You better watch out: US COVID-19 wave dynamics versus vaccination strategy
Giacomo Cacciapaglia, Corentin Cot, Anna Sigridur Islind, María ?skarsdóttir, Francesco Sannino
YYou better watch out : US COVID-19 wave dynamics versus vaccination strategy
Giacomo Cacciapaglia ∗ and Corentin Cot † Institut de Physique des deux Infinis de Lyon (IP2I), UMR5822, CNRS/IN2P3, F-69622, Villeurbanne, France andUniversity of Lyon, Universit´e Claude Bernard Lyon 1, F-69001, Lyon, France
Anna Sigridur Islind ‡ and Mar´ıa ´Oskarsd´ottir § Department of Computer Science, Reykjav´ık University, Menntavegur 1, 102 Reykjav´ık, Iceland
Francesco Sannino ¶ CP3-Origins & the Danish Institute for Advanced Study. University of Southern Denmark. Campusvej 55,DK-5230 Odense, Denmark;Dipartimento di Fisica E. Pancini, Universit`a di Napoli Federico II — INFN sezione di NapoliComplesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126 Napoli, Italy.
We employ the epidemic Renormalization Group (eRG) framework to understand, reproduceand predict the COVID-19 pandemic diffusion across the US. The human mobility across differentgeographical US divisions is modelled via open source flight data alongside the impact of socialdistancing for each such division. We analyse the impact of the vaccination strategy on the currentpandemic wave dynamics in the US. We observe that the ongoing vaccination campaign will notimpact the current pandemic wave and therefore strict social distancing measures must still beenacted. To curb the current and the next waves our results indisputably show that vaccinationsalone are not enough and strict social distancing measures are required until sufficient immunity isachieved. Our results are essential for a successful vaccination strategy in the US.
The United States (US), raged by the SARS-CoV-2virus, are paying an immense toll in terms of the lossof human lives and jobs, with a dreadful impact on soci-ety and economy. Understanding and predicting the timeevolution of the pandemic plays a key role in defining pre-vention and control strategies. Short-term forecasts havebeen obtained, since the early days, via effective methods[1–3]. Furthermore, time-honored mathematical modelscan be used, like compartmental models [4–8] of the SIRtype [9] or complex networks [10–12]. Nevertheless, itremains very hard to understand and forecast the wavepattern of pandemics like COVID-19 [13].In this work, we employ the epidemic RenormalizationGroup (eRG) framework, recently developed in [14, 15].It can be mapped [15, 16] into a time-dependent com-partmental model of the SIR type [9]. The eRG frame-work provides a single first order differential equation,apt to describing the time-evolution of the cumulativenumber of infected cases in an isolated region [14]. Ithas been extended in [15] to include interactions amongmultiple regions of the world. The main advantage overSIR models is its simplicity, and the fact that it relies onsymmetries of the system instead of a detailed descrip-tion. As a result, no computer simulation is needed inorder to understand the time-evolution of the epidemiceven at large scales [15]. Recently, the framework has ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ¶ [email protected] been extended to include the multi-wave pattern [17, 18]observed in the COVID-19 and other pandemics [19].The Renormalization Group approach [20, 21] has along history in physics with impact from particle to con-densed matter physics and beyond. Its application toepidemic dynamics is complementary to other approaches[10–12, 22–29]. Here we demonstrate that the frameworkis able to reproduce and predict the pandemic diffusionin the US taking into account the human mobility acrossdifferent geographical US divisions, as well as the impactof social distancing within each one. To gain an insightand to better monitor the human exchange we make useof open source flight data among the states. We calibratethe model on the first wave pandemic, raging from Marchto August, 2020. With these insights, we then analyseand understand the current second wave, raging in allthe divisions. The eRG framework can also be easilyadapted to take into account vaccinations [23]. We pro-pose a new framework and use it to quantify the impactof the vaccination campaign, started on December 14th,on the current and future wave dynamics. Our results arein agreement with previous work based on compartmen-tal models [30], and confirm that the current campaignwill have limited impact on the ongoing wave. I. METHODOLOGY
In this section we briefly review our methods that in-clude the open source flight data description, their inter-play with the eRG mathematical model framework and,last but not least, the interplay with vaccine deploymentand implementation. a r X i v : . [ phy s i c s . s o c - ph ] D ec A. Data description
The flight data comes from the OpenSky Network,which is a non-profit association that provides open ac-cess to real-world air traffic control dataset for researchpurposes [31]. The OpenSky COVID-19 Flight Dataset was made available in April 2020 and is currently up-dated on a monthly basis, with the purpose of supportingresearch on the spread of the pandemic and the associ-ated economic impact. This dataset has been used toinvestigate mobility in the early months of the pandemic[32] as well as the pandemic’s effect on economic indica-tors [33].The data provides information about the origin anddestination airports as well as the date and time of allflights worldwide. For our analyses we considered do-mestic flights in the US only. We aggregated the data, toobtain the number of flights between all pairs of airportsper day, from the beginning of April until the end of Oc-tober, 2020. Subsequently, the airports in each state andthe number of flights associated with them were com-bined, to give the number of within and between stateflights, on a day to day basis for the whole period.The number of daily infected cases, which is also usedfor analysis in this paper, is provided by the open sourceonline repository Ourworldindata [34] . Divisions included in our simulations
Geographical Divisions
NEMA
WNC: West North Central M: Mountain P: Pacific ESC: East South Central WSC: West South Central ENC: East North Central NE: New England MA: Mid-Atlantic SA: South Atlantic
SAWSC ESCWNC ENCMP
FIG. 1. Illustration of the geographical divisions of the USused in this study.
B. Mathematical modeling
The states within the US have different population anddemographic distribution. A state-by-state mathemati- opensky-network.org ourworldindata.org/coronavirus cal modeling, therefore, is challenged by statistical arti-facts. For these reasons we group the states followingthe census divisions , as summarized in Table I and il-lustrated in Fig.1. Note, that contrary to the officialdefinitions, we include Maryland and Delaware in Mid-Atlantic instead of South Atlantic. The main reason isthat the population of these two states is more connectedto states in Mid-Atlantic, as proven by the diffusion tim-ing of the virus.Building upon our successful understanding of theCOVID-19 temporal evolution [35] we apply our frame-work to the US case. Building on that framework weemploy the following eRG set of first order differentialequations [15] to describe the time-evolution of the cumu-lative number of infected cases within the US divisions: dα i dt = γ i α i (cid:18) − α i a i (cid:19) + (cid:88) j (cid:54) = i k ij n mi ( e α j − α i − , (1)where α i ( t ) = ln I i (t) , (2)with I i ( t ) being the cumulative number of infected cases per million inhabitants for the division i and ln indicatingits natural logarithm. These equations embody, withina small number of parameters, the pandemic spreadingdynamics across coupled regions of the world via the tem-poral evolution of α i ( t ). The parameters γ i and a i canbe extracted by the data within each single wave. Thefit methodology is described in [14, 15].In the US, it is well known that the COVID-19 pan-demic started in NE and MA (mainly in New York City)and then spread to the other divisions. Thus, we definethe US first wave period from March to the end of Augustas shown in Fig. 2. In particular, one observes a peak ofnew infected in NE and MA around April, while for theother divisions the main peak occurs around July. Wealso observe an initial feature in the latter divisions thatwe did not attempt to model except for ENC (mostly lo-cated in Chicago) and WNC. For the two latter divisions,we considered these as two independent first wave com-ponents. The
US second wave is thus associated with theepisode starting in October, 2020.As a first method, working under the assumption thatthe US pandemic indeed originated in New York (MA),we first determine the k ij matrix entries between the di-vision MA and the others. The values are chosen to rea-sonably reproduce the delay between the main peaks ofthe first wave in pairs of divisions (C.f. the top sectionin Table II). Interestingly, with the exception of NE, theentries of the k matrix are comparable to the ones weused for Europe [35]. For NE, a large coupling is neededdue to the tight connections between the two regions, inparticular New York City with the neighbouring statesand Massachusetts. US Census Bureau
Division compositionDivision names Division code States within the divisionNew England NE Massachusetts, Connecticut, New Hampshire, Maine, Rhode Island and VermontMid-Atlantic MA New York, Pennsylvania, New Jersey, Maryland and DelawareSouth Atlantic SA Florida, Georgia, North Carolina, Virginia, South Carolina and West VirginiaEast South Central ESC Tennessee, Alabama, Kentucky and MississippiWest South Central WSC Texas, Louisiana, Oklahoma and ArkansasEast North Central ENC Illinois, Ohio, Michigan, Indiana and WisconsinWest North Central WNC Missouri, Minnesota, Iowa, Kansas, Nebraska, South Dakota and North DakotaMountains M Arizona, Colorado, Utah, Nevada, New Mexico, Idaho, Montana and WyomingPacific P California, Washington, Oregon, Hawaii and AlaskaTABLE I. States of the US integrated into 9 divisions. Maryland and Delaware are moved from South Atlantic to Mid-Atlantic.
Mar Apr May Jun Jul Aug Sep Oct Nov Dec
NEMASA ( weeks ) N e w i n f e c t ed / m illi on Mar Apr May Jun Jul Aug Sep Oct Nov Dec
ESCWSCENC ( weeks ) N e w i n f e c t ed / m illi on Mar Apr May Jun Jul Aug Sep Oct Nov Dec
WNCMP ( weeks ) N e w i n f e c t ed / m illi on FIG. 2. Weekly new number of cases for all the 9 divisions.
As a second method, we used the flight data to esti-mate the number of travellers between different divisions,under the assumption that the k ij matrix entries are pro-portional to this set of data. To have a realistic matrixfor k ij , we first take the mean number of flights fromdivision i to division j during the period from April 1stto May 31st for the first wave, and from September 1stto October 31st for the second wave. Then, we multiplythe number of flights by an effective average number ofpassengers, and normalize it by 10 , following the defini-tion of k ij [15]. For the first wave, the optimal averagenumber of passenger is found to be 10, while for the sec-ond wave we find an optimal value of 5. Note that thesevalues do not correspond to the actual number of pas-sengers in the flights: in fact, the values of the couplings k ij also take into account the probability of the passen-gers to carry the infection as compared to the averagein the division of origin. When the value is low it mightsuggest that the sample of passengers in a flight is lessinfectious than average, as people with symptoms tendnot to travel. Controls at airports may also contribute tothis. The key information we extract from the flight datais the relative flux of infections among different divisions.The results are listed in the middle and bottom sec-tions of Table II. We keep the same value from the pre-vious fit only for MA-NE. The reason behind this choiceis the tight connection between the two divisions, wheremost of the human mobility is imputable to road trans- port.By the end of November, we clearly observe a newrise in the number of infections, signalling the onset of asecond wave pandemic in the US (see Fig. 2). Using ourframework, we model and then simulate the second waveacross the different US divisions.Finally, to check the geographical diffusion of the virusduring the various phases of the pandemic in the US,we define an indicator of the uniformity of the new caseincidence [18]. This indicator can be defined week byweek via a χ -like variable, given by: χ ( t ) = 19 (cid:88) i =1 (cid:18) I (cid:48) i ( t ) (cid:104)I (cid:48) ( t ) (cid:105) − (cid:19) , (3)where I (cid:48) i ( t ) is the number of new cases per week in divi-sion i at time t and (cid:104)I (cid:48) ( t ) (cid:105) the mean of the same quantityin the 9 divisions. The parameter χ quantifies the geo-graphical diffusion of the SARS-CoV-2 virus in the US:the smaller its value, the more uniform the pandemicspread within the whole country. The result is shown inFig. 3: during the first peak in April (light gray shade),the value of χ is large, signalling that the epidemic diffu-sion is localized in a few divisions; during the second peakof the first wave (gray shade), the value has dropped, sig-nalling that the epidemic has been spreading to all divi-sions. Finally, the data for the ongoing second wave (darkgray shade) shows that χ is dropping towards zero, as k ij values (1st wave fits)Division code NE MA SA ESC WSC ENC WNC M PMA 0 .
72 0 0 . . . . . .
002 0 . k ij values (Flight data, from April 1st to May 31st)Division code NE MA SA ESC WSC ENC WNC M PNE 0 0 .
72 0 . . . . . . . .
72 0 0 .
019 0 . . .
012 0 . . . . .
018 0 0 . .
013 0 .
019 0 . . . . . . . . . . . . . .
014 0 . . . .
011 0 . . .
012 0 .
018 0 . . . . . . . . . . . . . . . . . .
011 0 . . . . . . . .
010 0 . . .
030 0Second wave k ij values (Flight data, from September 1st to October 31st)Division code NE MA SA ESC WSC ENC WNC M PRegion-X 0 . .
028 0 .
029 0 .
013 0 .
019 0 .
027 0 .
014 0 .
03 0 . .
72 0 . . . . . . . .
72 0 . .
011 0 .
002 0 . . . . . . .
011 0 0 .
005 0 .
005 0 . .
003 0 . . . . . . . . . . . . . . . . .
004 0 . . . . . .
003 0 0 . . . . . . . . . . . . . . . . .
004 0 . . . . . . . . . .
015 0TABLE II. Values of the k ij entries among US divisions. In the top section, the values between Mid-Atlantic (MA) and theother divisions are obtained from fits of the first wave timing. In the central and bottom sections, the complete matrix (exceptthe entries between MA and NE) is obtained using flight data for the first wave (from April 1st to May 31st) and the second(from September 1st to October 31st), respectively. expected for a more diffuse incidence of infections. Mar Apr May Jun Jul Aug Sep Oct Nov Dec ( weeks ) χ FIG. 3. Evolution of the uniformity indicator χ over time(weekly basis). The shaded bands indicate the period whenepidemic peaks are recorded. C. Vaccine deployment and implementation
Various vaccines have been developed for the COVID-19 pandemic, and their deployment in the US has alreadystarted on December 14th . The effect of the immuniza-tion due to the vaccine has been studied in the contextof compartmental models, like SEIR [30]. In our mathe-matical model, the simplest and most intuitive effect is areduction of both the total number of infections during asingle wave, a i , and/or the effective diffusion rate of thevirus γ i , in each division.To validate this working hypothesis, and understandhow the vaccination of a portion of the population affectsthe values of a and γ in the eRG framework, we studiedthe effect of immunization in a simple percolation model,which has been shown to be in the same class of univer-sality as simple compartmental models [36]. To do so, weset up a Monte-Carlo simulation, consisting of a square grid whose nodes are associated to a susceptible individ-ual. Each node can be in four exclusive states: Suscep-tible (S), Infected (I), Recovered (R) or Vaccinated (V).At each step in time in the simulation, for each node wegenerate a random number r between 0 and 1: if the nodeis in state S and r < γ ∗ , we switch its state to I, else it re-mains S; if the node is in state I and r < (cid:15) ∗ , we switch itsstate to R, else it remains I; if the node is in state R or V,it will not change. This model reproduces the diffusionof the infection, where γ ∗ is the infection probability onthe lattice and (cid:15) ∗ is the recovery rate. Finally, we fit thedata from the simulation to the solution of a simple eRGequation to extract γ and a . The vaccination is imple-mented by setting a random fraction R v of nodes to thestate V before the simulation starts. The values of a and γ as a function of the fraction of vaccinations are shownin Fig.4: we observe that both parameters are reduced bythe same percentage as the vaccination up to R v (cid:46) a and γ proportionally reinforcing ourexpectation. % of vaccinated a % of vaccinated ga mm a FIG. 4. a and γ fit parameters versus initial percentage ofvaccinated nodes for γ ∗ = 0 . (cid:15) ∗ = 0 . In a realistic scenario, the vaccination of the popula-tion can only be implemented in a gradual way, so thatthe total vaccination campaign has a duration in time.We can thereby assume that a fraction R v of the pop-ulation is vaccinated in a time interval ∆ t . The rate ofvaccinations is therefore c = R v / ∆ t . This implies thatthe variation in γ , during the time interval from t v to t v + ∆ t , is given by: dγ ( t ) dt = − c γ ( t v ) , (4)where γ ( t v ) is the effective infection rate before the startof the vaccination campaign. The solution for the time-dependent effective infection rate is γ ( t ) = γ ( t v )[1 − c ( t − t v )] , (5)until t = t v + ∆ t , after which γ remains constant againat a reduced value γ ( t v ) (1 − R v ).To find the variation of a ( t ) within the vaccinationinterval t v to t v +∆ t , we assume that the not-yet-infectedindividuals are vaccinated at the same rate c as the totalpopulation. Thus, at any given time, the variation inthe number of individuals that will be exposed to theinfection, e a ( t ) , is proportional to the difference e a ( t ) − e α ( t ) . This leads to the following differential equation: da ( t ) dt = − c (1 − e α ( t ) − a ( t ) ) . (6)This equation needs to be solved in a coupled system withthe eRG one. Note that the derivative is zero outside ofthe time interval [ t v , t v + ∆ t ]. In the numerical solutionsfor the effect of the vaccine, we will add one equation foreach a i ( t ), assuming that the vaccination rate c is thesame in all divisions. II. RESULTSA. Validating the eRG on the first wave data
The epidemic data (C.f. Fig.2) shows that the MA di-vision (New York City) was first hit hard by the COVID-19 pandemic, and was followed closely by NE. The otherdivisions witnessed a comparable peak of new infections3-4 months later. Note that we are using cases normal-ized per million to facilitate the comparison between divi-sions with varying population. As a first study, we wantto test the eRG equations (1) against the hypothesis thatthe epidemic has been diffusing from MA to the other di-visions. The parameters a i and γ i are fixed by fittingthe data, as shown in Table III. Thus, the timing of thepeaks in the divisions is determined by the entries of the k ij matrix. Determining all 81 entries from the data isnot possible, as we only have 9 epidemiological curves.Thus, we assume that only the couplings between thesource MA and any other division are responsible. Theresults of the fits are shown in the top block of Table II,and will be used as a control benchmark.Except for k that links NE and MA, all the other k j are of order 10 − , thus confirming the range we foundfor the European second wave [35]. The value of k is of order unity, which implies that there is a stronger First wave parameters (fitted) Second wave parametersDivision Code a γ a γ
New England NE 9 . . .
189 0 . . . .
287 0 . . . .
195 0 . . . .
018 0 . . . .
713 0 . . . .
128 0 . . . .
060 0 . . . .
271 0 . . . .
944 0 . σ error. For the second wave, the values are chosen to reproduce the current data, adjournedto December 16th. connection between the two divisions. This may be ex-plained by the fact that there exist a significant flowof people between New York City and the neighbouringstates (including Massachusetts) in New England. Workcommutes and weekend travelling by car explains the re-quired high number of travellers per week. Another inter-esting feature is the presence of a small peak of infectionsfor ENC and WNC, around March. This feature cannotoriginate from the MA division, as that would imply a k -value of order 10, which is clearly unrealistic [15]. Theonly viable solution is that the epidemic hit these twodivisions from abroad. On the other hand, the secondpeak observed around August can be explained by theinteraction with MA.The values of k ij are, in principle, determined by theflow of people between different divisions. Thus, we coulduse any set of mobility data [37] to estimate the rela-tive numbers of the entries, while the normalization alsodepends on the effective infection power of the travel-ing individuals and it can be determined from the data.With the help of mobility data, we can reduce the 81parameters to a single one. Due to the large distancesacross divisions, we decided to focus on the flight data,as described in the methodology section. The values ofthe entries are reported in the middle section of Table II.Note that for MA-NE we used the same value obtainedfrom the previous fit, as the people’s flow is mostly dom-inated by land movements.Using this matrix of k ij to simulate the spread of thefirst wave across the country, as originating from MA, weobtain the curves in the left panels of Fig.5. For nearlyall divisions, we obtain the correct timing for the peak,with the exception of SA and ESC (for ENC and WNC,the anomaly may be linked to the presence of a mild earlypeak and the absence of a prominent second peak). Thisresult validates the method, as the diffusion of the virusseems to depend on the people travelling (by air) amongdivisions. For SA, the predicted curve is substantiallyanticipated compared to the data: this discrepancy maybe explained by the presence of an air hub in Atlanta, GE, so that many of the passengers of flights landing there donot stop in the division but instead take an immediateconnecting flight. B. Understanding the second wave
The US states are currently witnessing a second wave,which is ravaging in all the 9 divisions with comparableintensity. Previous studies in the eRG framework haveuncovered two possible origins for an epidemic wave tostart: one is the coupling with an external region witha raging epidemic [15], the second is the instability rep-resented by a strolling phase in between waves [17, 18].We have shown that the former mechanism can accountfor the peak structure during the first wave.As a first step, we will try to use the same method tounderstand the second wave. Since travelling to the USfrom abroad has been strongly reduced and regulated, wewill consider the divisions that witnessed a peak in July-August as source for the second wave. To this purpose,we define a Region-X [15] as an average sum of all thedivisions with a pandemic peak occurring in the July-August period. The parameters are chosen to reproducethe number of cases in the totality of the relevant 7 divi-sions (SA, ESC, NSC, ENC, WNC, M and P) normalizedby the total population. For each division, we optimized a i and γ i to reproduce the current data adjourned atDecember 16th (C.f. Table III). For the couplings k ij we use the flight data, except for the usual MA-NE cou-plings (C.f. bottom section of Table II). Finally, the k j connecting the 9 divisions to the source Region-X areoptimized to reproduce the correct timing of the secondwave. One might naively expect that their values wouldcorrespond to an average of their values of the first wave.The results of the eRG equations are shown in theright panels of Fig. 5, showing a good agreement. Incontrast to our naive expectation the values of the k j of the Region-X are one order of magnitude larger thanthe expected average of the first wave. This fact can First wave Second wave
Mar Apr May Jun Jul Aug Sep Oct
NEMASA ( weeks ) N e w i n f e c t e d / m illi o n Aug Sep Oct Nov Dec Jan Feb Mar Apr May ( weeks ) N e w i n f e c t e d / m illi o n Mar Apr May Jun Jul Aug Sep Oct
ESCWSCENC ( weeks ) N e w i n f e c t e d / m illi o n Aug Sep Oct Nov Dec Jan Feb Mar Apr May ( weeks ) N e w i n f e c t e d / m illi o n Mar Apr May Jun Jul Aug Sep Oct
WNCMP ( weeks ) N e w i n f e c t e d / m illi o n Aug Sep Oct Nov Dec Jan Feb Mar Apr May ( weeks ) N e w i n f e c t e d / m illi o n FIG. 5. Simulation of the spread of the first wave (left plots) and the second wave (right plots) using flight-data-derived kappamatrix. For the first wave, MA is used as a seed region, while for the second wave a combination of the first waves amongdivisions acts as the seed region (Region-X). be interpreted by the presence of hotspots in each divi-sion which also contribute significantly to the new wave.In other words, traveling among divisions cannot be theonly responsible factor for the onset of the second wave inthe US. This hypothesis can also be validated by studyingthe uniformity of the distribution of the new infections invarious states during the three peaks, as shown in Fig.3.Comparing the three peak regions, we see that the uni-formity indicator is systematically decreasing, thus in-dicating a more geographically uniform presence of thevirus.It is also interesting to notice that the value of γ i for the second wave is systematically smaller than the infectionrate during the first wave. This is in agreement withthe results we found in [17, 18], where we modelled themulti-wave structure of the pandemic via an instabilityinside each region. The result of this simple analysissupports the hypothesis that the virus is now endemicfor all states in the US, thus a multi-wave pattern willcontinue to emerge. Traveling among states (or divisions)is less relevant at this stage.The result of our eRG analysis shows that the cur-rent wave will end in March-April 2021. Note, however,that we have not taken into account the potential dis-astrous effect of the Christmas and New Year holidays,which could lead to an increase in the infection rates. Insome divisions there is a increase at the end of November,which can be attributed to the Thanksgiving holiday, sowe might see a similar effect due to the end-of-year trav-elling. C. Effect of the current vaccination strategy
Following the development of multiple vaccines forthe SARS-COV-2 virus , vaccination campaigns havestarted in many countries, including the US. This willinfluence the development of the current wave, and helpin curbing the future ones. The vaccination campaignstarted on December 14th in the US . We also knowthat the US has purchased 100 million doses from Pfizer(plus an additional 100 million from Moderna) , so thatat least 20% of the population may be vaccinated in thisfirst campaign. It is not known how many vaccinationsper week are being administered: thus, we will take somehints from the UK, for which data is available showing arate of 0 .
2% over the first week . The data listed abovedefines our starting benchmark for the current vaccina-tion campaign.To study the effect of the vaccinations, we have solvedthe eRG equations for the second wave, with the additionof the reduction of a i and γ i , as detailed in the methodol-ogy section. We show the result for two sample divisionsin Fig.6 (dashed curves) as compared to the same solu-tions without vaccines (solid curves). A vaccination at a0 .
2% rate per week does not affect the peak of new infec-tions. As a reference, we also increased the vaccinationrates to 1% and 2%: in these cases, an important flatten-ing of the epidemic curve can be observed for SA, wherethe vaccination started early compared to the peak of in-fections. This situation may be realized, as the vaccineis being administered to the population that is more atrisk of being infected by the virus. In the other extremecase, represented by WNC, the vaccine is ineffective inchanging the current wave because the peak has alreadybeen attained before the vaccination campaign started.Our results confirm that the current vaccination strat-egy, which is performed during a peak episode, is not ef-fective to substantially slow down the spread of the virus.On the other hand, the effectiveness for future waves isnot a question. It would be, in fact, very efficient to beable to administer the vaccine to a larger portion of thepopulation before the start of the next wave. https://ourworldindata.org III. DISCUSSION
In this paper we employ the epidemic RenormalizationGroup (eRG) framework in order to understand, repro-duce and predict the diffusion of the COVID-19 pandemicacross the US as well as the effect of vaccination strate-gies. By using flight data, we are able to see the changesin mobility across the divisions, and observe how thesechanges affect the spread of the virus. Furthermore, weshow that the impact of the vaccination campaign onthe current wave of the pandemic in the US is marginal.Based on that, the importance of social distancing is stillrelevant, especially now during the end-of-year holidays.Furthermore, we demonstrate that the current wave isdue to the endemic diffusion of the virus. Therefore,building upon our previous results [18], in order to con-trol the next pandemic wave the number of daily newcases per million must be around or less than 10-20 dur-ing the next inter-wave period. This conclusion is furthercorroborated in [38] for Europe.We learnt that the number of infected individuals inthe current wave will not be affected measurably by thevaccination campaign. However, it is foreseeable thatit will impact specific compartments such as the over-all number of deceased individuals. Our study includedan immunization rate between 0.2% to 2% of the totalpopulation each week. To curb the current and the nextwaves, our results indisputably show that vaccinationsalone are not enough and strict social distancing mea-sures are required until sufficient immunity is achieved.Our results should be employed by governments anddecision makers to implement local and global measuresand, most importantly, the results of this paper can beused as a foundation for vaccination campaign strategiesfor governments.Given that pandemics are recurrent events, our re-sults go beyond COVID-19 and are universally applica-ble. What we have seen in the data for the US is that itstarted in New York and, from there, it diffused to therest of the country. It is, therefore, important to containfuture pandemics at an early stage.
South Atlantic West North Central
Aug Sep Oct Nov Dec Jan Feb Mar Apr May c = = = = ( weeks ) N e w i n f e c t e d / m illi o n Aug Sep Oct Nov Dec Jan Feb Mar Apr May c = = = = ( weeks ) N e w i n f e c t e d / m illi o n FIG. 6. Evolution of the number of infections without vaccination ( c = 0) and with a vaccination rate of 0 . , 127 (2020).[2] M.-G. Hˆancean, M. Perc, and L. Juergen, R. Soc. opensci. , 200780 (2020).[3] T. Zhou, Q. Liu, Z. Yang, J. Liao, K. Yang, W. Bai, andW. Zhang, J Evid. Based Med. , 3 (2020).[4] K. Prem, Y. Liu, T. W. Russel, A. J. Kucharski, R. Eggo,and N. Davies, The Lancet Public Health
5, issue 5 ,E261 (2020).[5] A. Scala, A. Flori, A. Spelta, E. Brugnoli, M. Cinelli,W. Quattrociocchi, and F. Pammolli, Sci Rep , 13764(2020).[6] K. J. Friston, T. Parr, P. Zeidman, A. Razi, G. Flandin,J. Daunizeau, O. J. Hulme, A. J. Billig, V. Litvak, C. J.Price, R. J. Moran, and C. Lambert, “Second waves, so-cial distancing, and the spread of covid-19 across amer-ica,” (2020), arXiv:2004.13017 [q-bio.PE].[7] G. Sonnino, F. Mora, and P. Nardone, “A stochas-tic compartmental model for covid-19,” (2020),arXiv:2012.01869 [physics.med-ph].[8] A. Abou-Ismail, SN Compr Clin Med. , 1 (2020).[9] W. O. Kermack, A. McKendrick, and G. T. Walker,Proceedings of the Royal Society A , 700 (1927).[10] X.-X. Zhan, C. Liu, G. Zhou, Z.-K. Zhang, G.-Q. Sun,J. J. Zhu, and Z. Jin, Applied Mathematics and Com-putation , 437 (2018).[11] M. Perc, J. J. Jordan, D. G. Rand, Z. Wang, S. Boc-caletti, and A. Szolnoki, Physics Reports , 1 (2017).[12] Z. Wang, M. A. Andrews, Z.-X. Wu, L. Wang, and C. T.Bauch, Physics of Life Reviews , 1 (2015).[13] M. Scudellari, Nature , 22 (2020).[14] M. Della Morte, D. Orlando, and F. Sannino, Front. inPhys. , 144 (2020).[15] G. Cacciapaglia and F. Sannino, Sci Rep , 15828(2020), arXiv:2005.04956 [physics.soc-ph].[16] M. Della Morte and F. Sannino, “Renormalisation Groupapproach to pandemics as a time-dependent SIR model,”(2020), arXiv:2007.11296 [physics.soc-ph].[17] G. Cacciapaglia and F. Sannino, “Evidence for com-plex fixed points in pandemic data,” (2020),arXiv:2009.08861 [physics.soc-ph]. [18] G. Cacciapaglia, C. Cot, and F. Sannino, “Multiwavepandemic dynamics explained: How to tame the nextwave of infectious diseases,” (2020), arXiv:2011.12846[physics.soc-ph].[19] J. K. Taubenberger and D. M. Morens, Rev Biomed , 69 (2006).[20] K. G. Wilson, Phys. Rev. B , 3174 (1971).[21] K. G. Wilson, Phys. Rev. B , 3184 (1971).[22] L. Li, J. Zhang, C. Liu, H.-T. Zhang, Y. Wang, andZ. Wang, Applied Mathematics and Computation ,566 (2019).[23] Z. Wang, C. T. Bauch, S. Bhattacharyya, A. d’Onofrio,P. Manfredi, M. Perc, N. Perra, M. Salath´e, and D. Zhao,Physics Reports , 1 (2016).[24] J. M. A. Danby, Computing applications to differentialequations modelling in the physical and social sciences (Reston Publishing Company, Reston VA (USA), 1985).[25] F. Brauer, Journal of Biological Dynamics , 23 (2019).[26] J. C. Miller, Bulletin of mathematical biology , 2125(2012).[27] J. D. Murray, Mathematical biology , 3rd ed., Interdisci-plinary applied mathematics (Springer, New York (USA),2002).[28] D. Fishman, E. Khoo, and A. Tuite, PLOS CurrentsOutbreaks (2014).[29] B. Pell, Y. Kuang, C. Viboud, and G. Chowell, Epi-demics , 62 (2018), the RAPIDD Ebola ForecastingChallenge.[30] A. D. Paltiel, J. L. Schwartz, A. Zheng, and R. P. Walen-sky, Health Affairs (2020).[31] M. Sch¨afer, M. Strohmeier, V. Lenders, I. Martinovic,and M. Wilhelm, in IPSN-14 Proceedings of the 13thInternational Symposium on Information Processing inSensor Networks policy-report-financial-stability-report-may-2020 (2020).[34] E. O.-O. Max Roser, Hannah Ritchie andJ. Hasell, Our World in Data (2020),https://ourworldindata.org/coronavirus.[35] G. Cacciapaglia, C. Cot, and F. Sannino, Sci Rep ,15514 (2020), arXiv:2007.13100 [physics.soc-ph].[36] J. L. Cardy and P. Grassberger, Journal of Physics A:Mathematical and General , L267 (1985).[37] C. Yang, D. Sha, Q. Liu, Y. Li, H. Lan, W. W. Guan,T. Hu, Z. Li, Z. Zhang, J. H. Thompson, and et al., Inter-national Journal of Digital Earth , 1186–1211 (2020).[38] V. Priesemann, M. M. Brinkmann, S. Ciesek,S. Cuschieri, T. Czypionka, G. Giordano, D. Gur-dasani, C. Hanson, N. Hens, E. Iftekhar, et al. , TheLancet (2020). AUTHOR CONTRIBUTION
This work has been designed and performed conjointlyand equally by the authors, who have equally contributedto the writing of the article. ASI and MO have extractedand processed the data from flights; CC has worked onthe numerical results from the eRG equations and anal-ysed the epidemiological data.