Synchronization in Epidemic Growth and the Impossibility of Selective Containment

Synchronization in Epidemic Growth and the Impossibility of Selective Containment Jan Carl Budich 1∗ and Emil J. Bergholtz 2† 1Institute of Theoretical Physics , Technische Universit¨ at Dresden and W¨ urzburg-Dresden Cluster of Excellence ct.qmat, 01062 Dresden, Germany 2Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden (Dated: November 5, 2020) Containment, aiming to prevent the epidemic stage of community-spreading altogether, and mit- igation, aiming to merely ’flatten the curve’ of a wide-ranged outbreak, constitute two qualitatively different approaches to combating an epidemic through non-pharmaceutical interventions. Here, we study a simple model of epidemic dynamics separating the population into two groups, namely a low-risk group and a high-risk group, for which different strategies are pursued. Due to synchroniza- tion effects, we find that maintaining a slower epidemic growth behavior for the high-risk group is unstable against any finite coupling between the two groups. More precisely, the density of infected individuals in the two groups qualitatively evolves very similarly, apart from a small time delay and an overall scaling factor quantifying the coupling between the groups. Hence, selective containment of the epidemic in a targeted (high-risk) group is practically impossible whenever the surrounding society implements a mitigated community-spreading. We relate our general findings to the ongoing COVID-19 pandemic. The ongoing COVID-19 pandemic caused by the new Coronavirus SARS-CoV-2 is among the biggest global challenges of our time [1], and its quantitative analysis has thus been an intense focus of recent research [2–7]. A repeatedly debated [8, 9] mitigation strategy is based on selectively protecting vulnerable individuals that are at high risk to die or at least develop a severe condi- tion when contracting the virus. Such a strategy, in the following referred to as 2GROUPS, in practice amounts to defining (at least) two groups of individuals, a low- risk group (L) and a high-risk group (H), and then fo- cusing most of the available resources to try and pro- tect the group H from infection, while the larger low-risk group basically does “business as usual”, possibly com- bined with moderate general mitigation measures aimed at ’flattening the curve’ of infections for group L. Here, we analyze the expected qualitative outcome of such a 2GROUPS scenario [10]. Generally speaking, there are two crucial ingredients to a 2GROUPS strat- egy: First, efficient criteria to identify the high-risk and the low-risk individuals a priori. Regarding this aspect, earlier studies indicated that a large fraction of antici- pated severe cases may be concentrated in a relatively small group H, even if only the single criterion of age is used [2, 11]. Second, the isolation of the high-risk group aimed at strongly containing the prevalence of the dis- ease in group H. In this regard, we find that the selective isolation of the risk group is largely stymied by synchro- nization effects in the equations governing the pandemic dynamics (see Fig. 1 for an illustration). In particular, we analytically demonstrate within a minimal mathematical model that maintaining a qualitatively slower growth of infections in the high-risk group is unstable against any finite coupling between the two groups, and thus imprac- tical. Instead, the overall exposure of the group H is found to be proportional to that of group L, which may allow for a certain degree of mitigation within H, but FIG. 1: Syncronization of epidemic growth governed by Eq.1. Even if the high-risk group (H) isolates very efficiently, here quantified by a small basic reproduction numberRLH 0 = 0.5, a significant fraction ρH i (red) becomes infected if the low-risk group (L) exhibits a mitigated ’flatten the curve’ scenario, here at RH 0 = 1.5, as described by ρL i (blue). We show that this holds for any finite coupling between the groups due to a synchronization effect taking place in exponential growth phase of the epidemic. Specifically, during the synchroniza- tion the high and low-risk group infection rates are scaled by a simple constant factor, m = RL 0/RLH 0 and shifted by one time-step (cf. the black dashed curve). After the syn- chronization the infection rate in the high-risk group remains significant and bounded from below as mρH i (t + 1) ≥ ρL i (t). Initial conditions set to ρH i (t = 0) =ρL i (t = 0) = 10−4. largely rules out selective containment within the high- risk group. Minimal mathematical model of 2GROUPS. – We an- alyze a simple two-component model of SIR type [12], in which the qualitative synchronization between the epi- demic curves of the two groups (cf. Fig. 1) may be read- ily understood analytically. The corresponding equations . CC-BY-NC-ND 4.0 International licenseIt is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in(which was not certified by peer review)preprint The copyright holder for thisthis version posted November 7, 2020. ; https://doi.org/10.1101/2020.11.06.20226894doi: medRxiv preprint NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice. 2 for the low-risk group L read as ∆ρL s ∆t = −RHL 0 ρH i ρL s −RL 0ρL iρL s, ∆ρL i ∆t = RHL 0 ρL sρH i +RL 0ρL iρL s −ρL i, ∆ρL r ∆t = ρL i. (1) Here, ρα µ, µ =s,i,r, α =L,H stands for the density of individuals within group α that are susceptible (µ =s), infectious (µ = i), and recovered ( µ = r), respectively. The time-step of the discretized dynamics is denoted by ∆t and for SARS-CoV-2 roughly amounts to 5 days in real time [13]. The parameterRL 0 is the reproduction rate within group L, and the inter-group coupling RHL 0 quan- tifies the transfer of infections from group H to group L. The corresponding equations for the high-risk group H are identical to Eqs. (1) upon exchanging H andL in all instances. The resulting parameters RLH 0 and RH 0 then denote the transfer of infections from L to H, and the effective reproduction rate within H, respectively. The goal of 2GROUPs is of course to keep RLH 0 as low as possible, but it will still be non-zero in any realistic im- plementation. It is important to notice that the reproduction num- bers scale with the relative size of the groups: RL 0 and RLH 0 scale with the fraction of the population in the low- risk group while RH 0 and RHL 0 scale with the population fraction in the high-risk group. In a realistic scenario with the low-risk group being about five times as large as the high risk group this directly leads to a factor of five between RLH 0 as compared to RH 0 and RHL, as re- flected in the parameters used for the simulations shown in Fig. 1. Moreover, in our example in Fig. 1, we use RLH = 0.5 which corresponds to an additional three-fold reduction of contacts involving individuals from group H, compared to the already mitigated values (for SARS- CoV-2) within group L of RL 0 = 1.5. This, for the high- risk group, amounts to a rather strict isolation at a level of hard lockdowns in Europe during spring 2020, and even similar to the level reached during the lockdown in Wuhan [14]. To gain analytical insight, and motivated by the afore- mentioned scaling of parameters with relative group size, within a first approximation we neglect both RH 0 and RHL 0 , while keeping RLH 0 and RL 0 finite. This leads to a uni-directional decoupling of the two groups (note that H no longer appears in Eqs. (1) for the group L), which allows us to get an intuitive feeling for the synchroniza- tion of the two groups (see Fig.2(a)). Corrections arising from restoring the neglected couplings are discussed fur- ther below. Roughly speaking, this only adds additional channels of infection compared to the simplifying approx- imation, and at least does not make the situation more favorable for the H group. Within our approximation, we may first solve Eqs. (1) for group L and then plug the time-dependent solution ρL i (t) into the correspond- ing equations for H. Putting ∆ t = 1, i.e. measuring time in units of ∆t, we thus derive ρH i (t + 1) =RLH 0 ρL i (t)ρH s (t) =RLH 0 ρL i (t) ρH i (t)ρH i (t)ρH s (t) = ˜RH 0 (t)ρH i (t)ρH s (t), (2) where we have defined ˜RH 0 (t) = RLH 0 ρL i ρH i as an effective time-dependent reproduction rate for group H. Eq. (2) formally resembles a time-step in a simple epidemic dy- namics of a single group but with RH 0 replaced by the time-dependent ˜RH 0 (t). Importantly, since ˜RH 0 (t) is pro- portional to the quotient ρL i ρH i , a much lower density of infected in the H group (the main goal of 2GROUPS) increases the epidemic growth within H – a very unde- sirable but unavoidable effect of the non-linear coupling RLH 0 . During the initial exponential growth of infec- tions in L, a stable situation characterized by a time- independent ˜RH 0 only occurs if the condition RLH 0 RL 0 = ρH i ρL i (3) is satisfied. Then, the dynamics of ρH i simply follows the dynamics of ρL i with a delay of a single time step and an overall reduction in amplitude, effectively dividing it by m = RL 0 RLH 0 ≥ 1, (4) i.e. mρH i (t + 1) ∼ ρL i (t) (cf. Eq. (3)). Close to the peak and during the downward slope of the dynamics, the delay causes ρH i to grow even slightly above ρL i/m (see Fig. 1). Hence, a simple estimate for the overall number of infected IH within the high-risk group is given by IH ≥IL/m, (5) where IL denotes the overall number of infected within the low-risk group during the epidemic (see Fig. 2(b)). These general results are illustrated in Figs. 1 and 2(a). In fact, a stronger bound holds throughout the relevant parts of the epidemic, namely that mρH i (t + 1) ≥ ρL i (t) (cf. the black dashed and blue solid curves in Fig. 1). We stress that Fig. 1 shows data on the full model (1) with finite parameter values RH 0 = 0.1 and RHL 0 = 0.1, and the good agreement with our analytical predictions based on neglecting those couplings thus corroborates the ro- bustness of our analytical picture over a wider parameter range. More generally, we find that an undesirable outbreak within the high-risk group that, apart from a delay by one time-step, is qualitatively similar to the behavior of the low-risk group occurs as soon as RLH 0 is non-zero, even for very small values (see Fig. 2(a)). Furthermore, . CC-BY-NC-ND 4.0 International licenseIt is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in(which was not certified by peer review)preprint The copyright holder for thisthis version posted November 7, 2020. ; https://doi.org/10.1101/2020.11.06.20226894doi: medRxiv preprint 3 FIG. 2: Instability at weak inter-group coupling (a) and total infection during the epidemic (b), illustrating Eq. (5) for various inter-group coupling values RLH 0 (evaluated at t = 100, RL 0 = 1.5, RHL 0 =RH 0 = 0 for the initial condition ρH i (t = 0) =ρL i (t = 0) = 10−4). increasing RH 0 and RHL 0 is never found to reduce ρH i (t) (or IH for that matter). Hence, our analytical picture for RH 0 = RHL 0 = 0 may be seen as an optimistic lower bound for the infections within the high-risk group. Concluding remarks.– In this work, we have shown that the efficiency of a stratified epidemic strategy dividing the population into a low- and a high-risk group is dras- tically limited by synchronization effects occurring for any finite coupling between the groups. Specifically, the most optimistic hope to maintain a significantly slower effective reproduction rate for the high-risk group as com- pared to the low-risk group is largely ruled out. We have explicitly demonstrated this analytically in SIR-based 2GROUP models that give a coarse grained mean-field picture, noting that SIR models are microscopically more accurate for the spread of e.g. influenza viruses than for SARS-CoV-2, as clusters and superspreading events play an important role for the latter. Furthermore, certain quantitative aspects such as the delay in infections be- tween the two groups may differ significantly due to finer structures of a real society not captured by our simple modelling [15]. Finally, we note that our model assumes perfect immunity upon recovery. That may be a rea- sonable approximation during a specific wave of the epi- demic, and deviations from this assumption can for sure only make the situation worse for both groups. Despite our minimal modeling, we would find it very surprising, if the qualitative effects revealed in our present work, in particular the impossibility of sustaining a selective con- tainment for a risk-group only, could not be clearly iden- tified in more microscopic modelling scenarios. Along these lines, we hope that our findings stimulate future efforts to analyze more detailed COVID-19 specific mod- els from a viewpoint of epidemic curves corresponding to different groups, including effects such as partial immu- nity, time-dependent strategies, imperfect vaccinations, and relaxing NPIs. As a matter of fact, a sort of 2GROUPs strategy has been applied in Sweden for the first 7 months of the COVID-19 pandemic. While this has certainly con- tributed to keeping the fatalities lower than in an entirely unmitigated scenario, the comparably high fatality rates in Sweden may serve as a practical example of how hard it is to selectively protect high-risk groups [16,17]. While insufficient security measures naturally play a major role here, we note that the outcome is in agreement with our general analysis: the substantial impact on the high-risk group has been reflecting the high spread in society at large. This stands in contrast to the neighboring coun- tries that have applied a containment strategy directed to the society at large. Even without similarly stringent restrictions for the risk groups, this has resulted in a per capita fatality rate roughly an order of magnitude lower than in Sweden. This indicates that synchronization phe- nomena as the ones revealed in our present study might be more universal for epidemic dynamics, as long as in- fections occur between individuals (as opposed to disease spreading via agents such as mosquitos). Acknowledgments.— We would like to thank Marcus Carlsson, Thors Hans Hansson and Jan L¨ otvall for use- ful comments and discussions. EJB acknowledges the Science Forum Covid-19 (https://vetcov19.se/en/) for numerous discussions on epidemiology and COVID-19. EJB is supported by the Swedish Research Council (VR) and the Wallenberg Academy Fellows program of the Knut and Alice Wallenberg Foundation. ∗ Electronic address: [email protected] † Electronic address: [email protected] [1] WHO (2020)Coronavirus disease 2019 (COVID-19): Sit- uation Report - 72. April 1, 2020, World Health Organi- zation, Geneva (2020) , https://apps.who.int. . CC-BY-NC-ND 4.0 International licenseIt is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in(which was not certified by peer review)preprint The copyright holder for thisthis version posted November 7, 2020. ; https://doi.org/10.1101/2020.11.06.20226894doi: medRxiv preprint 4 [2] N. M. Ferguson et al., Impact of non- pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand , DOI: https://doi.org/10.25561/77482. [3] S. Flaxman et al., (2020) Estimating the number of infec- tions and the impact of non-pharmaceutical interventions on COVID-19 in 11 European countries (Imperial College London, 2020), DOI: https://doi.org/10.25561/77731. [4] T. Britton, F. Ball, and P. Trapman (2020) A mathe- matical model reveals the influence of population hetero- geneity on herd immunity to SARS-CoV-2 , Science 369, 6505, 846-849. [5] H. Salje et al., Estimating the burden of SARS-CoV-2 in France, Science 369, 6500, 208-211. [6] A. J. Kucharski et al., Early dynamics of transmission and control of COVID-19: a mathematical modelling study, The Lancet Infectious Diseases 20, 5, 553-558. [7] nCoV-2019 Data Working Group (2020) Epi- demiological data from the nCoV-2019 outbreak: early descriptions from publicly available data , http://virological.org/t/epidemiological-data-from-the- ncov-2019-outbreak-early-descriptions-from-publicly- available-data/337. [8] M. Kulldorf et al. (2020), The Great Barrington Decla- ration, https://gbdeclaration.org. [9] K. Alexanderson et al. (2020), THE JOHN SNOW MEM- ORANDUM , https://www.johnsnowmemo.com. [10] We do not attempt a detailed modeling with many pa- rameters specific to a given country, but rather aim at distilling robust qualitative properties of the considered scenarios that do not depend on any fine-tuning. Quan- titative predictions will be about orders of magnitude rather than changes of a few percent. [11] ourworldindata.org, Case fatality rate of COVID- 19 by age , https://ourworldindata.org/mortality-risk- covid#case-fatality-rate-of-covid-19-by-age. [12] W. O. Kermack, and A. G. McKendrick (1927), A Con- tribution to the Mathematical Theory of Epidemics , Pro- ceedings of the Royal Society A. 115 (772): 700–721. [13] H. Nishiura, N. M. Linton, A. R. Akhmetzhanov (2020). Serial interval for novel coronavirus (COVID-19) infec- tions, International Journal of Infectious Diseases. 93, 284-286. [14] B. Rahman, E. Sadraddin and A. Porreca (2020). The basic reproduction number of SARS?CoV?2 in Wuhan is about to die out, how about the rest of the World? , Rev Med Virol., 10.1002/rmv.2111. [15] An example of this is that the infection may be intro- duced to society in a specific subpopulation of the low- risk groups, such as rich and healthy tourists that go on skiing vaccations, while it takes a few time steps (genera- tions of transmission) until it has reached e.g. the usually poorer care home workers that are in direct contact with the high-risk group. [16] M. Brand´ en et. al. (2020), Residential context and COVID-19 mortality among adults aged 70 years and older in Stockholm: a population-based, observational study using individual-level data , Lancet Healthy Longev 2020, https://doi.org/10.1016/ S2666-7568(20)30016-7. [17] A.C. Roxby and T.R. Gure (2020), Lessons from Swe- den: where can older adults shelter from COVID-19? , Lancet Healthy Longev 2020, https://doi.org/10.1016/ S2666-7568(20)30035-0. . CC-BY-NC-ND 4.0 International licenseIt is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in(which was not certified by peer review)preprint The copyright holder for thisthis version posted November 7, 2020. ; https://doi.org/10.1101/2020.11.06.20226894doi: medRxiv preprint