OPTIMAL CONTROL OF A COMMUNITY VIOLENCE MODEL: COMMUNITY VIOLENCE TREATED AS A CONTAGIOUS DISEASE

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. Violence was, for a long time, misunderstood and misdiagnosed. This misdiagnosis led to ineffective and counterproductive treatments and control strategies. Recent advances in neuroscience and epidemiology show that violence is a contagious disease. In this paper, community violence is treated as an infection that spreads from person to person through victimization or through witnessing violence. A compartmental model is used for formulating the spread of cummunity violence as a system of differential equations. The distribution of treated individuals is considered as a control variable. Our objective is to characterize an optimal control (treatment) that minimizes the number of individuals who use violence and the cost associated with this treatment. A numerical simulation analysis is used to confirm the effectiveness of our control.


INTRODUCTION
Violence is the utilization of actual power to harm, misuse, harm, or destroy [1]. Community violence covers viciousness among colleagues and outsiders, and incorporates youth brutality; attack by outsiders; violence related with vandalism related misdemeanors; and violence in working environments and different institutions [2]. There is a solid connection between degrees of violence and modifiable factors in a society, for example, concentrated poverty, earning and gender inequality, the destructive utilization of alcohol, and the absence of safe, stable, and supporting family climate. Moreover, violence frequently has deep rooted ramifications for physical and emotional wellness and social working and can slow economic and social development [1]. Violence meets the word reference meaning of a disease, and many studies currently affirm that violence is contagious [3]. The specific contagion of violence is started by victimization or visual exposure and intervened by the brain, similarly as the lungs intercede replication of tuberculosis or the intestines cholera. The brain processes violence exposure into scripts, or replicated practices, and oblivious social assumptions. This processing can likewise prompt a few situationally versatile reactions including aggression, impulsivity, depression, stress, excessive startle reactions, and changes in neurochemistry [4]. [5] used compartmental model to describe the spread of criminal gang membership, and [6] used compartmental model for the analysis of domestic violence. But none of them incorporated the spatial diffusion which plays an important role into the spread of violence. In this paper, we treat community violence as an infection that multiplies through exposure (victimization or witnessing violence). The violence is controlled by identifying and helping individuals at the highest risk factors (drugs, alcohol, poverty, poor education, family structure, . . . [4]) to make them more averse to submit viciousness by talking in their terms, examining the expenses of utilizing violence, and assisting them with getting the help and social services (e.g., education, job training, drug treatment) towards behavior change and changes in life course [7]. The rest of this paper is structured as follows: In section 2, we show the mathematical model. Section 3 is about the associated optimal control problem. In section 4 we prove the existence of an optimal solution. Then we formulate the necessary optimality conditions in section 5. The numerical results are showed in section 6. Finally, we give the conclusion of the paper in the 7th section.

MATHEMATICAL MODEL
In this work, the total population (T) is divided into three compartments: (S) susceptible individuals, (H) susceptible individuals with high risk factors to use violence and (V) individuals who use violence. Let T (t) be the total population at an instant t ∈ [0, τ].
We assume that: • Violence is contageous, • An individual can only be infected by violence through contact with violent individuals, • The population is uniformly mixed, • The population tend to move to regions, • The densities depend on time and position in space since the population tend to move to regions (leading to the notations S(t, x), H(t, x) and V (t, x)).
Let us define some parameters used in this model: r : Recovery rate (recovering from violence) β : Rehabilitation rate (from (V) to (S)) (some individuals who recover from violence return to the (S) compartment) c : Rehabilitation rate (from (H) to (S)) (some (H) individuals return to the (S) compartment) The susceptible individuals can become violent at a rate α 2 V S T . The rate at which susceptible individuals with high risk factors become violent is α 3 V H T . Some violent individuals return to the previous states at a rate rV and the proportion of those who become susceptible with high risk factors is (1 − β ). Some susceptible individuals with high risk factors return to the susceptible state at a rate cH (without the intervention seen above) and in the other direction the rate at which a susceptible individual move to the second state is α 1 S.
We obtain the following system of reaction-diffusion equations as a spatiotemporal SHV model for the spread of community violence: (1) with the homogeneous Neumann boundary conditions where Ω is a fixed and bounded domain in R 2 with smooth boundary ∂ Ω , the time t belongs to a finite interval [0, τ], while x varies in Ω .
The initial distribution of the three populations is supposed to be:

OPTIMAL CONTROL PROBLEM
In this paper, the community violence is controlled by identifying and helping susceptible individuals with high risk factors to obtain the support and social services, so we include a control u in model (1) where u(t, x) represents the density of beneficiaries per time unit and space, and we assume that they are transfered directly and immediatly to the susceptible class.
The controlled system is given by: In this current work, we want to minimize the density of the violent individuals and the cost of treating susceptible individuals. The objective functional can be given by: where u belongs to the set U ad of admissible controls

EXISTENCE OF GLOBAL SOLUTION
In this section, we will prove the existence of a global strong solution of the problem (4)-(6). and let A be the linear operator defined as follows: If we consider the function f defined by : then the problem (4)-(6) can be written in the space H(Ω ) under the form: Theorem 1.

EXISTENCE OF OPTIMAL SOLUTION
In this section, we will prove the existence of an optimal solution of the problem (4)-(8).
Since u n is bounded in L 2 (Q) then it has a weakly convergent subsequence, denoted again u n , so u n → u * weakly in L 2 (Q) and u n y n 2 → u * y * 2 weakly in L 2 (Q). We also know that U ad is a closed and convex set in L 2 (Q). It follows that U ad is weakly closed, so u * ∈ U ad . By passing to the limit in L 2 (Q) as n → ∞ in (18)-(21), we deduce that y * is the solution of (1) -(3) corresponding to u * . And since J(y * , u * ) ≤ inf u∈U ad J(y, u), then (y * , u * ) minimizes (7).

NECESSARY OPTIMALITY CONDITIONS
Let us prove first, that the mapping u → y(u) is Gateaux sifferentiable with respect to u * ( where y(u) is the corresponding solution of (1) -(3) corresponding to u).

S(t − s)Fu(s)ds
Since the coefficients of the matrix E ε are bounded uniformly with respect to ε and using Gronwall's inequality, we deduce that there exists a constant C 1 > 0 such that: Thus, z ε i are bounded in L 2 (Q) uniformly with respect to ε. And since we have: then y ε i → ε→0 y * i in L 2 (Q) for i = 1, 2, 3. By denoting : admits a unique solution given by : By subtracting (38) from (32) we obtain :

s)) z(s)]ds
Since all the the elements of the matrix E ε tend to the corresponding elements of the matrix E in L 2 (Q) and by using Gronwall's inequality, it follows that: Thus, we have proved the following result: Proposition 3. The mapping y : U → W 1,2 (0, τ; H(Ω)), with the conditions of the theorem 1, is Gateaux differentiable with respect to u * . For u ∈ U, z = y (u * )u is the unique solution in W 1,2 (0, τ; H(Ω)) of the following problem : Moreover, the dual system associated to the system (4)-(8) is : where p = (p 1 , p 2 , p 3 ) is the adjoint variable, (y * , u * ) is the optimal pair, and the matrix is Under hypotheses of Theorem 1, if (y * , u * ) is an optimal pair, then the system (38) admits a unique solution p ∈ W 1,2 (0, τ; H(Ω)) with p i ∈ L(τ, Ω) for i = 1, 2, 3. Moreover, Proof. By making the change of variable s = τ −t and the change of functions q i (s, x) = p i (τ − s, x) = p i (t, x) for (t, x) ∈ Q,we can prove, with the same method used in the proof of Theorem 1, that the system (38) admits a unique solution p ∈ W 1,2 (0, τ; H(Ω)) with p i ∈ L(τ, Ω) for i = 1, 2, 3.
Let us now prove the second part of the theorem. Let (y * , u * ) be an optimal pair, u ε = u * + εh ∈ U (ε > 0) and y ε = (y ε 1 , y ε 2 , y ε 3 ) the state solution corresponding to u ε . Then : Since J is Gateaux differentiable at u * , U ad is convex and the minimum of the objective functional is attained at u * , we conclude that: By standard arguments varying u, wo obtain:

NUMERICAL RESULTS
We used forward-backward sweep method (FBSM) to simulate the state system (4) − (6), the dual system (42) and the characterization of the control (43). We wrote a code in MAT LAB where, using a finite difference method, we solved the system (4) − (6) forward in time and the system (43) backward in time. A 40km × 30km grid Ω represents the population's habitat. We start by considering that the population density is 80 per square kilometer at t = 1. We suppose that violence spread starts from the middle Ω 1 . We consider two situations: the presence of intervention and the absence of the intervention where, in both cases, the spread of violence is displayed over a period of 24 months. Table1, resume the values of the initial conditions and parameters used in our numerical simulation.
In Figures 1 − 3     To confirm this effectiveness, we simulate the spread of community violence in the second case where the intervention starts after 6 months. In Figures 7 − 9, we see clearly that again, the number of violent inviduals remains low, and the density of violent inviduals is around 12 per square kilometer. The intervention starts after 180 days FIGURE 9. Violent behavior within Ω with control. The intervention starts after Finally, let us consider a third case where the intervention starts after 16 months. Figure 12 shows that the number number of violent individuals decreases significantly and immediately after the the intervention and the density goes down from 60 per square kilometer at the beginning of the intervention (16th month) to 49 per square kilometer by the end of the simulation which is still high. This relative high number of violent individuals can be explained by Figure 11 where we see that the number of susceptible individuals with high risk factors reaches its peak by the 4th month and that at the start of the intervention (16th month) the vast majority of those individuals beacame already violent. We conclude that, to eliminate violence by using only this intervention method, we need to intervene in the first months. In the other case other intervention methods are needed ( social norms, prison, ...).

CONCLUSION
In this paper, we presented a model (SHV ) for treating community violence described by a system of partial differential equations. We treated community violence as an epidemic disease controlled by identifying and helping individuals at the highest risk. To this purpose, we used optimal control theory and we proved the existence and characterization of the optimal control.
The numerical results show that the spread of community violence is quick in absence of any intervention. Our simulation proved that the control strategy used in this work is highly effective at stopping community violence from spreading and that we can eliminate violence if we start the intervention in the first months.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.