Investigating the link between influenza propagation and commuting: spatial autocorrelation and model design 

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Propagation at urban scale

At a smaller scale, [63] modelled the behavior and daily travels of Portland inhabitants, using an individual-based second by second microsimulation. A static network of interactions was constructed using dynamical movements of individuals, themselves generated from data about transportation networks, location of work or leisure places and composition of the population. A contact was created between 2 individuals if they stayed in the same location, even for a brief time : the more they remained in the same place, the more the contact was considered intense. The obtained contact network was a small world one, but the number of contacts might have been overestimated, as no difference was made between different environments, assuming that people interact in the same way at home, at work or during their leisure. Aggregating the contacts occurring at different times in one graph might also have induced an overestimation of epidemic speed.

Comparison of different models

The introduction of IBM models in 2004 has not met full support from epidemiologists : the high complexity of this model, which needs multiple assumptions on human behavior and are highly parameterized, seemed unnecessary to many, who tried to show that simple compartment models could be used instead to study the same situations [64]. Indeed, despite their realistic possibility of modelling each individual separately, the complexity of IBM models, and the high computational time they ask for, make them maladjusted to some problematics, for which metapopulation models can be a better choice.
They are thus mainly used to study small scales propagations : in a city, a country or sometimes a continent, where gathering complete data about human behavior can be easier, and where the smaller number of individual makes extensive simulations easier. On the other hand, metapopulation models can be used to model wider areas and study worldwide propagation, as their smaller time of execution allows to execute more replications of stochastic simulations. However, the results obtained with these models are less detailed than previous ones.
The 2 approaches have been compared by [65], with a confrontation of influenza propagation in Italy simulated with GLEaM and an agent-based model. Starting from the same initial conditions, the 2 models simulated very similar epidemics, exhibiting the same timing of epidemic peaks in different cities. However, due to its lack of precise contact structure, the metapopulation model induced more interaction between individuals and the epidemic it generated affected more people than the one simulated with agent-based model. 4.4 Confrontation of models to data A large range of situations have been addressed by epidemiological models : from airplane travels to the internal mobility of the city of Portland, many different networks of movements have been modelled to simulate the propagation of different infectious diseases, including influenza, SARS or smallpox. In some cases, surveillance data describing the propagation of the disease studied on the scale considered are available : the confrontation of simulation results to these data have permitted to confirm the interest of using mobility movements to simulated disease diffusion.
The role of airplane travel on the propagation of several infectious diseases, including SARS and influenza has been showed by several comparasons between simualtion results and surveillance data. To confirm the predictions of GLEaM on influenza international diffusion, [67] compared their simulation results to the number of cases registered in 7 countries and 9 american states during the
2001-2002 influenza season, and to the number of acute respiratory infection cases of tje same period. They found the imulations to be a good predictor of the epidemic timing, which confirmed the pertinence of using airplen movements to predict international propagation of influenza. During the 2009 H1N1 pandemic, [42] used data on the ongoing epidemic to compare the date on which first cases of influenza where declaired in several countries to the intensity of air travel between these countries and Mexico. A strong correlation was found between the two variables. Similar results have been obtained on SARS : [68] compared the number of cases predicted in 20 countries by a model based on airpalne flows to the number of cases reported in these countries in the WHO database : once again, simulations results gave a good prediction of propagation. Despite their common use in epidemiological models, less studies have been done to prove the pertinence of commuting movements to predict disease propagation. Such a study was realized by [46] , who performed a Mantel test to evaluate the correlation between the synchrony of influenza temporal series in american states to commuting flows and other measures of distance, like airflows or geographical distance. Among all distances used to perform the test, commuting flows were the best predictor of similarity of epidemic timing between states. This analysis confirmed that commuting movements have an influence on influenza propagation. To our knowledge, no other study has been done to compare surveillance data to commuting flows.

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Investigating the link between influenza propagation and commuting: spatial autocorrelation and model design

The Sentinelles network gathers over 1300 physicians distributed between French territory: as the network covers regularly the territory, the influenza incidenceit records can be expressed at NUTS3 level (which corresponds to the French administrative division of department). We compared able to compare the incidence temporal series of the 26 epidemics to the commuting flows between departments. In order to account for the different commuting behavior of workers and students, we used two separated networks, each of them describing one type of commuting flows.
To perform the analysis of the spatial autocorrelation of Sentinelles data, we performed Mantel’s test, as presented in [137] and calculated Moran’s index for both networks of commuting. Moran’s index, which evaluates spatial autocorrelation at a specific time, was found significantly positive throughout the epidemic, with both network of commuting. In both cases, the index was positive 2 to 3 weeks before the national peak of the epidemic. Mantel’s test, which compared the synchrony between the temporal series of incidence in the departments to the matrices of commuting, was also significant. Both results confirmed the existence of a spatial structure in influenza incidences. The progression of Moran’s index (increasing during the first phase of the epidemic, when influenza was transmitted from department to department and decreasing when the incidence started decreasing in some departments) and the correlation between the synchrony of incidence evolution in different departments and commuting flows, established the existence of a correlation between epidemic spread and commuting flows.

Influence of the network structure on the propagation

The networks shaped by commuting movements are complex structures, composed of 495891 edges for the work network and 282883 for the school one (Figure 4.4-a,b). They both have a strongly clustered structure, with a significantly high local clustering coefficient for both networks (0.46 for school commuting and 0.38 for work commuting). As most individuals commute in the vicinity of their home (Figure 4.4), clustering is mainly local and communities gather neighboring districts. Given this structure of commuting networks, we ventured the hypothesis that the similarity of patterns observed between some epidemics at their beginning and during their course was linked to the structure of the networks’ clusters. To test this hypothesis, we developed two criterions, based on the measure of similarity between epidemic propagations, that will be presented in next part of this chapter. In the following part, will be exposed our investigation of the correlation between the distribution of these criterions and the structure of the networks. A part of this analysis was exposed in the article ”Commuter mobility and the spread of infectious diseases: application to
influenza in France”.

Table of contents :

1 Preface 
2 Background & definitions 
2.1 Why do we use models ?
2.2 The SIR model
2.2.1 Transmission rate
2.2.2 Recovery rate
2.2.3 Basic reproduction ratio
2.2.4 Extensions of the SIR model
2.3 Spatial models
2.3.1 Patch models
2.3.2 Distance transmission
2.3.3 Network models
3 Disease propagation in the light of human mobility 
3.1 Article
3.2 Observations
4 Influence of commuting movements on influenza propagation 
4.1 Introduction
4.2 Spatial autocorrelation of influenza incidence
4.2.1 Investigating the link between influenza propagation and commuting: spatial autocorrelation and model design
4.2.2 Observations and Perspectives
4.3 Influence of the network structure on the propagation
4.3.1 Relation between the structure of commuting networks and similarities between epidemic courses
4.3.2 Article
4.3.3 Observations, Supplementary informations and Perspectives
4.3.4 Structure of the network and similarities between epidemic propagation
4.3.5 Interpretation and perspectives
4.4 Conclusion
5 Districts role in the system dynamics 
5.1 Linearized model
5.1.1 Linearization
5.2 Kernels
5.2.1 Kernel definition
5.2.2 Kernel analysis
5.3 Perspectives of use for the kernels
5.4 Conclusion
6 Analysis of the system global dynamics 
6.1 Next generation operator
6.2 Eigenvalues and eigenvectors of the next generation operator
6.2.1 Isolated districts
6.2.2 Partial analyses for isolated areas
6.2.3 Perspectives
6.3 Conclusion
7 Technical development 
7.1 Management of the simulations
7.1.1 OpenMOLE and simulations on grid
7.1.2 From stochastic to deterministic
7.2 Management of the results
8 Conclusion 
8.1 Influence of commuting structure on influenza propagation .
8.1.1 Role of commuting in the propagation
8.1.2 Age related commuting
8.1.3 Underlying structure: attractors and basins of attraction
8.2 New methodology for network analysis
8.2.1 Early propagation
8.2.2 Global dynamics


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