SOCIAL HETEROGENEITY AND THE COVID-19 LOCKDOWN IN A MULTI-GROUP SEIR MODEL

. The goal of the lockdown is to mitigate and if possible prevent the spread of an epidemic. It consists in reducing social interactions. This is taken into account by the introduction of a factor of reduction of social interactions q , and by decreasing the transmission coe(cid:30)cient of the disease accordingly. Evaluating q is a di(cid:30)cult question and one can ask if it makes sense to compute an average coe(cid:30)cient q for a given population, in order to make predictions on the basic reproduction rate R 0 , the dynamics of the epidemic or the fraction of the population that will have been infected by the end of the epidemic. On a very simple example, we show that the computation of R 0 in a heterogeneous population is not reduced to the computation of an average q but rather to the direct computation of an average coe(cid:30)cient R 0 . Even more interesting is the fact that, in a range of data compatible with the Covid-19 outbreak, the size of the epidemic is deeply modi(cid:28)ed by social heterogeneity, as is the height of the epidemic peak, while the date at which it is reached mainly depends on the average R 0 coe(cid:30)cient. This short note is an application and a summary of the technical results that can be found in [4].

We consider a compartmental model based on the SEIR equations (for Susceptible, Exposed, Infected, Recovered ) with n categories of susceptible individuals The average incubation period is 1/α, the parameter β k is the product of the average number of contacts per person and per unit time by the probability of disease transmission in a contact between a susceptible individual in the group k and any infectious individual, γ is a transition rate so that 1/γ measures the duration of the infection of an individual (or actually how long he is infectious and able to contaminate other people before being isolated), and N is the total population size. An individual in the group k is characterized by his transmission rate β k . In simple models of lockdown, a single group is considered (n = 1) and it has been proposed for instance in [5] to introduce a factor of reduction of social interactions q so that the eect of the lockdown is to reduce the transmission rate β before lockdown to β/q after lockdown. Our goal is to study what happens if there are, after lockdown, n ≥ 2 groups with dierent factors of reduction of social interactions q k , so that We shall speak of social groups and social heterogeneity because each group has its own transmission rate β k . This rate does not interfere with the dynamics of the disease once the corresponding individual is infected. In a period of lockdown, the reduction factor is not the same for a health worker, a supermarket cashier or an employee working from home by internet. If we focus on β k , we can also take into account other characteristics, like age groups, which are not directly disease related, but apparently play a role in the risk of becoming infected.
Reduction of social interactions and basic reproduction ratio. Let us consider the probability distribution among the groups such that In a disease free equilibrium corresponding to S = N , it would be very natural consider an average factor of reduction of social interactions q = n k=1 p k q k and this is actually what is done when a single compartment of susceptible individuals is considered. In that case, the basic reproduction ratio computed for instance by the next generation matrix method (see [3] and references therein) is β q γ .
However, if we apply the next generation matrix method to (1), we nd that the basic reproduction ratio is . CC-BY-NC-ND 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 May 20, 2020. . https://doi.org/10.1101/2020.05.15.20103010 doi: medRxiv preprint Although this is a relatively elementary property, there are already important consequences, for instance in the case β q γ < 1 < R 0 . While a model with a single compartment and an average factor of reduction of social interactions predicts the extinction of the epidemic, a small group with high transmission factor β k , even if numerically not very large (that is, with p k small), can spread the disease and trigger an outbreak. This is one of the possible explanations, probably among many other, why the statistical data were suggesting a pattern corresponding to a basic reproduction ratio larger than 1 in the initial stage of the lockdown rather than an exponential decay of the number of cases.
Limitations and choice of the numerical parameters. System (1) is an extremely simplied model, in which Undetected or asymptomatic cases are not taken into account, although this seems an important issue in the Covid-19 pandemic.
It is also a model for short term (say of the order of three months) so that the evolution of the structure as well as natural birth and death rates are not taken into account. If Covid-19 becomes endemic or last a year or more, it will not be possible to keep ignoring such issues anymore. Feedback mechanisms and changes in the values of the parameters due to the evolution of the disease or the changes in social habits are certainly going to play a role. More important is the fact that the parameters are tted on the basis of ocial data subject to revision and only for the very early stage of the disease.
System (1) is homogeneous so that we can simply consider the fractions of the Susceptible, Exposed, Infected and Recovered individuals among the whole population. Numerical values are taken from [2]: for the initial data, we consider a perturbation of the disease free equilibrium given by based on data available on March 15, 2020 in France from [6], and choose in (1) the parameters β = 2.33 , α = 0.25 , γ = 1 , which gives us a basic reproduction ratio (corresponding to a single group) of 2.33.
The reader interested in further details and comparisons is invited to refer to [4], where the choice of the numerical parameters is discussed in greater details. This set of initial data and parameters is however not a key issue for our discussion.
Our contribution is focused on understanding the theoretical implications of social heterogeneity and remains valid for other sets of numerical choices. Numerical choices (3)-(5) are given only for an illustrative purpose.
In [2], N. Bacaër is able to t the curve of the number of infected individuals with a single group and q 1 = 1.7, at the beginning of the lockdown. In [4], it is shown that with two groups, p 1 = 98%, p 2 = 2%, q 1 = 2.35 and q 2 = 0.117, we can explain the reproduction ratio of 1.37 found in [2]. In order to x ideas, we shall take q 1 = 2.4 which corresponds to a reduction of the transmission rate of 1/q 1 ≈ 60% in the group k = 1. We emphasize that we have no empirical basis to determine this . CC-BY-NC-ND 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 May 20, 2020. . https://doi.org/10.1101/2020.05.15.20103010 doi: medRxiv preprint parameter. The idea is to pick a value larger than 2.33 so that, with a single group, the epidemic would rapidly extinguish without spreading in the whole population.
However, a second group with a moderate or high transmission coecient β 2 is enough to produce an outbreak corresponding to a R 0 > 1 and we study how various features of the epidemic curve depend on β 2 . See Fig. (1 Numerical values of R 0 : we take p small but assume that the individuals in the group k = 2 may have a moderate or high transmission coecient β 2 . From now on, we shall assume in all our numerical results that n = 2 (two groups) and vary p = p 2 (so that p 1 = 1 − p) at t = 0, which is the time of the beginning of the lockdown. Consistently with the idea that the initial stage of the outbreak is independent of the groups, we take as initial data s 1 (0) = (1−p) s(0) and s 2 (0) = p s(0). We keep p small and take it in the range 1 to 5% in our examples. It is not dicult to understand that the destabilization due to a small fraction p of the population (the k = 2 group) in a large group (the k = 1 group) in which the epidemic would be under control if it were isolated, is all the stronger when the basic reproduction ratio (inside the k = 1 group, considered as isolated) is close to 1: this is the reason why we arbitrarily choose q 1 = 2.4. The case of q 2 small, corresponding to individuals in the group k = 2 with high transmission rates, is of particular interest.
. CC-BY-NC-ND 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.   Epidemic size ζ as a function of β 2 for p = 0.01, 0.02,. . . 0.05 (blue; n = 2) and p = 1 (red, n = 1). This red curve corresponds to an outbreak with a basic reproduction ratio β 2 /γ. This is however not so instructive as the k = 2 group is the one with highest transmission rate.
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The copyright holder for this preprint this version posted May 20, 2020. . https://doi.org/10.1101/2020.05.15.20103010 doi: medRxiv preprint  Epidemic size ζ as a function of R 0 for p = 0.01, 0.02,. . . 0.05 (blue) and p = 1 (red), which corresponds to the case of a single group. With the same R 0 , we see that ζ is drastically decreased. For instance, R 0 = 1.37 is achieved in a single group with q ≈ 1.70 as in [2], which gives an epidemic size ζ ≈ 49%. The same value R 0 = 1.37, which is obtained in two groups with p = 1%, q 1 = 2.4 (and q 2 ≈ 0.06), gives an epidemic size ζ ≈ 12%.   Figure 4. Fraction of exposed and infected individuals e + i as a function of time (in days) for p = 0.01, 0.02,. . . 0.05, with β 2 = 11.65 corresponding to q 2 = 0.2. Our convention is that the epidemic peak is the maximum of t → e(t) + i(t).
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The copyright holder for this preprint this version posted May 20, 2020.  Figure 5. A striking eect of the heterogeneity is the attening of the curve. For the same value of R 0 ≈ 1.5, the curve t → e(t) + i(t) is represented in red for a single group (this corresponds here to a factor of reduction of social interactions q ≈ 1.55) and in blue in the case of two groups (with parameters q 1 = 2.4, q 2 = 0.2 and p = 0.05). Note that the blue curve is the same as in Fig. 4 with p = 5%. Now, let us describe how the epidemic peak depends on the parameters in our two groups example. For a given value of β 2 , larger values of p mean a larger R 0 , with a linear dependence given by (2), a larger epidemic size (see Fig. 2), but also a larger epidemic peak (see Fig. 6).
Let us denote by t the date of the peak and let m := e(t ) + i(t ) = max t≥0 e(t) + i(t) .
We observe that p → m (p) is increasing with p and p → t (p) is non-increasing with p for suciently large values of p.
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It is also interesting to make comparisons with the same value of R 0 . In our model case for n = 2, by inverting (2), we can for instance choose q 2 as a function of p to achieve a xed value of R 0 and get See Fig. 7 for some qualitative results on the dependence of m and t in R 0 , for various values of p. Left: size of the epidemic peak as a function of R 0 . Although not linear, it is a clearly increasing function of p in the outbreak regime. Right: date of the epidemic peak as a function of R 0 . It is an empirical but remarkable fact that, for our set of data, the date almost does not depend on p. Plots correspond to p = 0.01, 0.02,. . . 0.05.
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The copyright holder for this preprint this version posted May 20, 2020. . https://doi.org/10.1101/2020.05.15.20103010 doi: medRxiv preprint The R 0 is not the message. As seen in Fig. 7, R 0 does not retain all relevant information and for a given q 1 and R 0 , one may still vary either p or q 2 and eliminate the other one using (2). Here we shall rely on (6) for a few more plots which summarize our ndings.   . CC-BY-NC-ND 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 May 20, 2020. . https://doi.org/10.1101/2020.05.15.20103010 doi: medRxiv preprint Discussion. The goal of a lockdown is to reduce the basic reproduction ratio of the epidemic to a value less than 1 and drive the disease to extinction, or at least decrease it in such a way that the curve is attened. The eciency of the lockdown is achieved by a reduction of social interactions, which is measured by a factor q. Here we question whether such an average factor makes sense or not. Beyond the dicult issue of giving realistic values to q, we study the impact of social heterogeneities when the population is divided into groups for which q takes dierent values.
We observe that the basic reproduction ratio R 0 is computed as an average of the basic reproduction ratios for each group, or on the basis of an average transmission rate, and not as a global ratio based on an average q factor. A small group with a high transmission rate eventually triggers an outbreak even if the basic reproduction ratio R 0 of the majority would be below 1.
A second message is the following: the equilibrium in presence of heterogenous groups is not the same as when considering a single group with the same (averaged) basic reproduction ratio. The dynamics of the outbreak, for instance the height of the epidemic peak, is also changed. A model with only one group and tting the observed data in the initial phase of the outbreak will be more pessimistic concerning the epidemic outcomes than a heterogeneous model. This is even more true after lockdown when social distancing measures have been enforced, the lockdown being by its nature a creator of heterogeneity. In terms of public health, this also underlines the importance of targeting prevention measures on individuals with a high level of social interactions. Further details and mathematical proofs can be found in [4].