Continuous Time Models

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Lecture 1

• Continuous Time Models

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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

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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/

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Mathematical Modelling

• What do we do?

Simplification
Analysis
Verification
Interpretation
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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

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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

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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

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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

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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

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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

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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

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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

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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)

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Differential Equations

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

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Differential Equations

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

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Systems of Linear DEs

• Consider a system of n 1st order linear odes

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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

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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 ?

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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

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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

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Systems of Linear DEs

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

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Stability of First Order equations

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

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Stability of Linear Systems

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

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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

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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

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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

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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

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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