Estimation: Informing models with data

Estimation and inference

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)

Tasks for estimation

  • 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)

Some approaches to measuring R0

  1. Final outbreak size estimator
  2. Mean age of infection
  3. Epidemic takeoff
  4. Doubling time

Final outbreak size estimator

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.

Worked example

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\( N=764 \)

\( X(0) = 763 \)

\( Z(\infty) = 512 \)

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

Final outbreak size estimator

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.