Models come in many types…

The purpose of a model…

  • A model captures the essence of an object or process
  • A model should be appropriate for the specific task or questions

The purpose of a model…

“A model should be as simple as possible but no simpler…”

Albert Einstein

Why build mathematical models?

Uses of epidemiological models

  • Understand disease dynamics
  • Estimate important parameters
  • Identify where we need more data
  • Explore control options
  • Make predictions

Implementing a model…

Implementing a model…

Simple population dynamics

Exponential growth or decline

  • Rate of increase (or decrease) is proportional to current population size

Exponential growth or decline

  • Rate of increase (or decrease) is proportional to current population size

Exponential growth or decline

  • Rate of increase (or decrease) is proportional to current population size

Exponential growth or decline

  • Rate of increase (or decrease) is proportional to current population size

Hamster population explosion

Human population growth

  • Define a variable to represent the number of individuals in the population at time \(t\)
  • \(N(t)\), where \(N\) is a dependent variable and \(t\) is an independent variable
  • is the same as \(N_t\)
  • or just \(N\) (shorthand)

Initial conditions

  • Define the value of the population at time \(t=0\)

    \[\begin{eqnarray} N(t = 0) &=& N_0 \\ &=& 7\ 000\ 000\ 000 \end{eqnarray}\]

Parameters

  • Specify the rate of increase
    • Human population increasing at approximately 1.5% each year
    • Rate of increase \(\lambda = 0.015\) per year

The difference equation model

  • Incrementing the human population in 1 year time steps

    \[N(t + 1) = \lambda \times N(t) + N(t)\]

  • where \(\lambda = 0.015\)

Changing the time step

  • Incrementing the human population in 1 month time steps

    \[N(t + 1) = (\lambda / 12) \times N(t) + N(t)\]

  • where \(\lambda = 0.015\)

Return to time steps of a year

  • Incrementing the human population in 1 year time steps

    \[N(t + 1) = \lambda \times N(t) + N(t)\]

  • where \(\lambda = 0.015\)

How long until population doubles?

\[N(t + 1) = \lambda \times N(t) + N(t)\]

\[ \mathit{i.e.\quad} N(t + 1) = (\lambda + 1) \times N(t)\]

How long until population doubles?

\[\begin{eqnarray} N(t + 2) &=& (\lambda + 1) \times N(t + 1) \\ &=& (\lambda + 1)\times(\lambda + 1) \times N(t) \\ &=& (\lambda + 1)^2 \times N(t) \end{eqnarray}\]

How long until population doubles?

\[N(t + n) = (\lambda + 1)^n \times N(t)\]

So, the number of years (\(n\)) taken to double is the \(n\) that satisfies

\[(\lambda + 1)^n = 2\]

Solving the equation

\[\begin{eqnarray} (\lambda + 1)^n &=& 2 \\ \log((\lambda + 1)^n) &=& \log(2) \\ n \times \log(\lambda + 1) &=& \log(2) \\ n &=& \log(2) / \log(\lambda + 1) \\ n &=& 46.6 \end{eqnarray}\]

Doubling time

  • The doubling time (\(n\)) does not depend on the starting population size
  • This is always true of an exponentially growing population

Fixed doubling time

Model types

  • Difference equation models
    • Describe updating of population in chunks of time
    • e.g. \(N(t + 1) = \lambda \times N(t) + N(t)\)

Model types

  • Difference equation models
    • Describe updating of population in chunks of time
    • e.g. \(N(t + 1) = \lambda \times N(t) + N(t)\)
  • Differential equation models
    • Describe continuous updating of population
    • e.g. \(\frac{dN}{dt} = \lambda \times N\)
    • where \(\frac{dN}{dt}\) is the rate of change of \(N\)

Model types

  • Difference equation models
    • Describe updating of population in chunks of time
    • e.g. \(N(t + 1) = \lambda \times N(t) + N(t)\)
  • Differential equation models
    • Describe continuous updating of population
    • e.g. \(\frac{dN}{dt} = \lambda \times N\)
    • where \(\frac{dN}{dt}\) is the rate of change of \(N\)
  • Difference equations
    • Good approximations to continuous models when time chunks are small
    • No need to worry about differential equations today :)

Practicals

  • Programming in R Practical:
    • Learn some programming skills
    • Build simple population dynamics models in R