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Continuous Time Models With Linear Preferences

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Continuous Time Models - PowerPoint PPT Presentation

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Continuous Time Models

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Similar modelling tolls and techniques are used in different fields ... Example Drug concentration, a metabolic process. Nonlinear Systems ... – PowerPoint PPT presentation

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Title: Continuous Time Models

1
Lecture 1

  • Continuous Time Models

2
Mathematical Modelling

  • Help us better understand things
  • We may describe it mathematically
  • Idealization of the real world
  • Never completely accurate
  • But can get conclusions that are meaningful
  • Want to predict something in the future
  • A population, real estate value, of people with
    a disease, stock market
  • A mathematical model can help us understand a
    behaviour, and help us plan for the future

3
Mathematical Modelling

  • H1N1 modelling
  • WHO seeks H1N1 math model
  • http//www.straitstimes.com/Breaking2BNews/World/
    Story/STIStory_397977.html
  • In Canada
  • http//www.eurekalert.org/pub_releases/2009-06/cio
    h-goc060509.php
  • Also at York University
  • http//www.cdm.yorku.ca/

4
Mathematical Modelling

  • What do we do?

Simplification
Analysis
Verification
Interpretation
5
Mathematical Modelling

  • Models simplify reality
  • Use proportionality
  • Process x is proportional to process y
  • xky
  • Or. the change in x is proportional to y
  • dx/dt ky
  • X(t) kyx(t-?t) future value
    present value change
  • ?ana1-a0

6
Mathematical Modelling

  • Steps in modelling
  • Identify the problem
  • Make assumptions
  • Classify the variables
  • Determine interrelationships among variables
    selected for study
  • Simplification vs complexity
  • Solve or interpret the model
  • Do the results make sense? verification
  • Implement the model tell the world!
  • Maintain the model or increase its complexity

7
Mathematical Modelling

  • Interesting tidbits
  • Similar modelling tolls and techniques are used
    in different fields
  • Sometimes a complex process can have a very
    simple answer
  • Central limit theorem
  • Even a very simple model can produce meaningful
    conclusions
  • Models help those involved visualize what is
    happening
  • I say I am a mathematical immunologist ACK!!
  • But after describing what I do please are amazed
    that they can understand my work

8
Mathematical Modelling

  • Quite often we have information relating a rate
    of change of a dependent variable with respect to
    one or more independent variables and are
    interested in discovering the function relating
    the variables
  • Giordano et al.
  • Example
  • Suppose we want to model the growth of a
    population

9
Mathematical Modelling

  • Example
  • Suppose we want to model the growth of a
    population
  • Let P(t) the number of people in a large
    population at some time t
  • What is the relationship between P(t) and t?
  • The change in population is given by the
    difference between the population at time t?t
    and the present population
  • Lets assume that ?P is proportional to P

10
Mathematical Modelling

  • Lets assume that ?P is proportional to P
  • This is a difference equation
  • Discrete time periods rather than varying time t
    continuously over some interval
  • Assume that t varies continuously
  • ?P / ?t the average rate of change in P during
    the time period ?t

11
Mathematical Modelling

  • ?P / ?t the average rate of change in P during
    the time period ?t
  • Where dP/dt instantaneous rate of change
  • Now that we have a derivative we can use calculus
  • We have tools from calculus i.e. derivatives,
    integrals
  • Where P0 is the population at time t0

12
Mathematical Modelling

  • Many differential equation cannot be solved so
    easily using analytic techniques
  • In these cases we can approximate the solution
    using a discrete method
  • The derivative is an instantaneous rate of change
  • The derivative is the slope of the line tangent
    to the curve
  • Geometrical interpretation

13
Differential Equations

  • Linear DE
  • An nth order ordinary differential equation (ode)
  • ai constant coefficients
  • Homogeneous if g(t) 0
  • Independent var t
  • Unknown function y(t)

14
Differential Equations

  • Solving a linear ode
  • Assume that
  • We find our eigenvalues and we write the solution
    as a linear combination
  • Example

15
Differential Equations

  • Example contd
  • So the two solutions are
  • And the general solution is

16
Systems of Linear DEs

  • Consider a system of n 1st order linear odes

17
Systems of Linear DEs

  • When fi(t)0 we have a homogeneous system
  • Otherwise it is nonhomogeneous (have to find a
    particular solution and a solution to the
    homogeneous system)
  • The linear system can be written in matrix form

18
Systems of Linear DEs

  • How do we solve these systems?
  • Find solution to homogeneous system
  • Use x(t) e?t where ? is an unknown constant
  • Find the eigenvectors v a nonzero constant
    unknown vector
  • Find ? and then use the following to find v for
    each ?

19
Systems of Linear DEs

  • Case I all eigenvalues are distinct and real
  • Case II repeated real eigenvalues
  • If there are m linearly independent eigenvectors
  • If there is only one eigenvector vij are column
    vectors

20
Systems of Linear DEs

  • Case III complex eigenvalues
  • If the real matrix A has complex conjugate
    eigenvalues abi with corresponding eigenvectors
    B1B2, then two linearly independent real vector
    solutions to

21
Systems of Linear DEs

  • How do we solve these systems?
  • Find particular solution

22
Stability of First Order equations

  • Steady state
  • Stability
  • So,
  • Stable when alt0
  • Unstable when agt0

23
Stability of Linear Systems

  • We do much the same thing
  • Suppose we have the system
  • Steady state ? x0, y0
  • Stability

24
Stability of Linear Systems

  • Stability depends on ?1,2
  • Lets find the stability at pt (0,0)
  • Case 1 real distinct roots of the same sign
  • Case 2 real distinct with opposite signs
  • Case 3 real equal roots
  • Case 4 complex conjugate roots
  • Case 5 put imaginary roots

25
Compartmental Analysis

  • Many complicated processes can be broken down
    into distinct stages and the entire system
    modelled by describing the interaction between
    the various stages
  • Example Drug concentration, a metabolic process

26
Nonlinear Systems

  • The study of nonlinear systems is very important!
  • Since most real systems are nonlinear in nature
  • Disadvantage
  • There are no analytic solutions for most
    non-linear systems
  • Even numerical solutions are difficult to obtain
  • Se we use qualitative analysis in order to
    extract the most important features of the system
    without having to solve it

27
Nonlinear systems

  • In this section we focus on autonomous systems
  • Where f, g are real valued functions tht do not
    depend explicitly on t
  • We also assume that f and g are continuous and
    differentiable in some region R in the xy plane

28
Nonlinear systems

  • In some cases autonomous systems can be
    approximated in a region about a critical point
    by a certain linear system
  • In these cases we analyze the stability of the
    critical point of the corresponding linear system
  • Here we use the jacobian and eigenvalues
  • We also need to find the fixed points and
    calculate the jacobian for these
  • Then find eigenvalues
  • Get threshold conditions for stability of the
    critical points

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