CALCULUS FOR MODELLING 11
Numerical Approximation
Numerical methods approximate model trajectories when an exact analytical solution is unavailable or inconvenient.
The main idea
For
\[X'=f(t,X),\]Euler's method uses
1. Why this works
Over a short interval,
\[\Delta X\approx\frac{dX}{dt}\Delta t.\]Euler's method repeatedly applies this local linear approximation.
2. Time grid
Choose
\[t_n=t_0+n\Delta t.\]The method produces approximate values \(X_0,X_1,X_2,\ldots\).
3. Simple example
For \(X'=0.1X\), \(X_0=100\), and \(\Delta t=1\),
\[X_1=100+0.1(100)(1)=110.\]4. Systems
For a vector state \(\mathbf X\),
This makes Euler's method directly applicable to compartmental epidemic models.
5. Step size
A smaller \(\Delta t\) usually improves accuracy but requires more computational steps. A step that is too large can distort the dynamics or become unstable.
6. Numerical error
The numerical trajectory is an approximation to the solution of the mathematical model. Numerical error is therefore different from model error or parameter uncertainty.
7. More accurate methods
Runge-Kutta methods estimate change more accurately within each step and are widely used for deterministic ODE systems.
8. Checking numerical solutions
Useful checks include reducing the step size, verifying conserved quantities, comparing with an exact solution when available, and ensuring states remain physically meaningful.
9. Simulation versus prediction
A numerical solution tells us what the chosen model predicts for specified parameters and initial conditions. It does not by itself establish that the model accurately represents reality.