Summary and Discussion �Lab 6: �MLE fitting of a dynamic model

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Reshma Kassanjee and Seth Blumberg

University of Cape Town �SACEMA

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MMED 2024

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Some slides by Juliet Pulliam

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Goals

• Understand how to simulate cross-sectional prevalence data around a derived and assumed epidemic trajectory
• Understand that the likelihood is a function of the hypothesized model parameters (and data-generating process)
• Understand that “fitting” the dynamic model involves maximizing this likelihood
• Understand why we transform parameters for fitting
• Be aware that the choice of optimization algorithms affects the outcome of the optimization
• Create 95% confidence regions for a multivariate model fit

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Boxcar model

β × exp(-α×I/N)

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Lines 1-99

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The ‘true’ prevalence over time

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Simulate prevalence data

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No longer know the true prevalence

🡪 Fit the model to data by estimating unknown parameters by � maximizing the likelihood function (using optimization algorithms)

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Lines 100-199

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Boxcar model

β × exp(-α×I/N)

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β × exp(-α×I/N)

The model world = real world

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The ‘fitted’ prevalence over time

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Parameter transformations

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• Used log of the parameters when optimizing

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Why?

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• Most optimization algorithms assume the inputs are defined on a scale from –∞ to ∞

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• More efficiently explore parameter space

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Parameter transformations

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Parameter transformations

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Optimization algorithms

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• Carefully choose and modify the settings of your optimization algorithms

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• Sometimes you may want to use multiple algorithms (and multiple starting values)

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Simulated annealing

DOI:10.1155/2022/7363279

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Nelder-Mead optimization

https://codesachin.wordpress.com/2016/01/16/nelder-mead-optimization/

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Lines 200-318

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Parameter estimates

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Parameter estimates

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Parameter estimates

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Uncertainties??

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Normal distribution

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Likelihood ratio

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Hessian matrix

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20 vs 80 samples per surveillance study

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Changing the frequency of surveillance

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Take-away points

• Always fit fake data (e.g., simulated by your model) before attempting to fit real data: to validate the fitting approach and identify any errors

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• Think about your optimization algorithms and what they are doing, and choose an approach that will be both efficient and accurate – this may include transforming parameters to scales that span the real numbers

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This presentation is made available through a Creative Commons Attribution-Noncommercial license. Details of the license and permitted uses are available at� http://creativecommons.org/licenses/by-nc/3.0/

Title: Lab 6 Summary: MLE fitting of a dynamic model to prevalence data

Attribution: Juliet R.C. Pulliam & Reshma Kassanjee & Steve E. Bellan & Seth Blumberg, Clinic on the Meaningful Modeling of Epidemiological Data

Source URL: http://www.ici3d.org/MMED/tutorials/Lab6_summary.pdf

For further information please contact admin@ici3d.org.

© 2024 International Clinics on Infectious Disease Dynamics and Data

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