- The document summarizes a mathematical model of crime hotspots presented in a student's senior thesis. It describes a discrete model where houses are represented on a lattice and criminals move between houses according to attractiveness. House attractiveness is determined by both intrinsic and dynamic factors related to past burglaries and crime in nearby areas. The model aims to explain how crime hotspots form and evolve over time due to phenomena like repeat victimization and broken windows. The document outlines the key components and equations of the discrete model.

1. Pomona College
Senior Thesis in Mathematics
Mathematical Modeling of Crime
Hotspots
Kevin Kannappan
advised by
´Angel Ch´avez
May 8, 2016

2. 1 Introduction
Throughout history, even though political institutions are built upon laws and regula-
tions, there are always a subset of individuals that do not abide by the rules outlined for them.
Unfortunately, this behavior is inherently criminal; it is disruptive and counterproductive
for the institution that holds them. Formally, we will define crime or criminal behavior to be
an action that constitutes an offense that may be prosecuted by the state and is punishable
by law. Crime has been pervasive throughout time and manifests itself today, regardless of
the generally accepted perspective of higher qualities of life and education. Crime can take
many forms, such as violent crimes like battery and homicide, or smaller crimes like petty
theft or shoplifting. Although crime itself is occurring in any given period in time, there
have been numerous studies indicating that crime is not uniformly distributed throughout
space and time [7, 8]. Specifically, while some residential neighborhoods can be far more safe
and friendly, others can be more dangerous and have higher levels of crime. Additionally,
these areas may not always remain the way that they have been: some neighborhoods may
improve in safety and see a reduction in crime rates, while others may see the opposite.
Crime “hotspots” will be the term that this paper will define as aggregates of crime in
a given period of space and time. Specifically, crime hotspots will change in nature based
on some given geographic, economic, seasonal conditions, or even the type of crime being
observed. Law enforcement has largely struggled in trying to prevent the formation of these
hotspots, even though substantial progress has been made in collecting relevant data. Thus,
by using this wealth of data on how these crime hotspots are formed, mathematical modeling
may provide the solution towards preventing crime hotspots from occurring in the first place.
In this work, we will only look at the evolution of crime hotspots due to residential burglary,
and we will study the mathematical model used in [1]. As indicated in [1], in order to
effectively explain the dynamics behind the evolution of crime hotspots, it is important
to understand the sociological factors driving them. In this paper, we will look at repeat
victimization and the “broken windows” effect as notable driving factors in the formation of
residential burglary hotspots.
Repeat victimization will be formally defined as the process by which a residential bur-
glar will return to a previously burgled site because they have good information about the
inhabitants and potential property to be stolen [3, 4, 5, 6]. In essence, victims of crime are
more likely to be repeatedly targeted within short periods of time after the original crime
was committed. Based on the concept of repeat victimization as a driving force towards
the proliferation of crime in a specific location, it is easy to observe at how this can affect
the dynamics of crime hotspots. Furthermore, the so-called “broken windows” effect can
influence the formation of crime hotspots. It is the notion that past crimes in a given area
create the image of a crime-tolerant area and subsequently lead to the proliferation of illicit
activity [5, 6]. Hence, criminals are more likely to be drawn to worse-off areas in order to
commit crimes as the likelihood of getting caught has decreased in these areas. As it has
been demonstrated in the concepts introduced thus far, the primary factors driving criminals
1

3. towards houses are contained in the environment surrounding those houses: past-burglary,
or the prevalence of crime in the surrounding area. In [1], there is no effort to expand on
the environment serving as the driving factor in motivating residential burglary. We will
therefore discuss possible questions to consider in this direction.
Figure 1: Simple version of the Lattice described in Section 2
The structure of the paper is as follows. We take the same structure as [1]. The first
section will describe a discrete model for criminal dynamics. A basic version of this model
is seen in Figure 1. The next section will cover a continuum derivation based on the dis-
crete model. Additionally, this section will feature two coupled reaction-diffusion equations
covering the formation of hotspots. The corresponding linear stability analysis is performed
in Section 4. Lastly, the paper will conclude by presenting the findings of our model and
propose future directions of research, specifically with incorporating inhomogeneity into the
attractiveness field A0
s. We propose applying this updated attractiveness field to try to fit the
residential burglary hotspots that were observed in San Francisco in 2009. More complicated
mathematical processes not directly related to the models presented will be covered in three
appendices.
2

4. 2 The Discrete Model
2.1 Overview
In this section, we present an overview of the discrete crime model presented in [1]. The
discrete model in [1] is used to model residential burglary. As touched on earlier, for the sake
of simplicity for the system, there are only two components to the discrete model: houses
and criminals. Houses will be represented by points on a two-dimensional square lattice
whereas criminals will be agents that are able to move from lattice site to lattice site, or
from house to house. In particular, criminal agents will move from house to house according
to a biased random walk. In order to create an environment for the simulations, we need
to simulate a representative residential neighborhood. This is why we agree to work on a
two-dimensional rectangular lattice, or in more colloquial terms, a common rectangular grid.
Explicitly, we shall fix a positive real number and consider the following two-dimensional
rectangular lattice defined below:
Z2
= {( i, j) ∈ R2
i, j ∈ Z}. (2.0.1)
We impose periodic boundary conditions on Z2
to reduce the infinitely defined lattice
above to a finite set. Periodic boundary conditions are often used in computer simulations
and mathematical models to reduce the complexity of calculations. In order to accomplish
this, we first fix any two given positive integers N and M. From there, we shall declare any
two points (x, y) and (r, s) in Z2
to be equivalent if x − r = N and y − s = M. We let
Λ denote the periodic lattice given by taking the quotient of Z2
by this equivalence. To
shorten this notation, we will simply let s = (i, j) denote a given lattice site (δi, δj) ∈ Λ .
We assume that criminal actions (residential burglaries) occur over a discrete time unit,
which we denote as δt. It could be helpful to think about this discrete time unit in terms of
hours or days in subsequent analysis. As indicated before, each site s is representative of a
house. In addition to each site s, we will now associate a quantity As(t). As(t) is meant to
measure a burglar’s perception of the attractiveness of the house at site s. In this section, we
will not attempt to add in a level of criminal behavior independent of house attractiveness.
Criminals will rob houses purely based on the attractiveness of houses at a given point in
time. To begin, we will assume that:
As(t) = A0
s + Bs(t), (2.0.2)
where A0
s represents a static (not varying with time) component of the attractiveness of site
s and Bs(t) is a dynamic (varying with time) component associated to the sociological phe-
nomena of repeat, near-repeat victimization and the “broken windows” effect. Essentially,
A0
s, or the intrinsic attractiveness of a particular house, can be thought of as how likely the
house is to have valuable items, how concealed the house is from plain sight, or how much
security is used to monitor the house. Intrinsic attractiveness is purely based on the qualities
of the house that make it desirable to burgle and the assumption that we are making is that
3

5. houses do not change in this way...a house will always maintain its endowed level of intrinsic
attractiveness.
Repeat victimization is the belief that residences experience an elevated risk of being bur-
gled again in a short period to time after a prior burglary [4]. This is a relevant phenomenon
to include in our model because criminals are more inclined to return to a previously burgled
site because they have better information about the inhabitants and/or property. Hence,
crime hotspots are likely to form as a result of repeat victimization because a house that is
burgled will incentivize more crimes to occur in the future. However, repeat victimization
is largely a temporary phenomenon post-burgle, and therefore will decrease as subsequent
time periods pass. Therefore, dynamic attractiveness Bs(t) should depend upon previous
burglary events at site s within a certain time period threshold. Concretely, each time
a house is burglarized, we increase Bs(t) by a fixed number θ according to the following
equation:
Bs(t + δt) = Bs(t)(1 − ωδt) + θEs(t), (2.0.3)
where ω is a fixed constant as well as a decay rate, and Es(t) is the number of burglaries at
site s that occur in time interval [t, t + δt].
The “broken windows” effect is modeled in our next equation. As indicated in earlier
sections, the “broken windows” effect is when past crimes in a certain area create the image
of a crime-tolerant area and then lead to the proliferation of illegal activity. Essentially,
areas that have visibly more signs of crime will invite more crime due to the perception that
criminals will be safe to commit crimes there. Near-repeat victimization and the broken
windows effect are both modeled using a relatively straightforward idea: Suppose s is a
neighbor of s. If site s is burglarized, then Bs (t) should increase as a result. We modify
(2.0.3) to read:
Bs(t + δt) = (1 − η)Bs(t) +
η
4 s ∼s
Bs (t) (1 − ωδt) + θEs(t), (2.0.4)
where η ∈ [0, 1]. As observed in the equation above, higher values of η lead to a larger spread
of attractiveness of site s to each of its neighboring sites.
Remark 2.1. We can note that if η = 0, then there will be no broken windows effect. If
η = 1, then the dynamic attractiveness of a given site, aside from repeat victimization, is
purely determined by the attractiveness of its neighbors.
While the equation introduced thus far captures the relevant phenomena seen in the
empirical literature on criminal activity, it is quite complex but can be simplified and still be
representative of the system. In order to simplify our model, we will introduce an element
of graph theory to 2.0.4:
4

6. Definition 2.2. Suppose φ is a function (real or complex-valued) on Λ . We define a discrete
Laplacian operator by setting:
(∆φ)(s) =
1
2
s ∼s
φ(s ) − 4φ(s) , (2.2.1)
where s ∼ s if and only if s is a neighbor of s.
Appendix B contains a brief note on why expression (2.2.1) can in fact be thought of as
a discrete version of the classical Laplacian on R2
:
2
=
∂2
∂x2
+
∂2
∂y2
. (2.2.2)
Applying ∆ to the function Bs(t) gives:
η 2
4
∆Bs(t) =
η
4 s ∼s
Bs (t) − 4Bs(t) = −ηBs(t)
s ∼s
1 +
η
4 s ∼s
Bs (t), (2.2.3)
which implies:
−ηBs(t) =
η 2
4
∆Bs(t) −
η
4 s ∼s
Bs (t). (2.2.4)
Therefore, we can write (2.0.4) as:
Bs(t + δt) = Bs(t) −
η 2
4
∆Bs(t) (1 − ωδt) + θEs(t). (2.2.5)
While we have discussed the potential phenomena that drive the burglar to select a given
house, we have not discussed how the burglar moves from house to house or the probability
that a burglar will rob a house. We will explicitly define the movement and probability that
a burglar will rob a particular house here. The burglar at site s can perform one of two
actions during a given time interval: burglarize the house at site s or move to one of the four
neighboring houses. The probability that a burglar at site s will burglarize the house at site
s sometime in the interval [t, t + δt] is given by ps(t):
ps(t) = 1 − e−As(t)δt
. (2.2.6)
Remark 2.3. Let us observe the probability that a burglary will occur at a given point
in time, as well as the expected number of burglaries in a given time interval. In order to
observe this, let us fix a site s and let Ns(t) be the number of burglaries that occur in the
time interval [0, t]. We are assuming that {Ns(t) : t ≥ 0} is a Poisson Process with rate
function λ characterized by:
As(t) δt =
t+δt
t
λ(τ) dτ.
5

7. Figure 2: Visual Diagram of the Model used for the Discrete Simulations
In fact, P[Ns(t + δt) − Ns(t) ≥ 1] denotes the probability that a burglary occurs at some
point in the time interval [t, t + δt]. One may see that
P[Ns(t + δt) − Ns(t) ≥ 1] =
∞
k=1
P[Ns(t + δt) − Ns(t) = k]
= e−As(t)δt
∞
k=1
As(t)δt
k
k!
= 1 − e−As(t)δt
.
We also note the expected number of burglaries in the interval [t, t + δt] is given by:
E[Ns(t + δt) − Ns(t)] = As(t)δt.
Also observe that in the notation of (2.0.3), we have Es(t) = Ns(t + δt) − Ns(t). Please see
Appendix A for background on the Poisson process.
Based on [6] and a degree of common sense, we are aware that burglars do not like to stay
the houses that they burgle. Therefore, in our computer simulations, a burglar is removed
from the lattice if a site is burglarized. The reason for removing the criminal entirely from the
system is because we are assuming that a criminal leaves the scene of the crime immediately
after a burglary and is assumed to return home and to halt illegal activity. In other words,
even though the burglar has committed a previous crime, they are no longer acting as a
burglar and should not belong in the system. In the computer simulations, criminals will
be generated at each lattice site at a constant rate Γ as to signify criminals returning to
active status. The reason for making burglar production constant is that the discrete model
6

8. is purely being used as a tool to show crime hotspot formation, and is not intended to model
what is occurring in the real world. Criminals will be generated at different rates according
to data containing a given set of socio-economic conditions in later sections.
However, if a burglar at site s decides not burgle the house, then they will move to a
neighboring site on the lattice. Specifically, the criminal will only move a length of along
the lattice to one of the four points available surrounding site s. It subsequently follows that
this process will be treated as a biased random walk. In particular, the probability that a
criminal in site s at time t will move to neighboring site r at time t + δt is given by
qs→r(t) =
Ar(t)
s ∼s As (t)
, (2.3.1)
where s ∼ s means that s is a neighbor of s. Figure 2 provides a completed flowchart that
covers this overview section and all of its equations in visual form.
Remark 2.4. The transition probability in (2.3.1) is biased toward high values of attrac-
tiveness: If Ar(t) is high, then the probability that a criminal will walk from site s to site r
is also high. In an unbiased random walk, a criminal chooses a neighboring site uniformly.
For example, given site s has 4 neighbors, then qs→r(t) = 1/4.
Table 1: Summary of Parameters in Discrete Model
Parameter Name Meaning
Grid Spacing in Lattice
δt Discrete Time Step
ω Dynamic Attractiveness Decay Rate
η “Broken Windows” Effect Parameter
θ Attractiveness Increase due to Burglary
A0
s Intrinsic Attractiveness of Site s
Γ Burglar Generation Rate of Site s
Es(t) Number of Burglaries at site s in [t, t + δt]
∆ Discrete Laplacian Operator
ρs(t) Probability of Criminal Burglarizing at Site s
When using mathematical models as a tool for observing a system’s behavior, the goal
of the model is to evaluate the equilibrium solution. The least mathematically strenuous
scenario for the discrete model presented in this section would be the spatially homogeneous
equilibrium solution. In this case, all sites will have the same attractiveness, denoted as ¯A,
and on average, the same number of criminals as well, denoted as ¯n. Now, let us think about
7

9. this logically. In order for the sites to maintain both the same level of attractiveness and the
same amount of burglars, “what comes in must equal what comes out.” Explicitly, for the
level of attractiveness at any given house to stay fixed, the amount by which attractiveness
increases due to burglaries must equal the amount by which attractiveness decays in a time
step:
ω ¯Bδt = θ¯n¯p. (2.4.1)
Furthermore, for the number of criminals at any given site to stay fixed, the number of
criminals produced at rate Γ must be equal to the number of criminals removed from that
site in one time-step (equal to the number of burglaries at that time-step):
¯n¯p = Γδt. (2.4.2)
From here, we now have a system of equations that will allow us to solve for the homoge-
neous equilibrium values for the level of attractiveness and amount of criminals respectively:
¯B =
θΓ
ω
, ¯n =
Γδt
1 − e− ¯Aδt
. (2.4.3)
We will investigate whether or not the system in this homogeneous equilibrium state will
remain in it in the following section. In other words, we will observe if the mathematically
derived equilibrium solutions will be validated by computer simulations.
2.2 Computer Simulations: Discrete Equations
Following the visual representation presented in Figure 2, computer simulations were
created in this section to follow the overview of the discrete model. By use of computer
simulation, we are able to effectively see how different combinations of parameters affect the
system. The parameters of interest are contained in Table 1.
By changing the values for different parameters, we can see three distinct phenomena
that are occurring with respect to the attractiveness field As(t) that has developed on the
lattice of houses:
1. Stationary Hotspots. Stationary hotspots are a result of a system in which there are
spots of high attractiveness that remain in a given location, throughout time. They
can be thought of as a steady state solution, and the size of the spots will vary based
on parameter selection.
2. Dynamic Hotspots. The more realistic scenario of hotspot formation, dynamic hotspots
result in areas of increased attractiveness that change throughout time. Dynamic
hotspots can form and remain present or dissipate entirely. The size as well as duration
of the spots seen will vary based on parameter selection.
8

10. 3. Spatial Homogeneity. Spatial homogeneity refers to an attractiveness field in which
there is no hotspot formation, rather the level of attractiveness remains relatively
constant throughout time.
These phenomena are demonstrated visually below, where we have shaded the attrac-
tiveness field relative to the homogeneous equilibrium value for the level of attractiveness,
¯B which is shaded in green. Other values of attractiveness correspond to a “rainbow” spec-
trum, in which the lower-bound case where Bs = 0 is violet, and the upper-bound case where
Bs ≥ 2 ¯B.
Figure 3: Output from the discrete simulations using the parameters outlined in the text.
We see dynamic hotspots, spatial homogeneity, and stationary hotspots forming in a), b),
and c) respectively.
9

11. All images have been produced on a lattice of size 128 x 128, with initial conditions
Bs(0) = ¯B and, on average, the number of criminals at each site ns(0) = ¯n. Furthermore,
all images have used the parameter values of = 1, δt = 0.1, ω = 1/15, and A0
s = 1/30.
These parameter choices are consistent with those used in [1], and they have been chosen
to be representative of potentially realistic values which lend themselves well to estimation.
The grid spacing in the lattice, , can be contextualized as the distance between houses. The
discrete time-step, δt, can be interpreted in units of days. We can see the system exhibit
the cases of stationary hotspots, dynamic hotspots, and spatial homogeneity by changing η,
θ, and Γ. Figure 3 a) uses η = 0.03, θ = 0.56, and Γ = 0.019. Figure 3 b) uses η = 0.03,
θ = 5.6, and Γ = 0.002. Lastly, Figure 3 c) uses η = 0.2, θ = 0.56, and Γ = 0.019.
Based on the computer simulations run, there seems to be a tendency for the system
to move towards either both of stationary hotspots and spatial homogeneity or dynamic
hotspots. Essentially, the system will move towards dynamic hotspots with a lesser amount
of criminals, whereas with a greater amount of criminals, stationary hotspots or spatial
homogeneity are likely to form. We believe that this makes sense intuitively because if there
are greater amounts of criminals, then we would expect crime hotspots to form either in set
locations consistently or not at all. Whereas if there are not that many criminals, we expect
them to operate in select locations, that may increase or decrease in perceived attractiveness
across time. In order to appropriately understand the dynamics of the model, we will now
take a continuum limit of the discrete model.
3 Continuum Limit
Interested in the spatial inclinations of a burglar in a particular moment in time, we
will take a continuum limit of the discrete model outlined in Section 2.1. In order to begin
the process of taking a continuum limit, we will need to look closely at Bs(t), or dynamic
attractiveness. Recall the latest equation modeling Bs(t) in (2.2.5):
Bs(t + δt) = Bs(t) −
η 2
4
∆Bs(t) (1 − ωδt) + θEs(t). (2.2.5)
Now, let us adjust this equation by using the expectation of dynamic attractiveness after
one time-step. Specifically, we see that the expected value of Es(t), the number of burglaries
at a site s during the period [t, t + δt], is given by the product ns(t)ps(t). This new equation
is given by
Bs(t + δt) = Bs(t) −
η 2
4
∆Bs(t) (1 − ωδt) + θns(t)ps(t). (3.0.1)
This new equation makes sense because the expected number of burglaries in this time-
step is equal to the product of the probability of a burglar burgling and the amount of
burglars present.
10

12. Here, let us determine the density of criminals by dividing ns(t) by 2
. We will rename
this density as ρ(x, t). The equation will be simplified by using the new density formula
(obtained by multiplying the last term by 2
/ 2
):
Bs(t + δt) = Bs(t) −
η 2
4
∆Bs(t) (1 − ωδt) + θ 2
ρ(x, t)ps(t). (3.0.2)
We will now subtract Bs(t) from both sides and divide by δt. This should leave us with
almost the definition of the derivative on the left side, and expanding out the right side will
result in:
Bs(t + δt) − Bs(t)
δt
=
Bs(t) − ωBs(t)δt + η 2
4
δBs(t) − ωη 2
4
δBs(t)δt + θ 2
ρ(x, t)ps(t) − Bs(t)
δt
(3.0.3)
We can simplify the above equation some by canceling the Bs(t)/δt terms on the right
hand side. But before we take the continuum limit, we need to apply some constraints. The
ratio of 2
/δt will remain fixed with a value defined as D, and the product of θδt will also
remain fixed with a value defined as . Now we are ready to take the continuum limit, in
which we take the limit as both δt and tend to zero. The simplified equation yields the
dynamics of the continuum version of attractiveness:
∂B
∂t
=
ηD
4
2
B − ωB + DρA (3.0.4)
where 2
is the standard Laplacian operator on R2
. As mentioned previously, the discrete
Laplacian converges to 2
, and this is outlined in Appendix B.
The derivation of the continuum limit for ns(t), or the amount of burglars at site s at
time t, is more complicated than the derivation of the continuum limit for attractiveness. As
done with attractiveness, we will begin the process by noting the number of burglars that
we expect to see at site s after one time-step, t → t + δt. Beginning theoretically, we know
that the number of burglars at a given site must be equal to the sum of the burglars coming
to the site after deciding not to burgle a given neighbor and the number of burglars being
generated at that site over that time-step:
ns(t + δt) = Burglars Generated at s + Burglars Expected from Neighbors. (3.0.5)
Now, we know that the burglar generation rate at site s is the constant rate of Γ. Hence,
we subsequently know the first part of the equation, or the number of burglars created at
site s after one time-step:
Burglars Generated at s = Γδt. (3.0.6)
From here, it is a relatively straightforward exercise to determine the number of burglars
that are expected to arrive at site s after deciding to not burgle a neighboring site, r.
11

13. Recall equation (2.3.1), the transition probability, where we have given the probability that
a criminal in site s at time t will move to neighboring site r at time t + δt. Therefore, the
second part of the equation, or the number of burglars expected to arrive from a neighboring
site r, must be the product of the transition probability and the expected number of criminals
at a neighboring site that will not burgle:
Burglars Expected from a Given Neighbor r = nr(t)(1 − pr(t))qr→s(t). (3.0.7)
Thus, we must update this equation to account for the fact that multiple neighbors might
be contributing burglars to site s:
Burglars Expected from Neighbors =
r∼s
nr(t)(1 − pr(t))qr→s(t). (3.0.8)
We can now re-write (3.0.5) mathematically as
ns(t + δt) = Γδt +
r∼s
nr(t)(1 − pr(t))qr→s(t)
= Γδt + As(t)
r∼s
nr(t)(1 − pr(t))
Tr(t)
= Γδt + As(t)
r∼s
nr(t)e−Ar(t)δt
Tr(t)
,
where we have defined:
Tr(t) =
r ∼r
Ar(t).
From here, we now have an equation expressed in a similar form to that of the previous
equation in (2.0.4). Following a similar procedure to simplify the equation prior, we begin
with the definition of the discrete spatial Laplacian to write the following expression:
r∼s
ns(t)e−As(t)δt
Ts(t)
= 2
∆
nr(t)e−Ar(t)δt
Tr(t)
+ z
ns(t)e−As(t)δt
Ts(t)
.
From here, we substitute this relationship into the above revision of equation (3.0.5):
ns(t + δt) = Γδt + 2
As(t)∆
ns(t)e−As(t)δt
Ts(t)
+ zAs(t)
ns(t)e−As(t)δt
Ts(t)
. (3.0.9)
As we did before with the previous continuum limit, we will re-express ns(t) in terms of
ρ(x, t). Additionally, we will subtract ns(t) from both sides of the equation, divide by 2
and divide by δt. The almost complete form of the derivative remains on the left side, and
expanding out the right side yields
ρ(x, t + δt) − ρ(x, t)
δt
= γ + DAs(t)∆
ρ(x, t)e−As(t)δt
Ts(t)
+ z
As(t)ρ(x, t)e−As(t)δt
Ts(t)δt
−
ρ(x, t)
δt
.
(3.0.10)
12

14. Now, we can use the definition of the discrete spatial Laplacian again to write
Ts(t) = 2
∆As(t) + zAs(t).
We insert this relationship into equation (3.0.11) to get
DAs(t)∆
ρ(x, t)e−As(t)δt
Ts(t)
= DAs(t)∆
ρ(x, t)e−As(t)δt
2∆As(t) + zAs(t)
. (3.0.11)
Before we take the following continuum limit, we need to apply the additional constraint
that the ratio of Γ/ 2
will remain fixed with a value defined as γ. The subsequent equation
results after taking the limits of and δt to zero as described previously:
D
z
A(x, t) 2 ρ(x, t)
A(x, t)
. (3.0.12)
However, we know that the Laplacian satisfies the identity
2 f
g
=
1
g
2
f −
2
g
f
g
· g −
f
g2
2
g
for any appropriate functions f, g. We can let f = ρ and g = A in this formula and then
take the scaling limit of
z
As(t)ρ(x, t)e−As(t)δt
Ts(t)δt
−
ρ(x, t)
δt
(3.0.13)
to get the continuum equation for burglar density:
∂ρ
∂t
=
D
4
· ρ −
2ρ
A
A − ρA + γ. (3.0.14)
Equations (3.0.4) and (3.0.14) are the unaltered equations that result from the same
process taken in [1]. It is clear to see that these equations are the general form of a reaction-
diffusion system, which can lead to pattern formation [9]. Reaction-diffusion systems are
generally characterized by some physical phenomena: a change in space and time of the
concentration of one or more chemical substances. Specifically, substances will transform into
each other based on chemical reactions, and diffusion will cause the substances to spread out.
Applying this reasoning to our model (a non-chemical system), one can see that burglars are
created at a constant rate and are diminished by interactions with the attractiveness field. It
is also important to note that burglars move at a speed that is inversely proportional to the
local attractiveness field. This makes intuitive sense because a burglar that is in a particular
neighborhood that has many ideal houses to burgle will likely not travel very quickly away
toward less ideal houses. With regards to the attractiveness field, attractiveness will diffuse
throughout the lattice as it decays in time and reacts with the burglars, resulting in even
more attractiveness.
13

15. Looking at (3.0.4) and (3.0.14) more closely, we find two additional interesting character-
istics of the system. After integrating the steady-state of (3.0.14), we find that the spatially
averaged crime rate density is equal to γ, given that the criminal flux is either zero or periodic
at the boundaries. Based on this information, we know that all systems in the steady-state
will have the same overall crime rate according to the given γ, even if we observe the crime
concentration to be higher in the hotspots. After integrating the steady-state of (3.0.4), the
homogeneous equilibrium attractiveness value, ¯B as given in (2.4.3), is also the spatially av-
eraged attractiveness value for the steady-state system. Additionally, we can re-write (2.4.3)
with our new continuum parameters in the following way:
¯B =
Dγ
ω
. (3.0.15)
Following from this information, the computer simulations based on the system with the
stationary hotspots are better explained. Explicitly, in the stationary hotspot discrete sim-
ulations, the areas of high attractiveness are surrounded by areas of very low attractiveness.
This phenomenon makes sense intuitively because the average attractiveness of the system
is fixed.
The next step in the math modeling process is to re-write equations (3.0.4) and (3.0.14)
in dimensionless form. The reasons for doing this is are as follows:
1. To reduce computational complexity through means of variable reduction.
2. To analyze the behavior of the system regardless of the units used to measure the
variables.
3. To re-scale parameters and variables such that all computations are of the same order.
We begin this process by giving the natural time scale for our model, τ ≡ 1/ω, as noted
in prior sections. From here, we need to assign a characteristic length scale:
c ≡
D
ω
. (3.0.16)
This value can be interpreted as approximately the distance over which burglars diffuse in the
time τ it takes for the attractiveness of a newly burgled house to return to its original value.
Hence, we will scale variables accordingly, providing the following dimensionless, variables
denoted with a tilde:
B = B/ω, A = A/ω, ρ = 2
cρ, x = 2x/ c, t = ωt. (3.0.17)
Inserting the dimensionless variables into the original equations and simplifying accord-
ingly, we get the following two dimensionless continuum equations for (3.0.4) and (3.0.14)
14

16. respectively. The re-dimensionalization of these continuum equations is outlined in Appendix
C. In particular, we obtain the equations
∂B
∂t
= η 2
B − B + ρA, (3.0.18)
∂ρ
∂t
= · ρ −
2ρ
A
A − ρA + ¯B, (3.0.19)
where we have dropped the tilde and ¯B is given as the dimensionless version of (3.0.15).
At this time, we have taken the discrete system with seven different parameters and have
transformed our system into a dimensionless continuum version with three “free” parameters.
As the nature of the system of continuum equations is quite complicated, we will not
implore the use of computer simulations on these equations. The computer simulations of
the continuum equations help to provide validity for the equations as well as serve as a
comparison to those in the discrete model. Therefore, if one wishes to seek out the actual
simulations, they exist in [1]. However, we will summarize the conclusions of [1] here. They
find that the continuum simulation output is quite similar to the outputs in the discrete
equations where the amount of burglars is larger. From there, they conclude that the the
continuum equations are good approximations of the discrete system under these conditions.
Unfortunately, dynamic hotspots were not observed in the continuum simulations because
of the aforementioned stochasticity and finite size effects.
Boiling it down, we still see that there remains a question unanswered: why do some
systems exhibit hotspots while others do not? This question will be answered by means of
a linear stability analysis.
4 Linear Stability Analysis
Much of the derivations that take place in this section are omitted and can be referenced
in [1], yet we still go through the process here. Prior to performing the linear stability analysis
of the system of continuum equations, we must first consider the homogeneous equilibrium
solutions of (3.0.18) and (3.0.19). For the sake of simplicity, we still take a spatially uniform
value for A0
in order to find these values. These solutions are
¯A = A0
+ ¯B, ¯p =
¯B
A0 + ¯B
. (4.0.1)
In order to observe the system behavior, we now examine the behavior of the solutions of
the form
A(x, t) = ¯A + δAeσt
eik·x
, (4.0.2)
ρ(x, t) = ¯ρ + δρeσt
eik·x
. (4.0.3)
As indicated thus far, we will only consider deviations of A and ρ with the same wavenum-
ber k. This decision was made because it can be easily shown that all deviations of differing
15

17. wavenumber will decay in time [1]. After substituting these above equations into (3.0.18)
and (3.0.18) respectively, we obtain the following linear system:
−η|k|2
− 1 + ¯ρ ¯A
2¯ρ
¯A
|k|2
− ¯ρ −|k|2
− ¯A
δA
δρ
= σ
δA
δρ
in which we attempt to solve for the value of σ. Analyzing this equation, we know that
the system will be linearly unstable for the value of wavenumber k given the determinant is
negative:
η|k|4
− (3¯ρ − η ¯A − 1)|k|2
+ ¯A < 0. (4.0.4)
The inequality above will hold true for a finite amount of wavenumbers, and if the parameters
hold true to the following inequality:
3¯ρ − η ¯A − 1 > 2 η ¯A. (4.0.5)
This inequality can be used to differentiate between systems that either exhibit instabilities
or systems that do not based on the quantities of η, A0
, and ¯B. Further simplifying the
equation, one can see that the condition below must hold for there to be instability:
¯B >
A0
2
. (4.0.6)
Hence, if the inequality holds, then the following must be true for η:
η <
3¯ρ + 1 −
√
12¯ρ
¯A
. (4.0.7)
A special case of (4.0.5), in which A0
= 0, lends a deeper understanding of the system.
Explicitly, under this condition, the inequality may be re-written as follows:
ηD
ω
γ
ω
< 4 − 2
√
3. (4.0.8)
Here, the first fraction on the left-hand side of the inequality can be interpreted as the
span of influence of a singular burglary event, or in other words, the area over which the
increase in B from a burglary event can be measurably felt before it decays. The second
fraction on the left-hand side of the inequality can be interpreted as the average number
of events per area in time τ at the steady-state. The inverse of this fraction is the average
area per event at the steady-state. According to dimensional analysis, the inequality as a
whole indicates that isolated burglary hotspots will only occur at steady-state if the average
area per event is greater than the span of influence of a singular burglary event. In essence,
isolated hotspots of high B can exist if and only if the hotspots are far enough away from
each other so that they are unable to interact.
16

18. Figure 4: Hotspot separation as indicated by the Linear Stability Analysis in this section.
Note that η runs along the x-axis and λ∗
= 2 ∗ π/|k∗
| runs along the y-axis.
If, however, the system is unstable, there exists a wavenumber k∗
that will exhibit the
fastest growth rate of all the unstable modes. This maximally growing mode is given by
|k∗
|2
= (1 − ¯A)/(1 − η) − ¯ρ(5 − η)/(1 − η)2
+ η(1 + η)2 ¯ρ ( ¯A(3 − η) − 2)(1 − η) + 2¯ρ(3 − η) /η(1 − η)2
.
Additionally, this equation should set the scale where λ∗
= 2π/|k∗
| for hotspot separation at
the steady-state. We have observed the behavior of this equation by varying η in Figure 3.
5 Conclusion
Mathematical modeling can be applied to an array of processes in order to understand
how the modeled process works. In the case of this paper, we have used a math model to
try to understand the dynamics behind crime hotspot formation. Specifically, we were able
to create a system that responded to the implementation of certain empirically supported
17

19. phenomena: repeat victimization and the “broken windows” effect. Hence, even though our
modeling techniques are primarily used to be a “dummy” city or neighborhood to try to
simulate crime hotspot formation, we are motivated to provide future research with the path
to apply this model to be representative of an actual city. In order to do so, we needed
to obtain relevant data to serve as a target city for the model. We decided to select San
Francisco, based on the way the city is explicitly defined in a certain area. Explicitly, we
attempt to indicate how this model could develop hotspots similar to the hotspots actually
formed in San Francisco during 2009.
Using a data-set from the San Francisco Police Department (SFPD) Crime Incident
Reporting System developed by Kaggle Inc. for one of its online competitions, we have
aggregated a subset of residential burglaries. The data-set contains the time and location of
each residential burglary incident from 1/1/2003 to 5/13/2015 in the greater San Francisco
area. After some data-wrangling in which we were able to separate variables and make them
usable for visual purposes, we used tools provided by ArcGis.com. As we assumed that crime
hotspots increased after the economic turmoil felt in 2008, we used ArcMaps to graph the
hotspots observed in 2009, partitioned into three-month periods in each picture. Figure 4
displays these results.
Fitting the parameters in our paper to a city is a surprisingly difficult task. As there is no
way for researchers to actually know what the spread of attractiveness from house to house is,
or the corresponding attractiveness increase in a house after a burglary is, we propose trying
to fit the model to what is empirically observed. Some possible ideas that we propose to
get San Francisco’s attractiveness field include finding data on neighborhood housing prices,
crime rates, security systems, proximity to police stations, etc. While it would be ideal to
use a combination of these measures, unfortunately there is no appropriate way to assign
weights to each variable as there is substantial heterogeneity in each. Therefore, we propose
future research to develop the attractiveness field according to housing prices.
In conclusion, we were able to effectively simulate crime hotspot formation based on a
model that was built on empirically supported literature of criminal behavior. Furthermore,
the crime hotspots that were formed in the model are also similar to the ones observed in
actual cities. Hence, we note that sociological factors such as repeat victimization and the
“broken windows” effect are subsequently quite influential in hotspot formation. In the end,
applying the knowledge of the underlying dynamics from the work presented here could be
paramount to actually fitting the model to the real-world. Future research will benefit from
creating a non-homogeneous attractiveness field that fits to an actual city, in which case
we believe San Francisco is a promising choice. Additionally, we have presented a detailed
outline for how following research could apply the results of the paper to San Francisco.
Other ideas include molding the model to correspond to given socioeconomic conditions at
the time. Regardless, law enforcement will benefit from the results of this paper, specifically
why hotspots form and the dynamics behind them. This is the most important conclusion
as crime research will only be important if we can provide the tools for law enforcement to
contain it.
18

20. Figure 5: Dynamic changes in residential burglary hotspots in San Francisco during three
month periods throughout 2009. Top-left is months January-March, top-right is April-June,
bottom-left is July-September, and bottom-right is October-December.
A Poisson Process
A Poisson process with rate λ is a collection of random variables indexed by a real pa-
rameter (which is regarded as time): {Nt : t ≥ 0}. This family of random variables musty
satisfy the following properties.
19

21. (i) For 0 < t1 < t2 < . . . < tk, the increments
Nt1 , Nt2−t1 , · · · , Ntk−tk−1
are independent.
(ii) The individual increments have the Poisson distribution:
P[Nt − Ns = n] = e−λ(t−s) λ(t − s)
n
n!
,
where n = 0, 1, . . . and 0 ≤ s < t.
Proposition A.1. If {Nt : t ≥ 0} is a Poisson process, then
E[Nt − Ns] = λ(t − s),
for s < t.
Proof. We have
E[Nt − Ns] =
∞
k=1
kP[Nt − Ns = k]
=
∞
k=1
k e−λ(t−s) λ(t − s)
k
k!
= e−λ(t−s)
∞
k=1
k
λ(t − s)
k
k!
= e−λ(t−s)
∞
k=1
λ(t − s)
k
(k − 1)!
= e−λ(t−s)
∞
k=0
λ(t − s)
k+1
k!
= λ(t − s)e−λ(t−s)
eλ(t−s)
= λ(t − s).
Example A.2. Characteristic examples of Poisson processes are similar to the concept of
an arrival time. The number of arrival times during one time interval may be statistically
independent of the number of arrivals during another non-overlapping time interval. A
telephone hot-line would be a great example of a Poisson process in that the calls may be
statistically independent of each other in separate time intervals.
20

22. An non-homogeneous Poisson process with rate function λ is a collection of random
variables {Nt : t ≥ 0} that still satisfies (i), but with the condition that
P[Nt − Ns = n] = e−Λs,t
Λn
s,t
n!
,
where
Λs,t =
t
s
λ(τ) dτ.
Remark A.3. If λ(t) = λ is a constant function, then this new definition of a Poisson
process agrees with the old one.
Proposition A.4. If {Nt : t ≥ 0} is a Poisson process with rate function λ, then
E[Nt − Ns] = Λs,t.
Proof. The proof is similar to that of Proposition A.1.
21

23. B Discrete Laplacian Operator
The Discrete Laplacian Operator was used to describe how a criminal moves to neigh-
boring sites and effectively simplify our equation used. Specifically, we want to know the
inclination of a given criminal towards a given house in the two dimensional lattice. The
Discrete Laplacian Operator can help.
We shall begin this process by taking a one dimensional lattice. A diagram representation
of this lattice is given below:
It follows from the diagram that we can approximate the derivative of a function f(x)
by showing that
f (x) ≈
f(x + h) − f(x − h)
2h
.
In other words, we are taking the derivative of f at any arbitrary point, x, by evaluating
f at each neighbor, on the left and on the right. In order to prove this mathematically, let
us recall the standard form of the Taylor Series Expansion for the function g(h) = f(x + h)
at h = 0:
g(h) = g(0) + g (0)h +
g (0)
2!
h2
+
g (0)
3!
h3
+ · · · .
From here, it is easy to explicitly define f(x + h) using the Taylor Series Expansion
f(x + h) = f(x) + f (x)h +
f (x)
2!
h2
+
f (x)
3!
h3
+ · · · .
And f(x − h) is just a change in sign on the values with odd degree:
f(x − h) = f(x) − f (x)h +
f (x)
2!
h2
−
f (x)
3!
h3
+ · · · .
22

24. To approximate f (x), let us evaluate f(x + h) − f(x − h):
f(x + h) − f(x − h) = f(x) − f(x) + f (x)h + f (x)h +
f (x)
2!
h2
−
f (x)
2!
h2
+ · · · .
Canceling like terms, we get
f(x + h) − f(x − h) = 2f (x)h + 2
f (x)
3!
h3
· · · .
Rearranging the equation, we see that
−2f (x)h ≈ f(x − h) − f(x + h).
Finally, by algebraic manipulation, we have proven that the derivative of f(x) as given by
the diagram satisfies
f (x) ≈
f(x + h) − f(x − h)
2h
.
Remark B.1. Please note that the second derivative of f, f (x), can easily be obtained by
taking the sum of f(x + h) and f(x − h). The result is
f (x) =
f(x + h) + f(x − h) − 2f(x)
h2
.
We use a two dimensional lattice in our model, and therefore, we can apply this same ap-
proach to evaluating the two dimensional lattice f(x, y) and subsequently define our Discrete
Laplacian Operator. Let us evaluate the second derivative based on this picture:
23

25. Following from the one dimension lattice example, it is easy to see that we can partition
this graph into two one dimensional planes: x and y. Thus, the second derivative of f(x, y)
with respect to x, fxx(x, y), satisfies
fxx(x, y) ≈
f(x + h, y) + f(x − h, y) − 2f(x, y)
h2
.
And the second derivative of f(x, y) with respect to y, fyy(x, y), satisifes
fyy(x, y) ≈
f(x, y + h) + f(x, y − h) − 2f(x, y)
h2
.
From here, we can see that the Laplacian of f is approximated by the Discrete Laplacian
Operator on a square lattice, specifically with mesh size h:
fxx(x, y)+fyy(x, y) ≈
[(f(x − h, y) + f(x + h, y) + f(x, y − h) + f(x, y + h) − 4f(x, y)]
h2
= ∆f(x, y).
This can also be adapted to show rigorously that the Discrete Laplacian approaches the
Laplacian on R2
by taking mesh size to zero.
24

26. C Re-Dimensionalization of Continuum Equations
Here, we show our work for the development of our dimensionless equations in the Con-
tinuum Limit section of this paper. Before we begin with the equations, let us list the
dimensionless relationships that we have established, beginning with the natural time scale
and characteristic length scale, respectively:
τ ≡ 1/ω, c ≡
D
ω
(C.0.1)
Now, let us provide the following dimensionless variables, denoted with a tilde:
B = B/ω, A = A/ω, p = 2
cp, x = 2x/ c, t = ωt (C.0.2)
With all of the relevant information at hand, let us begin the re-dimensionalization of
our system by starting with the equation for the continuum version of attractiveness:
∂B
∂t
=
ηD
4
2
B − ωB + DpA. (C.0.3)
Specifically, let us break up the ∂t. This can be expressed as
∂
∂t
=
∂t
∂t
·
∂
∂t
. (C.0.4)
Using (C.0.2), we can re-write this expression inserting the partial of t with respect to the
partial of t:
∂
∂t
= ω ·
∂
∂t
. (C.0.5)
Hence, now we can evaluate the left-hand side of the original equation by applying the
chain-rule and using (C.0.2):
∂B
∂t
= ω ·
∂
∂t
ωB
= ω2
·
∂B
∂t
.
From here, let us begin with the first term on the right-hand side. By definition of the
Laplacian,
2
B =
∂2
∂2x1
B +
∂2
∂2x2
B, (C.0.6)
where we must transform the set of (x1, x2) to the set of (x1, x2). Applying (C.0.2) to the
transformation we get
(x1, x2) =
2
c
x1,
2
c
x2 . (C.0.7)
25

27. Since we know that both of the terms will maintain the same operations and coefficients, let
us begin with ∂B
∂x1
and apply (C.0.2):
∂B
∂x1
=
∂B
∂x1
·
∂x1
∂x1
=
∂(ωB)
∂x1
·
∂x1
∂x1
=
2ω
c
·
∂B
∂x1
Recall that we are after ∂2B
∂2x1
, so let us use the same process and the same dimensionless
variables in our simplification:
∂2
B
∂2x1
=
2ω
c
·
∂
∂x1
∂B
∂x1
·
∂x1
∂x1
=
2ω
c
·
∂2
B
∂2x1
·
2
c
=
4ω
2
c
·
∂2
B
∂2x1
.
Using (C.0.1), we can write the last equation as
∂2
B
∂2x1
=
4ω2
D
·
∂2
B
∂2x1
(C.0.8)
which gives
∂2
B
∂2x2
=
4ω2
D
·
∂2
B
∂2x2
. (C.0.9)
In essence, we now have the following relationship that we can use for the first term in the
right-hand side of the equation:
2
B =
4ω2
D
2B. (C.0.10)
Combining the coefficient to this term, we now have
ηD
4
2
B =
ηD
4
4ω2
D
2B
= ηω2 2B.
Luckily, the conversion of the other terms in this equation to dimensionless forms is a little
more straightforward. We immediately note that by using (C.0.2), the second term of the
right-hand side of the equation is
ωB = ω2
B. (C.0.11)
26

28. The last term on the right-hand side of the equation can also be simplified by using both
(C.0.2) and (C.0.1):
DpA = D
p
2
c
A
=
pDA
2
c
= p ω A
= p ω2
A
= pω2
A.
Combining it all together and properly substituting the new values term by term, we
get the following equation, where we want to isolate the dimensionless equation for the
continuum version of attractiveness:
ω2
·
∂B
∂t
= ηω2 2B − ω2
B + pω2
A (C.0.12)
Canceling the ω2
from both sides and dropping the tilde notation, we get the same equation
as stated in (3.0.18):
∂B
∂t
= η 2
B − B + pA. (C.0.13)
Let us now finish the re-dimensionalization of our system by concluding with the equation
for the continuum version of burglar density:
∂p
∂t
=
D
4
· p −
2p
A
A − pA + γ. (C.0.14)
We still will use the same dimensionless scales and variables in the re-working of this equation,
(C.0.1) and (C.0.2) respectively. Additionally, much of the relationships observed in the
previous equation will carry over to use in this equation, shortening some of the work. The
process will start similarly to the previous equation, with the left-hand side of the equation:
∂p
∂t
= ω
∂
∂t
p
2
c
=
ω
2
c
·
∂p
∂t
.
We begin re-writing the right-hand side by removing the dimensions associated with :
=
∂
∂x1
,
∂
∂x2
. (C.0.15)
27

29. similar to the process used in the previous equation, we have
∂
∂x1
=
∂x1
∂x1
·
∂
∂x1
=
2
c
·
∂
∂x1
.
Now, let us re-write (C.0.15) as
=
2
c
·
∂
∂x1
,
2
c
·
∂
∂x2
. (C.0.16)
Finally, we have a relationship for the dimensionless discrete spatial Laplacian:
=
2
c
. (C.0.17)
Let us apply this to the first term in the brackets:
p =
2
c
p
2
c
=
2
3
c
p.
The second term in the brackets follows the same logic:
A =
2
c
ωA
=
2ω
c
A.
Now, let us solve for the dimensionless expression within the brackets:
p −
2p
A
A =
2
3
c
p −
2
ωA
p
2
c
2ω
c
A
=
2
3
c
p −
2p
A
A .
Adding the coefficient to this expression, we get
D
4
· p −
2p
A
A =
D
4
2
c
·
2
3
c
p −
2p
A
A
=
D
4
c
· p −
2p
A
A
=
ω
2
c
· p −
2p
A
A .
28

30. From here, let us look at the remaining two terms on the right-hand side, beginning with
pA:
pA =
p
2
c
ωA
=
ω
2
c
pA.
Now, let us look at γ, specifically we will use the relation given in (3.0.15):
γ =
ω
D
ω ¯B
=
ω
2
c
¯B.
Combining it all together and properly substituting the new values term by term, we
get the following equation, where we want to isolate the dimensionless equation for the
continuum version of burglar density:
ω
2
c
·
∂p
∂t
=
ω
2
c
p −
2p
A
A −
ω
2
c
pA +
ω
2
c
¯B. (C.0.18)
Canceling the ω/ 2
c from both sides and dropping the tilde notation, we get the same
equation as stated in (3.0.19):
∂p
∂t
= · −
2p
A
A − pA + ¯B. (C.0.19)
29

31. References
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[2] Colvin, Mark. Crime and coercion: An integrated theory of chronic criminality. New
York: St. Martin’s Press, 2000.
[3] L. Anselin, J. Cohen, D. Cook, W. Gorr and G. Tita, Criminal Justice 2000, Vol. 4
(National Institute of Justice, 2000), pp. 213-262.
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