A x = A^ .

Let us present [14]

G(10 + iAx, 10 + jA^ = Gij,  D(10 + iAx) = Di, p (10 + iAx) = pi (i, j = 1... k).(9)

Let us replace the integral tend j G(x, £)D(Qd^ with the sum £ GijDjA^, tj

0                   (i = 1... k).(10)

Then, instead of integral equation (7), we get a system of linear algebraic equations

£ GijDjA^ = pA.

j = 1

The system (11) in the matrix form looks like

• [ G ](D }T={ P }T.(12)

The elements gij of the matrix [G] are determined by the information on various negative social processes and negative char‑ acteristics of a person. The elements di of the vector {D} determine the unknown dis‑ crete meaning of the danger of the person in different time intervals of observation of a person. On the right side of the equation the elements pi of vector {P} determine an unknown discrete meaning of the probabil‑ ity of committing a crime in different time intervals. Let us define the probability that the crime will occur in the last time interval of observation as pk = p.                  (13)

Let us determine the probability of com‑ mitting a crime in the time interval preceding the crime. The probability that the event will not occur in the last time interval k , but will occur in the preceding time interval k – 1 can be defined as [15]

p k - 1 = (1 — p ) p .             (14)

The probability of committing a crime in the first time interval of observation is equal to [15]

p 1 = (1 - p ) ^{k - 1} p .            (15)

Taking into account the relations (13)–(15), the system (12) can be represented as

’ g 11	g 12    -	g 1 k ’		’ d 1 "		(1-p)k-1 p
g 21	g 22    •••	g 2 k	X	d 2	=	(1-p)k-2 p
gk - 1,1	g k - 1,2   •	g k - 1, k		• dk - 1		• (1-p) p
[ gk. 1	g k 2   •	g kk ]		_ d k ]		_ p _

Since we limit the period of observation of a person to time intervals k , it seems that the ratio (4) may not be fulfilled exactly, but by some probability p^* < 1.

To solve the system (16), it is necessary to find the probabilities of committing a crime by time intervals from the equation p+(1 - p) p+(1 - p )2 p+....+(1 - p)k-1 p = p

Thus, after the solution of the system (16), the value of the personality dangerousness function d i will be determined at different time intervals and at the time of committing the crime.

Potential Function of Personality Dangerousness.

The value determined by the proposed math‑ ematical model d k quantitatively characterizes the dangerousness of a person, i. e. the pro‑ pensity to commit a crime over the observed period of time. Consequently, the higher the value d i , the more dangerous the person under study. And if the negative processes in society and the negative characteristics of a person were smaller, perhaps the crime would not be committed. An offence is known to be charac‑ terized by the gravity of the crime committed. Therefore, in order to determine the poten‑ tial (maximum possible) value of the danger‑ ousness function of a particular person it is necessary to take into account the gravity of the crime committed by the person. We de‑ termine the gravity of a crime based on four indicators, although there may be other ways to differentiate these indicators. In case of us‑ ing mathematical methods in the study and crime prevention, the following quantitative indicators of the gravity of crimes have been used in this work: m = 1 — of little gravity; m = 2 — of medium gravity; m = 3 — grave crimes; m = 4 — especially grave crimes.

Applying the linear law and using the indi‑ cator m for persons who have already commit‑ ted an offence, it is possible to determine the potential value of personality dangerousness function D c as

D c = d_k x m ¹ x 4. (18)

Thus, value D c characterizes the potential danger of any person, while value D c can be used for description of and comparison with the dangerousness of other offenders.

On the basis of social determinants ex‑ trapolation as well as the knowledge of kinet‑ ics of the personal characteristics in a certain time period after an offence, for example, the term of a conditional sentence, the risk of re‑ cidivism can be assessed by the value of a per‑ son’s dangerousness function D c .

The choice of the time period for which the system of equations (16) is drawn up and the number of equations (discretization is the time interval of the selected period) will af‑ fect the final result of the solution and, conse‑ quently, the found personality dangerousness function. It seems that the choice of the time period of life for which the system of equa‑ tions (16) is drawn up can be limited to 2–5 years before the crime.

An Example of Determining the Function of Potential Danger of Two Personalities1

Let us determine the personality dangerous‑ ness function for two persons who committed crimes in 2014 and 2019 and received condi‑ tional sentence. Let us limit ourselves to five equations ( k = 5) in the system of equations (16). Therefore, the observation period of the subject will be from 2010 to 2014 and from 2015 to 2019 (the time period for collecting statistical information is one year t i = 1).

In order to solve the system (16), it is nec‑ essary to define the elements g ij of the matrix [ G ] that depend on C c and S d .

To determine the function of personality dangerousness Cc, let us assume that the per‑ sonality traits are characterized by a certain number of diskette values n1 obtained from the observation over the person. For the above mentioned cases let us assume that a person‑ ality can be characterized by 20 (n1 = 20) val‑ ues that contribute to committing a crime. Since the maximum value of Cc = 1, and as‑ suming that each factor has the same impact on committing a crime, each component has the “weight” of CCi = CC x n1-1 = 0.05. This assumption implies that if a person has all 20 negative factors, the probability of committing a crime is maximized and is equal to one.

The negative factors are listed below for criminal cases 1 and 2, respectively.

• 1.    Psychopathy and psychopathic states: 0 and 0.05

• 2.    Organic lesions of the central nervous system, brain contusions, mental anomalies and neurotic diseases: 0 and 0.05

• 3.    Alcoholism and narcotization: 0.05 and 0.05

• 4.    Emotional instability: 0 and 0

• 5.    Social degradation, marginalization: 0 and 0.05

• 6.    Ethnic, religious intolerance: 0 and 0

• 7.    Belonging to a criminal subculture: 0 and 0.05

• 8.    Belonging to other marginal groups, in‑ formal groups with illegal interests: 0 and 0.05

• 9.    Low level of material security: 0.05 and 0.05

• 10.    Lack of work: 0.05 and 0.05

• 11.    Unskilled employment: 0.05 and 0

• 12.    Unmarried: 0.05 and 0

• 13.    Lack of secondary education: 0.05 and 0

• 14.    Criminal record: 0 and 0.05

• 15.    Bringing to criminal liability: 0 and 0.05

• 16.    Lack of housing: 0 and 0

• 17.    Belonging to urban population: 0.05and 0

• 18.    Cynicism, legal nihilism: 0 and 0.05

• 19.    Unresolved life conflicts: 0 and 0.05

• 20.    Facts of domestic violence: 0 and 0

Therefore, we assume that the personal characteristics during the five years of obser‑ vation are constant and equal to С c = 0,35 and С_c = 0,6.

The function of the influence of nega‑ tive social processes on the personality un‑ der study Sd is also characterized by a certain number of diskette values n2 obtained from statistics. Let us assume that a person can be influenced by four (n2= 4) negative so‑ cial factors, which in a known way correlate with general crime. They are: international migration, deficit of income, the number of unemployed aged 15–72 and urbanization. Information on these processes for the obser‑ vation period from 2010 to 2019 is available on the website

Statistical information on internation‑ al migration S d ^* in the number of people for 2010–2019 is calculated according to the for‑ mula

S di = -i —min , i = 1....5,     (19)

max min where si, smax, smin, current, maximum and min‑ imum values of international migration.

Below is the normalized distribution of international migration S d ^* calculated using the formula (19).

For case 1: 2010 — 0; 2011 — 0,413; 2012 — 0,566; 2013 — 0,727; 2014 — 1.

For case 2: 2015 — 0,242; 2016 — 0,069; 2017 — 0,172; 2018 — 0; 2019 — 1.

Similar to international migration, statis‑ tical information is processed on the deficit of income (in billion rubles), the number of unemployed aged 15–72 years (in thousands) and urbanization (in thousands). Below is in‑ formation on the normalized distribution of income deficit S_d ^** :

For case 1: 2010 — 0,4; 2011 — 0,477; 2012 — 0; 2013 — 0,415; 2014 — 1.

For case 2: 2015 — 0,119; 2016 — 0,123; 2017 — 0,154; 2018 — 0; 2019 — 1.

the number of unemployed S_d ^*** :

For case 1: 2010 — 1; 2011 — 0,624; 2012 — 0,145; 2013 — 0,149; 2014 — 0.

For case 2: 2015 — 1; 2016 — 0,974; 2017 — 0,629; 2018 — 0,243; 2019 — 0.

And urbanization S_d ^**** (total urban pop‑ ulation growth):

For case 1: 2010 — 0,355; 2011 — 0; 2012 — 0,5050; 2013 — 1; 2014 — 1.

For case 2: 2015 — 1; 2016 — 0,999; 2017 — 0,696; 2018 — 0,066; 2019 — 0.

Thus, the coefficients of the matrix [G] are determined, i. e. the functions of influence gij = Ccig + Sig x Sdig .          (20)

However, since the argument in equation (7) is time t , and the system under study is not closed, the theorems of Maxwell and Betty‑ Rayleigh on the reciprocity of displacements and works for mechanical systems are violat‑ ed [20]. Therefore, the coefficients in the sys‑ tem (16) are g ij * g ji . This is due to the fact that for the task under consideration, when an event (the function of influence g ij ) affects only its own time interval and subsequent time intervals the system (16) therefore takes the form of

’ g 11	0	.     0 ’	Г d 1 '			(1-p) k-1 p
g 11	g 22	0	x	d 2	=	(1-p) k-2 p
g .. L g 11	g 22 g 22	0 • gkk _		d k - 1 L d k J		(1-p) p L p J (21)

Thus, thus, after solving the system (21) the personality dangerousness function for the first criminal case by years for the period of 2010–2014 is distributed as follows:

d 1 = 0.0136, d 2 = 0.039, d 3 = 0.0166, d 4 = 0.5713, d 5 = 2.086.

The personality dangerousness function for the second criminal case by years for the period of 2015–2019 is distributed as follows:

d 1 = 0.00797, d 2 = 0.02417, d 3 = 0.10065, d 4 = 0.45489, d 5 = 1.5468.

Thus, taking into account the little gravity of the crime for both criminal cases ( m = 1), the potential danger of the persons in ques‑ tion is D c = 8.344 and D c =6.187 respectively. As mentioned above, value D c is a quantitative characteristic of a person by which it is possi‑ ble to assess their potential danger, and com‑ pare the potential danger with other persons who have committed a crime.

Conclusions

The authors of this work have determined the personality dangerousness functions for two offenders who committed crimes and received conditional sentence. From the obtained val‑ ues it can be seen that in the last period of life (5 years) the dangerousness function of the studied personalities dynamically increases. And in the year of committing the crime it reaches a certain critical value for these per‑ sons. If the established individual character‑ istic for the studied persons increases, it can lead to relapses.

The values of the potential dangerousness function of the studied personalities have been determined, which take into account the grav‑ ity of the crime committed by them. By these values it is possible to estimate the possible dangerousness of the person and compare it with the dangerousness of other convicts.

The accuracy of a certain value of the per‑ sonality dangerousness function (actual and potential) strongly depends on the amount of information about personal qualities and so‑ cial determinants for a certain period of time, as well as the time period of observation and its discretization.

The proposed model will make it possible to model behavior and identify the possible risk of recidivism among persons who have received conditional sentence and, thus, to effectively regulate social intervention and support.

There are many factors characterizing negative social processes and negative per‑ sonality traits that contribute to the commis‑ sion of crimes, including motives and even unconscious attitude. The more factors we take into account in the model, the shorter the time intervals are taken for observation, the more reliable indicators of the personality dangerousness we will have.