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.

Mean age at infection

Mean age of infection is equivalent to the amount of time spent in the susceptible class

For a disease near it's endemic equilibrium this is given by \( 1/\beta Y^* \), which leads to

\[ A = \frac{1}{\mu (R_0-1)} \]

Since \( 1/\mu \) is life expectancy (\( L \)), this can be rewritten to obtain

\[ R_0 = L/A +1 \]

which is a well-known formula.

Mean age at infection

This relationship was historically very important in the derivation of estimates of \( R_0 \).

Mean age of infection \( A=4.5 \)

If we assume \( L\approx75 \) we have \( R_0 \approx 17.6 \)

Mean age at infection

Anderson & May. 1982. Science.

Epidemic takeoff

For outbreaks initiated at the disease free equilibrium we can perform a linear stability analysis of the SIR model to obtain

\[ Y(t) \approx Y(0) e^{(R_0 - 1)\gamma t} \]

This formula holds during the exponential phase of an epidemic

Taking logarithms we have

\[ \log Y(t) \approx \log Y(0) + (R_0 - 1)\gamma t \]

suggesting that the slope of a linear regression on a logarithmic scale contains information about \( R_0 \)

Epidemic takeoff

Boarding school example looks like classical exponential growth

Applying this method is one focus of the following exercise

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Doubling time

From the exponential growth equation

\[ \log Y(t) \approx \log Y(0) + (R_0 - 1)\gamma t \]

we can related the “doubling time” (\( T_d \)) to \( R_0 \) to obtain

\[ R_0 = \frac{\log (2)}{T_d \gamma} +1 \]

Effect of confinement on transmission

Vynnycky, E. et al. 2007. Estimates of the reproduction numbers of Spanish influenza using morbidity data. International Journal of Epidemiology 36:881-889.

Effect of confinement on transmission

Vynnycky, E. et al. 2007. Estimates of the reproduction numbers of Spanish influenza using morbidity data. International Journal of Epidemiology 36:881-889.

Estimating model parameters through trajectory matching



Estimating model parameters through trajectory matching


Estimating model parameters through trajectory matching


Acknowledgements

Presentations and exercises draw significantly from materials developed with Pej Rohani, Ben Bolker, Matt Ferrari, Aaron King, and Dave Smith used during the 2009-2011 Ecology and Evolution of Infectious Diseases workshops and the 2009-2019 Summer Institutes in Statistics and Modeling of Infectious Diseases.

Licensed under the Creative Commons attribution-noncommercial license, http://creativecommons.org/licenses/bync/3.0/. Please share and remix noncommercially, mentioning its origin.