Emerging dynamics from high-resolution spatial numerical epidemics

Mathematical modelling is a powerful tool to describe infectious disease epidemics, for example when combined to statistical modelling, and also to understand ongoing processes (Kermack and McKendrick, 1927; Keeling and Rohani, 2008). The recent COVID-19 pandemic has put in the spotlights the importance of mathematical epidemiology models to elaborate intervention strategies (Adam, 2020). These models face important challenges such as stochasticity, spatial structure, or individual heterogeneity. In the initial stages of an outbreak, the effect of spatial structure is minimal because transmission chains are largely independent, but individual heterogeneity and stochasticity have major effects (Trapman et al., 2016; Britton and Scalia Tomba, 2019). As the epidemic unfolds and host become immune, accounting for the exact shape of local contact networks matters increasingly (Keeling and Rohani, 2008). Several approaches have been developed in epidemiological models to capture spatial structures, for example meta-populations (Grenfell and Harwood, 1997), moment equations (Lion, 2016), or contact networks (Pellis et al., 2015). However, one of their limitation is that they simplify the geographical structure (sometimes ignoring it completely), which can lead to overlooking emerging patterns and complicate the practical implementation of local policies.

Agent-based simulation (ABS), where individuals are modelled explicitly, represent a seducing option to achieve a high degree of realism, from a biological and environmental point of view (Abar et al., 2017), but their routine use in public health faces three major limitations. First, ABS are very computationally demanding (Eubank et al., 2004), which restrains the total number of agents that can be simulated. The second limitation comes from the model dimensionality and the introduction of numerous parameters, many of which are poorly informed and set in an ad hoc manner, to capture individual heterogeneity. A third limit resides in the way the geographic structure is implemented into the simulation.

The recent SARS-CoV-2 pandemic has led to the creation or the re-implementation of ABS that alleviate some of the limitations. For instance, some of these simulations were used to introduce biological details into the model that would require numerous equations in an analytical approach (see e.g. Kerr et al., 2021). Other ABS introduced some spatial structure to tackle generic questions, such as the impact of individual movement on epidemic spread. Given the nature of the questions asked and also given the additional types of structures in the model, for instance in terms of individual ages of households, these simulations typically simplified the spatial structure using contact networks (see e.g. Hinch et al., 2021, Kerr et al., 2021 at a city level, or Aleta et al., 2020 for unstructured network regenerated by contact tracing). To our knowledge, few ABS feature high-resolution geographical scale. There are exceptions and, for instance, Smieszek et al., 2011 analyse the spread of influenza virus in Switzerland using an ABS using a grid with 500x500m resolution. Rockett et al., 2020 perform a similar analysis in Australia using the 2310 level 2 statistical areas Also, the recent work by The GAMA platform allows one to combine a detailed epidemiological model with a high geographical resolution (Taillandier et al., 2019). In general, achieving a high degree of geographical realism for a large number of agents is an open challenge in epidemiology.

Here, we introduce Epidemap, a novel numerically efficient agent-based method that addresses many limitations of current ABS platforms. In particular, it can simulate infectious diseases epidemic scenarios at the scale of a whole country by combining high-resolution geographical structure, demographic information, and mobility statistics. Practically, this is achieved by using high-performance computing (HPC) techniques, building-level spatial data from the OpenStreetMap (OSM) project (Haklay and Weber, 2008), and sophisticated mobility models (Barbosa et al., 2018). Overall, in addition to the number and age of the hosts (which is informed by the demography) and the initial conditions (number of infected hosts at $t=0$), these simulations only require seven parameters (see Table 2).

To illustrate the power and flexibility of the platform, we study the transmission dynamics of an (uncontrolled) epidemic of respiratory infections in France. The biological features of the epidemiological model originate from a discrete-time SARS-CoV-2 transmission model parameterised with national hospital data (Sofonea et al., 2021). Given the general focus of the study, we voluntarily focus more on the transmission dynamics than on the clinical dynamics but both are implemented. We analyse the output of 100 stochastic simulations from epidemic emergence to extinction, that is, approximately 1 year. A typical simulation generates daily mobility patterns for 66 million individuals at the scale of buildings and lasts less than 2 hr. Further details about the simulation specifications and comparisons with existing platforms can be found in the Materials and methods.

In the simulations, which are summarised in Figure 1 and detailed in the Materials and methods, each individual is assigned to a residency building and can visit two other buildings every day. These buildings are chosen at random based on a distance kernel (see the Materials and methods). If more than one individual visits the building the same day, transmission can occur. The life-history of the infection is parameterised using data from the COVID-19 epidemic in France (Sofonea et al., 2021). Note that most of the parameters are required to simulate Intensive Care Unit (ICU) bed occupancy dynamics, which do not affect the transmission dynamics. Given the general scope of this study, we assume that recovered individuals have perfect immunity for the duration of the simulation (i.e. a classical Susceptible Infected Recovered (SIR) epidemiological formalism) and that all simulations are initialised with 15 infected individuals in Paris to avoid premature random epidemic extinction. Similarly, although the simulation tracks the age of the individuals, which is distributed geographically according to national demographics data (INSEE, 2020), following a parsimony principle, we assume that it does not affect the mobility or the transmission model (but age does affect the clinical model).

Figure 1

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Epidemap simulations do not attempt to reconstruct a past epidemic. They are not conceived either to perform statistical parameter inference. Their main goal is to simulate realistic scenarios to better understand how mobility patterns, geographical structure, and infectious disease biology interact to shape epidemic spread and to optimise public health responses.

Figure 2 shows the output of the dynamics at the national level for 100 stochastic simulations. For optimal readability, the dates are aligned based on the day where ICU-occupancy reaches a value of 700. With our minimal parameterisation, we see that the basic reproduction number, which is denoted (R₀) and corresponds to the average number of secondary infections caused by an infected individual (Anderson and May, 1991), is of the order of 3, which is consistent with estimates for the French epidemic (Salje et al., 2020; Sofonea et al., 2021). Note that this value is here computed directly at the individual level, by counting how many infections an individual causes. Furthermore, the daily estimates for the temporal reproduction number calculated in the same manner are very similar to those estimated using the daily case incidence data (dashed blue curve) and the method from Wallinga and Lipsitch, 2007. These uncontrolled epidemics last 308 days (95% confidence interval (CI): [286;345] days) and the final total epidemic size is $q=61.1\%$ (95% CI [60.1%;61.9%]) of the initial susceptible population. As described by earlier studies (Keeling, 1999), this proportion is lower than the prediction from a mean-field model that is given by the well-known equation $q{R}_0+\log (1-q)=0$ from Kermack and McKendrick, 1927, which yields 93% for $R_0=3$. This shows that geographical structure greatly impacts the unfolding of the epidemic.

Figure 2 with 1 supplement see all

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The national prevalence data uncovers a bimodal structure of the epidemic, which can be understood by moving to the regional scale (Figure 2b). The first peak corresponds to the spread in the region where the outbreak emerges (here the Ile-de-France), whereas the second corresponds to the sum of the epidemic peaks in the other regions. This bimodal structure is particularly pronounced because of the high population density in the region of origin of the outbreak (Ile-de-France), but it is a direct consequence of the detailed geographical information in the simulation platform. As expected, in a well-mixed setting the parameters used for the transmission model yield a single peak (Figure 2—figure supplement 1).

The resolution of the Epidemap simulations allows us to perform analyses at the district level (see Videos 1 and 2). In Figure 3a, we show that the date of onset of the epidemic in an area strongly depends on its distance from the origin (here assumed to be Paris). Furthermore, there is an additional effect of density such that denser areas are infected first. Interestingly, at the departmental level, these trends are not significant, further showing the importance of a fine-grain simulation level. Furthermore, the total proportion of inhabitants of a district who have been infected at the end of the epidemic strongly increases with density. The pandemic propagation velocity also increases with time. This can be explained by the fact that, when incidence is high, long-distance dispersion events are not rare anymore, which biases the mean distance of contamination towards higher values. For densely populated districts (Figure 3b), this proportion converges towards the mean-field prediction from well-mixed models mentioned above (Kermack and McKendrick, 1927). However, if the population density is low, this proportion is more variable showing the limit of classical assumptions.

Figure 3

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Video 1

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Video 2

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The individual-based nature of our simulations allows us to follow transmission chains (Video 3), which has direct applications. For instance, we can count, at the end of each infection, how many secondary infections were caused. The distribution of these individual reproduction numbers is particularly important in the context of emerging epidemics because the more disperse, the more the spread relies on super-spreading events (Lloyd-Smith et al., 2005). Early in the epidemic, the distribution is tightly centred around the R₀ value (Figure 4a). As the epidemic unfolds, the distribution changes with a mode that decreases towards 0, and a wider dispersal. This pattern can be formalised by assuming that the distribution of individual reproduction numbers follows a negative binomial distribution (Lloyd-Smith et al., 2005). In Figure 4b, we show that the mean of this distribution (in blue) follows the pattern estimated in Figure 2a. The dispersal parameter ($k$) indicates that super-spreading events reach a peak at the end of the first national epidemic wave. During the end of the epidemic, stochasticity is strong but there is a general trend towards a decrease in super-spreading events. Note that in these simulations we do not introduce host heterogeneity, which means we only capture the dimension of superspreading that originates from spatial heterogeneity. This is already sufficient to show that studies attempting to quantify the importance of superspreading events should account for the stage of the epidemic they analyse.

Video 3

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Figure 4

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Finally, we focused so far on the infection spread but by adding a clinical part to the infection model, and therefore additional parameters (Table 3), we can also capture the hospital epidemic wave dynamics. For instance, Video 1 shows the detailed geographical dynamics of the density of residents in ICU. As expected in the case of SARS-CoV-2, where clinical dynamics have little effect on transmission dynamics due to the life-history of the infection (see the Materials and methods), these dynamics closely follow that of infection cases.

Simulating the daily activity of millions of individuals at a build-level resolution with a realistic mobility model significantly improves our understanding of how epidemics unfold. Early stages appear to be consistent with stochastic and deterministic mean-field models. However, once local saturation effects cannot be neglected anymore, Epidemap reveals striking patterns with an unprecedented resolution. First, a two-waves epidemic pattern emerges at the French national level, which is largely driven by temporally shifted dynamics at the regional level. Second, we find that districts are affected by the epidemic depending on how far they are from the epicentre, but also depending on their density. The latter effect is absent at the departmental level, which illustrates the added value of a detailed geographical structure. Furthermore, as expected (Keeling, 1999), the simplistic estimate of the final epidemic size as a function of R₀ does not apply at the national level. Conversely, this estimate does yield relevant results at the district level if the density is sufficiently high. Finally, being able to perform individual follow-ups allow us to see that super-spreading events become increasingly important as the epidemic unfolds, but that their role decreases as the importance of stochastic processes increase again at the end of the epidemic wave. In general, many insights can also the gained from the ability to follow individual trajectories and transmission chains (Video 3), and superspreading events.

Replacing these results in the broader context of infectious disease epidemiological modelling helps to better identify the originality of the approach. For instance, the link we identify between the final size of the epidemic in a district and its population density is not reported in the other ABS we discussed here, but appears quite strongly in SARS-CoV-2 data (Smith et al., 2021). Similarly, superspreading events are known to be an important target for public policies because they increase the risk of stochastic extinction but also to fuel the speed of spread of epidemics that do emerge (Lloyd-Smith et al., 2005). However, although field studies find that their importance may vary over the course of an epidemic (Lau et al., 2017), we are not aware of an in-depth analysis of these emerging trends using ABS. Note that since Epidemap can store transmission chains, which resemble infection phylogenies, we could further investigate the possibility to detect superspreading events using various kinds of data (Alizon, 2021).

Being able to perform such detailed simulations for so many agents at a national scale has to be traded-off against some simplifying assumptions. The major one is that individual movements are based on a distance kernel centered in their residency home. As shown by earlier work on influenza dynamics, although this assumption is relevant for France, it might be too simplistic for countries like the United States, where air traffic represents a greater proportion of the travels (Crépey and Barthélemy, 2007). A second limitation is that, because of the lack of relevant data, we assumed all individuals to have the same average behaviour, e.g. in terms of the number of buildings visited per day. This assumption was motivated by the will to develop a parsimonious study and not by a constraint of the Epidemap formalism. In fact, as mentioned in the Materials and methods, we already follow the age or the infectious status of the agents and could easily have mobility that depend on these parameters.

Focusing more specifically on the importance of individual age, there are several ways in which accounting for this data in the simulations could be of interest. First, we could investigate how patterns observed for some ‘childhood’ infectious diseases emerge in the simulations. Indeed, one possibility is that these are only due to children having specific mobility patterns, which could be modelled here by imposing that instead of visiting two buildings at random, they only visit the nearest school. Another possibility could be that explaining childhood infectious disease dynamics require children-specific patterns in the infection model, that is susceptibility to the infection and/or contagiousness. Finally, both could also be needed. By separating the mobility model and the infection model, and by introducing a high-resolution and realistic mobility model, Epidemap could yield original insights into such observed epidemiological patterns.

These results have immediate applications for public health. For instance, they allow authorities to derive a risk factor per district to help control an epidemic and prevent outbreaks. From an even more applied perspective, Epidemap can readily simulation hospitalisations by sending an individual to the nearest hospital, thereby allowing to anticipate the saturation of ICU at a detailed geographical level. In the context of the SARS-CoV-2 pandemic, this simulation platform can also be instrumental to optimise vaccine coverage but also compare the effect of specific Non-Pharmaceutical Intervention (NPI) (e.g. mask-wearing vs. stay-at-home requirements).

We focused here on respiratory infection similar to SARS-CoV-2 but an asset of this platform is its versatility. We already mentioned how it could be modified to investigate childhood respiratory infections. Simulating sexually transmitted infections could prove to be challenging with such a framework because these require to model partnerships and a fine-grain geographical resolution seems less crucial (Althaus et al., 2012). However, a more feasible extension could be the study of vector-borne diseases. Indeed, in Epidemap, the vector density could be directly informed by field data and used to implement infection risk. This could even be an opportunity to interact with the general public to benefit from local signalling of specific vectors (Pernat et al., 2021). Independently of the infection followed, Epidemap can then be used to explore a variety of control scenarios with high resolution at a national level.

The approach used in Epidemap couples three models. The first model dispatches each agent to a building, depending on the national demographic distribution (INSEE, 2019) and the properties of the buildings (from OSM). The second model determines the buildings that each agent will daily visit and socials interactions with other agents located in the same distant buildings, and is based on that from Barbosa et al., 2018. Finally, the third model captures the life-history properties of the infectious disease in infected agents and is based on that from Sofonea et al., 2021.

Infection model

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Epidemap is versatile and can be adapted to simulate many infectious disease epidemics. Here, we simulate the case of a respiratory infection, focusing in particular on SARS-CoV-2. Given the life-cycle of the infection, schematised in Figure 5 we can separate the transmission and clinical dynamics because the latter has little to no effect on the former. Indeed, data shows that patients are typically hospitalised 2 weeks after infection. Of the secondary infections, 95% occur within the first 11 days post-infection. Furthermore, only a small fraction of the hosts are hospitalised (less than 1% in France O'Driscoll et al., 2021). The parameterisation was done using data from the French COVID-19 epidemic (Sofonea et al., 2021; Tables 2 and 3).

Figure 5

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Table 2

Name	Description	Value	Reference
$N$	Number of agents (population size)	6.6E7	INSEE, 2019
$H$	Number of buildings visited during the day (+ home)	3	Schneider et al., 2013
$l$	Individual dispersal kernel distance	$PDF(0.5l)=\text{LN}(2,0.88)$	fit of INSEE, 2016
N1	Maximum daily number of agent met (distant)	17	user-defined
N2	Maximum daily number of agent met (at home)	5	user-defined
$\zeta (t)$	Contagiousness $t$ days after infection	$PDF(\zeta )=\text{We}(2.24,5.42)$	Nishiura et al., 2020
$b(t)$	Transmission probability per contact	$5\%\zeta (t)$	Variant Technical group, 2021

Table 3

Name	Description	Value	Reference
µ	Hospital mortality rate	56%	Santé Publique France, 2020
$\eta (t)$	ICU admission daily probability for a severely infected patient	$PDF(\eta )=\text{We}(1.77,6.52)$	Linton et al., 2020
$\upsilon (t)$	ICU departure daily probability	$PDF(\upsilon )=\text{We}(2,10)$	Linton et al., 2020
$\theta (a)$	Probability for a severely infected patient of age $a$ to die	$\text{IFR}(a)/\mu$	Verity et al., 2020

For each interaction between a susceptible and an infected agent, we assume a constant probability of contamination $b=5\%$ (Variant Technical group, 2021) multiplied by normalised daily infectivity, or generation time, $\zeta (t)$ which follows a Weibull distribution with parameters $k=2.24$ and $\lambda =5.42$, with $t$ the number of day since contamination (Nishiura et al., 2020). We can simulate the effect of non-pharmaceutical interventions (e.g. mask-wearing) by decreasing this probability. Therefore, the transmission model only requires three parameters, two of which can be informed from contact-tracing data (Nishiura et al., 2020).

A fraction ${\theta }_a$ of infections, where $a$ is the age of the host, become critical (i.e. leading to intensive care unit (ICU) admission and/or death). These individuals have a daily probability $\eta (t)$ to be hospitalised and then a daily probability $\upsilon (t)$ to either recover or die, with probability $1-{\mu }_a$ and $1{\mu }_a$ respectively. All recovered hosts are assumed to be immune to the infection until the end of the simulation.

The raw data associated with the 100 simulations performed and the R scripts used to generate the figures are available from the Zenodo repository at https://zenodo.org/record/5542171 (results.zip).

1. 1. Thomine O
   2. Alizon S
   3. Barthelemy M
   4. Boennec C
   5. Sofonea MT

   (2021) Zenodo

   Emerging dynamics from high-resolution spatial numerical epidemics.

   https://doi.org/10.5281/zenodo.5542171

1. 1. Abar S
   2. Theodoropoulos GK
   3. Lemarinier P
   4. O’Hare GMP

   (2017) Agent based modelling and simulation tools: a review of the state-of-art software

   Computer Science Review 24:13–33.

   https://doi.org/10.1016/j.cosrev.2017.03.001

   • Google Scholar
2. 1. Adam D

   (2020) Special report: The simulations driving the world's response to COVID-19

   Nature 580:316–318.

   https://doi.org/10.1038/d41586-020-01003-6

   • PubMed
   • Google Scholar
3. 1. Aleta A
   2. Martín-Corral D
   3. Pastore Y Piontti A
   4. Ajelli M
   5. Litvinova M
   6. Chinazzi M
   7. Dean NE
   8. Halloran ME
   9. Longini IM
   10. Merler S
   11. Pentland A
   12. Vespignani A
   13. Moro E
   14. Moreno Y

   (2020) Modelling the impact of testing, contact tracing and household quarantine on second waves of COVID-19

   Nature Human Behaviour 4:964–971.

   https://doi.org/10.1038/s41562-020-0931-9

   • PubMed
   • Google Scholar
4. 1. Alizon S

   (2021) Superspreading genomes

   Science 371:574–575.

   https://doi.org/10.1126/science.abg0100

   • PubMed
   • Google Scholar
5. 1. Althaus CL
   2. Turner KM
   3. Schmid BV
   4. Heijne JC
   5. Kretzschmar M
   6. Low N

   (2012) Transmission of Chlamydia trachomatis through sexual partnerships: a comparison between three individual-based models and empirical data

   Journal of The Royal Society Interface 9:136–146.

   https://doi.org/10.1098/rsif.2011.0131

   • PubMed
   • Google Scholar
6. Book

   1. Anderson RM
   2. May RM

   (1991) Infectious Diseases of Humans Dynamics and Control

   Oxford: Oxford University Press.

   https://doi.org/10.1001/jama.1992.03490230111047

   • Google Scholar
7. 1. Barbosa H
   2. Barthelemy M
   3. Ghoshal G
   4. James CR
   5. Lenormand M
   6. Louail T
   7. Menezes R
   8. Ramasco JJ
   9. Simini F
   10. Tomasini M

   (2018) Human mobility: Models and applications

   Physics Reports 734:1–74.

   https://doi.org/10.1016/j.physrep.2018.01.001

   • Google Scholar
8. 1. Britton T
   2. Scalia Tomba G

   (2019) Estimation in emerging epidemics: biases and remedies

   Journal of The Royal Society Interface 16:20180670.

   https://doi.org/10.1098/rsif.2018.0670

   • PubMed
   • Google Scholar
9. 1. Crépey P
   2. Barthélemy M

   (2007) Detecting robust patterns in the spread of epidemics: a case study of influenza in the United States and France

   American Journal of Epidemiology 166:1244–1251.

   https://doi.org/10.1093/aje/kwm266

   • PubMed
   • Google Scholar
10. 1. Eubank S
    2. Guclu H
    3. Kumar VS
    4. Marathe MV
    5. Srinivasan A
    6. Toroczkai Z
    7. Wang N

    (2004) Modelling disease outbreaks in realistic urban social networks

    Nature 429:180–184.

    https://doi.org/10.1038/nature02541

    • PubMed
    • Google Scholar
11. 1. Grenfell B
    2. Harwood J

    (1997) (Meta)population dynamics of infectious diseases

    Trends in Ecology & Evolution 12:395–399.

    https://doi.org/10.1016/S0169-5347(97)01174-9

    • PubMed
    • Google Scholar
12. 1. Haklay M
    2. Weber P

    (2008) OpenStreetMap: User-Generated Street Maps

    IEEE Pervasive Computing 7:12–18.

    https://doi.org/10.1109/MPRV.2008.80

    • Google Scholar
13. 1. Hinch R
    2. Probert WJM
    3. Nurtay A
    4. Kendall M
    5. Wymant C
    6. Hall M
    7. Lythgoe K
    8. Bulas Cruz A
    9. Zhao L
    10. Stewart A
    11. Ferretti L
    12. Montero D
    13. Warren J
    14. Mather N
    15. Abueg M
    16. Wu N
    17. Legat O
    18. Bentley K
    19. Mead T
    20. Van-Vuuren K
    21. Feldner-Busztin D
    22. Ristori T
    23. Finkelstein A
    24. Bonsall DG
    25. Abeler-Dörner L
    26. Fraser C

    (2021) OpenABM-Covid19-An agent-based model for non-pharmaceutical interventions against COVID-19 including contact tracing

    PLOS Computational Biology 17:e1009146.

    https://doi.org/10.1371/journal.pcbi.1009146

    • PubMed
    • Google Scholar
14. Website

    1. INSEE

    (2016) De plus en plus de personnes travaillent en dehors de leur commune de résidence

    Bilan Démographique. Accessed October 29, 2021.

    https://www.insee.fr/fr/statistiques/2019022
15. Website

    1. INSEE

    (2019) Populations légales 2017. recensement de la population régions, départements, arrondissements, cantons et communes

    Bilan Démographique. Accessed October 29, 2021.

    https://www.insee.fr/fr/statistiques/4265511
16. Website

    1. INSEE

    (2020) Population totale par sexe et âge au 1er janvier 2020

    Bilan Démographique. Accessed October 29, 2021.

    https://www.insee.fr/fr/statistiques/1892088?sommaire=1912926
17. 1. Keeling MJ

    (1999) The effects of local spatial structure on epidemiological invasions

    Proceedings of the Royal Society of London. Series B: Biological Sciences 266:859–867.

    https://doi.org/10.1098/rspb.1999.0716

    • PubMed
    • Google Scholar
18. Book

    1. Keeling MJ
    2. Rohani P

    (2008) Modeling Infectious Diseases in Humans and Animals

    Princeton University Press.

    https://doi.org/10.2307/j.ctvcm4gk0

    • Google Scholar
19. 1. Kermack WO
    2. McKendrick AG

    (1927) A contribution to the mathematical theory of epidemics

    Proceedings of the Royal Society of London 115:700–721.

    https://doi.org/10.1098/rspa.1927.0118

    • Google Scholar
20. 1. Kerr CC
    2. Stuart RM
    3. Mistry D
    4. Abeysuriya RG
    5. Rosenfeld K
    6. Hart GR
    7. Núñez RC
    8. Cohen JA
    9. Selvaraj P
    10. Hagedorn B
    11. George L
    12. Jastrzębski M
    13. Izzo AS
    14. Fowler G
    15. Palmer A
    16. Delport D
    17. Scott N
    18. Kelly SL
    19. Bennette CS
    20. Wagner BG
    21. Chang ST
    22. Oron AP
    23. Wenger EA
    24. Panovska-Griffiths J
    25. Famulare M
    26. Klein DJ

    (2021) Covasim: An agent-based model of COVID-19 dynamics and interventions

    PLOS Computational Biology 17:e1009149.

    https://doi.org/10.1371/journal.pcbi.1009149

    • PubMed
    • Google Scholar
21. 1. Lau MS
    2. Dalziel BD
    3. Funk S
    4. McClelland A
    5. Tiffany A
    6. Riley S
    7. Metcalf CJ
    8. Grenfell BT

    (2017) Spatial and temporal dynamics of superspreading events in the 2014-2015 West Africa Ebola epidemic

    PNAS 114:2337–2342.

    https://doi.org/10.1073/pnas.1614595114

    • PubMed
    • Google Scholar
22. 1. Linton NM
    2. Kobayashi T
    3. Yang Y
    4. Hayashi K
    5. Akhmetzhanov AR
    6. Jung SM
    7. Yuan B
    8. Kinoshita R
    9. Nishiura H

    (2020) Incubation Period and Other Epidemiological Characteristics of 2019 Novel Coronavirus Infections with Right Truncation: A Statistical Analysis of Publicly Available Case Data

    Journal of Clinical Medicine 9:538.

    https://doi.org/10.3390/jcm9020538

    • PubMed
    • Google Scholar
23. 1. Lion S

    (2016) Moment equations in spatial evolutionary ecology

    Journal of Theoretical Biology 405:46–57.

    https://doi.org/10.1016/j.jtbi.2015.10.014

    • PubMed
    • Google Scholar
24. 1. Lloyd-Smith JO
    2. Schreiber SJ
    3. Kopp PE
    4. Getz WM

    (2005) Superspreading and the effect of individual variation on disease emergence

    Nature 438:355–359.

    https://doi.org/10.1038/nature04153

    • PubMed
    • Google Scholar
25. 1. Nishiura H
    2. Linton NM
    3. Akhmetzhanov AR

    (2020) Serial interval of novel coronavirus (COVID-19) infections

    International Journal of Infectious Diseases 93:284–286.

    https://doi.org/10.1016/j.ijid.2020.02.060

    • PubMed
    • Google Scholar
26. 1. O'Driscoll M
    2. Ribeiro Dos Santos G
    3. Wang L
    4. Cummings DAT
    5. Azman AS
    6. Paireau J
    7. Fontanet A
    8. Cauchemez S
    9. Salje H

    (2021) Age-specific mortality and immunity patterns of SARS-CoV-2

    Nature 590:140–145.

    https://doi.org/10.1038/s41586-020-2918-0

    • PubMed
    • Google Scholar
27. 1. Pellis L
    2. Ball F
    3. Bansal S
    4. Eames K
    5. House T
    6. Isham V
    7. Trapman P

    (2015) Eight challenges for network epidemic models

    Epidemics 10:58–62.

    https://doi.org/10.1016/j.epidem.2014.07.003

    • PubMed
    • Google Scholar
28. 1. Pernat N
    2. Kampen H
    3. Jeschke JM
    4. Werner D

    (2021) Citizen science versus professional data collection: Comparison of approaches to mosquito monitoring in Germany

    Journal of Applied Ecology 58:214–223.

    https://doi.org/10.1111/1365-2664.13767

    • Google Scholar
29. 1. Rockett RJ
    2. Arnott A
    3. Lam C
    4. Sadsad R
    5. Timms V
    6. Gray KA
    7. Eden JS
    8. Chang S
    9. Gall M
    10. Draper J
    11. Sim EM
    12. Bachmann NL
    13. Carter I
    14. Basile K
    15. Byun R
    16. O'Sullivan MV
    17. Chen SC
    18. Maddocks S
    19. Sorrell TC
    20. Dwyer DE
    21. Holmes EC
    22. Kok J
    23. Prokopenko M
    24. Sintchenko V

    (2020) Revealing COVID-19 transmission in Australia by SARS-CoV-2 genome sequencing and agent-based modeling

    Nature Medicine 26:1398–1404.

    https://doi.org/10.1038/s41591-020-1000-7

    • PubMed
    • Google Scholar
30. 1. Salje H
    2. Tran Kiem C
    3. Lefrancq N
    4. Courtejoie N
    5. Bosetti P
    6. Paireau J
    7. Andronico A
    8. Hozé N
    9. Richet J
    10. Dubost CL
    11. Le Strat Y
    12. Lessler J
    13. Levy-Bruhl D
    14. Fontanet A
    15. Opatowski L
    16. Boelle PY
    17. Cauchemez S

    (2020) Estimating the burden of SARS-CoV-2 in France

    Science 369:208–211.

    https://doi.org/10.1126/science.abc3517

    • PubMed
    • Google Scholar
31. Book

    1. Santé Publique France

    (2020)

    Covid-19 : Point Épidémiologique Hebdomadaire Du 15 Mars 2020

    Santé Publique France.

    • Google Scholar
32. 1. Schneider CM
    2. Belik V
    3. Couronné T
    4. Smoreda Z
    5. González MC

    (2013) Unravelling daily human mobility motifs

    Journal of The Royal Society Interface 10:20130246.

    https://doi.org/10.1098/rsif.2013.0246

    • Google Scholar
33. 1. Smieszek T
    2. Balmer M
    3. Hattendorf J
    4. Axhausen KW
    5. Zinsstag J
    6. Scholz RW

    (2011) Reconstructing the 2003/2004 H3N2 influenza epidemic in Switzerland with a spatially explicit, individual-based model

    BMC Infectious Diseases 11:115.

    https://doi.org/10.1186/1471-2334-11-115

    • PubMed
    • Google Scholar
34. 1. Smith TP
    2. Flaxman S
    3. Gallinat AS
    4. Kinosian SP
    5. Stemkovski M
    6. Unwin HJT
    7. Watson OJ
    8. Whittaker C
    9. Cattarino L
    10. Dorigatti I
    11. Tristem M
    12. Pearse WD

    (2021) Temperature and population density influence SARS-CoV-2 transmission in the absence of nonpharmaceutical interventions

    PNAS 118:e2019284118.

    https://doi.org/10.1073/pnas.2019284118

    • PubMed
    • Google Scholar
35. 1. Sofonea MT
    2. Reyné B
    3. Elie B
    4. Djidjou-Demasse R
    5. Selinger C
    6. Michalakis Y
    7. Alizon S

    (2021) Epidemiological monitoring and control perspectives: application of a parsimonious modelling framework to the COVID-19 dynamics in France

    Epidemics 100459.

    https://doi.org/10.1101/2020.05.22.20110593

    • Google Scholar
36. 1. Taillandier P
    2. Gaudou B
    3. Grignard A
    4. Huynh Q-N
    5. Marilleau N
    6. Caillou P
    7. Philippon D
    8. Drogoul A

    (2019) Building, composing and experimenting complex spatial models with the GAMA platform

    GeoInformatica 23:299–322.

    https://doi.org/10.1007/s10707-018-00339-6

    • Google Scholar
37. 1. Trapman P
    2. Ball F
    3. Dhersin JS
    4. Tran VC
    5. Wallinga J
    6. Britton T

    (2016) Inferring R0 in emerging epidemics-the effect of common population structure is small

    Journal of The Royal Society Interface 13:20160288.

    https://doi.org/10.1098/rsif.2016.0288

    • PubMed
    • Google Scholar
38. Book

    1. Variant Technical group

    (2021)

    Investigation of novel SARS-COV-2 variant: Variant of concern 202012/01 Technical Briefing 3

    Public Health England.

    • Google Scholar
39. 1. Verity R
    2. Okell LC
    3. Dorigatti I
    4. Winskill P
    5. Whittaker C
    6. Imai N
    7. Cuomo-Dannenburg G
    8. Thompson H
    9. Walker PGT
    10. Fu H
    11. Dighe A
    12. Griffin JT
    13. Baguelin M
    14. Bhatia S
    15. Boonyasiri A
    16. Cori A
    17. Cucunubá Z
    18. FitzJohn R
    19. Gaythorpe K
    20. Green W
    21. Hamlet A
    22. Hinsley W
    23. Laydon D
    24. Nedjati-Gilani G
    25. Riley S
    26. van Elsland S
    27. Volz E
    28. Wang H
    29. Wang Y
    30. Xi X
    31. Donnelly CA
    32. Ghani AC
    33. Ferguson NM

    (2020) Estimates of the severity of coronavirus disease 2019: a model-based analysis

    The Lancet Infectious Diseases 20:669–677.

    https://doi.org/10.1016/S1473-3099(20)30243-7

    • PubMed
    • Google Scholar
40. 1. Wallinga J
    2. Lipsitch M

    (2007) How generation intervals shape the relationship between growth rates and reproductive numbers

    Proceedings of the Royal Society B: Biological Sciences 274:599–604.

    https://doi.org/10.1098/rspb.2006.3754

    • PubMed
    • Google Scholar

Author details

1. Olivier Thomine

   LIS UMR 7020 CNRS, Aix Marseille University, Marseille, France

   Present address

   CNRS-LIS, Marseille, France

   Contribution

   Data curation, Software, Formal analysis, Investigation, Visualization, Methodology

   For correspondence

   olivier.thomine@protonmail.com

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0002-4847-3224

2. Samuel Alizon

   MIVEGEC, Université de Montpellier, CNRS, IRD, Montpellier, France

   Contribution

   Resources, Formal analysis, Validation, Methodology, Writing - original draft, Writing - review and editing

   Competing interests

   No competing interests declared

3. Corentin Boennec

   MIVEGEC, Université de Montpellier, CNRS, IRD, Montpellier, France

   Contribution

   Formal analysis, Validation, Visualization, Methodology, Writing - original draft, Writing - review and editing

   Competing interests

   No competing interests declared

4. Marc Barthelemy

   Institut de Physique Théorique, CEA, Saclay, France

   Contribution

   Data curation, Formal analysis, Methodology, Writing - original draft, Writing - review and editing

   Competing interests

   No competing interests declared

5. Mircea Sofonea

   MIVEGEC, Université de Montpellier, CNRS, IRD, Montpellier, France

   Contribution

   Resources, Formal analysis, Validation, Visualization, Methodology, Writing - original draft, Writing - review and editing

   Competing interests

   No competing interests declared

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