2014-03-03ETH Zürich

Modeling and Simulating Social Systems with MATLAB

Lecture 3 – Dynamical Systems

Chair of Sociology, in particular of

Modeling and Simulation

Olivia Woolley, Tobias Kuhn, Dario Biasini, Dirk Helbing

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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) straightforward

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

Ordinary (ODE) :

Partial (PDE) :

df (x)

dx= f (x)

∂f (x,y)

∂x+ ∂f (x,y)

∂y= f (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 Method:

t

x

Δt

x(t)

x(t-Δt)

,..)( )()(

,...)()()(

,...)(

xftttxtx

xft

ttxtx

xfdt

dx

∆+∆−=

=∆

∆−−

=

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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:

)sin(θθL

g−=′′

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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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Set limits of window

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Pause for 10ms

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Pendulum: Executing MATLAB code

The file pendulum.m can be found on the website

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Lorenz attractor

The Lorenz attractor defines a 3-dimensional trajectory by the differential equations:

σ, r, b are parameters.

dxdt

= σ(y− x)

dydt

= rx− y− xz

dzdt

= xy−bz

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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

The file lorenzattractor.m can be found on the website

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Food chain

29

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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 preyy: number of predatorsα, β, γ, δ: parameters

)(

)(

xydt

dy

yxdt

dx

δγ

βα

−−=

−=

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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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SIR model

A general model for epidemics is the SIR model, which describes the interaction between Susceptible, Infected and Removed (Recovered) 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: Constant population size. No incubation period. The duration of infectivity is as long as the duration of

the clinical disease.

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Kermack-McKendrick model

35

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Kermack-McKendrick model

The Kermack-McKendrick model is specified as:

S: Susceptible I: InfectedR: Removed/recovered β: Infection/contact rateγ: Immunity/recovery rate

)(

)( )()(

)()(

tIdt

dR

tItStIdt

dI

tStIdt

dS

γ

γβ

β

=

−=

−=

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Kermack-McKendrick model

The Kermack-McKendrick model is specified as:

S: Susceptible I: InfectedR: Removed/recovered β: Contact rateγ: Recovery rate

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Kermack-McKendrick model

38

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Exercise 1

Implement and simulate the Kermack-McKendrick model in MATLAB.

Use the values: S(0)=I(0)=500, R(0)=0, β=0.0001, γ =0.01

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Exercise 2

A key parameter for the Kermack-McKendrick model is the reproductive number R0= β/γ.

Plot the time evolution of the model and investigate the epidemiological threshold, in particular the cases:

1. R0S(0) < 12. R0S(0) > 1

S(0)=I(0)=500, R(0)=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/Lotka-VolterraEquations.html

http://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations

"Lorenz Attractor”. http://mathworld.wolfram.com/LorenzAttractor.html