 Open Access
 Total Downloads : 43
 Authors : Ruhul Amin
 Paper ID : IJERTV8IS090045
 Volume & Issue : Volume 08, Issue 09 (September 2019)
 Published (First Online): 14092019
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Mathematical Model of Crime and Unemployment
Ruhul Amin
Associate Professor
Department of Mathematics, West Goalpara College, Assam, India.
Abstract: In this paper, we are interested in possible contribution of mathematical modeling of crime. The concentration of criminal activities is not proportional in every area. Because criminal activities depend on socioeconomic factors like population densities, unemployment, literacy rate, per capita income, schedule castes and schedule tribes etc. There is a correlation between the volume of crime and these socioeconomic factors. The equation of the line of regression is established to interpret the nature of relationship between crimes and unemployment.
Keywords: Volume of crime, Correlation, Regression
INTRODUCTION
This is the simplest mathematical model which can be applied in many cases where relationship among variables actually exists. As for example, the relationship between the number of criminal convictions and the number of unemployed in a particular span of time in a society can be established.
Similarly, volume of crime and literacy, volume of crime and scheduled castes and tribes and volume of crime and per capita income in a certain area also exhibit some relationship.
The above relationship can be expressed in the form of an equation connecting the dependent variable Y and one independent variable X. More precisely, the equation takes the form
Y C BX
(1)
This is called the simplest regression equation, where C and B are said to be the regression coefficients.
Similarly, if more than one variable are considered then the regression equation can take another form. In particular, we already know that criminal activities are somewhat related with population density, per capita income, literacy rate, unemployment and proportion of scheduled castes and tribes etc.
Then the regression equation takes the following form:
Y C B1X1 B2 X2 B3 X3 B4 X4 B5 X5
(2)
Where Y =Volume of crime per million population
X1 = Population density; X2 = Per capita income; X3 = Literacy rate;
X4 = Unemployment;
X5 = Percentage of scheduled castes and tribes; and C and Bi s are regression coefficients.
It is remarkable that the negative sign before the third and fourth terms in relation (2) indicates that the volume of crime reduces for increase of per capita income and literacy rate. Hence there is a negative relationship.
Per capita income and literacy have apparently an inverse relation (negative relation) with crime which suggests that as income levels and literacy rise, crime tends to decrease. The hypothesis is supported to the extent that the bulk of reported crime can be traced to the economically deprived sections and the illiterate on whom the full impact of lawenforcement is felt. It does not necessarily absolve the affluent and the literate from criminality which may assume more subtle forms which do not form part of Penal Code and also have the capacity to defy conventional lawenforcement [7]. On the other hand, unemployment has significantly positive correlation with crime, followed by population density. Although not very significant, the percentage of scheduled castes and tribes appears to have some positive relationship. The marginal significance of this factor can be ascribed to the fairly uniform proportion of this segment in all states. The relationship between the major socioeconomic variables hold good for nearly high percentage of crime under the Indian Penal Code and establishes unemployment as the most significant criminogenic factor.
The above equation no. (2) suggests that the relationship between two variables is such as a change in one variable results in a positive or negative change in the other, also greater change in one variable results in a corresponding greater change in the other, is known as correlation.
LITERATURE REVIEW
The modern mathematical model on crime was initiated by G.S.Beckers model of rational criminal activity[1]. Becker assumed a social loss function which includes costs and benefits of crime. Its minimization determines how many resources and how much punishment should be used to enforce the law.
Isaac Ehrlich developed a model where crime as considered as goods and individuals make rational decisions in the market of crime with a hypothesis a person commits a crime if his expected utility exceeds the utility he could get with legal activities [2][3][4].
Cambel et al. offer a differential methodological approach to the process by which crime rates changes over time [5]. Their approaches is similar that used in mathematical biology to describe how potential epidemics are either spread or contained in a population [8] . Cambel at al. considered the criminal activity as an epidemic problem. They described the dynamic of the crime rate growth by some differential equations.
Another model describing the interaction of three sociological species, termed as Owners, Criminals and Security Guards [6].In this model [Juan C. Nuno et al.] Criminal is the predator for the species Owners and Security Guards is the predator for the species Criminals. On the basis of prepredator model they propose a system of three ordinary differential equations to account for the dynamics of Owners, Criminals and Security Guards.
Some modeler tried to relate crime rates to possible explicative variables through linear regressions [9]. The models assume that crime rate = f(explicative variables), where f(.) is a linear function and the explicative variables considered as average income, gender inequality, age, education level, race etc.
Preliminaries

If x and y are two random variables then the correlation coefficients between x and y is denoted by r or rxy and is defined by
x y xi yi
r
i
i
x2
i i
i
i
x 2
N
N
i
i
y2
y 2
i
i
N
where, 1 r 1, r has not units and is a mere number
If r = 1, then there exist a perfect and positive correlation between the variables x and y. if r 1, then there exist a perfect
and negative correlation between the variables, x and y. The above relation is known as Karl Pearsons correlation coefficients.

The equation of the line of regression of y over x is
y y r y (x x )
x
Similarly, the equation of the line of regression of x over y is
x x r x y y
y
where x and y are the means of the values of x and y respectively. These two relations are known as Equation of line of regressions.
We have already discussed that the criminal activities are related with several factors such as population density, per capita income, literacy rate, unemployment etc.
These factors can be correlated positively or negatively or partially with the help of the regression equation (2). We can apply these mathematical or statistical concepts for the analysis of crime pattern.
Relationship between crime and unemployment
The figure in the following Table 1 gives the number of unemployed and volume of crime in the states of India for the year 1971. We have to find out the coefficient of correlation for the given data. Also we shall find the equation of the line of regression to interpret the nature of relationship between crime and unemployment.
Volume of crime and number of unemployment of India, 1971 Table 1
Sl. No. 
Nae of States 
Number of Unemployed = x (Per Thousand Population) 
Volume of Crime (Per One Lakh Population) 
1 
Andhra Pradesh 
336 
106 
2 
Orissa 
135 
138 
3 
Karnataka 
270 
124 
4 
Tamil Nadu 
459 
144 
5 
Bihar 
420 
147 
6 
Uttar Pradesh 
531 
166 
7 
Gujarat 
171 
121 
8 
Maharashtra 
430 
195 
9 
Assam 
789 
175 
10 
Kerala 
357 
139 
11 
West Bengal 
868 
176 
12 
Haryana 
100 
82 
13 
Punjab 
118 
84 
14 
Rajasthan 
139 
142 
15 
Madhya Pradesh 
315 
211 
16 
Himachal Pradesh 
45 
73 
17 
Jammu & Kashmir 
25 
119 
18 
Tripura 
30 
114 
19 
Manipur 
38 
180 
Source:

Crime in India,Ministry of Home Affairs, New Delhi

Statistical Abstracts, Central Statistical Organization, Government of India, New Delhi.

Labour Bureau, Government of India.
Table 2
Calculation for correlation coefficient
Sl. No. 
No. of Unemployment = x 
Volume of Crime = y (Per One Lakh Population) 

x 
u 
u 2 
y 
v 
v 2 
uv 

1 
336 
21 
441 
106 
33 
1089 
693 
2 
135 
222 
49284 
138 
1 
1 
222 
3 
270 
87 
7569 
124 
15 
225 
1305 
4 
459 
102 
10404 
144 
5 
25 
510 
5 
420 
63 
3969 
147 
8 
64 
504 
6 
531 
174 
30276 
166 
27 
729 
4698 
7 
171 
186 
34596 
121 
18 
324 
3348 
8 
430 
73 
5329 
195 
56 
3136 
4088 
9 
789 
432 
186624 
175 
36 
1296 
15552 
10 
357 
0 
0 
139 
0 
0 
0 
11 
868 
511 
261121 
176 
37 
1369 
18907 
12 
100 
257 
66049 
82 
57 
3249 
14649 
13 
118 
239 
57121 
84 
55 
3025 
13145 
14 
139 
218 
47524 
142 
3 
9 
654 
15 
315 
42 
1764 
211 
72 
5184 
3024 
16 
45 
312 
97344 
73 
66 
4356 
20592 
17 
25 
332 
110224 
119 
20 
400 
6640 
18 
30 
327 
106929 
114 
25 
625 
8175 
19 
38 
319 
101761 
180 
41 
1681 
13079 
u 1207 
u2 1078329 
v 5 
v2 26787 
uv 96271 
Correlation coefficients
uv uv
r n
u2
u 2
v2
v2
n n
96271 (1207) (5)
19
0.586
1078329
(1207)2
26787
(5)2
19 19
Now, Regression coefficient of x on y b r x
r u
xy
uv u v
n
v2
y v
3.58
v2
n
Similarly, Regression coefficient of y on x
b r y r v
yx
x u
uv u v
n
u 2
0.096
0.10
u2
n
Now, the equation to the line of regression of x over y is
x x r x ( y y )
y
x x 3.58( y y)
(3)
Arithmetic average of unemployment

assumed average u
n
357 1207 293.47
19
and Arithmatic average of volume of crime

assumed average v
n
139 5
19
138.74
So the equation (2.3) becomes
x 293.47 3.58( y 138.74)
x 203.22 3.58y
(4)
and the equation to the line of Regression of y over x is
y y r y (x x )
x
y 138.74 0.10(x 293.47)
y 109.39 0.10x
(5)
150
150
Volume of crime(yaxis)
Volume of crime(yaxis)
These two regression equations show that as the unemployment increases the volume of crime also increases.
Correlation between unemployment and crime
Correlation between unemployment and crime
250
250
200
200
50
50
0
0
0
100
200
300
400
500
600
700
800
900
1000
0
100
200
300
400
500
600
700
800
900
1000
Nos. of unemployed(xaxis)
Nos. of unemployed(xaxis)
100
100
Fig. 1 Correlation between unemployment and crime
Fig. 1 Shows r 0 for standard data given in the Table 1
The Fig.1 exhibits that as unemployment increases the volume of crime also increases.
CONCLUSION
The positive correlation coefficient r 0 shows that, the volume o crime increases as the unemployment increases. The two
equations of regression (4) and (5) represent straight line which exhibit that as unemployment increases the volume of crime also increases. The Fig.1 also exhibits the same interpretation.
ACKNOWLEDGMENTS
The author thanks Dr. Atowar Rahman,Associate Professor of the Department of mathematics, B.P. Chaliha College, Nagarbera for many suggestions.
REFERENCES

Becker, G.S. (1968) Crime and Punishment: An Economic Approach, Journal of Political Economy 76: 169217.

Ehrlich, I. (1973) Participation in Illegitimate Activities: A Theoretical and Empirical Investigation. Journal of political Economy, Vol. 18, 521565.

Ehrlich. I. (1975) The Deterrent Effect of Capital Punishment: A Question of Life and Death. American Economic Review 65: 397417.

Ehrlich, I. (1996) Crime, Punishment and the Market for Offences, Journal of Economic Perspectives, Vol.10, No.1, pp. 4367.

Cambel, M. & Ormerod, P. (1998) Social interaction and the dynamic of crime.

Nuno, J. C., Herrero, M. A. & Primcerrio, M. (2008) A triangle model of criminality. PACS: 387, 29262936.

Rao, S. Venu Gopal (1981), Dynamics of Crime Spatial and SocioEconomic Aspects of Crime in India IIPA, New Delhi.

Murray, J.D.(1990) Mathematical Biology: Springer Verlag, Barlin

Gordon, M.B.(2010) Random walk in literature on criminality: A partial and critical view on some statistical analyses and modeling approaches Euro. Jnl. Of Applied Mathematics, Vol.21, pp.325348