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Basic reproduction number R0
With the SIR model, it can be further demonstrated that if the number of S individuals is smaller than ν/β, then dI(t)/dt < 0 and the epidemic will die out. ν/β, called the rel-ative removal rate, is inverse to the basic reproduction number R0, a key parameter used to describe the ability of a pathogen to spread within a population. R0 can be defined as the expected number of secondary infected individuals resulting from introducing one infected individual into an entirely susceptible population. Figure 1.3 illustrates the calculation of R0. In the upper figure, one infected individual introduced at time t in an entirely susceptible population infects R0=4 individuals at time t + 1 who in turn infects R0=4 individuals at time t + 2. But if 75% of the population is immune (lower figure), then only 25% of the contacts infect individuals (the proportion of susceptible s = 0.25). Thereby, the effective reproduction number -namely, the expected number of secondary infected individuals resulting from introducing one infected individual into a partially immune population- is Re = R0 × s = R0 × 0.25 = 1.
Therefore, an epidemic can occur only if R0 > 1. An obvious implication is that a thresh-old value can be defined for vaccine coverage to reach a proportion of non-vaccinated individuals (the Susceptible compartment) less than the removal rate. The minimum coverage to achieve eradication should be > 1 − 1/R0, defined as the herd immunity threshold. It clearly appears that the higher R0, the higher the herd immunity threshold and consequently the efforts necessary to control the epidemic. Nonetheless, estimation of R0 can be difficult, and is variable according to the type of data available and the methodology used. However, with R0 being a summary of both the height of elements of the matrix -contact rates- and of it structure, it can be used to summarize the WAIFW matrix. When an infectious individual of age a’ is introduced into an entirely susceptible population, the average number of individuals consequently infected can be expressed by the next generation operator, and R0 is the dominant eigenvalue of the latter: N D +∞ ′ ′ R0ℓ(a ) = ℓ(a)m(a)β(a, a )da, L A
where ℓ(a) denotes the leading right eigenfunction of the next generation operator [Diek-mann et al., 1990], and with population size N stratified by age, mean duration of infec-tiousness D and life expectancy L. The relative incidence by age can then be calculated from the leading right eigenvector of the specific next generation matrix. However R0 is rather theoretical as it is exceptional for all individuals of a population to be susceptible to an emerging infection. The effective reproduction number Re then represents the actual average number of secondary cases.
Generalized Estimating Equations
The basic linear regression model requires observations to be independent, and inde-pendence is not achieved with repeated measures or longitudinal data. Two approaches have been derived from the Generalized Linear Model (GLM), the Generalized Linear Mixed Model (GLMM) and Generalized Estimating Equations (GEE) [Twisk, 2013]. GLMM enables to use random effects in addition to fixed effects and deals easily with non-normal data. GEE is an alternative to GLMM, which proposes to deal with corre-lated data, even when the correlation is unknown [Hardin and Hilbe, 2003]. GEEs are aimed at estimating population-averaged effects and are less sensitive to the variance structure than GLMM. We have used the GEE in Chapter 3 to assess the influence of various factors on the number of contacts. GEE and GLMM can deal both with re-peated measures but where GLMM take it fully into account and model it, GEE treats correlation as a nuisance. Consequently, GEEs were chosen as we had no particular interest in studying the correlations. We also used a censored negative binomial model to reproduce the methodology used in Mossong et al. [2008]
Smoothing
When fitting a model to data, one tries to find a function that would fit the data correctly, in other words to minimize the difference between the observed response values and the response value that is predicted by our model. However, without constraints, a null difference can be obtained with a function that would interpolate all response values and thus overfit the data. Therefore, it is necessary to find a function that would render the difference small but smooth. Smoothing splines offer a trade-off using on the one hand a loss function compelling the model to fit the data well, and on the other hand a penalty term that penalizes variability from the model [James et al., 2013]. Generalized Additive Models were initially developed to mix GLMs with additive models [Hastie and Tibshirani, 1990], but in this thesis they were used only for smoothing purpose [Wood, 2006].
Maximum-likelihood estimation
Modelling usually requires estimating the parameters included in the model. Among different methodologies, maximum likelihood aims at estimating the optimum parame-ters of a model. A likelihood is a function built on the parameters of a statistical model. The likelihood of a set of parameters, given an outcome, is the probability of having this outcome given these parameters. In other words, the likelihood is the probability that a model with a given set of parameters will be able to predict the given outcome accu-rately. Under the condition that a maximum exists and given a data set, the maximum of the likelihood function will correspond to the set of values for the parameters that will be optimum to predict the observed dataset. For convenience, the natural logarithm of the likelihood is widely used, and the maximum-likelihood estimation is based on the log-likelihood. Formally, the likelihood is expressed as follows:
Table of contents :
Acknowledgements
Contents
List of Figures
List of Tables
Abbreviations
List of publications
1 General Introduction
1.1 Infectious diseases
1.2 Social contact studies
1.3 Outline of the thesis
1.4 Basic Concepts
1.4.1 Modelling concepts
1.4.1.1 Basic deterministic model
1.4.1.2 Who Acquires Infection From Whom
1.4.1.3 Basic reproduction number R0
1.4.2 Statistical tools
1.4.2.1 Generalized Estimating Equations
1.4.2.2 Smoothing
1.4.2.3 Maximum-likelihood estimation
1.4.2.4 Model selection
1.4.2.5 Bootstrap
2 Datasets
2.1 France
2.1.1 The country
2.1.2 The population
2.2 The Comes-F study: Methodology of the survey
2.3 Measles cases
2.4 Seroprevalence survey for Measles-Mumps-Rubella
2.4.1 The Saturn-Inf study
2.4.2 The Sero-Inf study
2.4.3 The Sero-RR study
2.5 Vaccine coverage
2.6 Weather data
3 French contact matrices
3.1 Introduction
3.2 Methods
3.2.1 Study design
3.2.2 Data analysis
3.2.3 Number of contacts
3.2.4 Who mixes with whom?
3.2.5 Professional contacts
3.3 Results
3.3.1 Number of Contacts
3.3.2 Who mixes with whom?
3.3.3 Where do people mix?
3.3.4 How long do people mix?
3.3.5 Professional contacts
3.4 Discussion
4 Measles, Mumps and Rubella
4.1 Introduction
4.2 Methods
4.2.1 Cohorts
4.2.2 Modelling the serology for the year of data collection
4.2.3 Deriving age-dependent susceptibility by department to the year of interest
4.2.4 Estimating the effective reproduction number and age-dependent relative incidence
4.2.5 Escape probability
4.3 Results
4.3.1 Measles risk in 2010 based on data from 2009
4.3.2 The French measles outbreak
4.3.3 Measles risk in 2016 based on data from 2013
4.3.4 Mumps
4.3.5 Rubella risk in 2016 based on data from 2009 and 2013
4.3.6 Holidays, Age of onset, Escape probability and sensitivity analysis
4.4 Discussion
5 Weather impact on mixing patterns
5.1 Introduction
5.2 Methodology
5.2.1 Matching weather data
5.2.2 Preparing weather data
5.2.3 Number of contacts and mixing patterns
5.3 Results
5.3.1 Influence of temperature, absolute humidity and rain
5.3.2 Impact of weather on the number of contacts with regard to other variables
5.4 Discussion
6 Gender differences
6.1 A literature review
6.1.1 Gender differences in immune response
6.1.2 Gender differences in behaviour
6.1.3 Gender differences in influenza
6.1.4 Gender differences in measles, mumps and rubella
6.2 Gender differences and contact matrices
6.2.1 Descriptive analysis according to gender
6.2.2 Contact matrices according to gender
6.2.3 Difference in ratio for France and POLYMOD
6.3 Taking gender into account
7 General discussion
7.1 Summary of our findings
7.2 Perspectives
A Contact diaries
B Matrices
C R code
D Gender Matrixes
E Weather analysis
E.1 Analysis by weather variable
E.1.1 Temperature
E.1.2 Absolute Humidity
E.1.3 Rain
E.1.4 Fog
E.1.5 Wind speed
E.1.6 Atmospheric pressure
E.1.7 Visibility
E.2 Impact of weather according to the type and place of contacts
E.3 Quantile regression
Bibliography
