CALCULUS FOR MODELLING 06
Epidemic Models
Compartmental epidemic models describe movement between states such as susceptible, infectious and recovered.
The SIR model
\[S(t),\qquad I(t),\qquad R(t),\qquad S+I+R=N.\]1. Infection and recovery flows
Infection flow:
\[\boxed{\beta\frac{SI}{N}}\]Recovery flow:
\[\boxed{\gamma I}.\]2. Equations
\[\boxed{\frac{dS}{dt}=-\beta\frac{SI}{N}}\]\[\boxed{\frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I}\]\[\boxed{\frac{dR}{dt}=\gamma I}\]
3. Conservation
\[\frac{d}{dt}(S+I+R)=0.\]4. Early epidemic growth
If most people are susceptible,
\[I'\approx(\beta-\gamma)I.\]5. Basic reproduction number
\[\boxed{R_0=\frac{\beta}{\gamma}}\]
If \(R_0\gt1\), infection can initially grow; if \(R_0\lt1\), it tends to decline.
6. SEIR extension
An exposed compartment \(E\) can represent a latent period between infection and infectiousness.
7. Deterministic interpretation
The ODE model describes continuous population-level flows, not individual random events.
8. Assumptions
The basic model commonly assumes homogeneous mixing, constant parameters and a closed population.