Operationalizing models requires calibrating them to specific uses

- Calibration can be
**ad hoc**or involve**estimation**of parameters from data (or a combination) - When statistical techniques are used for estimation, then one can conduct
**inference**(testing of hypotheses, theory confirmation)

- Measuring “aggregate parameters” (e.g. \( R_0 \))
- Estimation of parameters (\( \beta \), \( \gamma \), etc.)
- Estimation of unknown state variables (e.g. \( X(0) \), \( Y(0) \), \( Z(0) \), asymptomatic cases)

- Final outbreak size estimator
- Mean age of infection
- Epidemic takeoff
- Doubling time

Recall from presentation on deterministic models the **final size relation**
\[
1-R(\infty)-e^{-R(\infty)R_0} = 0
\]

This can be rearranged to give a formula for \( R_0 \) in terms of observables

\[ R_0 = -\frac{\log(1-R(\infty))}{R(\infty)} \]

This formula is valid even when numerous assumptions underlying the simple SIR model are relaxed.

Ma, J. & D. Earn. 2006. Generality of the final size formula for an epidemic of a newly invading infectious disease. *Bulletin of Mathematical Biology* 68:679-702.

\( N=764 \)

\( X(0) = 763 \)

\( Z(\infty) = 512 \)

\( R(\infty) = Z(\infty)/X(0) \approx 0.671 \implies R_0 \approx 1.65 \)

A related estimator based on cases (\( C \)):

\[ R_0 = \frac{N-1}{C} \sum_{i=X_f+1}^{X_0} \frac{1}{i} \]

with standard error

\[ SE = \frac{N-1}{C} \sqrt{ \sum_{i=X_f+1}^{X_0} \frac{1}{i^2} + \frac{CR_0^2}{(N-1)^2}} \]

Becker, N. 1989. *Analysis of Infectious Disease Data*. London: Chapman & Hall.