pp. X–XX COPS ON THE DOTS IN A MATHEMATICAL MODEL OF URBAN CRIME AND POLICE RESPONSE

Hotspots of crime localized in space and time are well documented. Previous mathematical models of urban crime have exhibited these hotspots but considered a static or otherwise suboptimal police response to them. We introduce a program of police response to hotspots of crime in which the police adapt dynamically to changing crime patterns. In particular, they choose their deployment to solve an optimal control problem at every time. This gives rise to a free boundary problem for the police deployment's spatial support. We present an efficient algorithm for solving this problem numerically and show that police presence can prompt surprising interactions among adjacent hotspots.

present an agent-based model of criminal behavior grounded in routine activity theory.
The model incorporates repeat and near-repeat victimization: Criminals are known to return to areas they have struck previously.
In the agent-based model, criminals move around in a spatial environment. Each location in this environment has an attractiveness value A that increases if a crime has been committed recently nearby.
Criminals are more likely to move to and strike at more attractive locations (where A is greater).  Short et al. (M3AS, 2008) present an agent-based model of criminal behavior grounded in routine activity theory.
The model incorporates repeat and near-repeat victimization: Criminals are known to return to areas they have struck previously.
In the agent-based model, criminals move around in a spatial environment. Each location in this environment has an attractiveness value A that increases if a crime has been committed recently nearby.
Criminals are more likely to move to and strike at more attractive locations (where A is greater).  Short et al. (M3AS, 2008) present an agent-based model of criminal behavior grounded in routine activity theory.
The model incorporates repeat and near-repeat victimization: Criminals are known to return to areas they have struck previously.
In the agent-based model, criminals move around in a spatial environment. Each location in this environment has an attractiveness value A that increases if a crime has been committed recently nearby.
Criminals are more likely to move to and strike at more attractive locations (where A is greater).  Short et al. (M3AS, 2008) present an agent-based model of criminal behavior grounded in routine activity theory.
The model incorporates repeat and near-repeat victimization: Criminals are known to return to areas they have struck previously.
In the agent-based model, criminals move around in a spatial environment. Each location in this environment has an attractiveness value A that increases if a crime has been committed recently nearby.
Criminals are more likely to move to and strike at more attractive locations (where A is greater).

Continuum equations
Though this is a discrete model, it has a continuum limit as a system of coupled PDEs: Criminal density: Attractiveness:

Continuum equations
Though this is a discrete model, it has a continuum limit as a system of coupled PDEs: Criminal density: Attractiveness:

Continuum equations
Though this is a discrete model, it has a continuum limit as a system of coupled PDEs: Criminal density: Attractiveness:

Continuum equations
This is a system of reaction-diffusion equations similar to the Keller-Segel model of chemotaxis.
A linear instability in these equations gives rise to hot spots in solutions, which accords with real crime data.

Continuum equations
This is a system of reaction-diffusion equations similar to the Keller-Segel model of chemotaxis.
A linear instability in these equations gives rise to hot spots in solutions, which accords with real crime data.

Adding police
The original model assumes that police won't react to the emergence of hot spots.
In two papers in 2010 (PNAS, SIADS), Short et al. introduce a "cops on the dots" approach.
d is a radial function centered at the center of a hot spot that is 1 outside the hot spot and less than 1 inside.

Adding police
The original model assumes that police won't react to the emergence of hot spots.
In two papers in 2010 (PNAS, SIADS), Short et al. introduce a "cops on the dots" approach.
d is a radial function centered at the center of a hot spot that is 1 outside the hot spot and less than 1 inside.

Adding police
The original model assumes that police won't react to the emergence of hot spots.
In two papers in 2010 (PNAS, SIADS), Short et al. introduce a "cops on the dots" approach.
d is a radial function centered at the center of a hot spot that is 1 outside the hot spot and less than 1 inside.

Adding police
The original model assumes that police won't react to the emergence of hot spots.
In two papers in 2010 (PNAS, SIADS), Short et al. introduce a "cops on the dots" approach.
d is a radial function centered at the center of a hot spot that is 1 outside the hot spot and less than 1 inside.

Adding police
The original model assumes that police won't react to the emergence of hot spots.
In two papers in 2010 (PNAS, SIADS), Short et al. introduce a "cops on the dots" approach.
d is a radial function centered at the center of a hot spot that is 1 outside the hot spot and less than 1 inside.

Dynamic policing
If deterrence displaces the hot spot, the natural question to ask is what happens when the police pursue the hot spot. Rather than prescribing the deterrence factor d with a specific form constant in time, assume the deterrence arises out of a dynamic deployment of police forces. Let κ(x, y ) represent the amount of resources the police choose to deploy to point (x, y ). Think of cops walking a beat or squad cars patrolling. The effect on the criminals is d(κ(x, y )), where d : R + → (0, 1] is a deterrence function specifying the impact of having a certain amount of police at a given point. d is a smooth, convex, decreasing function; think of d(k) = e −k .
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability

Dynamic policing
If deterrence displaces the hot spot, the natural question to ask is what happens when the police pursue the hot spot. Rather than prescribing the deterrence factor d with a specific form constant in time, assume the deterrence arises out of a dynamic deployment of police forces. Let κ(x, y ) represent the amount of resources the police choose to deploy to point (x, y ). Think of cops walking a beat or squad cars patrolling. The effect on the criminals is d(κ(x, y )), where d : R + → (0, 1] is a deterrence function specifying the impact of having a certain amount of police at a given point. d is a smooth, convex, decreasing function; think of d(k) = e −k .
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability

Dynamic policing
If deterrence displaces the hot spot, the natural question to ask is what happens when the police pursue the hot spot. Rather than prescribing the deterrence factor d with a specific form constant in time, assume the deterrence arises out of a dynamic deployment of police forces. Let κ(x, y ) represent the amount of resources the police choose to deploy to point (x, y ). Think of cops walking a beat or squad cars patrolling. The effect on the criminals is d(κ(x, y )), where d : R + → (0, 1] is a deterrence function specifying the impact of having a certain amount of police at a given point. d is a smooth, convex, decreasing function; think of d(k) = e −k .
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability

Dynamic policing
If deterrence displaces the hot spot, the natural question to ask is what happens when the police pursue the hot spot. Rather than prescribing the deterrence factor d with a specific form constant in time, assume the deterrence arises out of a dynamic deployment of police forces. Let κ(x, y ) represent the amount of resources the police choose to deploy to point (x, y ). Think of cops walking a beat or squad cars patrolling. The effect on the criminals is d(κ(x, y )), where d : R + → (0, 1] is a deterrence function specifying the impact of having a certain amount of police at a given point. d is a smooth, convex, decreasing function; think of d(k) = e −k .
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability

Dynamic policing
If deterrence displaces the hot spot, the natural question to ask is what happens when the police pursue the hot spot. Rather than prescribing the deterrence factor d with a specific form constant in time, assume the deterrence arises out of a dynamic deployment of police forces. Let κ(x, y ) represent the amount of resources the police choose to deploy to point (x, y ). Think of cops walking a beat or squad cars patrolling. The effect on the criminals is d(κ(x, y )), where d : R + → (0, 1] is a deterrence function specifying the impact of having a certain amount of police at a given point. d is a smooth, convex, decreasing function; think of Cops on the dots in a mathematical model of urban crime and po Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability The new system The PDE system becomes At each time t, the police minimize the total crime occurring in the domain Ω at t, They face a resource constraint Ω κ dx = K for fixed K > 0 and a positivity constraint κ ≥ 0.
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability The new system The PDE system becomes At each time t, the police minimize the total crime occurring in the domain Ω at t, They face a resource constraint Ω κ dx = K for fixed K > 0 and a positivity constraint κ ≥ 0.
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability The new system The PDE system becomes At each time t, the police minimize the total crime occurring in the domain Ω at t, namely Ω d(κ(x, t))ρ(x, t)A(x, t)dx.
They face a resource constraint Ω κ dx = K for fixed K > 0 and a positivity constraint κ ≥ 0.
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability The new system The PDE system becomes At each time t, the police minimize the total crime occurring in the domain Ω at t, namely Ω d(κ(x, t))ρ(x, t)A(x, t)dx.
They face a resource constraint Ω κ dx = K for fixed K > 0 and a positivity constraint κ ≥ 0.
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability

Dual formulation of the optimization problem
The optimization problem is convex, so we expect a unique solution to exist.
Theorem: This unique solution is where λ is the Lagrange multiplier associated to the L 1 constraint Ω κ dx = K .

Dual formulation of the optimization problem
The optimization problem is convex, so we expect a unique solution to exist.
Theorem: This unique solution is where λ is the Lagrange multiplier associated to the L 1 constraint Ω κ dx = K . d is decreasing, so κ is an increasing function of ρA.
Thus there exists some threshold value C of ρA so that κ > 0 where ρA > C but κ = 0 where ρA < C .
If κ is not continuous across the level set {ρA = C }, then it would be advantageous to take some police in {C < ρA < C + δ} and redeploy them to {C − δ < ρA < C }. Thus κ must be continuous. d is decreasing, so κ is an increasing function of ρA.
Thus there exists some threshold value C of ρA so that κ > 0 where ρA > C but κ = 0 where ρA < C .
If κ is not continuous across the level set {ρA = C }, then it would be advantageous to take some police in {C < ρA < C + δ} and redeploy them to {C − δ < ρA < C }. Thus κ must be continuous. d is decreasing, so κ is an increasing function of ρA.
Thus there exists some threshold value C of ρA so that κ > 0 where ρA > C but κ = 0 where ρA < C .
If κ is not continuous across the level set {ρA = C }, then it would be advantageous to take some police in {C < ρA < C + δ} and redeploy them to {C − δ < ρA < C }. Thus κ must be continuous. d is decreasing, so κ is an increasing function of ρA.
Thus there exists some threshold value C of ρA so that κ > 0 where ρA > C but κ = 0 where ρA < C .
If κ is not continuous across the level set {ρA = C }, then it would be advantageous to take some police in {C < ρA < C + δ} and redeploy them to {C − δ < ρA < C }. Thus κ must be continuous. d is decreasing, so κ is an increasing function of ρA.
Thus there exists some threshold value C of ρA so that κ > 0 where ρA > C but κ = 0 where ρA < C .
If κ is not continuous across the level set {ρA = C }, then it would be advantageous to take some police in {C < ρA < C + δ} and redeploy them to {C − δ < ρA < C }. Thus κ must be continuous. d is decreasing, so κ is an increasing function of ρA.
Thus there exists some threshold value C of ρA so that κ > 0 where ρA > C but κ = 0 where ρA < C .
If κ is not continuous across the level set {ρA = C }, then it would be advantageous to take some police in {C < ρA < C + δ} and redeploy them to {C − δ < ρA < C }. Thus κ must be continuous.

Free-boundary problem
The system is now This is a free-boundary problem, with the boundary being the edge of the support of κ.

Free-boundary problem
The system is now This is a free-boundary problem, with the boundary being the edge of the support of κ.

Free-boundary problem
The system is now This is a free-boundary problem, with the boundary being the edge of the support of κ.

Linear stability
A central finding of Short et al. (2008) was that the homogeneous steady state solution (without police) is linearly unstable in certain parameter regimes. This instability gives rise to hot spots.
Our model also has a homogeneous steady state solution:

Is this linearly stable?
Joseph R. Zipkin, Martin B. Short, Andrea L. Bertozzi Cops on the dots in a mathematical model of urban crime and po Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability

Linear stability
A central finding of Short et al. (2008) was that the homogeneous steady state solution (without police) is linearly unstable in certain parameter regimes. This instability gives rise to hot spots.
Our model also has a homogeneous steady state solution:

Is this linearly stable?
Joseph R. Zipkin, Martin B. Short, Andrea L. Bertozzi Cops on the dots in a mathematical model of urban crime and po Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability

Linear stability
A central finding of Short et al. (2008) was that the homogeneous steady state solution (without police) is linearly unstable in certain parameter regimes. This instability gives rise to hot spots.
Our model also has a homogeneous steady state solution:

Is this linearly stable?
Joseph R. Zipkin, Martin B. Short, Andrea L. Bertozzi Cops on the dots in a mathematical model of urban crime and po Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability
We do not perturb κ. Instead, let it be the solution to the optimal control problem given this A and ρ. Calculus of variations lets us separate O( ) and o( ) terms.
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability
We do not perturb κ. Instead, let it be the solution to the optimal control problem given this A and ρ. Calculus of variations lets us separate O( ) and o( ) terms.
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability

The linearized system
The linearized system is The system is stable when both eigenvalues σ are negative.
Joseph R. Zipkin, Martin B. Short, Andrea L. Bertozzi Cops on the dots in a mathematical model of urban crime and po Prior work Dynamic policing: an optimal control approach Numerics Results Future work Model specification Dual formulation of the optimization problem Linear stability

The linearized system
The linearized system is The system is stable when both eigenvalues σ are negative. For example, if d(k) = e −k , then D(k) = 0 identically, so the system becomes The matrix is triangular, so the eigenvalues are the diagonal entries.
Contrast with the original Short model. For example, if d(k) = e −k , then D(k) = 0 identically, so the system becomes The matrix is triangular, so the eigenvalues are the diagonal entries.
Contrast with the original Short model. For example, if d(k) = e −k , then D(k) = 0 identically, so the system becomes The matrix is triangular, so the eigenvalues are the diagonal entries.
Contrast with the original Short model. For example, if d(k) = e −k , then D(k) = 0 identically, so the system becomes The matrix is triangular, so the eigenvalues are the diagonal entries.
Contrast with the original Short model.

Overview
We discretize space into a square grid of length N with periodic boundary conditions. We discretize time into equally spaced steps.
We use standard spectral methods to get A n+1 and ρ n+1 from (A n , ρ n , κ n ).
Prior work Dynamic policing: an optimal control approach Numerics Results Future work

Overview
We discretize space into a square grid of length N with periodic boundary conditions. We discretize time into equally spaced steps.
We use standard spectral methods to get A n+1 and ρ n+1 from (A n , ρ n , κ n ).
Prior work Dynamic policing: an optimal control approach Numerics Results Future work

Overview
We discretize space into a square grid of length N with periodic boundary conditions. We discretize time into equally spaced steps.
We use standard spectral methods to get A n+1 and ρ n+1 from (A n , ρ n , κ n ).
Prior work Dynamic policing: an optimal control approach Numerics Results Future work A dual method for κ As before in the continuous theory, κ n+1 i,j is positive if ρ n+1 i,j A n+1 i,j exceeds some threshold value and 0 otherwise.
Prior work Dynamic policing: an optimal control approach Numerics Results Future work A dual method for κ As before in the continuous theory, κ n+1 i,j is positive if ρ n+1 i,j A n+1 i,j exceeds some threshold value and 0 otherwise.
Prior work Dynamic policing: an optimal control approach Numerics Results Future work Easy case: Police deploy everywhere If −λ n+1 /d (0) is less than all of the values of ρ n+1 i,j A n+1 i,j , then the police deploy everywhere.
Standard iterative methods suffice for this.
Unfortunately you must compute H at every iteration. Dynamic policing: an optimal control approach Numerics Results Future work Less easy case: Police do not deploy everywhere Sort the values of ρ n+1 A n+1 in descending order. Where we had a 2D N × N vector we now have f , a 1D vector of length N 2 .
Finding the cutoff crime level −λ/d (0) is equivalent to finding the cutoff index J so that G is increasing, so this has a unique solution.
We use a discrete false position (linear interpolation) method to find J iteratively. Dynamic policing: an optimal control approach Numerics Results Future work