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Compartmental models in epidemiology
The spread of infectious diseases is a complex phenomenon that depends on many factors. A category of mathematical models used to study the dynamics of outbreaks is the case of compartmental models. In this class of models, the population is divided into comparments describing the health state of the individuals. The number of nodes in each compartment fluctuates, whereas the total number of nodes remains constant (when studying a closed population). We give two examples of compartmental models: the SI model and the SIR model.
In the simplest epidemic model, the nodes can have two different states: Suscep-tible (S) or Infectious (I). The number of individuals in each state is denoted by S and I. The total number of nodes remains constant: N = S + I. A sus-ceptible node in contact with an Infectious node changes its state to Infectious β with probability β: S + I −→ 2I. The Infectious nodes cannot return to the Susceptible state. We use the mean field approximation i.e., we consider that all nodes are linked to < k > (average degree) neighbors. In this approximation, each Infectious individual (I) has < k > S/N Susceptible neighbors, thus the number of new Infectious individuals per unit time is βI < k > S/N . The differential equations describing the evolution of the number of individuals in each compartment are given here (note that here the variables S, I and the time are continuous):
dSdt = −β < k > ISN
dIdt = β < k > ISN
In the SIR model, besides the S and I states, the Infectious nodes transfer µ to the immune Recovered state at rate µ: I −→ R (N = S + I + R). The Recovered individuals cannot infect other individuals and cannot be infected anymore. As a consequence, the outbreak can stop before infecting all the nodes. The final fraction of Recovered nodes depends on the parameters β and
(A) R0 = 1.3 (B) R0 = 10
Figure 1.1: Numerical solution of the differential equations: evolution of the numbers of Susceptible, Infectious and Recovered nodes over time for two dif-ferent R0. The dynamic stops when the number of Infectious nodes goes to zero.
µ. The basic reproduction number defined as R0 =< k > β/µ can be seen as the average number of individuals an Infectious node will infect before reaching the Recovered state, it describes the competition between the timescales of recovery and of transmission. If R0 < 1, the infection will die out before reaching a significant portion of the nodes, if R0 > 1, the infection is more likely to spread in a population. The corresponding differential equations are:
dSdt = −β < k > ISN
dIdt = β < k > ISN − µI
dRdt = µI.
The SocioPatterns collaboration
SocioPatterns is a collaboration for interdisciplinary research between researchers and developers from the following institutions and companies: ISI Foundation (Turin, Italy), CNRS – Centre de Physique Th´eorique (Marseille, France) and Bitmanufactory (Cambridge, UK). It was originally created by Alain Barrat (CNRS and ISI Foundation), Ciro Cattuto (ISI Foundation), Jean-Fran¸cois Pin-ton (ENS Lyon) and Wouter Van den Broeck (ISI Foundation) in 2008.
The SocioPatterns collaboration developed an infrastructure able to obtain accurate data on face-to-face contacts with high temporal and spatial resolu-tion. This infrastructure is based on wearable wireless sensors working on RFID technology. The participants to the deployment wear the sensors as badges. A contact is detected between two individuals when they are close enough (less than 1.5-2 meters) from each other and are facing each other (the signal is tuned so that the human body acts as an obstacle). The signal is then sent to an antenna (devices are shown in Appendix: Figure A.1). This process is schematized in Figure 1.2. The temporal resolution is 20 seconds. The system is used to gather data in various real-world environments as schools, conferences, workplaces etc.
Description of datasets
The most used datasets in this thesis are face-to-face contacts data collected by the SocioPatterns collaboration in a French high school in Marseille (Lyc´ee Thiers) over three years from 2011 to 2013. The number of participants involved and the duration of study vary from one year to another. The data were collected in the same environment over the three years but the students changed from one year to the next.
• In 2011 (Thiers11), the study lasted for 4 days (Tuesday to Friday in December 2011) and involved 118 students divided into three different classes.
• In 2012 (Thiers12), the study lasted for 7 days (from a Monday to the Tuesday of the following week in November 2012) and involved the same three classes already involved in the study of 2011 plus two other classes (gathering 180 students).
• Finally in December 2013 (Thiers13), the study lasted for 5 days (from Monday afternoon to Friday) and involved 327 students of nine classes (five classes of 2012 plus four other classes).
For each year, metadata about participants were also collected such as class or gender. Moreover, the dataset of 2013 contains data of different nature:
• At the end of the fourth day (the 5th of Dec.), students were asked to fill in paper contact diaries giving the list of other students they had had contact with (where contact was defined as close face-to-face proximity) during the day in the high school, and to give the approximate aggregated duration of the contacts with each nominated individual, to choose in one of four possible categories: at most 5 minutes, between 5 and 15 minutes, between 15 minutes and 1 hour, more than one hour. 120 students returned a filled in diary (Contact diaries).
• During the period of the deployment, students were asked to give the names of their friends within the high school. Such friendship surveys were obtained from 135 students (Friendship).
• Finally, students were asked to use the Netvizz application to create their local network of Facebook friendships (i.e., the use of the application by a student yields the network of Facebook friendship relations between this student’s Facebook friends). 17 students gave access to their local network, from which we removed all users who were not concerned by the data collection (Facebook). We will also use two other datasets collected by the SocioPatterns collabo-ration in two different settings.
• InVS (workplace): the study lasted for two weeks (24 June – 5 July 2013) and took place in the office building of the “Institut de Veille Sanitaire” (the French institute for public health surveillance). The study involved 100 individuals structured in 5 departments.
• SFHH: the study took place during the Congress of the “Soci´et´ Fran¸caise d’Hygi`ene Hospitali`ere” (3-4 July 2009) and involved 403 individuals.
Longitudinal analysis at daily scale
Let us now turn to the longitudinal analysis of the dynamic network. Actually, our data sets allow us to have an instantaneous picture of the contact network every 20 seconds. Then we can aggregate these snapshots over different time windows. The study of the similarities and differences of contact patterns de-pending on the length of the aggregation window is of particular interest as it can help us to understand for instance how much information is lost if data are gathered only during one single day or few days, and how much data gathering is needed to inform models of human behavior.
Temporal evolution of contact patterns
In this section, we look at the time evolution of properties of the network. Figure 2.11 reports the evolution of the number of contacts at two different temporal resolutions: on the left we show this evolution over the whole study duration per one-hour time windows, on the right we look at the evolution over the course of each day discretized in 10-minutes time windows.
The number of contacts fluctuates strongly over the course of each day. Class breaks and lunch breaks are determined by strong peaks of activity. As the students leave the area of deployment after the end of lectures (around 5PM), the activity drops to zero at night. However, the evolution pattern is very similar from one day to another with peaks at the same time of the day, a feature already observed in other contexts [12].
Figure 2.12 shows the time evolution of the average strength and degree during the five days of study. The total time spent in contact by an average student grows regularly over time, both with students of the same class and with students of different classes, showing that the average amount of time spent in contact each day by a student does not fluctuate strongly from one day to the next, as also observed in a primary school [10]. However, the average strength is much higher for intra-class edges than for inter-class edges. Notably, the average number of distinct individuals with whom a student has been in contact also displays a strictly increasing behavior over the whole study duration, with no clear saturation trend. Note that Figure 2.12 shows average values: at each time the distribution of degrees is similar to the one displayed in Figure 2.6(a), the average value of this distribution increases as time increases but the shape remains similar. This means that an average student continues to meet new persons each day, and that his/her neighborhoods in the contact network, i.e., his/her individual contact patterns, change from one day to the next. The figure also shows that students meet a larger number of distinct individuals of the same class than of other classes, as expected from the previous analysis of contact matrices and networks, and that both numbers continue to grow during the whole study.
Table of contents :
1 Introduction
1.1 How to describe networks ?
1.2 Epidemic spreading processes on networks
1.2.1 Compartmental models in epidemiology
1.2.2 How are simulations performed?
1.3 Data collection
1.3.1 The SocioPatterns collaboration
1.3.2 Description of datasets
1.4 Overview of the following chapters
2 Analysis of face-to-face proximity data
2.1 Study context
2.2 Number and durations of contacts
2.3 Contact matrices
2.4 Contact network
2.5 Gender homophily
2.6 Longitudinal analysis at daily scale
2.6.1 Temporal evolution of contact patterns
2.6.2 Comparison of daily patterns
2.7 Long-term stability of patterns
2.8 Comparison with another similar study
2.9 Conclusion
3 Comparison of methods of data collection
3.1 Data analysis
3.1.1 Contact diaries
3.1.2 Friendships
3.1.3 Facebook
3.2 Contact diaries
3.2.1 Analysis of the contact diaries network
3.2.2 Comparing contact diaries and sensors data
3.3 Multiplex network of students’ relationships
3.3.1 Contact network versus friendship-survey network
3.3.2 Face-to-face contacts and Facebook links
3.3.3 Contacts and friendship networks as a multiplex
3.4 Epidemic risk from different methods of data collection
3.5 Conclusion
4 Equivalence between friendship network and a non-uniform sam- pling of contact network
4.1 Methodology
4.1.1 Sampling methods
4.1.2 How are simulations performed ?
4.2 Properties of sampled networks and outcome of SIR simulations
4.2.1 Simple sampling methods
4.2.2 More refined methods of sampling
4.3 Sampling model exploration
4.4 Impact of weight assignment
4.4.1 Contact network and sampling procedures independent from weights
4.4.2 Friendship network
4.4.3 WRE sampling procedure
4.4.4 EGOref sampling procedure
4.5 The EGOref sampling in other contexts
4.6 Conclusion and outlook
5 Conclusion
Appendices
A Introduction
B Analysis of face-to-face proximity data
C Equivalence between friendship network and a non-uniform sampling
of contact network
List of publication
