L – Modelling and Simulating Social Systems with MATLAB © ETH Zürich | Lesson 3 – Dynamical Systems Anders Johansson and Wenjian Yu

A. Johansson & W. Yu / 2 Projects  Implementation of a model from the Social-Science literature in MATLAB.  Carried out in pairs.  The projects will be assigned next week: March 15, 2010

A. Johansson & W. Yu / 3 Lesson 3 - Contents  Differential Equations  Dynamical Systems  Pendulum  Lorenz attractor  Lotka-Volterra equations  Epidemics: Kermack-McKendrick model  Exercises

A. Johansson & W. Yu / 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)

A. Johansson & W. Yu / 5 Dynamical systems  A dynamical system is a mathematical description of the time dependence of a point in a space.  A dynamical system is described by a set of linear/non-linear differential equations.  Even though an analytical treatment of dynamical systems is often complicated, obtaining a numerical solution is straight forward.

A. Johansson & W. Yu / 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:

A. Johansson & W. Yu / 7 Pendulum: MATLAB code

A. Johansson & W. Yu / 8 Set time step

A. Johansson & W. Yu / 9 Set constants

A. Johansson & W. Yu / 10 Set starting point of pendulum

A. Johansson & W. Yu / 11 Time loop: Simulate the pendulum

A. Johansson & W. Yu / 12 Perform 1 st order Euler’s method

A. Johansson & W. Yu / 13 Plot pendulum

A. Johansson & W. Yu / 14 Set limits of window

A. Johansson & W. Yu / 15 Make a 10 ms pause

A. Johansson & W. Yu / 16 Pendulum: Executing MATLAB code

A. Johansson & W. Yu / 17 Lorenz attractor  The Lorenz attractor defines a 3-dimensional trajectory by the differential equations:  σ, r, b are parameters.

A. Johansson & W. Yu / 18 Lorenz attractor: MATLAB code

A. Johansson & W. Yu / 19 Set time step

A. Johansson & W. Yu / 20 Set number of iterations

A. Johansson & W. Yu / 21 Set initial values

A. Johansson & W. Yu / 22 Set parameters

A. Johansson & W. Yu / 23 Solve the Lorenz-attractor equations

A. Johansson & W. Yu / 24 Compute gradient

A. Johansson & W. Yu / 25 Perform 1 st order Euler’s method

A. Johansson & W. Yu / 26 Update time

A. Johansson & W. Yu / 27 Plot the results

A. Johansson & W. Yu / 28 Animation

A. Johansson & W. Yu / 29

A. Johansson & W. Yu / Food chain 30

A. Johansson & W. Yu / 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

A. Johansson & W. Yu / 32 Lotka-Volterra equations  The Lotka-Volterra equations describe the interaction between two species, prey vs. predators, e.g. rabbits vs. foxes.

A. Johansson & W. Yu / 33 Lotka-Volterra equations  The Lotka-Volterra equations describe the interaction between two species, prey vs. predators, e.g. rabbits vs. foxes.

A. Johansson & W. Yu / Epidemics 34 Source: Balcan, et al. 2009

A. Johansson & W. Yu / 35 SIR model  A model for epidemics is the SIR model, which describes the interaction between Susceptible, Infected and Removed (immune) persons, for a given disease.

A. Johansson & W. Yu / 36 Kermack-McKendrick model  of diseases like the plague and cholera. A popular SIR model is the Kermack-McKendrick model. The model was proposed for explaining the spreading  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.

A. Johansson & W. Yu / 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

A. Johansson & W. Yu / 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

A. Johansson & W. Yu / 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

A. Johansson & W. Yu / 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  Slides/exercises: (Download only possible with Firefox!)

A. Johansson & W. Yu / 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

A. Johansson & W. Yu / 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 Eulter‘s method to give reasonable accuracy?  Check in the MATLAB help how the functions ode23, ode45 etc, can be used for solving differential equations.