Random Walk Contaminant Spread
by Parrish Myers
DSES-6620 Simulation Modeling and Analysis
Spring 2002
Ernesto Gutierrez-Miravete
Random Walk Contaminant Spread
Parrish Myers
Introduction
The release of a deadly contaminant within a populated ecosystem can become
of great concern. This concern can come from the death toll associated with the
outbreak of a deadly virus within a city, or even the impact of a fish population
associated with chemical waste being emptied into local lakes and rivers. In
each case a localized problem can both devastate that local ecosystem and
spread beyond the boundaries of the localized system and become a much
larger problem. Simulations can answer some of that concern by predicting the
death toll, or providing future trends in fish population.
Project Overview and Objectives
To investigate the process of a contaminant spread, a first-order model has been
constructed to simulate a small and localized ecosystem. The simulation
attempts to mimic a small container (or exposure chamber) filled with water and a
limited number of organisms. This small ecosystem (or system) will then have a
contaminant introduced, modeled using a random walk process.
The organism chosen for this simulation was Daphnia Magna a small freshwater
crustacean that is extensively used as both a fish food and as a bio-indicator in
toxicity experiments. The contaminant chosen was Methyl Parathion, an
insecticide that has been made in the United States since 19521. Both the
organism and contaminant were primarily chosen because of readily available
information from the Environmental Protection Agency’s (EPA) web site 2.
The original objective of this project, determining the killing radius and time it
takes for a contaminant to become non-lethal, was modified slightly to provide a
clear method for validation. The current objective of this experiment is to
1
More information can be found at the ATSDR homepage at
http://www.atsdr.cdc.gov/tfacts48.html.
2 The Environmental Protection Agency is located at http://www.epa.gov.
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determine the median lethal dose3 (LD50) value of Methyl Parathion, using
Daphnia Magna as the test organisms.
While this simulation will not be sufficient to answer any real questions about the
mortality rate of an organism’s population in a real ecosystem, it will provide a
non-lethal introduction to the subject of acute toxicity.
Model Design
This experiment has a clearly defined organism and contaminant under study,
even so, this simulation was designed with enough generality to allow the
simulation to be extended to any organism and contaminant combination, and
the following description of the system will reflect that.
Early in development it was recognized a general purpose programming
language would provide the needed flexibility and control necessary to
successfully create this simulation. This is in contrast to simulation packages,
such as ProModel or Witness, which are geared towards manufacturing and
service systems. The programming language Java4 was selected for its robust
2D drawing capabilities, and cross-platform nature. The resulting application,
named “rwcs” (random walk contaminant spread), can be run anywhere the Java
Runtime Environment has been ported to.
The source code (seen in Appendix B) uses the object-oriented nature of the
Java language to compartmentalize the structure of the source code. This
simulation program contains 7 main classes or sections,
3
Casarett, L. J., p.19.
J2SE version 1.3.1_03 was used for this simulation. It can be downloaded from
http://java.sun.com.
4
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Class
Purpose
Main entry point / user
interface controls and
input
Simulation thread / main
Simulation drawing and
control
All functions related to
managing the organism
All functions related to
managing the
contaminant
Central repository for
data and data access
Basic data structure to
provide the (x, y) position
and life count
All functions related to
the reading of default
options
rwcs
Contaminant
FishControl
ContaminantControl
SystemDataStore
Ball
GlobalProps
with a separate file for each class. This structure insured the program could be
understood and extended in functionality with minimal effort.
The main application interface is show in Figure 1. The interface contains basic
controls to start, stop, pause, and resume the simulation. Text entries have been
added to control some of the available parameters of the simulation. Additional
parameters are available via the configuration file that accompanies this
program. The main area of the application is where the contaminant spread is
visualized. A zoom has been added to aid in the visualization detail shown
during the simulation run-time. The bottom edge of the interface includes
system status and parameters of the running simulation. The system
visualization area consists of a black 2 dimensional field containing both
organisms and contaminant drawn as circles. Organism and contaminant are
distinguished by color, with healthy organisms drawn in blue, dead organisms
drawn in yellow with a black “x” in the center, and the contaminant drawn in red.
A system boundary drawn in green is also drawn, but is not shown in Figure 1.
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Figure 1 - RWCS Program Interface
The original proposal stipulated a number of simplifying assumptions, which were
agreed upon. Those assumptions guided the construction of the simulation.
Simulation Assumptions:
1. The system is to be modeled in 2 dimensions.
2. The contaminant spread is modeled as a random walk of constant step
size.
3. All N molecules of the contaminant start a (0,0), which is denoted to be
the center of the system.
4. The contaminant is assumed discrete, containing N i grouping of
molecules: where i  0,1, 2,... and each group consists of x molecules (the
exact number to be determined at verification of the model)
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5. There are two exits for the N i molecule groups within the system: 1) the
molecule group wanders beyond the boundary set up at system start and
2) the molecule groups collide with an organism.
6. Each organism within the system has a stochastic susceptibility, which will
be calculated at simulation start. When a N i molecule group collides with
the organism the life of that organism is reduced by a fraction of that
susceptibility amount.
7. During the life of the simulation the organisms are stationary
The ecosystem consists of a 2 dimensional plane broken into equidistant lattice
points set upon the Cartesian coordinate system. Therefore any object in the
system can be located using a two-element coordinate (x, y). The origin (0, 0) is
defined to be the center of the system.
The random walk process is extensively used in statistical physics to
characterize diffusion of molecules in a substance5. The contaminant spread is
modeled as a random walk of constant step size with no drift. This corresponds
to each step taken by one of the N i molecule groups having an equal probability
of moving up, down, left, or right one lattice point per time step.
At simulation start time a known number of molecule groups, N , are placed at
(0,0) and are allowed to start wandering (at each time step all molecule groups
are moved one lattice point). The direction is randomly picked using a random
variate with a discrete uniform distribution between 0 and 3. The direction of
travel is then correlated to the number drawn, with
5
The random walk is normally used to describe diffusion of a molecule traveling in a gas. But,
this process is still valid for molecules traveling in a liquid, albeit slower, and due to its simplicity it
is used for the contaminant spread in water.
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Enumeration
Direction
0
RIGHT
1
UP
2
LEFT
3
DOWN
Due to the sheer number of molecules present in any quantity of contaminant,
keeping track of all the individual molecules is computationally expensive.
Hence, a known quantity6 of molecules is grouped together to form discrete
element N i , where i  1, 2,3,... . The number of molecule groups is then keep less
than 1000. This restriction greatly reduces the computational power required to
run the simulation and allow it to be run on any platform or computer that
supports a recent version of the Java runtime environment.
Two exits exist in the simulation for molecule groups of contaminant to leave the
system. One, the molecule wanders past the system area defined7 and two a
molecule group collides with an organism. If a group of molecules wanders past
the defined boundaries, the likelihood that that group will interact with any of the
remaining organisms is small. Therefore, rather can continue wasting computer
cycles tracking a non-interacting object, the group is removed from the system.
The second exit condition provides the mechanism for interaction between
organism and contaminant. When a collision occurs the contaminant is removed
from the system, to prohibit that molecule group from reacting with another
organism. At simulation start a discrete random-variate is calculated for each of
the organisms that exist within the system. This defines the susceptibility (or life
count) of that organism. If a molecule group collides with that organism the life
count of that organism is reduced by one
6
7
The precise quantity of molecules in the group is determined at Validation.
The exact system area is defined at validation.
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before collision
Si  Li
after collision
Si  Li  1
where, Si is the life count and Li is the susceptibility calculated for organism i at
simulation start. When, the life count Si reaches 0
the organism is considered dead. To illustrate the
decrement of the life counter, a color scale has
been created, seen in Figure 2. When life counter,
Si , equals the initial susceptibility value, Li , the
organism is drawn in blue. As the organism loses
life the color transitions to green, and finally to
Figure 2 - Life Count
yellow when the life count reaches 0.
Scale for Organisms
To reduce model complexity all organisms within the
system are stationary. At simulation start, two random-variates of uniform
distribution is calculated for each coordinate x and y8 for each organism Oi
where i  1, 2,3,... .
Data Collection
Though much effort and investigation, no substantial data on the physiological
effects of substances could be collected in terms of the susceptibility of the
Daphnia Manga to a general contaminant. The basic physiological effects of a
substance on any organism are a complicated question to answer; hence the
need for Toxicology studies. Most of the data available consists of median lethal
dose9 (LD50) data. Without any real data, the two choices are uniform and
triangular random-variates. The susceptibility is thought to happen around a
central or median value. A distribution that is centered around a median value is
8
In reality a uniform random-variate is drawn for the distance from the origin and a second
uniform random-variate is drawn for the angle around the origin. These two coordinate values
are then translated into an x, y position using a standard transform from a circular coordinate
system to Cartesian.
9 Casarett, L. J. p.19
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better suited than a uniform random-variate, and therefore a triangular randomvariate10 was selected for the susceptibility of the Daphnia Magna.
A uniform random-variate has been chosen for the initial position of the
organisms within the system. While a heterogeneous environment my exist the
organisms can and will move around the system. No distribution or clumping can
be predicted or known in advance (assuming no currents, uniform temperature,
ph and lighting). Therefore the organisms can and will occupy all areas of the
system. A uniform random-variate works well for this situation.
Verification / Validation
A source code debugger and visual inspections of the simulation animation11
were extensively used in reducing grammatical and logical errors. The
simulation heavily relies on the compartmentalization attained by object-oriented
design. This ensured the design of the simulation has logically breaks between
features for readability and maintainability.
The simulation assumptions not only provided a framework for construction, but a
method for verification as well. Close attention has been made to insure the
simulation conforms to the stated assumptions. The simulation has also been
designed to terminate under many conditions:

All organisms are dead,

All contaminant molecule groups have exited the system, and

The system time limit has expired,
to ensure reliability.
The simulation procedure uses a modified version of the EPA’s standard
operating procedure # 2024 “48-Hour Acute Toxicity Test Using Daphnia Magna
10
the form of the triangular random-variate was obtained from
http://www.itl.nist.gov/div898/software/dataplot/refman2/ch8/tripdf.pdf.
11 Banks J., p. 370
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and Daphnia Pulex”12. This procedure outlines the process (seen in Appendix A)
that was used to calibrate the simulation.
Model Results
In an effort to follow the EPA’s SOP #2024 a set of parameters were determined
by trial and error, to produce a realistic simulation. The number of Daphnia
Magna was limited to 10. The radius of the system was set to 50.0. The
simulation was set to end in 48-hours, with a time step of 1 minute. At each time
step the contaminant (Methyl Parathion) was advanced 0.4, the lattice step size.
Three alarms were set to display the number of dead organisms at 1, 24, and 48
hours. The susceptibility parameters for the triangular distribution had a
minimum value of 1 and a maximum value of 5 with a mean of 3. Once these
parameters were set, they were not changed for the entire data collection period.
The only parameter left unspecified was the number of molecule groups of
Methyl Parathion. This parameter simulates the amount of contaminant within
the system. The procedure for operation thus follows:
1. Start with an amount of contaminant small enough to have minimal effect
2. Record the number of dead organisms at: 1-Hour, 24-Hours, and 48Hours
3. Increase the amount of contaminant and repeat step 2.
4. When organism mortality becomes 80%-90% STOP!
5. Tally data and find LD50.
The following table shows the data collected from following this procedure:
12
http://www.epa.giv/epahome/index/nameindx.htm
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Molecule
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Mortality Count
Groups
1 Hour
24 Hours
48 Hours
40
2
3
4
50
2
3
4
60
2
3
5
70
2
3
5
80
2
5
6
90
2
4
5
100
2
4
7
110
2
4
7
120
2
4
7
130
2
4
5
140
2
6
6
It was determined from the data that the effective concentration 50% region
(EC50) was between 60 and 90 molecule groups. The data collected from the
EPA web site13 stipulates that Methyl parathion is “very highly toxic” to Daphnia
Magna with the LD50 value is 0.14 ppb and the EC50 range is 0.09 – 0.2 ppb.
Using a simple linear correlation,
(3 23 x103 )M g  0.13  C
where M g is the number of molecule groups and C is the resulting dose
concentration, the number of molecule groups was correlated to concentration of
Methyl parathion in units of part per billion. This resulting data is shown in Figure
13
http://www.epa.gov/pesticides/op/methyl_parathion/mp_12_1.pdf
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3. The graph shows both the calibrated EC50 region and the corresponding
LD50 value of 0.14.
Mortality of Delphia Magna
9
EC50 Area
Number of Organisms Dead
8
LC50
7
6
5
4
3
48 hours
24 hours
1 hour
2
1
0.
60
0.
57
0.
53
0.
49
0.
46
0.
42
0.
38
0.
31
0.
35
0.
27
0.
24
0.
20
0.
16
0.
13
0.
09
0.
05
0.
02
0
Concentration (ppb)
Figure 3 - Simulation Results
Conclusion
While this simulation is only a first-order representation of an LD50 toxicity
experiment, it does provide valuable (and non-lethal) information in the
interaction and conduct of a contaminant spread within a small ecosystem. Even
so, this simulation only studied one organism, Daphnia Magna. A real medial
lethal dose toxicity experiment includes many organisms of various sizes and
species before any real conclusion can be drawn.
There are many enhancements and features that missed inclusion in this
simulation due to time constraints. Some of these features include:

The organisms within the system move

The system modeled in 3 dimensions

The animations can be made more elaborate
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
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The contaminant cannot exceed the boundaries of the system (they
bounce off the walls... this simulated being in a exposure chamber).

Many additional organisms should be simulated to generate a better
LD50 determination

The inclusion of a configuration menu to reduce the need for modifying
the configuration file every time a parameter (that is not on the interface)
with a text editor.

The ability to queue many runs.

Automatic data recording.

Automatic data conversion from simulation units to real units.
A second order model should address some of these suggestions.
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References
SOP# 2024, 1994, 48-Hour Acute Toxicity Test Using Daphnia Magna and
Daphnia Pulex, EPA, http://www.epa.gov/epahome/index/nameindx.htm
Casarett, L. J., 1980, Casarette and Doull’s Toxicology, Macmillan Publishing
Co., New York, p.17-27.
Reif, F., 1965, Fundamentals of statistical and thermal physics, McGraww-Hill,
New York, p. 486-487.
Whitney, C. A., 1990, Random Processes in Physical Systems, Wiley
Interscience Publishing, New York, p. 37-46.
Banks, J., Carson, J. S. II, Nelson, B. L., Nicol, D. M., 2001, Discrete-Event
System Simulation, Prentice Hall, New Jersey, p. 18, 290-297, 367.
Ross, S. M., 1997, Introduction to Probability Models, Sixth Edition, Academic
Press, San Diego, p. 55-56.
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Appendix A – EPA SOP #2024
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Appendix B – Source Code
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