Assessment of effective mitigation and prediction of the spread of SARS-CoV-2 in Germany using demographic information and spatial resolution.

License CC-BY-NC-ND. This is he accep ed e sion o M. J. K¨uhn, D. Abele, T. Mi a, W. Koslow, M. Abedi, K. Rack, M. Siggel, S. Khailaie, M. Kli z, S. Binde , Luca Spa a o, J. Gilg, J. Kleine , M. H¨abe le, L. Pl¨o zke, C. D. Spinne , M. S eche , X. X. Zhu, A. Base mann, M. Meye -He mann, ”Assessmen o e ec i e mi iga ion and p edic ion o he sp ead o SARS-CoV-2 in Ge many using demog aphic in o ma ion and spa ial esolu ion”. Ma hema ical Biosciences 339, 108648 (2021) published by Published by Else ie Inc. The published jou nal a icle is a ailable ia h ps://www.sciencedi ec .com/science/a icle/ pii/S0025556421000845.Assessmen o e ec i e mi iga ion and p edic ion o he sp ead o SARS-CoV-2 in Ge many using demog aphic in o ma ion and spa ial esolu ion Ma in J. K¨uhna, Daniel Abelea, Tanmay Mi ab, Wadim Koslowa, Majid Abedib, Ka h in Racka, Ma in Siggela, Sahamoddin Khailaieb, Ma g i Kli za, Sebas ian Binde b, Luca Spa a oa, Jonas Gilga, Jan Kleine a, Ma hias H¨abe lec, Lena Pl¨o zkea, Ch is oph D. Spinne d, Melanie S eche e, Xiao Xiang Zhuc, Michael Meye -He mannb,1, Achim Base manna,1 aIns i u e o So wa e Technology, Depa men o High-Pe o mance Compu ing, Ge man Ae ospace Cen e , Cologne, Ge many bDepa men o Sys ems Immunology and B aunschweig In eg a ed Cen e o Sys ems Biology (BRICS), Helmhol z Cen e o In ec ion Resea ch, B aunschweig, Ge many cEa h Obse a ion Cen e , Depa men EO Da a Science, Ge man Ae ospace Cen e , Weßling, Ge many dTechnical Uni e si y o Munich, School o Medicine, Uni e si y Hospi al ech s de Isa , Depa men o In e nal Medicine II, Munich, Ge many eUni e si y Hospi al o Cologne, Depa men I o In e nal Medicine, Uni e si y o Cologne; Ge man Cen e o In ec ion Resea ch (DZIF), Cologne, Ge many Abs ac Non-pha maceu ical in e en ions (NPIs) a e impo an o mi iga e he sp ead o in ec ious dis- eases as long as no accina ion o ou s anding medical ea men s a e a ailable. We assess he e ec i eness o he se s o non-pha maceu ical in e en ions ha we e in place du ing he cou se o he Co ona i us disease 2019 (Co id-19) pandemic in Ge many. Ou esul s a e based on hyb id models, combining SIR- ype models on local scales wi h spa ial esolu ion. In o de o accoun o he age-dependence o he se e e acu e espi a o y synd ome co ona i us 2 (SARS-CoV-2), we include ealis ic p epandemic and ecen ly eco ded con ac pa e ns be ween age g oups. The implemen a ion o non-pha maceu ical in e en ions will occu on changed con ac pa e ns, im- p o ed isola ion, o educed in ec iousness when, e.g., wea ing masks. In o de o accoun o spa ial he e ogenei y, we use a g aph app oach and we include high-quali y in o ma ion on commu ing ac- i i ies combined wi h a eling in o ma ion om social ne wo ks. The emaining unce ain y will be accoun ed o by a la ge numbe o andomized simula ion uns. Based on he de i ed ac o s o he e ec i eness o di e en non-pha maceu ical in e en ions o e he pas mon hs, we p o ide di e en o ecas scena ios o he upcoming ime. Keywo ds: SARS-CoV-2, Co id-19, Co ona i us disease, Mi iga ion, Non-pha maceu ical in e en ions, Fo ecas 1. In oduc ion Wi h mo e han 2.2 million epo ed dea hs [1], he co ona i us disease 2019 (Co id-19) emains one o he mos p essing issues o he whole globe. Al eady in Oc obe , WHO o icials es ima ed ha 10 % o he wo ld’s popula ion had been in ec ed [2] and accina ion o he popula ion will s ill ake a conside able amoun o ime. Since exposing people is highly une hical [3], he only in e im solu ion is o mi iga e he sp ead o he disease by he applica ion o non-pha maceu ical in e en ions. Email add esses: [email p o ec ed] (Ma in J. K¨uhn), [email p o ec ed] (Michael Meye -He mann), [email p o ec ed] (Achim Base mann) 1Sha ed co esponding au ho s in alphabe ic o de . P ep in submi ed o Ma hema ical Biosciences Augus 19, 2021 The assessmen o non-pha maceu ical in e en ions and p edic ion by simula ion has o be based on eliable models; c . [4, 5, 6, 7, 8, 9, 10, 11] o he sp ead o SARS-CoV-2 (se e e acu e espi a o y synd ome co ona i us 2) and o he in ec ious diseases. Only hen, he mos e ec i e in e en ions can be de e mined as a basis o in o med poli ical decisions. The aim o ou s udy is o assess non-pha maceu ical in e en ions and o p o ide a eliable o ecas o he Co id-19 pandemic in Ge many based on ou p inciples. Fi s , we accoun o he age-dependence o SARS-CoV-2 [12, 13, 14]. Second, we include ealis ic con ac pa e ns be ween di e en age g oups [15, 16, 17, 18, 19]. Thi d, we include high-quali y, spa ially esol ed in o ma ion on commu ing ac i i ies [20, 21] combined wi h a eling in o ma ion based on he social ne wo k Twi e . Fou h, we combine all in o ma ion and accoun o he emaining unce ain y by Mon e-Ca lo Ensemble uns. To ou knowledge, such an in-dep h s udy is no accoun ed o in he li e a u e so a . The emainde o his pape is s uc u ed as ollows. We i s p esen ou ma hema ical model and i s nume ical solu ion app oach. Then, we p esen he social and non-pha maceu ical pa am- e e s used in ou model and we discuss a e wa ds he epidemiological pa ame e s ob ained by ex ensi e analyses. Ou esul s a e p esen ed and discussed in he ollowing. 2. Ma e ials and me hods 2.1. Model and sol e The e a e a ious models o o ecas he sp ead o in ec ious diseases ac oss a coun y o com- muni y. Besides he well known SIR- ype ODE models [22, 23], he e a e in eg o-di e en ial mod- els [22, 24], Bayesian Mon e Ca lo app oaches [11], o agen -based models [25, 26]. While SIR- ype models a e p aised o hei simplici y and unde s andabili y, hey lack o a good ep esen a ion o spa ial he e ogenei y. To combine he ad an ages o a SIR model wi hou loss o spa ial esolu ion, we use mul iple SIR- ype models on a ine local scale and connec he compa men s and age g oups by g aphs ha ep esen a eling; c . [27] among o he s. These SIR- ype models can be easily exchanged by agen -based models due o he gene ic implemen a ion o ou g aphs. This is in eg a ed as pa o ou high pe o mance modula epidemics simula ion so wa e MEMILIO ha is con inuously unde de elopmen [28]. 2.1.1. Age- esol ed SIR- ype model The base o ou SIR- ype model can be ound in he i s e sion o [9]. Ou model consis s o he compa men s Suscep ible (S), heal hy indi iduals wi hou immune memo y o SARS-CoV-2; Ex- posed (E), who ca y he i us bu a e no ye in ec ious o o he s; Ca ie (C), who ca y he i us and a e in ec ious o o he s bu do no ye show symp oms ( hey may be p e- o asymp oma ic); In- ec ed (I), who ca y he i us, a e in ec ious and show symp oms; Hospi alized (H), who expe ience a se e e de elopmen o he disease; In In ensi e Ca e Uni (U); Dead (D); and Reco e ed (R), who canno be in ec ed again. To esol e age-speci ic disease pa ame e s, we di ide he o ali y o people Nin o ndi e en age g oups. We hen ha e Z:= Sn i=1 Zi:= Sn i=1{Si, Ei, Ci, Ii, Hi, Ui, Ri, Di}. Fo each age g oup i= 1, . . . , n, he ansmission isk is deno ed by ρiand he p opo ion o in ec ed people no isola ed o qua an ined is deno ed by ˜ βi; see Tables 1 and 2 o de ails. In ec ion esul s om con ac wi h people om di e en age g oups. We in oduce he con ac equency ma ix Φ = (φi,j)i,j=1,...,n,(1) whe e φi,j ep esen s he (mean) daily con ac s o a pe son o age g oup iwi h people om age g oup j. We e e o [29] which s a es ha ” he esul ing ma ix is no symme ic due o he di e en numbe o indi iduals in each age-g oup”. So due o he pa icula ly chosen age g oups and he demog aphy o Ge many, hese con ac ma ices will be non-symme ic in ou case. The naming con en ion o he emaining pa ame e s can be unde s ood as ollows: We use he a iables T∗2 ∗1 o he ime spen in s a e ∗1∈ Zibe o e mo ing o s a e ∗2∈ Zi. Fo example, TRi Hi 2 Suscep ible S Exposed E Ca ie C In ec ed I Hospi alized H ICU U Reco e ed R Dead D φ ρ C+˜ βI N 1 TC E 1−µR C TI C µR C TR C 1−µH I TR I µH I TH I µU H TU H 1−µU H TR H µD U TD U 1−µD U TR U Figu e 1: SIR- ype model and s onges in e -coun y commu e ac i i ies. SIR- ype model o one Ge man coun y, based he on i s e sion o [9] (le ). We omi he age-dependence index i o cla i y; see Table 1 and 2 o a desc ip ion o he pa ame e s. G aph wi h cen e poin s o all Ge man coun ies as nodes and edges acco ding o commu e ac i i y ( igh ). Edges only shown whe e mo e han 10 000 wo ke s commu e on a daily basis. ep esen s he ime an indi idual in age g oup i= 1, . . . , n spen in he hospi al be o e e u ning home due o eco e y om he disease. Acco dingly, µ∗2 ∗1 ep esen s he p obabili y o a pa ien o ansi o s a e ∗2when ha pa ien is cu en ly in s a e ∗1. The model, as exp essed in Fig 1, is dSi d =−Siρi n X j=1 φi,j Cj+˜ βjIj Nj ,(2) dEi d =Siρi n X j=1 φi,j Cj+˜ βjIj Nj −1 TCi Ei Ei,(3) dCi d =1 TCi Ei Ei− 1−µRi Ci TIi Ci +µRi Ci TRi Ci!Ci,(4) dIi d =1−µRi Ci TIi Ci Ci− 1−µHi Ii TRi Ii +µHi Ii THi Ii!Ii,(5) dHi d =µHi Ii THi Ii Ii− 1−µUi Hi TRi Hi +µUi Hi TUi Hi!Hi,(6) dUi d =µUi Hi TUi Hi Hi− 1−µDi Ui TRi Ui +µDi Ui TDi Ui!Ui,(7) dRi d =µRi Ci TRi Ci Ci+1−µHi Ii TRi Ii Ii+1−µUi Hi TRi Hi Hi+1−µDi Ui TRi Ui Ui,(8) dDi d =µDi Ui TDi Ui Ui.(9) The equa ions (2)–(9) ep esen he ansi ion o people om one s a e o ano he . No e ha people in an age g oup Zicanno ansi o ano he g oup Zj o i6=j. In he sec ion o he epidemiological pa ame e s, we will discuss in de ail which o he pa ame e s we assume o be age-dependen and how his is included in ou model. These indings a e summa ized in Tables 1 and 2. 3 Pa am. Desc ip ion Re e ence Resou ces ρ(0) ρiage-dependen ansmission isk Eq. (14),(2),(3) [30, 31, 32, 33, 34, 35, 36] kseasonali y pa ame e Eq. (15), (14) [37, 38, 39, 40] ˜ βp opo ion o no isola ed o qua an ined symp oma ic indi iduals Eq. (2), (3) Assump ion. TC Epe iod o la en non-in ec ious s age Eq. (3), (4) [9, 41, 42] µR Cp opo ion o mild, asymp oma ic cases Eq. (4), (5), (8) [43, 44, 45, 35, 46] TR Cpe iod o asymp oma ic s age be o e eco e y Eq. (4), (8) [9] TI Cpe iod o la en in ec ious s age Eq. (4), (5) [9, 41, 42] µH Ip opo ion o symp oma ic cases needing hospi aliza ion Eq. (5), (6), (8) [14, 47],[48, Repo o Sep . 15] TH Ipe iod o mild symp oms o indi iduals equi ing hospi aliza ion la e on Eq. (5), (6), Suppl. Ma . [49, 50, 51] TR Ipe iod o mild symp oms o indi iduals no equi ing hospi aliza ion la e on Eq. (5), (8) [52, 9] µU Hp opo ion o hospi alized indi iduals ge ing ICU ea men Eq. (6), (7), (8) [53, 54, 47] TU Hpe iod o hospi aliza ion be o e ICU ea men (o c i ical cases) Eq. (6), (7), Suppl. Ma . [49, 50] TR Hpe iod o hospi aliza ion be o e eco e y (o non-c i ical cases) Eq. (6), (8), Suppl. Ma [9] µD Up opo ion o indi iduals in ICU ca e ha die Eq. (7), (8), (9), Fig 4 [54, 55] TR Upe iod o ICU ea men be o e eco e y Eq. (7), (8), Suppl. Ma [9, 56, 50] TD Upe iod o ICU ea men be o e dea h Eq. (7), (9), Suppl. Ma [50, 9] Table 1: Desc ip ion o pa ame e s and main esou ces o hei de i a ion. 2.1.2. Spa ial esolu ion While SIR- ype models a e s aigh o wa d o apply and in e p e , hey lack he possibili y o modeling local e ec s o spa ial he e ogenei y. In o de o a oid a e aging o e impo an e ec s such as in ec ion clus e s, we assign one pa icula age- esol ed model o each coun y. We ep esen each coun y by a node o a (di ec ed) g aph. The edges o he g aph ep esen he connec ions be ween he di e en coun ies and a e weigh ed wi h he numbe o people commu ing daily and a eling on a e age. The edges do no only hold single alues (weigh s) o how many people daily commu e be ween di e en coun ies bu also coe icien s o de e mine he p opo ion o people o di e en age g oups and compa men s ha commu e o a el. Doing so, we can es ic a el ac i i ies o heal hy o only mildly in ec ed indi iduals. Le nCbe he numbe o coun ies (nodes o he g aph). Then, o wo nodes akand al, 1≤k, l ≤nC, he weigh wk,l on edge ek,l ep esen s he p opo ion o people going daily om ak o al. 2.1.3. Nume ical sol e Common nume ical sol e s o he sys em o nonlinea o dina y di e en ial equa ions (2)-(9) a e semi-implici o adap i e explici . While he o me allow o la ge ime s eps, he la e allow adap i e ime s eps o p e en la ge nume ical e o s. We ha e implemen ed an adap i e Runge- Ku a-Fehlbe g45 (RKF45) me hod [57] ha uses me hods o 4 h and 5 h o de and sol es he equa ions wi hou excessi ely small ime s eps. The nume ical p ocedu e becomes mo e challenging when we also esol e he equa ions spa ially. Fo his, we de ine a commu e as a pe son who a els om coun y ak o al, 1 ≤k, l ≤nCand back again wi hin one day (whe he i is wo k o ee ime ela ed). Gi en s a alues om day , we 4 ange in age g oup pa am. 0-4 5-14 15-34 35-59 60-79 80+ ρ(0) [0.02,0.04] [0.05,0.07] [0.08,0.10] [0.15,0.20] k[0.1,0.3] ˜ βsigmoidal cu e om [0.1,0.3] o [0.3,0.5] TC E[2.67,4.00] µR C[0.20,0.30] [0.15,0.25] TR CTI C+ 0.5TR I TI Csampled wi h TC Eand (16), incuba ion pe iod = 5.2 µH I[0.006,0.009] [0.015,0.023] [0.049,0.074] [0.15,0.18] [0.20,0.25] TH I[9,12] [5,7] TR I[5.6,8.4] µU H[0.05,0.10] [0.10,0.20] [0.25,0.35] [0.35,0.45] TU H[3,7] TR H[4,6] [5,7] [7,9] [9,11] [13,17] µD U[0.00,0.10] [0.10,0.18] [0.3,0.5] [0.5,0.7] TR U[5,9] [14,21] [10,15] TD U[4,8] [15,18] [10,12] Table 2: Summa y o he age-dependency o pa ame e s and hei anges. ad ance ou adap i e RKF45 sol e o 0.5 days. Nex , we allow people o commu e o a el. Thei amoun is de ined by he commu e a e be ween wo coun ies, namely he weigh s wk,l in oduced in he p e ious sec ion and u he speci ied in ollowing sec ion. No e ha commu ing also depends on he in ec ion s a e since hospi alized indi iduals canno commu e and in ec ed indi iduals will a el less han heal hy ones. Fo he la e , we assume he same le el o isola ion o qua an ine as on coun y le el. Wi h he upda ed popula ion, we again ad ance ou adap i e sol e o 0.5 days. Addi ionally, we conduc an auxilia y s ep wi h s ep size o 0.5 days wi h an explici Eule sol e whe e we only conside he in-commu e s, using he coun y’s popula ion as con ac popula ion only. This s ep is execu ed since, a e he high p ecision scheme om + 0.5 o + 1, we do no know he upda ed s a e o ou commu e s (e.g., suscep ible may ha e become exposed o ca ie s ha e become symp oma ic). This is due o he na u e o he SIR model (2)-(9) ha does no keep ack o indi iduals. S ill, he commu e s ha e o go back o hei home coun y, and we need o know hei mos likely in ec ion s a e. We use he esul s om he explici Eule s ep, o quan i y he p opo ion o indi iduals o he di e en compa men s ha e u n. Wi h his es ima ion, we s a he e u ning p ocess. These conside a ions a e summa ized in Fig 2. 2.2. Social and non-pha maceu ical pa ame e s The sp ead o SARS-CoV-2 depends on many pa ame e s. While some o hese pa ame e s a e inhe en o he i us, o he s depend on social con ac pa e ns and non-pha maceu ical in e en ions in oduced by decision make s. In his sec ion, we will ocus on non-pha maceu ical in e en ions and hei in luence on con ac pa e ns and commu e a es in ou model. 2.2.1. In e -coun y a el Le us i s con inue wi h he spa ial esolu ion o ou model and ocus on how we speci y he a e o wo k o leisu e commu e s wk,l be ween di e en coun ies akand al. To es ima e wk,l on 5 Figu e 3: In e -age g oup con ac pa e ns. Combined p epandemic con ac pa e ns φGe Bo [17, 16] o Ge many in e pola ed o age in e als as p o ided by [62] ( op). Ex apola ed, pandemic con ac pa e ns φGe M o simula ed lockdown phase o Ge many (as o end Ma ch in he UK; based on con ac s udy [18]) (bo om). bigge ci ies like Hambu g, Hanno e , Cologne, o S u ga . Assuming ha he mobili y ob ained om Twi e accoun s o 20 % o all a el ac i i ies (wo k commu ing, s uden commu ing, leisu e a el e c.), we scale he wi e ma ix acco dingly. The esul ing alues a e di ided by he popula ion size and he esul will be deno ed by k,l, 1≤k, l ≤nC. The amoun o mobili y in ou model is gi en be he edge weigh s ek,l o he g aph. These weigh s a e de i ed om he ma ix ck,l o he wo k commu e s and om k,l o he ‘Twi e ’ ac i i ies. The weigh s also depend on he implemen a ion o non-pha maceu ical in e en ions. In pa icula , hey depend on he NPI ela ed pa ame e s (∗) W,i,i o wo k and (∗) O,i,i o ”o he ” places ela ed measu es ha mainly a ec ee- ime ac i i ies. He e, ∗ ∈ {1,2}and i= 1, . . . , n a e he co esponding age g oups; see he co esponding sec ion and Table 3 o de ails on hese pa ame e s. The commu e ma ices con ain many insigni ican coe icien s close o ze o. These a e due o loosely coupled egions whe e only a e y limi ed numbe o indi iduals commu e on a daily basis (e.g., 1 o 2). To educe he compu a ional e o , we elimina e edges ek,l whe e ck,l <4·10−5and k,l <1·10−5. The cu o alues a e chosen so only 1 % o in o ma ion is d opped and ha mo e han 99 % o a els a e included. We also paid a en ion o e lec he abo e assump ion ha Twi e da a ep esen s 20 % o a els. Wi h his p ocedu e, he numbe o edges is educed by app oxima ely 60 % and he compu a ional e iciency is inc eased signi ican ly. 2.2.2. Con ac pa e ns in Ge many In he ollowing, we ocus on he in a-coun y con ac pa e ns. In his sec ion, we de i e a baseline, p epandemic con ac ma ix φGe Band a minimum con ac ma ix φGe M o a simula ed s ic lockdown in Ge many. As SARS-CoV-2 ansmission occu s mainly du ing human- o-human in e ac ion, educing con- ac s can e icien ly slow down he sp ead o he disease; c . [63, 15, 16, 17] o li e a u e on con ac s and he sp ead o in ec ious diseases. Howe e , lockdowns which e ec i ely educe con ac s o a minimum should be a oided due o hei p o ound nega i e impac on many indi iduals and com- muni ies [3]. The e o e, he challenge o oday’s decision make s is o ind he mos app op ia e and e ec i e in e en ions o he ac ual de elopmen s. P epandemic pa e ns. In o de o quan i y he po en ial o ansmission educ ion by con ac 7 pa e n changes, good p epandemic as well as ecen da a is needed. F om [15] and i s p ojec- ions [17], we use ealis ic con ac pa e ns o Ge many spli up in o he ca ego ies “Home”, “School”, “Wo k”, and “O he ”. In [15], con ac s a e de ined as skin- o-skin con ac , o whe e a leas h ee wo ds we e exchanged. Fo he pa icula case o school con ac s, he mean numbe s o con ac s eco ded in [15] a e a he low o Ge many. Gi en he ac o ae osol ansmission isk in closed spaces, we sugges o assume sligh ly highe con ac a es o a conse a i e es ima e on he sp ead o he disease. Fu he in o ma ion is o e ed by he demog aphy-based school con ac ma ix in [16]. We use he quo ien o he maximum eigen alues be ween bo h ma ices o scale he con ac ma ix o [17] which hen esul s in a la ge numbe o school con ac s. The combina ion o baseline con ac s o “Home”, “Wo k”, and “O he ” om [15, 17] and o “School” based on he compa ison o [17, 16] esul s in he con ac ma ix φGe B; c . Fig 3 ( op). SARS-CoV-2- ela ed minimum pa e ns. The po en ial o possible con ac educ ion is lim- i ed by he minimum numbe o necessa y con ac s ha keep essen ial sec o s o he socie y unning. To assess his, we conside he con ac s udy [18], ha s a ed du ing he lockdown phase in he Uni ed Kingdom. By he end o Ma ch, many ‘non-essen ial’ pa s o he economy we e shu down and social in e ac ion was limi ed o a minimum [64]. This s udy yields he minimum con ac ma ix ΦUK M. The ma ix ΦUK Mis missing alues since only indi iduals aged 18 o olde pa icipa ed in [18]. In o de o ill ou he missing in o ma ion, we ollow a s a egy simila o [65]. We employ he p epandemic/baseline con ac ma ix ΦUK B om [17]. We scale his ma ix by he a io o he dominan eigen alues λBand λMo he lowe - igh , squa e ma ices (φUK ∗,i,j,)i,j≥3,∗ ∈ {M, B}. Then, we use his scaled e sion o ill ou he missing subse φUK M,i,j =φUK B,i,j ·λM λB ∀i∈ {1,2}, j ∈ {1, ..., 6}.(10) We aim a de i ing a minimum con ac ma ix φGe M o a simula ed s ic es lockdown in Ge many. We conside he numbe o con ac s in he UK by he end o Ma ch o be a minimum ha we can achie e in a SARS-CoV-2- ela ed lockdown. F om he UK da a, we conside he quo ien o con ac educ ion di,j =φUK B,i,j /φUK M,i,j ∀i, j ∈ {1,...,6},(11) and apply hese ac o s o he ma ices φGe B,i,j de i ed om [17, 16] φGe M,i,j =di,j ∗φGe B,i,j ∀i, j ∈ {1,...,6}.(12) In Fig 3, he minimum ΦGe Mis shown a he bo om. No e ha he single en ies gi en in he bo om ow o Fig 3 a e no equi ed o be smalle han he ones in he op ow. The minimum o con ac s du ing lockdown is o be unde s ood as he minimum o o al con ac s o all indi iduals. Locally, o one loca ion and he in e ac ion o wo age g oups, he mean con ac s could e en inc ease sligh ly. The di e ence be ween he op and bo om o Fig 3 de ines ealis ic bounda ies o all non-pha maceu ical in e en ions ha could possibly be implemen ed. To assess unce ain y, we allow o a 5-10 % de iance o he gi en alues in ou ensemble uns. F om [66], we ha e an es ima ed con ac educ ion du ing sp ing lockdown in Ge many o 63 %, aking he minimum alues he e, we could achie e a con ac educ ion o 76 %. 2.2.3. Con ac - ela ed in e en ions The e a e wo ways o educe po en ially dange ous con ac s, namely, o a oid he con ac s ( i s le el o educ ion) a all o o wea masks, keep dis ance and en ila e closed spaces (second le el). While he mean numbe o daily con ac s in Ge many is lowe han in many o he Eu opean coun ies, a ela i ely la ge pe cen age o con ac s happens a wo kplaces [15]. Hence, many ans- missions can be a oided by wo king om home whene e possible. F om a ecen analysis o 8 in e en ion implemen a ion ac o anges commen wo king weak (1) W,i,j ∈[0.0,0.1] om in e media e (1) W,i,j ∈[0.2,0.3] home s ong (1) W,i,j ∈[0.4,0.5] pa ial school none (1) S,i,j = 0.0 closu es and weak (1) S,i,j = 0.25 emo e in e media e (1) S,i,j = 0.5 schooling comple e (1) S,i,j = 1 ga he ing bans, weak (1) O,i,j ∈[0.0,0.2] addi ional inc ease o (pa ial) closing a he weak (1) O,i,j ∈[0.2,0.4] (1) W,i,j by 0.05 o 0.20, o ba s, in e media e (1) O,i,j ∈[0.4,0.6] acco ding o s ic ness; es au an s, s ong (1) O,i,j ∈[0.6,0.8] c . co esp. sec ion cinemas e c. e y s ong (1) O,i,j ∈[0.8,1.0] ace masks, weak (2) ∗,i,j ∈[0.0,0.2] dis ancing, a he weak (2) ∗,i,j ∈[0.2,0.4] egula in e media e (2) ∗,i,j ∈[0.4,0.6] ∗ ∈ {H, S, W, O} en ila ion o s ong (2) ∗,i,j ∈[0.6,0.8] closed spaces e y s ong (2) ∗,i,j ∈[0.8,1.0] Table 3: Summa y o di e en non-pha maceu ical in e en ions and implemen a ions in simula ions. Ge many [21], up o 40-50 % o he popula ion could wo k om home i necessa y. We suppose ha in he p epandemic phase “home o ice” was only used by 5 % wi h as much as 20-35 % wo king om home du ing di e en phases o he pandemic [21, pp. 96-101]. Fu he con ac educ ion is induced by people who s op wo king al oge he , so hese alues ha e o be sub ac ed om he p epandemic ma ix. We know ha abou 20 % o he popula ion s opped wo king in Ma ch and Ap il [21, p. 96]. Fo he less s ic in e en ions, we assume alues o 5-10 %. While wo king om home is easible o a la ge pa o he popula ion, global school closu es and he esul ing home schooling “p esen an unp eceden ed isk o child en’s educa ion, p o ec ion and well-being” [67]. Apa om school closu es, con ac s in schools can be educed by smalle classes whe e possible, ixed sea ing a angemen s, egula en ila ion, o pooled es ing [68]. The e a e a numbe o u he loca ions whe e con ac educ ions a e easible such as ba s, es au an s, supe ma ke s, o public anspo . We he e include he e ec o in e en ions such as ace masks, dis ancing e c. In many s udies such as [38, 69], ace co e ings and ejec ed ai lows while b ea hing, speaking, o coughing a e s udied. The me a analyses in [70] and [71] ind (la ge) p o ec i e e ec s o ace masks o SARS- CoV-2 ansmission such as 40 % o e en a pooled odds a io o 0.35. In pa icula , he p o ec i e e ec o communi y-wide masks is shown in [72]. We will conside di e en isk educ ion anges o wea ing masks combined wi h keeping dis ance and egula en ila ion o closed spaces. In he p edic i e analysis o he sp ead o in ec ious diseases, no only non-pha maceu ical in e en ions and one- ime con ac changes bu also adhe ence o in e en ions is impo an . While he e is a small dec ease in adhe ence o p e en i e measu es obse ed in [19, 73] a e mon hs o he pandemic, he adhe ence is s ill la ge and a he s able wi h an e en inc easing numbe o people wea ing masks [73] (e.g., 93 % wea hem o en o always). In he consequence, we do no include hese opposing e ec s in ou simula ions. In he esul s sec ion, we a y he s ic ness o he in e en ions acco ding o he poli ical decisions and Table 3. 9 ∗ ∈ {H, S, W, O}, which is a qui e s ong measu e o dis ancing, ace masks and o he in e en ions. Simula ion. Ou ini ial condi ions a e de i ed om he age- esol ed case da a p o ided by [62]. We ake con i med cases a ound he s a da e o ou simula ion om which we ex apola e he compa men s in eq. (2)-(9) by using he pa ame e s in Table 2. Based on [84] and ou pa ame e s, we ex apola e age- esol ed ICU da a. In o de o ob ain age- esol ed ICU da a, we sligh ly educe he ini ial ex apola ion o 80+ in ensi e ca e cases since oo la ge dea h a es occu in he beginning pa o he simula ion o he wise. We e ain om u he co ec ion o ini ial alues wi hou ha ing mo e eliable da a. Gi en posi i e a es o 1 % o less du ing summe [48, Repo o Aug. 26], we assume ha he numbe o unknown symp oma ic in ec ions was small. The e o e, we s a ou simula ions on June 1, July 15, and Sep embe 1 di ec ly om he numbe o con i med cases. Gi en he inc eased p opo ion o posi i e es s up o mid o Oc obe , we s a he he simula ions wi h a wo old o he con i med cases. Fo each scena io, we un 1000 Mon e Ca lo uns such ha we ha e a eliable se o pa ame e s sampled in he gi en anges. Fo he uns, we p o ide he median alues as well as he pe cen iles ob ained om he simula ion uns. We use a se en day mo ing a e age o eal wo ld da a and ex apola e he day o dea h, using he pa ame e s om Table 2. 2.4.3. Discussion o Re ospec i e Scena ios In he ollowing, we discuss he esul s o he ou di e en e ospec i e scena ios wi h he desc ibed se s o implemen ed NPIs o e ime. We compa e he o e all and he age- esol ed dea h a es wi h he ex apola ed eal da a. We also compa e he o e all in ec ion a es and he ICU occupancy. In he co esponding Figu es 5, 7, 9, and 11, we p esen he median (pe cen ile p50) and he pe cen iles anges om (p05 and p95) as well as p25 and p75 as explained abo e. Addi ionally, in he maps o Figu es 6, 8, 10, and 12, we show wo snapsho s o he egional sp ead o he in ec ion, whe e we compa e he ex apola ed eal da a on he le wi h ou simula ion on he igh . No e ha he scaling o he colo ba s di e s o he scena ios and ep esen s he ela i e numbe o in ec ions pe 100 000 inhabi an s. Scena io 1. Ou i s scena io (S1) is compu ed 45 days om June 1 onwa d. Du ing he summe mon hs, we obse e only a slow ise in he in ec ions in he RKI da a and a dec ease o ICU occupancy om DIVI (Fig 5). Bo h a e cap u ed well wi h he median o ou simula ions. The small ele a ion in he numbe o in ec ions in June is due o an ou b eak o Co id-19 in a slaugh e house in G¨u e sloh. As expec ed, we canno cap u e such a s ochas ic e en . Focusing on he i s pa o June (see Figu e 6, op), we obse e u he egions ha a e unde es ima ed by ou simula ion and some whe e i is he o he way a ound. O e all, howe e , a la ge incidence in he RKI da a mos ly co esponds o a la ge incidence in he simula ion da a. No e ha o e - and unde es ima ing also happens due o a di e en s ic ness o local in e en ions. Wo h men ioning is he sp ead o in ec ions in o neighbou ing egions which we can see o Be lin om he middle o he bo om. Due o he inclusion o mobili y, we see ha he in ec ion sp eads in o he nea egions in ou simula ion ( igh ) as in he ex apola ed eal da a (le ). Due o he longe simula ion ime, he esul s on he bo om show u he de ia ion om he da a. In Figu e 5, we plo he median dea h a e, he pe cen iles and he ex apola ed eal da a o he di e en age g oups. Ou model is qui e close o he o e all dea h a e. The age g oup 60-79 yea s is cap u ed e y well and he age g oups below a e also cap u ed well. We a e a li le less close o he age g oup 80+ yea s. The de ia ions a e o be expec ed since we aim o an o e all good i o ou model wi hou weaking i o he indi idual scena ios. Addi ionally, we lack age- esol ed ICU occupancy da a and ha e o ex apola e he in ensi e ca e inpu da a o he di e en age g oups. This hen also a ec s he simula ed dea h a es. No e ha he age g oups below 15 yea s only con ibu e ma ginally o he dea h a e. The de iance wi h simula ion seems la ge, bu due o he small numbe s (0 o 1), we a e ac ually close o he eal wo ld alue. Scena io 2. Ou second scena io (S2) is compu ed 45 days om July 15 onwa d. The esul s o his Scena io a e e y simila o he esul s o S1. This holds o he o e all dea h a es and he dea h a es in he di e en age g oups as depic ed in Figu e 7. A main di e ence is he in luence o a el e u ne s [48, Repo o Aug. 9] ha migh be esponsible o he ise o in ec ed in he 16 and Resea ch o he p ojec CoViDec (FKZ: 01KI20102). The unding bodies had no ole in he design o he s udy, collec ion, analysis, and in e p e a ion o he esul s, o w i ing he manusc ip . Con lic o in e es . D . Spinne epo s no con lic o in e es du ing he conduc o he s udy; bu pe sonal ees om AbbVie, g an s and pe sonal ees om Ape ion, g an s and pe sonal ees om Janssen- Cilag, g an s and pe sonal ees om Gilead Sciences, pe sonal ees om molecula pa ne s, g an s and pe sonal ees om MSD, g an s and pe sonal ees om ViiV Heal hca e/GSK, ou side he submi ed wo k. All o he au ho s decla e ha hey ha e no con lic o in e es . Acknowledgemen s. We hank Vale ie G appendo , a s uden a he Hochschule ¨u Ges al ung Schw¨abisch Gm¨und, o con ibu ing Fig 2. We exp ess ou deep g a i ude o all s udy eams suppo ing he LEOSS s udy. The LEOSS s udy g oup con ibu ed a leas 5 pe mille o he analyses o his s udy: C. Spinne , S. Rieg, F. Hanses, S. Bo gmann, M. Howe , M. Veh eschild, M. M. R¨u h ich, L. Tome en, C. Piepel, S. Dol , K. Wille, J. Lanzs e , M. on Be gwel -Baildon, U. Me le, C. R¨ommele, C. Degenha d , J. F¨u s , S. Dalin, N. Isbe ne , B. G ¨une , N. Jung, H. Haake, K. Hellwig, W. Rimili, C. Raichle, L. Ebe wein, S. G unwald, M. Ako a, A. F ied ichs, D. Rauschning, C. Wyen, B. Jensen, K. de Wi h, W. Guggemos, J. Kiels ein, B. Schul heis, J. T au h, R. Bals, P. Ma ka , S. S iegli z, I. A kin, M. Milo ano ic, K. Ro h uss, J. R¨uddel, J. Na e mann, D. Heigene , L. Wal e , J. Schube , J. Voig , G. M¨ulle -J¨o ge , C. Riedel, M. Wo m. The LEOSS s udy in as uc u e g oup: J. J. Veh eschild, L. Pilg am, M. S eche , M. Schons, C. E. M. Jakob, A. Claßen, S. M. Nunes de Mi anda, S. Fuh mann, B. F anke, N. Schulze, F. P aße and M. Lablans. The LEOSS s udy was suppo ed by he Ge man Cen e o In ec ion Resea ch (DZIF) and he Willy Robe Pi ze Founda ion. Re e ences [1] JHU Cen e o Sys ems Science and Enginee ing, A cGIS Dashboa ds, 2021-02-04. URL: h ps: //www.a cgis.com/apps/opsdashboa d/index.h ml#bda7594740 d40299423467b48e9ec 6. [2] Deu sche Welle, Co ona i us: WHO es ima es 10% o global pop- ula ion in ec ed wi h COVID-19, 2020. URL: h ps://www.dw.com/en/ co ona i us-who-es ima es-10-o -global-popula ion-in ec ed-wi h-co id-19/a-55162783. [3] Wo ld Heal h O ganiza ion, Co ona i us disease (COVID-19): He d immuni y, lock- downs and COVID-19, 2020. URL: h ps://www.who.in /news- oom/q-a-de ail/ he d-immuni y-lockdowns-and-co id-19. [4] N. Banholze , E. . Weenen, B. K a zwald, A. Seelige , D. Tsche nu e , P. Bo ighi, A. Cenedese, J. P. Salles, W. Vach, S. Feue iegel, Impac o non-pha maceu ical in e en ions on documen ed cases o COVID-19, medRxi (2020) 2020.04.16.20062141. doi:doi:10.1101/2020.04.16.20062141. [5] M. V. Ba ba ossa, J. Fuh mann, J. H. Meinke, S. K ieg, H. V. Va ma, N. Cas elle i, T. Lip- pe , Modeling he sp ead o co id-19 in ge many: Ea ly assessmen and possible sce- na ios, PLOS ONE 15 (2020) 1–22. URL: h ps://doi.o g/10.1371/jou nal.pone.0238559. doi:doi:10.1371/jou nal.pone.0238559. [6] S. Cauchemez, N. M. Fe guson, C. Wach el, A. Tegnell, G. Saou , B. Duncan, A. Nicoll, Clo- su e o schools du ing an in luenza pandemic, The Lance In ec ious Diseases 9 (2009) 473–481. doi:doi:10.1016/S1473-3099(09)70176-8. [7] A. Des a s-La i e, E. De ic, N. Haug, T. Niede k o en hale , J. Chen, A. Di Na ale, J. Lasse , D. S. Gliga, A. Roux, J. So ge , A. Chak abo y, A. Ten, A. De ic, A. Pacheco, A. Ju czak, D. Cse jan, D. Lede hilge , D. Bulska, D. Be ishaj, E. F. Tames, F. S. Al a ez, H. Tak i i, J. Ko bel, J. Red- dish, J. G zymala-Moszczynska, J. S angl, L. Hadzia dic, L. S oege , L. Goo iah, L. Gey ho e , M. R. Fe ei a, M. Ba oszek, R. Vie linge , S. Holde , S. Habe ellne , V. Ahne, V. Reisch, V. D. P. Se edio, X. Chen, X. M. Pocasang e-O ellana, Z. Ga nca ek, D. Ga cia, S. Thu ne , A s uc u ed open da ase o go e nmen in e en ions in esponse o COVID-19, Scien i ic Da a 7 (2020) 1–9. doi:doi:10.1038/s41597-020-00609-9. [8] L. Fe e i, C. Wyman , M. Kendall, L. Zhao, A. Nu ay, L. Abele -D¨o ne , M. Pa ke , D. Bon- sall, C. F ase , Quan i ying SARS-CoV-2 ansmission sugges s epidemic con ol wi h digi al con ac acing, Science 368 (2020). doi:doi:10.1126/science.abb6936. 25