851-0585-04L – Modeling and Simulating
Social Systems with MATLAB
Lecture 3 – Dynamical Systems
Karsten Donnay and Stefano Balietti
Chair of Sociology, in particular of
Modeling and Simulation
© ETH Zürich |
2011-03-07
Schedule of the course
Introduction to
MATLAB
21.02.
28.02.
07.03.
14.03.
21.03.
28.03.
Working on
projects
(seminar
theses)
04.04.
18.04.
02.05.
09.05.
16.05.
23.05.
30.05.
2011-03-07
Introduction to
social-science
modeling and
simulation
Handing in seminar thesis
and giving a presentation
(final deadlines to be communicated)
K. Donnay & S. Balietti / [email protected] [email protected]
2
Schedule of the course
Introduction to
MATLAB
21.02.
28.02.
07.03.
Create and Submit a
Research Plan
14.03.
21.03.
28.03.
Working on
projects
(seminar
theses)
04.04.
18.04.
02.05.
09.05.
16.05.
23.05.
30.05.
2011-03-07
Introduction to
social-science
modeling and
simulation
Handing in seminar thesis
and giving a presentation
(final deadlines to be communicated)
K. Donnay & S. Balietti / [email protected] [email protected]
3
Goals of Lecture 3: students will
1. Consolidate knowledge acquired during lecture 2, through
brief repetition of the main points and revision of the
exercises.
2. Understand fully the importance and the role of a
Research Plan and its main standard components.
3. Learn the basic definition of Dynamical Systems and
Differential Equations.
4. Implement Dynamical System in MATLAB a series of
examples from Physics and Social Science Literature
(Pendulum, Lorenz attractor, Lotka-Volterra equations,
Epidemics: Kermack-McKendrick model)
2011-03-07
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Repetition

plot(), loglog(), semilogy(), semilogx()
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
hist(), sort()

The amplitude of the sample matters! law of large
numbers – LLN. (Ex.1)

Outlier observations can significantly skew the
distribution, e.g. mean != median. (Ex. 2)

Cov() and corrcoef() can measure correlation
among variables. (Ex. 3)
subplot() vs figure()
Log-scale plots reduce the complexity of some functional
forms and fit well to draw large range of values
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Repetition – Ex. 4
>> loglog(germany(:,2),'LineWidth',3);
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Project
 Implementation of a model from the Social
Science literature in MATLAB
 During the next two weeks:




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Form a group of two or three persons
Choose a topic among the project suggestions
(available on line) or propose your own idea
Fill in the Research Plan (available on line) and send it
to [email protected] AND [email protected] by email
Subject line: [MATLAB FS11] name1,name2,name3
E.g. [MATLAB FS11] donnay, balietti
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Research Plan Structure
 1-2 (not more!) pages stating:
 Brief, general introduction to the problem
 Fundamental questions you want to try to answer
 Existing literature you will base your model on and
possible extensions
 Research methods you are planning to use
 Use a Word/Openoffice format, to easily track
changes
 It will become the first page of your report.
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Research Plan
 Upon submission, the Research Plan can:



be accepted
be accepted under revision
be modified later on only if change is justified (create
new version)
 Talk to us if you are not sure
 Remember: the sooner the better!
 Final deadline for signing-up for a project is:
March 14th 2011
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“Plans are useless,
but planning is
everything.”
Dwight David Eisenhower
(14 Oct 1890 – 28 Mar 1969)
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Dynamical systems
 Mathematical description of the time dependence of
variables that characterize a given problem/scenario in its
state space.
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Dynamical systems
 A dynamical system is described by a set of
linear/non-linear differential equations.
 Even though an analytical treatment of
dynamical systems is usually very complicated,
obtaining a numerical solution is (often) straight
forward.
 Differential equation and Difference equation are
two different tools for operating with Dynamical
Systems
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Differential Equations
 A differential equation is a mathematical
equation for an unknown function (dependent
variable) of one or more (independent)
explicative variables that relates the values of
the function itself and its derivatives of various
orders
df
(x)
 Ordinary (ODE) :
 f (x)
dx
f (x, y) f (x, y)
 Partial (PDE) :

 f (x, y)
x
y
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Numerical Solutions for Differential Equations
 Solving differential equations numerically can be
done by a number of schemes. The easiest way
is by the 1st order Euler’s Method:
dx
 f ( x,...)
x(t)
dt
x(t-Δt)
x(t )  x(t  t )
 f ( x,...)
t
x(t )  x(t  t )  t f ( x,..)
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x
t
Δt
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A famous example: Pendulum
 A pendulum is a simple dynamical system:
L = length of pendulum (m)
ϴ = angle of pendulum
g = acceleration due to
gravity (m/s2)
The motion is described by:
g
    sin(  )
L
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Pendulum: MATLAB code
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Set time step
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Set constants
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Set starting point of pendulum
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Time loop: Simulate the pendulum
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Perform 1st order Euler’s method
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Plot pendulum
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22
Set limits of window
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23
Make a 10 ms pause
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Pendulum: Executing MATLAB code
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Lorenz attractor
 The Lorenz attractor defines a 3-dimensional
trajectory by the differential equations:
 σ, r, b are parameters.
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dx
  (y  x)
dt
dy
 rx  y  xz
dt
dz
 xy  bz
dt
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Lorenz attractor: MATLAB code
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Set time step
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Set number of iterations
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Set initial values
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Set parameters
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Solve the Lorenz-attractor equations
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Compute gradient
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Perform 1st order Euler’s method
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Update time
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Plot the results
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Animation
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Food chain
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Lotka-Volterra equations
 The Lotka-Volterra equations describe the
interaction between two species, prey vs.
predators, e.g. rabbits vs. foxes.
x: number of prey
y: number of predators
α, β, γ, δ: parameters
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dx
 x(   y )
dt
dy
  y (  x)
dt
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Lotka-Volterra equations
 The Lotka-Volterra equations describe the
interaction between two species, prey vs.
predators, e.g. rabbits vs. foxes.
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Lotka-Volterra equations
 The Lotka-Volterra equations describe the
interaction between two species, prey vs.
predators, e.g. rabbits vs. foxes.
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Epidemics
Source: Balcan, et al. 2009
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SIR model
 A general model for epidemics is the SIR model,
which describes the interaction between
Susceptible, Infected and Removed (immune)
persons, for a given disease.
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Kermack-McKendrick model
 Spread of diseases like the plague and cholera?
A popular SIR model is the Kermack-McKendrick
model.
 The model assumes:


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A constant population size.
A zero incubation period.
The duration of infectivity is as long as the duration of
the clinical disease.
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Kermack-McKendrick model
 The Kermack-McKendrick model is specified as:
S: Susceptible persons
I: Infected persons
R: Removed (immune)
persons
β: Infection rate
γ: Immunity rate
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S
β transmission
R
γ
recovery
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I
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Kermack-McKendrick model
 The Kermack-McKendrick model is specified as:
S: Susceptible persons
I: Infected persons
R: Removed (immune)
persons
β: Infection rate
γ: Immunity rate
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dS
   I (t ) S (t )
dt
dI
  I (t ) S (t )   I (t )
dt
dR
  I (t )
dt
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Kermack-McKendrick model
 The Kermack-McKendrick model is specified as:
S: Susceptible persons
I: Infected persons
R: Removed (immune)
persons
β: Infection rate
γ: Immunity rate
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Exercise 1
 Implement and simulate the KermackMcKendrick model in MATLAB.
Use the starting values:
S=I=500, R=0, β=0.0001, γ =0.01
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Exercise 2
 A key parameter for the Kermack-McKendrick
model is the epidemiological threshold, βS/γ.
Plot the time evolution of the model and
investigate the influence of the epidemiological
threshold, in particular the cases:
1.
βS/γ < 1
2.
βS/γ > 1
Starting values: S=I=500, R=0, β=0.0001
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Exercise 3 - optional
 Implement the Lotka-Volterra model and
investigate the influence of the timestep, dt.
 How small must the timestep be in order for the
1st order Euler‘s method to give reasonable
accuracy?
 Check in the MATLAB help how the functions
ode23, ode45 etc, can be used for solving
differential equations.
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References

Matlab and Art:
http://vlab.ethz.ch/ROM/DBGT/MATLAB_and_art.html

Kermack, W.O. and McKendrick, A.G. "A Contribution to the
Mathematical Theory of Epidemics." Proc. Roy. Soc. Lond. A 115,
700-721, 1927.

"Lotka-Volterra Equations”.http://mathworld.wolfram.com/LotkaVolterraEquations.html


http://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations
"Lorenz Attractor”.
http://mathworld.wolfram.com/LorenzAttractor.html
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