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?
- To simplify
- To understand
- To predict
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