Math modeling

Mathematical modeling

César V. Munayco, MSc, MPHDoctoral student

Department of Preventive Medicine and BiometricsUniformed University of the Health Sciences

Outline• Introduction to mathematical models of

infectious diseases

• How to built a mathematical model

• How to fit a model to data

• Uncertainty and Sensitivity analysis

Introduction to mathematical models of infectious diseases

Mathematical model. Definition

• The process of applying mathematics to a real world problem with a view of understanding the latter.

• It is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling.

Why do we need mathematical models in infectious diseases epidemiology?

• Better understand the disease and its population-level dynamics

• Make predictions, explain system behavior

• Evaluate the population-level impact of interventions:

• Vaccination, antibiotic or antiviral treatment

• Quarantine,

• Bednet (ex: malaria),

• Mask (ex: SARS, influenza), …

Thierry Van Effelterre. Mathematical Models in Infectious Diseases Epidemiology and Semi-Algebraic Methods

Important concepts• The force of infection is the probability for a susceptible host to acquire the

infection.

• Basic reproduction number (R0) = average number of new infectious cases generated by one primary case during its entire period of infectiousness in a totally susceptible population

• 0< R0 < 1 No invasion of the infection within the population; only small epidemics.

• R0 = 1 Endemic infection.

• R0 >1 The bigger the value of R0 the bigger the potential for spread of the infection within the population.

Evaluation of the potential for spread of an infection

How to built a mathematical model

Process of mathematical modeling

Gerda de Vries. What is mathematical model?Gerda de Vries. What is mathematical model?

Two types of models

• Deterministic models: the same input will produce the same output. The only uncertainty in a deterministic model is generated by input variation.

• Stochastic models: model involves some randomness and will not produce the same output given the same input.

Deterministic model

• Input factors: parameter values, initial conditions

• The input factors are uncertain due to

• natural variation

• error in measurements

• lack of current measurement techniques

Types of component models

e

SS II RR

SS II RREE

SS II RREEMM

SS II RR

SIR

SEIR

MSEIR

SIRS

ß r

ß

ß

ß

r

r

r

eƒ

π

Complex model

Travis C. Porco, Sally M. Blower. Quantifying the Intrinsic Transmission Dynamics of Travis C. Porco, Sally M. Blower. Quantifying the Intrinsic Transmission Dynamics of Tuberculosis. Theoretical Population Biology 54, 117132 (1998)Tuberculosis. Theoretical Population Biology 54, 117132 (1998)

Building a model

force ofinfection, λ,

System of ordinary differential equations:

Compartmental model

R Coding

R Coding

Model output – Figure I

Model output – Figure II

How to fit a model to data

Creating a database with real data

Data available

Model fitting

Fitting the model to data

beta=2.4029,gamma=0.9093,

delta=0.4123

Uncertainty and Sensitivity analysis

Uncertainty(UA) and Sensitivity Analysis (SA)

• The goal of both UA and SA is to determine how influential parameter variation is on the final model output.

• Uncertainty analysis: qualitatively decide which parameters are most influential in the model output

• Sensitivity analysis: quantitatively decide which parameters are most influential in the model output

Marino S, Hogue IB, Ray CJ, Kirschner DE. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol. 2008 Sep 7;254(1):178-96.Anna Mummert. Parameter Sensitivity Analysis for Mathematical Modeling.

Uncertainty Analysis• The purpose of UA is to quantify the degree of

confidence in the existing experimental data and parameter estimates.

• Monte Carlo analysis: use the probability distributions for model inputs - separate the parameter space into "equal width" intervals according to the probability distributions and choose one value from each interval.

• Latin hypercube sampling (LHS): LHS allows an un-biased estimate of the average model output, with the advantage that it requires fewer samples than simple random sampling to achieve the same accuracy

Marino S, Hogue IB, Ray CJ, Kirschner DE. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol. 2008 Sep 7;254(1):178-96.

Probability Distributions

Latin Hypercube Sampling Matrix

Anna Mummert. Parameter Sensitivity Analysis for Mathematical Modeling.Anna Mummert. Parameter Sensitivity Analysis for Mathematical Modeling.

Uncertainty range coding for beta

Uncertainty range coding for gamma

Local uncertainty analysis for beta

Local uncertainty analysis for lambda

Coding for LHS

Coding for sensitivity function

Latin Hypercube Sampling

Sensitivity functions

MCMC parameter values per iteration

Pairs plot of MCMC results

Cumulative quantile plot from the MCMC run

Sensitivity Analysis• The objective of SA is to identify critical inputs

(parameters and initial conditions) of a model and quantifying how input uncertainty impacts model outcome(s).

• Local sensitivity analysis (LSA): examine change in output values based only on changes in one input factor.

• Global sensitivity analysis (GSA): examine change in output values when all parameter values change.


Global Sensitivity Analysis• Partial rank correlation coefficient (PRCC): used for linear, and

non-linear but monotonic relationships between model inputs and model outputs.

• PRCC provides a measure of monotonicity after the removal of the linear effects of all but one variable.

• Fourier amplitude sensitivity test (FAST): use for nonlinear and non-monotonic relationships between model inputs and model outputs.

• FAST provides a measure of fractional variance accounted for by individual variables and groups of variables.


Coding Partial rank correlation coefficient (PRCC)

Partial rank correlation coefficient (PRCC)

Gilles Pujol, Bertrand Iooss, Alexandre Janon. Package ‘sensitivity’

Fourier amplitude sensitivity test (FAST)

Marino S, Hogue IB, Ray CJ, Kirschner DE. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J Theor Biol. 2008 Sep 7;254(1):178-96.Anna Mummert. Parameter Sensitivity Analysis for Mathematical Modeling.


Conclusion• Always perform a sensitivity analysis on the

parameters.

• Global sensitivity should be performed - examine change in output values when all parameter values change.

• Both partial rank correlation coefficient (linear, non-linear and monotonic) and the Fourier amplitude sensitivity test (non-linear, non-monotonic) should be performed.

Programming and examples

• Karline Soetaert. R Package FME : Inverse Modelling, Sensitivity, Monte Carlo - Applied to a Dynamic Simulation Model.

• Aaron A. King. Fitting mechanistic models to epidemic curves via trajectory matching.

• Anonymous. 1978. Influenza in a boarding school. British Medical Journal, 1:587.

AcknowledgementAdvisor Dr. Dechang Chen. PhD for reviewing

the PPT

Note: you can find the R code in this link

https://www.dropbox.com/s/hjvts55ntfutxqn/SIRmodelUSUHS.R



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