Modeling and Simulating Social Systems with MATLAB · 2016-10-10 · 2016-10-10 Modeling and...

2016-10-10 © ETH Zürich |

Modeling and Simulating Social Systems with MATLAB

Lecture 3 – Dynamical Systems and plotting with MATLAB

© ETH Zürich |

Computational Social Science

Olivia Woolley, Lloyd Sanders, Dirk Helbing

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Schedule of the course 26.09. 03.10. 10.10. 17.10. 24.10. 31.10. 07.11. 14.11. 21.11. 28.12. 05.12. 12.12. 19.12.

Introduction to MATLAB

Introduction to social science modelling and simulations Working on projects

(seminar thesis)

Handing in seminar thesis and giving a presentation

Flash Talks

Modeling overview

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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

§  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’s Method:

,..)( )()(

,...)()()(

,...)(

xftttxtx

xft

ttxtx

xfdtdx

Δ+Δ−=

=Δ

Δ−−

=

t

x

Δt

x(t)

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:

)sin(θθLg

−=ʹ′ʹ′

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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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Make a 10 ms pause

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Pendulum: Executing MATLAB code

The file pendulum.m may 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 may be found on the website!

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Food chain

31

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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

)(

)(

xydtdy

yxdtdx

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

37

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Kermack-McKendrick model §  The Kermack-McKendrick model is specified as:

S: Susceptible I: Infected R: Removed/recovered β: Infection/contact rate γ: Immunity/recovery rate

)(

)( )()(

)()(

tIdtdR

tItStIdtdI

tStIdtdS

γ

γβ

β

=

−=

−=

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Kermack-McKendrick model §  The Kermack-McKendrick model is specified as:

S: Susceptible I: Infected R: Removed/recovered β: Contact rate γ: Recovery rate

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SIR model with interaction structure

40

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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) < 1 2.  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

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

§  Strogatz, Steven H. “Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering.” Westview press, 2014.