MATHEMATICAL MODEL ANALYSIS OF CRIME DYNAMICS INCORPORATING MEDIA COVERAGE AND POLICE FORCE

In this paper, a mathematical model is proposed to recognize the dynamics of crime. Unlike some of other previous model, we have taken into account the impact of media coverage, police force and moral/religious activity on crime. Some fundamental properties of the model including existence and positivity as well as boundedness of the solutions of the model are investigated. The model exhibits two equilibria: the crime-free and the persistent equilibrium points. We sufficiently analyze asymptotic behavior of the solutions which depends on the basic reproduction number. Numerical simulation is carried out using Ode45 of Matlab, sensitivity analysis of the basic reproduction number is also constructed.


INTRODUCTION
Several authors have presented empirical as well as practical study on the generation and prevention of crime. Crime is an illegal act for which someone can be punished by the government. The possible contributions of mathematical modeling of crime has been nicely reviewed in [17]. Iglesias et al., [38] presented an economic analysis of crime based on the idea that a crime results from a trade off between exported benefit and the risk of punishment. In particular, the trade off between crime and punishment has been studied. H. Zhao et al., [19] proposed a mathematical model to study the interplay between criminality and poverty, and explored the possibility of crime control via government interventions.Vito GF et al., [36] proposed the prevention of juvenile delinquency is an essential part of any crime prevention program in the society. Modeling of delinquent behavior using an infectious disease approach has been done for fanatical and violent ideology [4,7,30] and the transmission of violent crime and burglary in the United Kingdom [5]. Becker et al., [15], has suggested that people drift towards criminality when the benefits of crime are more than potential punishment. The presence of police increases the chances of punishment and hence the expected cost of committing crime is less. For effectual prevention of crime, the increment in the police force should be made with the upsurge in crime rates. It is evident from the literature that the increased level of police has a substantial effect to reduce crime in society [34]. Hence, making additional recruitment in the police force in accordance to the level of crime in the society is a rational strategy for crime control.
Many of the social problems are assumed to be contiguous like an epidemic [21,23]. The media coverage is obviously not the most important factor responsible for fighting the transmission of the infectious disease, but it is a very important issue which has to be considered seriously [1].
Voluminous literature is available on epidemic models comprising the transmission and control of infectious diseases [18,29]. Some of epidemiological studies have also focused on the effect of immigration of infectives on the dynamics of disease [13,14]. Using the same approach, a mathematical model to asses the effect of police force on the prevailing of crime in the society is proposed by A.K.Misra [2]. It is considered that criminals in the society increase when people living in that region get involved in criminal activities due to contact with criminals.
In this research [2] importance of media coverage, non criminal individuals, susceptible individuals those are not aware about crime, susceptible individuals those are aware about crime moral/religious situation in the society are not considered. Here, we try to fill this gap in our research. Crime free and persistent equilibrium points are also computed. In this paper, we seek to understand the effects of media coverage, informal learning and moral/religious activity in the transmission of crime. The rest of the paper is organized as follows: the model is formulated in Section 2, existence and stability analysis of the endemic equilibrium is done in Section 3, numerical simulations and discussion are carried out in Section 4. The paper ends with a conclusion in Section 5.

MATHEMATICAL MODEL DESCRIPTION
In this thesis we extended the A.K.Misra model SCR [2] to the form S u S a CRQ where the total population N is divided into unaware susceptible class(S u ), aware susceptible class(S a ), class of criminal(C), class of non-criminal(Q) and class of prisoner(R). The model assumes that unaware susceptible class(S u ) increases by a constant recruitment rate σ Λ, decreases by a rates β S u C and ηS u due to interaction with criminals and family advice awareness creation, respectively. The aware susceptible class (S a ) increases by a constant recruitment rate (1 − σ )λ which is the remaining fraction rate, by rate ηS u and by rate (1 − θ )vR which is the jail leaving fraction rate, and it decreases by rate τS a due to moral or religious case. The criminal class(C) increases by rate β S u C and decreases by efficacy ρ of media coverage, and by jail rate γCP using legal authority or police force. The prisoner class(R) increases by imprisonment rate γCP and decrease by jail leaving rate (vR) where 1 v is prisoner period in jail. The non-criminal class Q increases by rates θ vR and τS a where θ is a proportion. The natural death rate µ is the same for compartments S u , S a ,C, R and Q. The police force(P) is incorporated explicitly in the model. It increases with rate φC which is proportional to the criminal C and decreases by rate φ 0 (P − P 0 ) due to retirement or natural death.
In addition to the above, we also consider the following assumptions: i. An individual can be criminal only through contacts with criminal individuals.
ii. The non-criminal individuals do not prone to any criminal activities.
iii. The police force do not commit any crime.
The flow chart of the mathematical model is explained as follows:

FIGURE 1. Flow chart of mathematical model
Considering the assumptions, the dynamics of the crime is described by using the following system of differential equations(see table 1 for the description of the involved paramers): where S u (0) > 0, S a (0) > 0,C(0) ≥ 0, R(0) ≥ 0, Q(0) ≥ 0, P(0) > 0, and 0 < θ < 1, 0 < σ < 1, and ρδC m+C is the reduced rate of contact with criminals due to media coverage.      Proof. From the the first equation of the model system we have, After integration, which is positive for all time t ≥ 0.
The positivity of the remaining state variables can be proved in the same way. Let N be the total population, N (t) = S u + S a +C + R + Q. Consider the set The theorem below establishes the boundedness of the solution.
Proof. Here, one can show that if N (t) = S u + S a + C + R + Q, then it follows that N(t) ≤ max N(0), Λ µ . On the other hand dP dt = φC − φ 0 (P − P 0 ) implies that P(t) ≤ max {P 0 , P(0)}. Hence, the solution of the model system is bounded in the region Ω.

Equilibrium points i) Crime-Free
Equilibrium Point E c f e : Crime-free equilibrium point E c f e is steady state solution, where there is no crime in the society, i.e., C = 0.
Thus, the crime free equilibrium point of the model is obtained to be Basic Reproduction Number (R 0 ): The basic reproduction number (R 0 ), which is important for the qualitative analysis of the model, is obtained by using the next generation matrix [37].
Rewriting the system of the model starting with the criminal compartments for both population gives as For the model under consideration, using notation X = (C, R) we have the vector functions The Jacobian matrix at the crime-free equilibrium point J F E c f e , J V E c f e : This represents the average number of secondary criminal cases generated by a single criminal during his or her entire life as criminal when introduced into a completely susceptible population.
ii) Crime-Persistent Equilibrium Point (E cpe ): Crime persistent equilibrium point, E cpe , is steady state solution where the crime persists in the population. If E cpe (S * u , S * a ,C * , R * , Q * , P * ) is crime-persistent equilibrium point, it satisfies the algebraic equations , and P * = φC * +φ 0 P 0

STABILITY ANALYSIS
Theorem 3. The equilibrium solution E c f e of the model system of equation (1)-(6) is locally asymptotically stable if R 0 < 1.
Proof. The Jacobian matrix J at the state variables is given by After simplification the eigenvalues of the matrix, J(E c f e ), is found as All the eigenvalues have negative real part if R 0 < 1. Hence, the theorem.
Global stability of the crime free equilibrium point E c f e : Based on Iggider [22] we write our system (1)-(6) in the form where Y n = (S u , S a , Q, P),Y i = (C, R). The components of Y n denote the non criminal individuals and the components of Y i denote the criminal individuals, and Y E c f e ,n is vector at crime free equilibrium point E c f e of the same vector length as Y n . According to Iggider [22] the crime-free equilibrium point is globally asymptotically stable if the following conditions hold: (i) A should be a matrix with real negative eigenvalues.
(ii) A 2 should be a Metzler matrix.
Theorem 4. The crime free-equilibrium point, E c f e , is globally asymptotically stable if R 0 < 1.
Proof. Referring to system (1)-(6) we define Equation (13) together with the model system (1)-(6) is written to the form: where the matrices A, A 1 and A 2 make the above equations meaningful.
Using the non-criminal elements of the Jacobian matrix of system (1)-(6) and the representation in eq'n (13) we calculate Hence, the sufficient conditions are satisfied. Therefore, the crime free equilibrium point E c f e is globally asymptotically stable if R 0 < 1.
Local stability of crime persistent equilibrium point (E cpe ): Theorem 5. If R 0 > 1, then the persistent equilibrium point, E cp f , is locally asymptotically stable.
Proof. The stability of the persistent equilibrium point is determined based on the signs of the eigenvalues of the Jacobian matrix which is computed at the crime persistent equilibrium.
The coefficients of the characteristic polynomial equation a 0 , a 1 , a 2 , a 3 , a 4 , a 5 are real positive.
More over the first column of the Routh-Hurwitz array have the same positive sign. Hence, by Routh-Hurwitz criteria all eigenvalues of the characteristics polynomial equation are negative.
Therefore the crime persistent equilibrium point is locally asymptotically stable if R 0 > 1.
Bifurcation: For all values of R 0 , the E c f e always exists and is never destroyed. When R 0 < 1 the crime free equilibrium point is stable and there is no other equilibrium point. As soon as R 0 > 1, the E c f e becomes unstable and a new equilibrium point, the crime persistent equilibrium point, E cpe , is created and is expected to be stable. Moreover for R 0 = 1, E c f e = E cpe . Hence a transcritical bifurcation occurs in the model at the bifurcation point R 0 = 1.

NUMERICAL SIMULATION AND DISCUSSION
In this section, we present numerical simulations to validate our analytical findings and stabil- From figures 5a and 5b we observe aware susceptible population S a (t) decreases while the noncriminal population C(t) increase as τ increases and from figures 6a and 6b we observe as media coverage increases aware susceptible population S a (t) increases while criminal population C(t) decreases. The figure 7 describes R 0 decreases as the baseline police force P 0 increases. In this it is observed P 0 attains its critical value P c which is required for the eradication of crime. From here it is concluded that for the establishment of crime free society, the baseline police force P 0 must be greater than the critical value P c .   Numerical simulation of sensitivity analysis: In this part we observe the influence of parameters on the reproduction number R 0 . The absolute change of R 0 with respect to P 0 , τ and γ is given by ∂ R 0 ∂ P 0 < 0, ∂ R 0 ∂ τ < 0 and ∂ R 0 ∂ γ < 0, respectively. This implies R 0 decreases as P 0 , τ and γ increase.
The normalized forward sensitivity index M I of a variable R 0 that depends on a parameter S, as [32] is defined as Since we have explicit formula for reproduction number R 0 in equation (11) it follows that M I > 0 for the parameters: Λ, σ , β , δ and M I < 0 for the parameters: τ, µ, η, γ, P 0 and α. Increasing awareness on media coverage, incarceration rate and base line police force are best strategies to reduce R 0 . We have numerically computed the relative sensitivity of R 0 with respect to the above parameters (using the parameters values in Table 2) and have displayed the results in Table 3.

Parameter Sign Sensitivity indices
From table 3, the negative signs indicate parameters are inversely proportional to R 0 while the positive sign parameters are directly proportional to R 0 . Model parameters whose sensitivity index values are near −1 or 1 suggest that a change in their magnitude has a significant impact on either increasing or decreasing the size of R 0 , respectively. We noted also that R 0 is sensitive to the paramers γ, P 0 and τ. An increase(decrease) in γ, P 0 and τ by 10% will decrease (increase) R 0 by 9.247% and 9.371%, respectively. Further, an increase (decrease) δ by 10% will increase (decrease) R 0 by 9.98%. A similar change in β , α, µ, η, σ has its own effect on R 0 .

CONCLUSION
Crime affects everyone in one way or the other. It is observed that criminal behavior is contagious like epidemics and spreads through peer influence. Thus, the epidemic modeling approach can readily be applied to model the dynamics of crime in the society and its control. In view of this, unlike some of other previous model, we have taken into account the impact of media coverage, police force on crime and moral/religious activity and we also included additional compartments. We have considered that criminals in the society increase due to interaction of criminals with people having a tendency to commit a crime. The police force discourage crime in the society by thesaurussing criminals. Crucial mathematical features which include; Wel-posedness, positivity of the solution, existence and stability criteria of the crime-free(E c f e ), the persistent equilibria(E cpe ) have been derived in terms of the basic reproduction number, R 0 .
Stability of the disease free and endemic equilibrium is studied. The persistent equilibrium point is determined and shown to be locally asymptotically stable when the threshold parameter value, R 0 , is greater than unity. The results of the crime free equilibrium showed that the model is both locally and globally stable when R 0 < 1, thus reducing R 0 to less than unity reduces the crime spread. The numerical analysis shows that in the presence of media coverage, police force and moral/religious the crime dies out faster while lack of these reporting the presence of the crime and preventive measures greatly increases the number of criminal people in the population which is not encouraging for the eradication of the crime.

RECOMMENDATION
The model that we adjusted has not carried out optimal control and cost effectiveness of different crime intervention strategies, which can be investigated in future to find out which strategy is the top in the control of the crime.