How to do math modeling

1. Mathematical modeling
César V. Munayco, MSc, MPH
Doctoral student
Department of Preventive Medicine and Biometrics
Uniformed University of the Health Sciences

2. 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

3. Introduction to mathematical
models of infectious diseases

4. 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.

5. 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

6. 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.

7. Evaluation of the potential for
spread of an infection

8. How to built a mathematical
model

9. Process of mathematical
modeling
Gerda de Vries. What is mathematical model?

10. 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.

11. 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

12. Types of component models
SI
R
SEI
R
MSEI
R
M
M
S
S
ƒ
SIR
S
S
S
ß
S
S
S
S
ß
E
E
ß
ß
r
II
r
e
E
E
II
π
R
R
II
e
r
R
R
II
R
R
r
R
R

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

14. Building a model
Compartmental model
System of ordinary differential
equations:
force of
infection, λ,

15. R Coding

16. R Coding

17. Model output – Figure I
400
200
0
Number of children
600
Susceptible
Infectious
Bed
Recovered
0
5
10
time, days
15

18. 400
Model output – Figure II
200
100
0
Number of children
300
Infectious
Bed
0
5
10
time, days
15

19. How to fit a model to data

20. Creating a database
with real data

21. 0
50
100
150
200
Number of children in bed
250
300
Data available
2
4
6
8
time, day
10
12
14

22. Model fitting

23. Fitting the model to data
300
B
Data
fitted
150
100
50
0
Numbers of
200
250
beta=2.4029,
gamma=0.9093,
delta=0.4123
0
5
10
time, day
15

24. Uncertainty and Sensitivity
analysis

25. 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.

26. 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 unbiased 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.

27. Probability Distributions

28. Latin Hypercube Sampling
Matrix
Anna Mummert. Parameter Sensitivity Analysis for Mathematical Modeling.

29. Uncertainty range coding for beta

30. Uncertainty range coding for
gamma

31. 0
5
10
time, days
15
0
5
10
time, days
15
0
600
700
10
5
10
time, days
15
500
5
400
0
300
0
0
100
50
50
150
150
200
200
300
400
500
Number of children
100
Number of children
100
600
250
200
600
700
300
250
Sensitivity to beta
200
15
100
10
0
300
5
Number of children
250
0
200
0
Number of children
400
I
150
250
200
Number of children
S
100
200
15
Number of children
150
10
100
600
5
50
50
Number of children
400
0
0
0
200
Number of children
Local uncertainty analysis for
beta
B
R
15
Min-Max
Mean+-sd
0
0
5
10
time, days
time, days
time, days
time, days
S
I
B
5
10
time, days
15
R
q05-q95
q25-q75
15

32. Local uncertainty analysis
for lambda
Sensitivity to gamma
I
10
15
Min-Max
Mean+-sd
600
400
200
Number of children
200
150
0
5
10
15
0
50
0
0
5
R
100
Number of children
400
300
100
200
Number of children
600
400
200
0
Number of children
0
B
250
500
S
0
5
10
15
0
5
10
time, days
time, days
time, days
S
I
B
R
10
time, days
15
600
200
400
children
100
children
0
5
10
time, days
15
0
0
50
100
0
5
150
200
400
300
200
children
600
children
400
200
0
0
q05-q95
q25-q75
250
500
time, days
15
0
5
10
time, days
15
0
5
10
time, days
15

33. Coding for LHS
Coding for sensitivity function

34. Latin Hypercube Sampling
Min-Max
Mean+-sd
200
150
100
50
0
Number of children
250
300
350
Sensitivity beta, gamma, delta
0
5
10
time, days
15

35. Sensitivity functions
All variables
200
0
-200
sensitivity
400
600
beta
gamma
delta
0
5
10
time
15

36. MCMC parameter values per
iteration
lambda
0.6
0.8
1.0
1.2
2.2 2.3 2.4 2.5 2.6 2.7
beta
0
200
400
600
800
1000
0
200
400
600
800
1000
iter
SSR
1000
var_model
6000
600
400
200
8000 10000
variance
14000
1000 1600
iter
800
0
200
400
600
iter
800
1000
0
200
400
600
iter

37. Pairs plot of MCMC results
0.8
1.0
1.2
2.7
0.6
2.2
2.3
2.4
2.5
2.6
beta
1.0
1.2
lambda
0.6
0.8
0.86
2.2
2.3
2.4
2.5
2.6
2.7

38. Cumulative quantile plot from
the MCMC run
lambda
2.2
0.6
0.7
2.3
0.8
2.4
0.9
2.5
1.0
1.1
2.6
1.2
beta
0
200
400
600
Iterations
800
1000
0
200
400
600
Iterations
800
1000

39. 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.
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.

40. 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.
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.

41. Coding Partial rank correlation
coefficient (PRCC)

42. Partial rank correlation
coefficient (PRCC)
0.0
-1.0
-0.5
B
0.5
1.0
Partial Rank Correlation Coefficients (PRCC)
beta
lambda
delta
Gilles Pujol, Bertrand Iooss, Alexandre Janon. Package ‘sensitivity’

43. 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.

44. 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.

45. 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, nonlinear and monotonic) and the Fourier amplitude
sensitivity test (non-linear, non-monotonic) should be
performed.

46. 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.

47. Acknowledgement
Advisor 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