CALCULUS FOR MODELLING 09
Thresholds
A threshold is a critical value at which the qualitative behaviour predicted by a model changes.
The main idea
\[\boxed{\text{below threshold}\longleftrightarrow\text{critical value}\longleftrightarrow\text{above threshold}}\]
1. Growth threshold
For \(X'=rX\),
\[r\gt0\Rightarrow\text{growth},\qquad r\lt0\Rightarrow\text{decay}.\]The critical value is \(r=0\).
2. Epidemic threshold
\[\boxed{R_0=\frac{\beta}{\gamma}}\]
The value \(R_0=1\) separates initial growth and decline in the simple fully susceptible SIR setting.
3. Why \(R_0=1\)?
\[I'\approx\gamma(R_0-1)I.\]The sign changes as \(R_0\) crosses one.
4. State thresholds
During an epidemic, infection grows when
\[\beta\frac{S}{N}\gt\gamma,\]equivalently
\[\frac{S}{N}\gt\frac1{R_0}.\]5. Interventions and capacity
Interventions may move parameters or states across a threshold. Capacity thresholds can also be external decision levels.
6. Uncertainty
When an estimated parameter is close to a threshold, uncertainty can materially affect the conclusion.
7. Threshold versus equilibrium
An equilibrium is a state where net rates vanish. A threshold separates qualitatively different behaviours.