Highlighting the impact of social relationships on the propagation of respiratory viruses using percolation theory

We develop a site-bond percolation model, called PERCOVID, in order to describe the time evolution of all epidemics propagating through respiratory tract or by skin contacts in human populations. This model is based on a network of social relationships representing interconnected households experiencing governmental non-pharmaceutical interventions. As a very first testing ground, we apply our model to the understanding of the dynamics of the COVID-19 pandemic in France from December 2019 up to December 2021. Our model shows the impact of lockdowns and curfews, as well as the influence of the progressive vaccination campaign in order to keep COVID-19 pandemic under the percolation threshold. We illustrate the role played by social interactions by comparing two typical scenarios with low or high strengths of social relationships as compared to France during the first wave in March 2020. We investigate finally the role played by the α and δ variants in the evolution of the epidemic in France till autumn 2021, paying particular attention to the essential role played by the vaccination. Our model predicts that the rise of the epidemic observed in July and August 2021 would not result in a new major epidemic wave in France.

The parameter p associated to the density of social relationships is extracted from a population-based contact survey in various European countries 13,14 . For France, the parameter q associated to the intensity of the social relationships is fixed in order to get a consistent description of the COVID-19 pandemic for more than 20 months. These consistency constraints are strong given that q should take a value between 0 and 1 (maximum for intra-household social contact). We assume also that q should not be too small (less than 0.20 for instance) in order to keep the possibility to have a very strict limitation of social contacts, comparable to those taken by Chinese authorities in Wuhan at the beginning of the COVID-19 pandemic. These considerations do fix also the value we should take for the SARS-CoV2 infectiousness r, since the probability for an infected person to contaminate a susceptible one in a daily social interaction is given by q.r. To take into account a reduced virulence of the virus during summer season, we introduced a reduction factor fT, with a gaussian time dependence 15 .
The mobility parameters correspond to the changes in mobility trends apart from the daily social contacts within the first or second circle of social relationships. They are therefore difficult to catch from absolute data for overall travels, except when very strict mobility restrictions are imposed by the authorities both for traveling and for the access to the second circle of social relationships, like for instance during the first confinement period in France. During that period, Apple and Google data show that the reduction of mobility has been very large, in good qualitative agreement with our adjustement (see table S4) 21 . Regarding the vaccination campaign in metropolitan France, we adjust the vaccination rate to the monthly number of persons having full vaccination one week before. Vaccine efficiency, relevant for the ability of a vaccinated person to infect someone else, is set to 95% for the SARS-CoV-2 initial strain and 90% for its variants.

Lattice configuration and initial conditions
Supplementary Table S2 provides details on the configuration of our cubic lattice and the model initial conditions. The initially infected persons from the SARS-CoV2 strain are randomly generated as symptomatic or asymptomatic with probabilites ps and (1-ps) respectively. The infection time is also generated randomly within the period [-(tr-ti),0]. In order to account for the expected inhomogeneous distribution of the initially infected persons over the metropolitan French territory in December 2019, we divide our cubic lattice in 3x3x3 sub-regions of equal size. Infected persons are distributed randomly in only five of these 27 regions with a distribution ¼(1,0.9,0.8,0.7,0.6). The number of initial infected persons in our simulation is fixed at 14. For smaller values, large fluctuations in the epidemic time development reduce simulation accuracy. The start of the simulation is adjusted to reproduce the observed peak of hospital admissions during the first wave. Simulating the epidemic evolution in metropolitan France at a scale 1/10 from December 2019 to September 2021 requires about 15 minutes computing time on an Intel Core i3 processor at 1.1GHz clock frequency. As far as the propagation of SARS-CoV-2 is concerned, and in the absence of a stratification in age that will be considered in a forthcoming study, we assume that a maximum of three persons in each household are susceptible of being infected. This gives a mean number of 1.95 persons per household and corresponds to a population of about 55,500,000 persons susceptible of being infected at the scale of metropolitan France.

Lattice configuration
The propagation of all SARS-CoV2 virus, including the initial strain and the a and d variants, follows the same algorithm. Variant infectiousness has been increased to reflect their higher contagiosity as documented in Supplementary Table S1. The fraction of infected persons requiring hospital admission is however taken the same for the initial strain and the a and d variants in order to get an overall agreement with both the observed patterns for the weekly hospital admissions and the incidence rate. For the initial conditions of the variants propagation, we fix the number of daily contacts in order to get the variant propagating on a large scale, with a similar inhomogeneous distribution over the metropolitan French territory as compared to the SARS-CoV-2 initial strain. These initial infections are associated to foreigners visiting France. Given the higher infectiosity of the variants, the number of these infections for the a and

Time Evolution
Each time step represents one day. This elementary time scale is associated to the definition of the number of mean social contacts per day, as encoded in the parameter p. At each time step, we proceed through the following actions: 1. Possible infection in the 1 st and 2 nd circle as well as intra-household, with symptomatic and asymptomatic rates, for each susceptible person on the lattice.
2. Possible infection outside the 1 st and 2 nd circle, with symptomatic and asymptomatic rates, for each susceptible person on the lattice.
3. Possible infection all over the country, with symptomatic and asymptomatic rates, from infected foreigners visiting France.
4. Change of status for each person in each household on the lattice according to its internal clock.

Phase diagram associated to the d variant
The phase diagram associated to the percolation transition depends explicitely on the infectiousness of the virus. It is indicated, for the SARS-CoV-2 initial strain, on Figure 1 while Supplementary Figure S1 shows the phase diagram associated to the d variant, corresponding to a very high infectiousness. As expected, the percolation zone extends to a much larger domain in this case, and necessitates a higher reduction of the intensity of the social relationships, and/or a higher vaccination coverage of the population in order to escape from this percolation zone.

Changes of social behaviors in France
We detail in this section how the various changes in the social behavior of the French population, due for instance to the governmental NPIs, can be accounted for in PERCOVID for the period under study. We do not attempt to get a best fit to the data but rather to have a semiquantitative understanding of the full evolution of the pandemic in metropolitan France in terms of the changes in the strength of social relationships. We define in these tables drmob as the ratio of rmob with respect to its value at time t=0, as given in Supplementary Table S1.

Impact of confinement periods
The confinement periods correspond to strong restrictions in the access to the second circle of social relationships as introduced in our model, and in the mobility and contacts outside the first and second circle. These restrictions were very strict for the first confinement period (March-May 2020), and less strict for the second (November 2020) and third confinements (April-May 2021) in metropolitan France. The corresponding expected changes in drmob, Nmob and p2c are documented in Supplementary Table S4. The indicated time windows correspond to official annoucements 18 , as collected for instance in Ref. [32]. Epidemic resurgence from the 1 st to the 2 nd confinement period The spreading of the COVID-19 pandemic in France during summer and autumn 2020 is governed by the change of the social behavior of the french population after the first lockdown. This corresponds mainly to an increase of the mobility ouside the first and second circle, as given by the parameters drmob and Nmob in Supplementary Table S5. The access to the second circle of the social relationships on the other hand is kept the same, and slightly reduced as compared to 1 in order to account for the restriction in the contacts with elderly. In this study, the time windows corresponding to the change of social behavior should be understood as indicative of a change of social behavior during summer holidays, the start of the academic year, and the boost of the social as well as economic activities in autumn.

Curfew periods
Between the second and third confinements, the French government enforced curfews. The curfew periods correspond to slight restrictions in the access to the second circle of the (lessessential) social relationships and in the mobility and contacts outside the first and second circle. This translates in slight reductions in drmob, Nmob and p2c, the rate of these reductions depending on the curfew hour (8pm for the first curfew and 6pm for the second), as indicated in Supplementary Table S6. The indicated time windows correspond to official announcements 18 , as collected for instance in Ref. [32]. In view of the early spread of the COVID-19 pandemic in France 16,17 , the number of infected participants involved in this event is relatively small at the scale of France. In our simulation for instance, we consider that already 140 persons were infected on December 1, 2019. According to our study, this event had therefore no significant impact on the epidemic nationwide, but a very significant one on the local spread of the epidemic in Mulhouse and in the east of France during that period.