key: cord-0987040-ir32bf0q
authors: Eggli, Y.; Rousson, V.
title: Lessons from a pandemic
date: 2022-04-02
journal: nan
DOI: 10.1101/2022.03.30.22273190
sha: 3bfd80bc9bdd6183d7c6a32874477eb849ea5cba
doc_id: 987040
cord_uid: ir32bf0q

Objectives : Several interventions have been used around the world trying to contain the SARS-Cov-2 pandemic, such as quarantine, prohibition of mass demonstrations, isolation of sick people, tracing of virus carriers, semi-containment, promotion of barrier gestures, development of rapid auto-tests and vaccines among others. We propose a simple model to evaluate the potential impact of such interventions. Methods : A model for the reproduction number of an infectious disease including three main contexts of infection (indoor mass events, public indoor activities and household) and seven parameters is considered. We illustrate how these parameters could be obtained from the literature or from expert assumptions, and we apply the model to describe 20 scenarios that can typically occur during the different phases of a pandemic. Results : This model provides a useful framework for better understanding and communicating the effects of different (combinations of) possible interventions, while encouraging constant updating of expert assumptions to better match reality. Conclusion : This simple approach will bring more transparency and public support to help governments to think, decide, evaluate and adjust what to do during a pandemic.

this pandemic, in a context of great uncertainty. We missed a conceptual framework to 48 assess the potential impact of each intervention, taking into account the characteristics 49 of the virus and the affected population. 50 The aim of this paper is to lay the foundations for such a theory for the different phases 51 of a pandemic. Most of the elements of this theory are known and have already been . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 2, 2022. ; https://doi.org/10.1101/2022.03.30.22273190 doi: medRxiv preprint intuitively applied. The purpose here is only to clarify certain points and to set the 53 minimum parameters necessary to monitor and manage a pandemic. Everyone can then 54 refine the proposed model and make their own assumptions and adjustments. 55 

A formula has long been available for measuring the development of an epidemic 57 through its reproduction number R, defined as the average number of people infected by 58 a carrier of a specific infection [11, 12] . If the value of R exceeds 1.0 the epidemic will 59 grow up exponentially, if it is less than 1.0 it will die out. This formula involves the 60 product of three factors as follows: 61 R = D * C * N (Formula 1) 62 where D is the average duration of contagion (measured in days), C is the 63 contagiousness (probability that a person will be contaminated during a contact with a 64 carrier), and N is the average number of contacts per day that a carrier may have. Of 65 course, the contagiousness will not be the same for each contact, obviously depending 66 on the "intensity" of the contact. In this formula, C must therefore be considered as an 67 "average" probability calculated over all types of daily contacts. 68 Formula 1 applies at the start of an epidemic, where it is usually noted R 0 , and when 69 anyone in contact is likely to be infected. To take into account the appearance of 70 immunized people (following the disease or by vaccination [13]), a fourth factor is 71 introduced into the formula [14] , yielding: 72 R = D * C * N * (1-P) (Formula 2) 73 where P is the prevalence of immunized people. Formula 2 explains herd immunity, the 74 value of R falling below 1.0 if P is large enough, in fact as soon as P>1-1/R 0 .

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The copyright holder for this preprint this version posted April 2, 2022. ; https://doi.org/10.1101/2022.03. 30.22273190 doi: medRxiv preprint From there, one can immediately understand how the various measures taken by a 76 government may affect the course of an epidemic. Testing and isolating the patients 77 makes it possible to reduce the duration of the contagion (D), the barrier gestures will 78 influence the contagiousness (C), the containment reduces the number of contacts (N) 79 and the vaccination will act on the prevalence of immunized people (P). These 80 mechanisms were clearly in the minds of the governments that imposed these measures. 81 However, as far as we can ascertain, there has been little or no discussion of the 82 expected consequences of the actions taken with respect to these simple formulae. people in more than one stratum, depending on his/her activities. Again, we are . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 2, 2022. ; https://doi.org/10.1101/2022.03.30.22273190 doi: medRxiv preprint considering a stratification of contaminations, not of individuals, so our terminology 99 may differ somewhat from its traditional meaning in epidemiology. 100 To use Formula 3 in practice, it is necessary to simplify it. We could for example assume 101 that the duration of the contagion and the prevalence of immunized people is the same 102 in all strata. This is not always possible though. For instance, the prevalence of 103 immunized people will increase more quickly in contaminated strata, such as homes for 104 the elderly, than in the general population. In fact, we shall not consider below the 105 (important but peculiar) stratum of the homes for the elderly, which actually 106 necessitates separate actions and discussions. On the other hand, it is probably fair to 107 assume that contagiousness will be lower for outdoor than for indoor activities, being 108 close to zero and almost negligible for the former. Finally, the number of contacts will 109 strongly depend on the strata, a carrier obviously meeting more people in a crowd or in 110 a public transport than within his/her household. 111 In this paper, we consider the following simplified formula (or model) to express the 112 reproduction number: 113 R = D (1-P) (C*N 1 +C*N 2 +C 3 *N 3 ) (Formula 4) 114 It is a model with three strata and seven parameters. These three strata represent 115 respectively indoor mass events (in a closed space with at least 100 persons, e.g. 116 concerts, cinemas, bars, dancing, choirs), public indoor activities (including public 117 transport, schools, work, shopping or leisure) and households (i.e. families), thought to 118 be three main contexts where contaminations occur. We consider that the number of 119 contacts might be different in each stratum, and assume that contagiousness is identical 120 in the first two strata, but might be higher within households, where the contacts are . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted April 2, 2022. ; https://doi.org/10.1101/2022.03.30.22273190 doi: medRxiv preprint closer. On the basis of such a simplification of reality, it becomes possible to partly 122 answer some of the questions above, to quantify approximately what the consequences 123 of a government's measures will be, and to suggest appropriate decisions, as we shall 124 see in the next section.

In what follows, we examine 20 scenarios involving different parameter values in 127 Formula 4. Results are summarized in Table 1 .

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The copyright holder for this preprint this version posted April 2, 2022. ; https://doi.org/10.1101/2022.03.30.22273190 doi: medRxiv preprint C = contagiousness (probability of contamination during a contact with a carrier) C 3 = contagiousness in households N 1 = average number of contacts per day during indoor mass events N 2 = average number of contacts per day during public indoor activities N 3 = average number of contacts per day in households 134 135 Baseline scenario (onset of the SARS-Cov-2 epidemic in Switzerland) 136 Since we have 7 parameters in our model, we need 7 information to estimate them. We is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 2, 2022. ; https://doi.org/10.1101/2022.03.30.22273190 doi: medRxiv preprint household of N 2 =(R 0 -D*C*N 1 -D*C 3 *N 3 )/(D*C)=7.06. These are contacts likely to produce 156 contamination, often estimated by fifteen minutes at least of direct or close physical 157 contact [20] . Of note, the R 0 =2.21 contaminations due to a carrier can be decomposed 158 into D*C*N 1 =0.19 (9%) contaminations in the first stratum, D*C*N 2 =1.33 (60%) in the 159 second stratum and D*C 3 *N 3 =0.69 (31%) in the third stratum. Surveys could be used to 160 check whether this decomposition reasonably matches the reality or not, and in the 161 latter case, suggest some appropriate update of the values of parameters in our model. 162 These data represent Scenario A in Table 1 is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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organization, but these simple calculations show that the reproduction number can vary 179 considerably from one country to another, only depending on the size of the households. is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 2, 2022. ; https://doi.org/10.1101/2022.03.30.22273190 doi: medRxiv preprint together), and thus intra-household contagiousness (C 3 ). Assuming a reduction of 50% 226 of N 2 (while further assuming N 1 =0) and an increase of 50% of C 3 , we obtain R=1.70 227 (scenario L), which is not sufficient to fall below the value of 1. Full-containment consist 228 in completely banning any public contact (N 1 =N 2 =0) . If, as a result, intra-household 229 contagiousness is doubled compared to the baseline scenario (C 3 =0.162), this yields is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 2, 2022. ; https://doi.org/10.1101/2022.03.30.22273190 doi: medRxiv preprint attempt to evaluate the influence of most of the potential protective measures that could 295 be applied by governments, either alone or simultaneously. 296 Using debatable but transparent assumptions, we were able to identify some scenarios 297 that would allow us to reach a reproduction number of less than one, a necessary 298 condition for the stabilization of an epidemic. One of the most effective scenario would 299 be just to rely on seasonal variations, as a virus is generally less contagious in summer An important aspect of our model is to consider a specific stratum for households. The 314 focus on stratifying for household contagion is not new [29, 30] . Surprisingly, almost no 315 action has been recommended to reduce this factor during the pandemic. De facto, many 316 professionals separated their spouses during the acute phases of the pandemic, 317 especially when masks and other protective measures were lacking. This is likely to . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted April 2, 2022. Cov-2 epidemic in China. This is because the proportion of immunized persons in our 334 model is assumed to be stable and identical in all three strata. In particular, we do not 335 capture in our model that the proportion of immunized persons in an infected 336 household will increase rapidly in the event of a full-containment, so that there will be 337 no one left to infect, thus stopping the epidemics. This will also apply to a lesser extend 338 in a semi-containment [33, 34] . We are currently investigating ways to extend our model 339 in this respect, to take into account the fact that herd immunity is reached more quickly 340 in households than in the general population.

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The copyright holder for this preprint this version posted April 2, 2022. ; https://doi.org/10.1101/2022.03. 30.22273190 doi: medRxiv preprint Other limitations of our model include the following. We are not able to assess the role 342 of border controls to limit the circulation of virus variants. As mentioned in the Methods 343 section, we ignored the important stratum of the homes for the elderly, while we did not 344 consider the specific situation of children. A model with more than three strata might be 345 preferable, although this would imply a larger number of parameters to estimate. 346 In the end, our model is probably too simple. Note that the use of more sophisticated . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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The copyright holder for this preprint this version posted April 2, 2022. Haplotype networks of SARS-CoV-2 infections in the Diamond Princess cruise ship . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

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