This document summarizes a study that aimed to estimate parameters of a dynamic transmission model for seasonal influenza by fitting the model to age-specific influenza-like illness (ILI) incidence data from Belgium over multiple seasons. The study used both weighted least squares and maximum likelihood approaches to estimate parameters like transmission rate, seasonal forcing amplitude, and basic reproduction number from the ILI data. The results showed that the parameter estimates from the two approaches did not differ significantly and provided a good fit to the observed ILI incidence rates across different age groups and seasons. The study concluded that the choice of estimation method had little consequence and recommended future use of Bayesian techniques.

1. Likelihood-based estimation of dynamic transmission
model parameters for seasonal influenza by fitting to
age and season specific ILI data
Michael Waithaka
September 18, 2014
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2. Outline
1 Introduction
2 Study objective
3 Data
4 Methodology
5 Results
6 Conclusions and Recommendations
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3. Introduction Mathematical models
Introduction
Mathematical models
! A mathematical model is a description of a system using mathematical
concepts and language.
! Such models are said to be dynamic if they account for time-dependent
changes in the state of the system.
! They are widely used in the design of infection control programmes.
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4. Introduction Seasonal influenza
Introduction cont’d
Seasonal influenza
! Seasonal influenza is a contagious respiratory illness transmitted mainly
through social interactions and strikes every year.
! In Europe, influenza epidemics usually occur between week 40 of the
current year and week 20 of the following year.
! Vaccination is the most common and most effective public health
response.
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5. Study objective
Study Objective
! The project aimed at directly estimating the parameters of a dynamic
transmission model using likelihood-based estimation methods.
→ This was achieved by fitting the model to age-specific
influenza-like-illness (ILI) incidence over multiple influenza seasons.
Why is parameter estimation important?
1 There exists considerable uncertainty over the most appropriate
values for parameters of such models.
2 Projections based on the mathematical models heavily rely on the
assumed input parameter values.
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6. Data
Data
! ILI incidence data from week 40 of year 2003 to week 35 of year 2009.
→ The data had been collected from a network of general practitioners
in Belgium.
! Belgian demographic data for year 2009 obtained from Eurostat.
! Daily rates of close contacts >15 minutes estimated by Goeyvaerts et al.
(2010) were also used.
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7. Methodology The dynamic model structure
The dynamic model structure
Figure 1: Age-stratified SEIRS model with vaccination
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8. Methodology Model parameters
The model parameters
→ Seasonal force of infection considered given by:
λa(t) = Z(t)
a
βa,a Ia (t),
where
Z(t) = 1 + δ ∗ sin
2π(t − t0)
365
.
Z(t) reflects the relative change of the basic reproduction number at
time t from the average basic reproduction number measured at
reference time t0.
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9. Methodology Estimation of the model parameters
Estimation of the model parameters
Weighted Least Squares Approach
The epidemiological system was assumed to be described by the
dynamic model.
The observed data were assumed to arise from some deviation of
the output of this system by observational errors.
4
j=1 i
vaj (wi) Yaj (wi) − αZaj (wi)
2
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10. Methodology Estimation of the model parameters
Estimation of the model parameters
Maximum Likelihood approach
ML estimation seek the values of the parameters that maximize
the likelihood function.
Maximizing the likelihood function determines the parameters
that are most likely to produce the observed data.
The observations were assumed to be negative binomial
distributed.
To account for overdispersion in the data.
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11. Results Exploratory results
Exploratory results
(a) Daily close contact rates (b) Observed ILI incidence rates for the
total population
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12. Results Exploratory results
Exploratory results cont’d
(a) 0 − 4 years (b) 5 − 14 years
(c) 15 − 64 years (d) ≥65 years
Figure 2: Observed ILI incidence rates stratified by age groups
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13. Results Parameters estimation results
Weighted least squares estimation
Season
Parameters
δ wv = wi α t0 tseed R0
2003 - 2004 0.201 0.440 0.212 Oct 05 Sept 21 5.002
2004 - 2005 Sept 17 Sept 14 3.530
2005 - 2006 Sept 01 Oct 11 2.942
2006 - 2007 Sept 30 Sept 02 4.119
2007 - 2008 Sept 07 Nov 15 3.236
2008 - 2009 Oct 26 Sept 04 4.707
Table 1: Weighted least squares estimates for the dynamic transmission model
parameters
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14. Results Parameters estimation results
Weighted least squares estimation
(a) 0 − 4 years (b) 5 − 14 years
(c) 15 − 64 years (d) ≥65 years
(e) Total population
Figure 3: Observed incidence rates & corresponding model-based estimates
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15. Results Parameters estimation results
Maximum likelihood estimation
Season
Parameters
δ wv = wi α t0 tseed R0
2003 - 2004 0.210 0.439 0.230 Oct 05 Sept 24 4.943
2004 - 2005 Sept 13 Sept 27 3.413
2005 - 2006 Sept 02 Oct 29 2.942
2006 - 2007 Oct 05 Sept 03 4.119
2007 - 2008 Sept 11 Dec 05 3.236
2008 - 2009 Oct 26 Sept 05 4.737
Table 2: Maximum likelihood estimates for the dynamic transmission model
parameters
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16. Results Parameters estimation results
Maximum likelihood estimation
(a) 0 − 4 years (b) 5 − 14 years
(c) 15 − 64 years (d) ≥65 years
(e) Total population
Figure 4: Observed incidence rates & corresponding model-based estimates
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17. Conclusions and Recommendations
Conclusions and Recommendations
Conclusions
Since the parameter estimates obtained using the two approaches do
not differ much, the choice between the two estimation methods have
trivial consequences.
Future study
Use of Bayesian approaches such as Markov Chain Monte Carlo
techniques.
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18. Thank you for your attention!
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19. References
J. Bilcke, P. Beutels, M. Brisson, and M. Jit.
Accounting for methodological, structural, and parameter uncertainty in decision-analytic models: a
practical guide.
Medical Decision Making, 31(4):675–692, 2011.
Eurostat.
Population table for Belgium, 2009.
Eurostat: Luxembourg, 2010.
N. Goeyvaerts, N. Hens, B. Ogunjimi, M. Aerts, Z. Shkedy, P. van Damme, and P. Beutels.
Estimating infectious disease parameters from data on social contacts and serological status.
Journal of the Royal Statistical Society: series C: applied statistics / Royal Statistical Society [London]
- ISSN 0035-9254, 59(2):255–277, 2010.
N. Goeyvaerts, L. Willem, K. V. Kerckhove, Y. Vandendijck, G. Hanquet, P. Beutels, and N. Hens.
Estimating dynamic transmission model parameters for seasonal influenza by fitting to age and
season specific Influenza-Like-Illness.
Unpublished, 2014.
E. Vynnycky, R. Pitman, R. Siddiqui, N. Gay, and W. J. Edmunds.
Estimating the impact of childhood influenza vaccination programmes in England and Wales.
Vaccine, 26(41):5321–5330, September 2008.
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