2010-10-11 © ETH Zürich | 851-0585-04L – Modeling and Simulating Social Systems with MATLAB Lecture 3 – Dynamical Systems © ETH Zürich | Giovanni Luca.

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Presentation transcript:

© ETH Zürich | L – Modeling and Simulating Social Systems with MATLAB Lecture 3 – Dynamical Systems © ETH Zürich | Giovanni Luca Ciampaglia, Stefano Balietti and Karsten Donnay Chair of Sociology, in particular of Modeling and Simulation

G. L. Ciampaglia, S. Balietti & K. Donnay / 2 Projects  Implementation of a model from the Social-Science literature in MATLAB  Carried out in pairs  The projects will be assigned next week: October 18, 2010  Project suggestions are online now; please visit the website and read through the material!

G. L. Ciampaglia, S. Balietti & K. Donnay / 3 Lesson 3 - Contents  Differential Equations  Dynamical Systems  Pendulum  Lorenz attractor  Lotka-Volterra equations  Epidemics: Kermack-McKendrick model  Exercises

G. L. Ciampaglia, S. Balietti & K. Donnay / 4 Differential equations  Solving differential equations numerically can be done by a number of schemes. The easiest way is by the 1 st order Euler’s Method: t x ΔtΔt x(t) x(t-Δt)

G. L. Ciampaglia, S. Balietti & K. Donnay / 5 Dynamical systems  Mathematical description of the time dependence of variables that characterize a given problem/ scenario in its state space  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.

G. L. Ciampaglia, S. Balietti & K. Donnay / 6 Pendulum  A pendulum is a simple dynamical system: L = length of pendulum (m) = angle of pendulum g = acceleration due to gravity (m/s 2 ) The motion is described by:

G. L. Ciampaglia, S. Balietti & K. Donnay / 7 Pendulum: MATLAB code

G. L. Ciampaglia, S. Balietti & K. Donnay / 8 Set time step

G. L. Ciampaglia, S. Balietti & K. Donnay / 9 Set constants

G. L. Ciampaglia, S. Balietti & K. Donnay / 10 Set starting point of pendulum

G. L. Ciampaglia, S. Balietti & K. Donnay / 11 Time loop: Simulate the pendulum

G. L. Ciampaglia, S. Balietti & K. Donnay / 12 Perform 1 st order Euler’s method

G. L. Ciampaglia, S. Balietti & K. Donnay / 13 Plot pendulum

G. L. Ciampaglia, S. Balietti & K. Donnay / 14 Set limits of window

G. L. Ciampaglia, S. Balietti & K. Donnay / 15 Make a 10 ms pause

G. L. Ciampaglia, S. Balietti & K. Donnay / 16 Pendulum: Executing MATLAB code

G. L. Ciampaglia, S. Balietti & K. Donnay / 17 Lorenz attractor  The Lorenz attractor defines a 3-dimensional trajectory by the differential equations:  σ, r, b are parameters.

G. L. Ciampaglia, S. Balietti & K. Donnay / 18 Lorenz attractor: MATLAB code

G. L. Ciampaglia, S. Balietti & K. Donnay / 19 Set time step

G. L. Ciampaglia, S. Balietti & K. Donnay / 20 Set number of iterations

G. L. Ciampaglia, S. Balietti & K. Donnay / 21 Set initial values

G. L. Ciampaglia, S. Balietti & K. Donnay / 22 Set parameters

G. L. Ciampaglia, S. Balietti & K. Donnay / 23 Solve the Lorenz-attractor equations

G. L. Ciampaglia, S. Balietti & K. Donnay / 24 Compute gradient

G. L. Ciampaglia, S. Balietti & K. Donnay / 25 Perform 1 st order Euler’s method

G. L. Ciampaglia, S. Balietti & K. Donnay / 26 Update time

G. L. Ciampaglia, S. Balietti & K. Donnay / 27 Plot the results

G. L. Ciampaglia, S. Balietti & K. Donnay / 28 Animation

G. L. Ciampaglia, S. Balietti & K. Donnay / 29

G. L. Ciampaglia, S. Balietti & K. Donnay / Food chain 30

G. L. Ciampaglia, S. Balietti & K. Donnay / 31 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

G. L. Ciampaglia, S. Balietti & K. Donnay / 32 Lotka-Volterra equations  The Lotka-Volterra equations describe the interaction between two species, prey vs. predators, e.g. rabbits vs. foxes.

G. L. Ciampaglia, S. Balietti & K. Donnay / 33 Lotka-Volterra equations  The Lotka-Volterra equations describe the interaction between two species, prey vs. predators, e.g. rabbits vs. foxes.

G. L. Ciampaglia, S. Balietti & K. Donnay / Epidemics 34 Source: Balcan, et al. 2009

G. L. Ciampaglia, S. Balietti & K. Donnay / 35 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.

G. L. Ciampaglia, S. Balietti & K. Donnay / 36 Kermack-McKendrick model  Spread of diseases like the plague and cholera? A popular SIR model is the Kermack-McKendrick model.  The model assumes:  A constant population size.  A zero incubation period.  The duration of infectivity is as long as the duration of the clinical disease.

G. L. Ciampaglia, S. Balietti & K. Donnay / 37 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 S I R β transmission γ recovery

G. L. Ciampaglia, S. Balietti & K. Donnay / 38 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

G. L. Ciampaglia, S. Balietti & K. Donnay / 39 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

G. L. Ciampaglia, S. Balietti & K. Donnay / 40 Exercise 1  Implement and simulate the Kermack- McKendrick model in MATLAB. Use the starting values: S=I=500, R=0, β =0.0001, γ =0.01

G. L. Ciampaglia, S. Balietti & K. Donnay / 41 Exercise 2  A key parameter for the Kermack-McKendrick model is the epidemiological threshold, β S/ γ. Plot the time evolution of the model and investigate the influence of the epidemiological threshold, in particular the cases: 1. β S/ γ < 1 2. β S/ γ > 1 Starting values: S=I=500, R=0, β =0.0001

G. L. Ciampaglia, S. Balietti & K. Donnay / 42 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 1 st 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.