{"paper_id":"f7257d9e-f2c3-458d-82bc-817e4d6d0ee3","body_text":"1 \n \nCOVID-19 PANDEMIC ANALYSIS \nVictor A.J. van Lint, Ph.D., retired \nAbstract \nThe development of coronavirus disease (COVID-19) deaths in selected nations and states is compared to \nthe result of calculations using a conventional SEIR model of pandemic development.  The model is \nbased on the infection multiplier, R0, defined as the number of people infected by each infectious person.  \nThe infection rate increases exponentially when R0 >1.0; it remains constant at R0 = 1.0 and decreases for \nR0 < 1.0.  R0 is determined by population behavior (frequency and proximity of interactions) and the ease \nby which a victim is infected by an infectious person (virulence of the virus).  It is reduced by herd \nimmunity when a large fraction of the population acquires immunity by vaccination or by recovering \nfrom infection. \nThe daily death rates in the U.S. and northern Europe exhibited peaks in April/May 2020 and Dec. \n2020/Jan. 2021 with more a modest rate during the summer of 2020 and a gradually decreasing rate since \nJan. 2021.  The model produces this type of oscillatory response if it assumes that the population’s R0 \nresponds to information reported about the pandemic, but with a delay between infections and resulting \nbehavioral adjustments.  Oscillatory behavior is typical of a control loop with delay in its feedback. \nThe analysis concludes that: \nGiven the history of R0 the model predicts the development of pandemic deaths.  However, since \nR0 is determined by the population’s behavior, control of the pandemic in democracies depends \nprimarily on preparation and the persuasive power of political and scientific authorities.  Data for \nS. Korea and New Zealand demonstrate the effectiveness of such methods. \nFor each death in the U.S. about 169 persons were infected, but fewer than half of them were \nidentified as cases. \nThe pandemic was prolonged in the U.S. because the population chose to keep R0 near 1.0 by \nrelaxing restrictions once the death rate subsided.  \nInitial values of R0 as high as 5.0 were observed, leading to infections doubling about every 2 \ndays.  If unabated, the resulting exponential growth increases the infected population by a factor \nof about 5000 before the death from the first infections is recorded. \nArrival from Italy probably initiated the pandemic in the eastern U.S., but, by the time the first \ndeath was recorded the number of domestic infections exceeded by far those that were imported.  \nImport restrictions beyond this point are ineffective except in delaying the arrival of more virulent \nmutations.  \nIf no social restrictions had been adopted, approximately 1.6 million deaths would have resulted \nin the U.S.  The vaccine, although developed and deployed at record speed, was too late to \nameliorate this result. \nA third peak in death rate in Sept. 2021 may be prevented if more than 80% of the population is \nvaccinated. \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \nNOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice.\n\n2 \n \nIntroduction \nThe objective of this analysis is to produce a model to guide the response to future pandemic threats.  \nThe coronavirus disease (COVID-19) pandemic stressed the United States and the world in general more \nseverely than any event since World War II. As of May 31, 2021, more than 560,000 people died in the \nU.S. (1700 deaths per million inhabitants, dpM), and the peak death rates in some states had exceeded 10 \ndpM/day. This report analyzes data on accumulated deaths for various geographic regions since March 1, \n2020, and compares the data to the calculations with the SEIR (Susceptible, Exposed, Infectious, \nRecovered) mathematical model presented in Appendix A.  \nIn most western nations and U.S. states the pandemic outpaced initial attempts at control, progressed \nrapidly until about 0.1% of the population had died, and then continued to spread at a lower rate. This \nbehavior is consistent with a model in which the public adjusts its average interaction rate in response to \npandemic information, such as data on testing, hospitalizations and deaths.  \nWe believe the most reliable data are the cumulative deaths attributed to the virus; hence, we define: \nd = d(t) = Cumulative total deaths at time t divided by the total population: 0 ≤ d ≤ 1. \nIt is assumed that the effectiveness of treatment does not change during the pandemic, such that d(t) is \nproportional to the fraction of the population that were infected at an earlier time, most of whom have \nrecovered. Because deaths must be reported, the primary uncertainty lies in the identified cause of death. \nDespite the numerous politically motivated attacks on the Internet, we are reasonably sure that these data \nare reliable. We have not relied on data for the reported number of “cases” because they depend on the \nintensity of testing, which has clearly changed during the pandemic. The disadvantage of using deaths is \nthat they lag the infection events by about one month.  \nOur approach selects and analyzes published data on accumulated COVID-19 deaths during the period \nfrom March 1, 2020, through May 31, 2021, for representative nations and states.  Fortunately, Wikipedia \nhas maintained the appropriate data logs, presumably recording data reported by official government \nsources, such as the U.S. Center for Disease Control.  The data are presented as deaths per million (dpM) \npopulation to provide a meaningful comparison of severity between areas of different populations.  \nResults \nA range of histories of the accumulation of COVID-19 deaths are shown in Figure 1.*  \n The data all start with exponential growth.  The growth rates (i.e., slopes on the semilogarithmic plots) \ngradually decrease, presumably as the population adjusts its behavior.  The data appear to fall into three \nclasses:  \n \n* All such historical data were taken from Wikipedia summaries.  They can be accessed on the Internet by \na search for “COVID-19, Wiki,” with the name of state or nation.  \n \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n3 \n \n \nFigure 1: COVID-19 dpM: a range of histories \nThose that appear to have stopped the pandemic at an early stage and maintained control; \nexamples include South Korea, New Zealand, Alaska and Hawaii, although Hawaii experienced a \nreemergence during the summer and early fall of 2020.     \nThose that are continuing to experience an increase at a modest rate; California is an example. \n \nThose that appeared to level off in the summer of 2020 after an early dramatic increase; examples \ninclude New York, Italy, Sweden, Michigan, and the United Kingdom.  Most of them resumed \ndramatic growth in the fall. \nThe U.S. is a composite of states at different stages of development. \nIt might appear from Fig. 1 that the pandemic development after the summer 2020 was mild compared to \nits March/April growths, but that is an illusion produced by the semilogarithmic plot.  Figure 2 presents \nthe daily deaths for the same nations/states as compared to the daily deaths during the 2017-18 annual \ninfluenza. It is clear that there were two waves with comparable intensities that are much more severe \nthan the annual flu.  Since daily data are inherently noisy and are distorted by reporting delays on the \nweekends, all daily data were calculated as averages over 7 days. \n \n0.1\n1.0\n10.0\n100.0\n1,000.0\n10,000.0\n2 4 6 8 10 12 14 16 18\nDeaths per Milion\nMonths after Jan. 1, 2020 \nCumulative Deaths per Million Inhabitants\nUS\nCalifornia\nHawaii\nNew York\nBrazil\nItaly\nNew Zealand\nS. Korea\nSweden\nUK\n2020 2021\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n4 \n \n \nFigure 2. Daily deaths per million for same states/nations as Figure 1  (7-day averages) \nIn many areas (e.g., California, Brazil) the peak death rates in January 2021 exceeded those encountered \nduring the earlier peak.   \nInception \nThe death rates soon after the onset of the pandemic in an area provides a measure of the threat.  The \nevolution of the death rates in the earliest onsets is illustrated in Figure 3.  \nThe time sequence of onsets reflects the transmission of the infection from its initial injection into Europe \nin Italy to Spain, and then to UK and Sweden, and in parallel to New York, Michigan and Illinois.  Even \nthough S. Korea and New Zealand were nearest to the source of the pandemic in China, their readiness \nenabled them to delay and suppress infections.  \nThe slopes of the semilogarithmic plots of death rates in Figure 3 measure the exponentiation times (1.44 \ntimes the doubling time) of the exponential growths, which depend on R0, the infection multiplier (i.e., the \nnumber of people infected by each infectious person), albeit the value of R0 at the time they were infected \nrather than the time they died.  These slopes, again averaged over 7 days, are shown in Figure 4.  These \ncurves are noisy because they depend on differences between daily deaths that fluctuate due to reporting \ndelays even when averaged over 7 days. \nThe numbers along the left side are values of R0 deduced from the approximate relationship  \nR0 ≈ exp (1.5/κ), where the exponentiation time is κTdelay and we assume Tdelay = 3 days.  This relationship \nwas calculated in Appendix A and shown in Figure A2.  \n \n0\n5\n10\n15\n20\n25\n30\n35\n40\n2 4 6 8 10 12 14 16 18\nDeaths  per Million per day\nMonths after Jan. 1, 2020\nDaily Deaths per Million\nUS\nCalifornia\nHawaii\nNew York\nBrazil\nItaly\nS. Korea\nSweden\nUK\n2017-18 Annual\nFlu\n2020 2021\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n5 \n \n \nFigure 3. Daily deaths per million after the earlier inceptions (7-day averages)  \n \nFigure 4. Exponentiating times at pandemic inception (7-day averages)  \n0.1\n1.0\n10.0\n100.0\n0 10 20 30 40 50 60 70\nDeaths per Million per day\nDays starting March 1, 2020\nDaily Deaths at Inception of Pandemic\nItaly\nSpain\nUK\nSweden\nNew York\nMichigan\nIllinois\nS. Korea\nNew\nZealand\n-0.2\n-0.1\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0 10 20 30 40 50 60 70\nExponent (1/days)\nDays from March 1, 2020\nExponential Growth Exponent at Start of Pandemic\nItaly\nSpain\nUK\nSweden\nNew York\nMichigan\nIllinois\n1.5\n6.05\nRo\n3.86\n2.4\n1.00\n0.63\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n6 \n \nThe falling curves show how the population in each area reacts to information about the pandemic.  The \nreaction is clear, but there is no evidence that what happened earliest in Italy had any effect on the later \nresponse elsewhere.  Two weeks later, the initial R0 in New York was even larger than the initial value in \nItaly.   \nThe rate of public reaction is measured by the slope of the curves in Figure 4.  In Italy R0 decreased by a \nfactor of 2 in about two weeks; in New York the response was faster, decreasing by a factor of 2 in about \nseven days.  Yet, the subsequent peak death rate, shown in Figure 2, was a factor of 2.7 higher in New \nYork.  The primary cause was the initial value of R0, which was about 2.5 in Italy and 5 in New York.  \nThe exponential growth produced by the shorter initial exponentiation time outweighed the faster \nadjustment.     \nThe Second Wave \nAs shown in Figure 2, a second wave of infections and deaths was encountered around the end of the \nyear.  The exponentiation times for some of the same nations/states during Dec. 2020 and Jan. 2021 are \nshown in Figure 5.  \nThe R0 at the pandemic onset shown in Figure 4 decreased from their initial high values in one to two \nweeks.  The R0 values in the second wave are mostly within 25% of unity and oscillate with periods of \none to two weeks.  These behaviors are characteristic of a control system in which the sensory feedback \nsignal is delayed by a week or so.  The section on Analysis of Community Response in Appendix A \nmodeled such a system by assuming that the community’s average R0 responded to information about \ntests, hospitalizations and deaths provided by government and the media.  That produced oscillations with \na much longer period.  Oscillations in Fig. 5 must be produced by shorter-term stimuli, e.g., effect of \nweekends.  \n \n Figure 5. Reciprocal exponentiation times - Dec. 1, 2020, to Jan. 31, 2021 (7-day averages) \n-0.08\n-0.06\n-0.04\n-0.02\n0.00\n0.02\n0.04\n0.06\n0.08\n-10 0 10 20 30 40 50 60 70\nExponent (1/days)\nDays from Dec. 1, 2020 \nExponentiation Times - Second Wave\nItaly\nNew York\nUK\nSweden\nRo\n1.31\n1.20\n1.09\n1.00\n0.91\n0.84\n0.76\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n7 \n \nIt should be noted that R0 remained above 0.75 during the December/January second wave, even when the \nabsolute death rate was still above 5 dpM/day.  In all the states and nations reviewed for this analysis, \nonly in Virginia in early March 2021 did R0 decrease to 0.5.  Apparently, the community relaxes its social \nrestraints on receiving reports of a decrease in daily positive tests, hospitalizations or deaths, even when \nthe absolute death rate remains at alarming levels. \nStates and Nations Reaching an apparent Limit \nIn June 2020 the worst appeared to be over.  Figure 6 shows the accumulated death toll in nations and \nstates where it appeared to have reached an asymptote at about 700 dpM, and for New York State, where \nit reached about 1250 dpM.  The daily death rate had subsided from a peak of 37 dpM/day in New York \nto less than 2 dpM/day in these areas.   \nThis behavior mimics the expected effect of herd immunity illustrated in Appendix A, Figure A1. The \ncredibility of herd immunity being achieved with deaths of about 0.1% of the population depends on Nrec, \nthe number of people who survive the infection for each person who dies.  Publicized data provided the \ncumulative number of people with positive test† results for the infection, usually listed as “cases.”  \n \nFigure 6. Accumulated dpM for countries and states reaching an apparent asymptote in summer 2020 \n \n† We must distinguish between two types of tests: virus and antivirus.  A positive result on a virus test implies that \nthere is active virus in that the person, i.e., he/she is infected and possibly infectious.  A positive antivirus test \nestablishes that the person has developed antibodies, but the test does not distinguish between active cases and those \nwho have recovered from the infection.  Hopefully, a positive antivirus result means that the person is immune from \nre-infection, at least for some time.  \n0.1\n1\n10\n100\n1000\n10000\n0 2 4 6 8 10 12 14 16 18\nDeaths per Million \nMonths after Jan. 1, 2020\nStates/Nations Apparently Reaching an Asymptote\nIllinois\nItaly\nMichigan\nNew York\nSpain\nSweden\nUK\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n8 \n \nHowever, the number of confirmed “cases” depends on the number of tests administered and the criteria \nused for testing, both of which have clearly varied over the course of the pandemic.  During the summer \nof 2020 the ratio of reported “cases” to deaths varied among states from 37 in New Jersey to 196 in \nAlaska.  It is not credible that the quality of medical care varies that much between states, so the \ndifference between these ratios probably reflects differences in local testing policies and the availability \nof testing resources.  Therefore, these values represent lower limits to Nrec.  \nA better estimate of Nrec can be deduced from seroprevalence surveys, which consist of tests for COVID-\n19 antivirus administered to randomly selected persons in a designated area.  Each test determines \nwhether a person is or has been infected.  A search of the Centers for Disease Control and Prevention \nwebsite for seroprevalence data described three types of surveys: large-scale geographic, community-\nlevel, and special population [1].  Only the community-level surveys use random sampling to choose the \nsubjects, so only they can produce a reliable estimate of f, the total fraction of the population that is \nrecovered and immune in the chosen area.  f can then be compared with the fraction of the population that \ndied, d, for the same area to estimate Nrec.  Because d(t) reflects f at an earlier time (approximately a \nmonth earlier), the comparison is best made after an asymptote has been reached. \nAlthough many serological surveys must have been performed, very few were reported.  Data were \navailable for ten areas tested prior to May 1.  6.9% of the tests proved positive in the New York City area \nsurveyed between March 23 and April 1 [1], although the number of confirmed “cases” at the same time \nwas only 53,803.  Thus, the total number of inhabitants estimated to be immune in New York City on \nApril 1 was 12 times the number of confirmed “cases” reported for the surveyed area at the same time.  \nThis proves that most victims of COVID-19 in New York City recovered without being counted as \n“cases.”   \nTo estimate Nrec, we need to know the number of deaths that resulted from these infections.  However, the \ndeath rate in New York was increasing rapidly on April 1, 2020, and some of those who tested positive \nwould die during the subsequent interval between infection and death, Tdie.  We estimate Tdie to be \napproximately 25 days based on fragmentary data, such as the first death attributed to the Sturgis \nmotorcycle rally [2] held in South Dakota from August 7 to 16.  It is a reasonable number since many \npatients had been admitted to a hospital for weeks before recovering or expiring.  \nThe report for the entire New York City area [3] cites 70,637 confirmed “cases” and 2632 deaths on April \n1 and 12,781 deaths on April 22.  Assuming the measured ratio of 12 between “cases” and infections \napplies to all of New York City, between 2,632 and 12,781 deaths were caused by the 848,000 infections \ndeduced for April 1, yielding a probability of death after infection between 0.31% and 1.5%.  \nA more accurate estimate was provided later by Dr. Anthony Fauci during his congressional testimony on \nSeptember 23, 2020 [4].  He quoted a value of 22% for the fractional immunity in New York City at its \ntemporarily asymptotic death level.  Combining this value with the asymptotic death fraction of 0.13% \n(Figure 6), we deduce that Nrec = 169, for an average probability of death after infection of 0.6%, which \ncompares with a typical value of 0.13% for the seasonal influenza.  Dr. Fauci’s estimate implies that by \nSeptember 23, 1.8 million of the residents of New York City had acquired immunity, while the number of \nconfirmed “cases” at that time was only 0.24 million for a ratio of 7.5.  Clearly, most people who \nacquired immunity never displayed symptoms and were never tested for the virus.  Whether they were \ninfectious at any time was not known.  \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n9 \n \nThis presented a dilemma: although the shape of the accumulated death curves in Figure 6 strongly \nsuggested herd immunity, the 22% infected fraction reported by Dr. Fauci was clearly insufficient to \nproduce it, as illustrated by the calculations in Appendix A.  All the curves in Figure 6 exhibit the same \nbehavior, yet none appear to have reached a high enough value of f to achieve herd immunity.  \nTo resolve this inconsistency, we postulated that the populations in these areas did not respond \nhomogeneously to the pandemic.  Rather, they could be approximated as a hybrid population composed \nof two sub-populations: those that comply with social restrictions (compliers) and those that deny that \nthere is a problem and defy social restrictions (deniers).  That most people were compliers was obvious \nfrom viewing the streets during the height of the pandemic.  Evidence for the existence of deniers \nappeared each time the formal restrictions were relaxed: some groups participated in dense gatherings at \nbars, church celebrations, political rallies, motorcycle rallies, etc.  Since these people ignored the risks of \ncontracting the virus by defying guidance when the restrictions were relaxed, it was reasonable to assume \nthat their private behavior prior to the lifting of formal restrictions also ignored the risk: that is, they \ngathered in dense groups in private quarters instead of public locations.  The example set by the President \nand the associated political polarization also encouraged deniers to ignore restrictions, with many even \nrefusing to wear face masks to protect those they encountered.  \nThe model developed in the Hybrid Population section of Appendix A demonstrates that in a hybrid \npopulation with 80% compliers and 20% deniers an asymptotic immunity of 22% could produce a \ndramatic decrease in the infection rate as the denier sub-population achieves herd immunity.  It could \neven extinguish the pandemic altogether if all remaining susceptible people controlled their behavior so \nthat \n0(1 ) 1.0Rf − .  Nevertheless, while a few active infections are sustained, relaxation of restrictions \ncan relight the fire.   \nStates with Various Reponses‡ \nFigure 7 presents the accumulated COVID-19 deaths per million inhabitants in states with various \npandemic growth histories.  The earliest onsets were in New York and Louisiana.  New York has the \nlargest influx of traffic from Europe.  New Orleans, LA hosted the Mardi Gras celebration on Feb. 25, \n2020.  The death rate in Louisiana reached 10 dpM/day on April 1, 2020, after a delay about equal to the \ntypical time between COVID-19 infection and death.  The latest onset was in Hawaii, the state farthest \nremoved from Europe.  It has the most interchange with Asian nations, but travel from China had been \ncurtailed by the President on Jan. 31, 2020.   \nNew York suffered the most from the first wave of the pandemic despite the “New York State on Pause” \norder issued by Gov. Cuomo on March 22, 2020, to shut down non-essential businesses.  By this time \nNew York already had 58 deaths.  The President had issued an order on March 12 to restrict travel from \nUK and Ireland, but not Italy, and it was not applied to U.S. citizens.   \n \n \n‡ CAUTION: This section combines objective data with context based on the author’s interpretation of \ncontemporary news reports.  While the implied subjectivity violates normal scientific standards, there should be no \ndoubt that political context was a major causal factor in the development of the COVID-19 pandemic in the U.S.  \nThe reader is encouraged to apply his/her own interpretation to the data. \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n10 \n \n \nFigure 7. Accumulated COVID-19 dpM for selected U.S. states with different pandemic growth histories.  \nNew York’s experience, which taxed its medical care and mortuary systems to their limits, served as a \nwarning to other states.  Its initial exponentiation rate of 0.36 day-1, its eventual peak death rate of 38 \ndpM/day and its first-wave asymptotic deaths at about 1200 dpM were all among the highest in the world.  \nA slower initial growth rate enabled Louisiana to slow down the pandemic initiated by the Mardi Gras \ncelebration near the 600 dpM asymptote, but persistent death rates between 2 dpM/day and 10 dpM/day \naccumulated an eventual total of 2270 dpM, which surpassed even that of New York. \nThe Michigan curve mimics New York, but with smaller first-wave asymptote at 600 dpM and eventual \ndeaths at 1900 dpM. \nGeorgia is an outstanding example of politics overriding good sense. It was among the leading U.S. states \nin deaths from the outset, but officials still fought over restrictions; there was even a legal battle between \nthe Governor of Georgia and the Mayor of Atlanta over face-mask requirements.  Defenders could argue \nthat their eventual toll of 1800 dpM is no worse than many other states.  \nCalifornia provides an example of early control followed by relaxation after many months of sacrifice.  \nGov. Newsom issued a “Stay at Home” order on March 19 to restrict the size of the gatherings.  The \nPresident’s cutting off travel from China with a two-week quarantine for citizen returnees helped prevent \nthe rapid onset experienced by New York.  However, the number of deaths continued to increase \ngradually until the Thanksgiving, Christmas and New Year celebrations caused the accumulated death toll \nto reach 1570 dpM.  It appears that the principal effect of the early restrictions was to postpone the \neventual deaths.  The resulting impact on the economy remains to be evaluated. \n10\n100\n1000\n10000\n0 2 4 6 8 10 12 14 16 18\nMonths after Jan. 1, 2020\nStates with Various Responses\nAlaska\nArizona\nCalifornia\nFlorida\nGeorgia\nHawaii\nLouisiana\nMichigan\nNew York\nTexas\nVirginia\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n11 \n \nFlorida, Texas and Arizona have similar histories.  Initially, they were favored with relative low rates of \ninfection.  However, overconfidence and political controversy produced an ongoing high death rate \nduring the summer of 2020 and the second wave at year’s end.  The eventual accumulated deaths in \nArizona at 2420 dpM exceeded even New York’s. \nAfter an initial surge, Virginia has managed to maintain a relatively low death rate except for a surge at \nthe end of February 2021.  Its final death toll was 1315 dpM, 55% of Arizona’s value. \nHawaii and Alaska demonstrated how their natural separation could be used to protect their population \nfrom the pandemic.  They limited the eventual deaths to 355 dpM and 495 dpM, respectively.  Both \ninsisted that entrants maintain self-quarantine for two weeks after arrival in the state, while strongly \nrecommending that people wear face marks and maintain social separation.    \nSelected Nations \nFigure 8 presents the accumulated COVID-19 death fraction for various nations. \n \nFigure 8. Accumulated COVID-19 dpM for various nations \nS. Korea imported the COVID-19 virus directly from China, but it clearly applied the most effective \nresponse, as discussed above.  New Zealand banned travel from China pre-emptively in early February \n2020.  The first confirmed case arrived in New Zealand from Iran on Feb. 26; the second case arrived on \nMarch 4 from Italy.  The borders were closed to non-residents on March 19 and a general lockdown \nimposed on March 25.  As a result, the death rate subsided to a negligible level by the end of April and \ndomestic restrictions were gradually relaxed in the fall 2020. \nItaly was the first western-hemisphere nation to be infected, but most others followed a similar course \nwith similar results.  The two waves discussed above are evident in all nations, while the onset was \n0\n1\n10\n100\n1000\n10000\n2 4 6 8 10 12 14 16 18\nDeaths per Million\nMonths after Jan. 1, 2020\nSelected Nations\nUS\nBrazil\nItaly\nMexico\nNew Zealand\nS. Korea\nSpain\nSweden\nUK\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n12 \n \ngenerally delayed by an amount determined by the intensity of travel from those nations with prior \npandemics.  Transmission to all northern nations in the western hemisphere occurred within a month.  \nTransmission to Brazil and Mexico was delayed by about two months, but their deaths eventually caught \nup.  Brazil accumulated the highest death toll of 2190 dpM on May 31, 2021. \nSweden, which applied the least formal restrictions on public behavior, produced the lowest eventual \ndeath toll in these nations, i.e., 1410 dpM.  The government’s approach appeared to be to inform and \nadvise the public without applying legal restraints.  \nCurrent Status \nEffective vaccines against COVID-19 were introduced in Dec. 2020, but their deployment was too late to \nprevent the second wave of infections and deaths.  Fortunately, they are available to prevent a third wave, \nbut only if a sufficiently large fraction of the population is vaccinated. \nAccording to Fig. 1, total deaths in New York reached almost 1900 dpM by the end of May 2021.  Using \nthe inferred infections/deaths of 169, about 32% of the New York population has recovered from the \ninfection.  Meanwhile, about 45% were vaccinated.  Thus, about 63% of the New York state community \nare presumed to be immune and the effective R0 is reduced to 37% of its normal-behavior value.  As \nshown in Figure 4, the initial value of R0 in New York was about 5.  Therefore, a return to normal \nbehavior would result in R0 ≈ 1.9, or a pandemic increasing with an exponentiation time of about 6 days.  \nMeanwhile, R0 has increased by an unknown factor by the introduction of the more infectious delta \nvariant.  Similar analysis of other states reveals a common conclusion: ongoing restraint in interacting \nwith others is essential.  \nIf R0 = 5.0 is typical of near-normal behavior, more than 80% of the population must be immune by a \ncombination of recovery and vaccination before it is safe to return to normal life.  The political climate in \nthe U.S. appears to be limiting vaccinations to about 50% of the population; therefore, ongoing infection \nwaves are likely until more than 60% of the unvaccinated part of the population has been infected and \nrecovered.  Since the ongoing pandemic has enabled more virulent strains of COVID-19 virus to develop, \nthe current normal-behavior R0 nay be even larger and refusing vaccination may be even more dangerous.  \nAllowing the pandemic to continue anywhere in the world also risks allowing a mutation to develop that \nresists the vaccines and the antivirus in persons recovered from previous infection, which could reignite \nan even worse disaster.  \nSummary  \nThe COVID-19 infection is transmitted primarily by human-to-human encounters, especially inhalation of \nair exhaled by an infected person.  The infectious agent is carried along with the air on miniscule particles \nthat are too small to be influenced by gravity.  Face-to-face conversations are probably the most effective \nmeans to transmit the infection. \nAs demonstrated in Appendix A, the development of a pandemic is determined by the infection \nmultiplier, R0.  If R0 < 1.0, the rate of infection, hospitalization and death decreases; if R0 > 1.0 they \nincrease at an exponentially increasing rate.  R0 is under the direct control of the members of a \ncommunity: it is determined by the frequency of person-to-person encounters and the probability of \npassing on an infection (e.g., proximity and masking) during an encounter.  It is also influenced by the \nvirus’ virulence, especially the amount of virus that needs to be passed to a recipient to produce an \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n13 \n \ninfection.  Halving that amount by a mutation in the virus automatically doubles R0 for a fixed encounter \nsituation.   \nAnalysis of the data on COVID-19 deaths during 2020 and 2021 demonstrates: \nThe observed behavior of the COVID-19 pandemic is consistent with results of SEIR model \ncalculations, recognizing that R0 is adjusted by the population in response to data and guidance. \nThe initial values of R0 ranged from about 3 to 5. \n The populations adjusted their behavior so that R0 decreased by a factor of 2 in one to two weeks \nuntil it was slightly below 1.0.  Thereafter, it tended to oscillate between values slightly below \nand above 1.0 with periods of 1 to 2 weeks.  \nInformation about the pandemic from the nations affected earlier (e.g., Italy) had negligible \neffects on the initial response in areas infected later (e.g., UK and New York).§ \nOn average, each death was associated with an estimated average of 169 infections, but the \nreported number of “cases” accounted for less than half of these infections in almost all nations \nand U.S. states except Alaska. \nThe apparent long-term periodicity (about 6 months) in the death rates are likely to be caused by \ndelays in the “control loop”, e.g., the population’s reaction to pandemic information.  The six-\nmonth period is consistent with model calculations using reasonable assumptions about the delay \nin providing information to the public and in the population’s response to that information.   \nThe short-term periodicity (1 – 2 weeks) must be due to some other factor, perhaps the influence \nof weekend activities. \nDiscussion \nConsider the day the first person in New York died of COVID-19, presumably because he/she was \ninfected an estimated 25 days earlier.  An average of 168 other people were infected at the same time.  By \nthe time we were able to measure the COVID-19 exponentiation rate it was 0.36 day-1 (i.e., the \ncumulative deaths were e-folding every 2.8 days).  If this earliest measured rate represents the rate during \nthe previous 25 days, for each of these 169 persons infected on day 1 another 8100 persons had been \ninfected by day 25.  Therefore, on the day that the first COVID-19 death occurred in New York, 1.4 \nmillion people had already been infected in New York (about 7% of the population) and 8100 of them \nwere fated to die.  This could not have been prevented unless the population responded to warnings: e.g., \npredictions based on Italy’s experience three weeks earlier. \nThe development of a pandemic such as COVID-19 is determined by the population’s behavior, i.e., the \nrate at which people encounter each other and how closely they interact. The infection multiplier, R0, can \nbe halved simply by halving the encounter frequency, or maintaining enough separation to halve the \nprobability of transmitting an infection.  It can be doubled by a virus mutation that doubles the probability \nof infection in each encounter. \n \n§ Author’s note: The data used to formulate these first three conclusions were available on the Internet by May 1, \n2020.   \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n14 \n \nA particularly dangerous aspect of COVID-19 is that most are infected, and presumably infectious, while \nnot displaying any symptoms, i.e., most infections are transmitted unwittingly!  Many who believe they \nhave been responsible citizens and avoided infection may, nevertheless, be infectious and potentially \ndeadly to more vulnerable citizens.  \nThe analysis presented above demonstrates a typical behavior of citizens in democracies: \nThey adjust their behavior only to a minor degree until the threat becomes apparent in their \ninteraction area.  Some deliberately defy instructions as a demonstration of their “freedom”. \nEventually most respond to convincing evidence of the threat, but that response is delayed \nenough that the exponential growth continues for a while, producing an overshoot in the infection \nrate.  \nThey re-adjust their behavior when the threat appears to be abating, but do not persistt until it is \nsuppressed.  As a result, there are sufficient infectious people in the population to re-ignite the \nexponential growth as soon as R0 > 1.0.  \nThe natural result of such behavior is oscillation in the infection rate, i.e., repeated waves of \ninfection and death rates separated by months, depending on the specific reaction times and rates. \nThe political climate in the US contributed additional reluctance and delay into behavior adjustment.  It is \nincomprehensible to us that political loyalty would persuade people to increase their contribution to R0 by \na significant factor by eschewing masks and attending rallies. \nAutocracies can prevent such oscillations, as demonstrated in China during the spring of 2020.  Wuhan \nwas clearly the source of the COVID-19 virus, and its epidemic initially grew rapidly.  Apparently, the \ngovernment imposed strict discipline, effectively imposing total isolation on buildings in the city and \nplacing a cordon around the city.  As a result, the total death toll, even adjusted for suspected \nmanipulation of data by the Chinese government, was far below that reached eventually in the major \ndemocracies.   \nThere are alternative approaches to controlling the pandemic, such as that demonstrated by S. Korea and \nNew Zealand.  S. Korea received the earliest cases exported from Wuhan into a closely interacting church \ncommunity. Yet, it managed to limit its accumulated death toll to 37 dpM, a factor of 46 less than the \nU.S.  It had prepared itself with test kits and trained personnel to detect and track all those who might \nhave been infected by the initial cases.  It had the determination and authority to isolate all who were \ninfected.  This approach can be highly effective, but it must be applied before exponential growth \novercomes the available testing and tracking resources.  \n The U.S. was not prepared.  Our only recourse was to slow the importation of the infection by quarantine \nof arrivals from infected areas.  The President suspended entry of non-citizens from China on Jan. 31, \n2020, thereby delaying the onset of the pandemic in the U.S. west coast, but entry from Europe was not \ncurtailed until March 2020. The resulting death rate in California during the first wave peaked at less than \n2 dpM/day, whereas in New York it exceeded 37 dpM/day. \nThe states of Hawaii and Alaska made use of their relative isolation to limit deaths by applying moderate \nrestrictions on public behavior and two-week quarantines on arrivals from elsewhere.  As a result, they \naccumulated 355 and 494 dpM, respectively, as compared to 1700 dpM for the U.S. average.  For \ncomparison, deaths in the U.S. from the annual influenza season ranged from 70 to 185 dpM during the \nseven winters prior to 2020.  \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n15 \n \nAnyone who doubts the effectiveness of social distancing should review the data on annual influenza \ndeaths.  They were reduced essentially to zero everywhere in the northern and southern hemispheres \nduring the 2020-2021 season, presumably by the social distancing measures taken to control COVID-19.  \nLessons Learned \nThe important question is, “How should we prepare to meet future challenges similar to COVID-19?” \nGuide Nationally, Manage Locally \nThe variety of responses to the pandemic in different areas and the differences in response to the first and \nsecond waves show that applying remedies (e.g., social and business restrictions) nationally is wasteful \nand demoralizing.  There are vital tasks to be performed for the nation, but managing the pandemic \nshould be more local, preferably by response area (e.g., area with established management in which \npeople interact frequently).  The response area could be as large as a state (e.g., Wyoming) or as small as \na borough in New York City.   \nTasks requiring national leadership include: \nAt all times, stockpile equipment and supplies required to control pandemics at their inception. \nIn preparation, provide training for personnel needed to control pandemics at their inception. \nIn anticipation, provide warning and general advice to the public.  Firmly worded advice to wear \nmasks is appropriate.  Masking impacts little discomfort and decreases R0 by a significant factor, \nwhich is particularly beneficial at its onset.  Otherwise, avoid imposing severe constraints on \nareas that have not been infected for fear of producing resistance later when they become \nimportant.  \nMonitor closely and publicize the development of infections, hospitalizations and deaths in all \nareas.  Perform and publicize seroprevalence surveys to guide the model calculations.  \nDuring the pandemic, provide ongoing guidance and data to local managers. \nProvide data, such as models and key parameters to support analysis and predictions, such as the \nmodel provided in Appendix A.  \nPerform research to resolve uncertainties in models, such as the parameters used in the model. \nConduct seroprevalence surveys to monitor the progress of the pandemic and publish the results. \nProvide up-to-date data on test results, hospitalizations, deaths, etc.  \nProduce and distribute important equipment and supplies to the most needful areas, precluding a \nbidding war between desperate users.  \nWhen necessary, invoke the Defense Production Act to accelerate the availability of essential \nitems.  \nTasks that are more appropriately performed by the local (response area) management are: \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n16 \n \nInform the local population of the intensity of the threat.  Use data but avoid large numbers that \nhave little meaning to most people.  Use comparisons, such as deaths per million compared to \nannual traffic fatalities (about 100 dpM).  Offer predictions of the future as it depends on whether \npublic behavior (e.g., R0) is modified.  Build public confidence by showing that the prior \npredictions were correct.  Avoid political issues where possible.  \nUse prediction results to provide guidance.  \nIssue enforceable orders restricting activities to prevent an increase in R0  above acceptable levels \nPrepare to Respond Promptly \nThe data demonstrate that there are two distinct types of responses in each response area (e.g., area in \nwhich people interact frequently): control the spread at its inception or minimize it after it has escaped \ninitial control.  They also show that long-term, large-scale oscillations are the natural result of delay in \nfeedback control, so the feedback (public reaction by adjusting R0) must be anticipatory instead of \nreactive. \n Control at inception requires: \nA plan with personnel trained to execute it. \nCapability to control ingress into the response area.  \nAdequate supply of test kits and personnel trained to track possible infections. \nLegal authority to track infections and resources to quarantine infectious persons. \nThe same techniques can be applied later once the number of infectious persons has been suppressed by \nsocial isolation and vaccination to a level consistent with the available resources.  \nManage Pre-emptively \nMinimizing the pandemic in a democracy requires the ability to predict the pandemic’s future and to \npersuade the public to alter its behavior accordingly.   \nThe tools to predict exist; they only require determining the appropriate value of a few parameters and an \nestimate of the future value of R0.  Data from other nations/states with earlier onsets can also serve to \nwarn the public.  \nPersuading is difficult in a democracy.  The U.S. experience in 2020 proves that persuasion by edict is \nineffective and that persuasion by political loyalty is dangerous.  Scientists try to persuade each other by \nformulating a consistent logical argument, but its reception requires an educated, logic-receptive \naudience.  Improving U.S. education could go a long way in this direction, but that is a long-term \nsolution.   \nIn the near term, persuasion by experience may be the best available.  Certainly, the U.S. suffered \nseverely enough from the 2020-2021 COVID-19 pandemic to get people’s attention.  Yet, a large fraction \nof the population still resists vaccination, in opposition to all quantitative data.  Nevertheless, a rational \napproach by leaders, one that makes verifiable quantitative predictions, should eventually prevail over \nemotional appeals.  The formula is a simple repetitive cycle:  \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n17 \n \n1. The model is used to predict the future effects of the pandemic based on current behavior. \n2. The leader informs the public of the consequences of maintaining current behavior and \nrecommends modifications. \n3. The public ignores the recommendations. \n4. The public suffers the consequences, as predicted. \n5. The model is recalculated with updated information. \n6. The leader informs the public of the consequences of the new situation and recommends \nbehavior modifications.  \nEventually, this process should generate confidence in the model and the leader, while creating enough \nsocial pressure on the dissenters to persuade them to comply.   \nLimitations \nThe data chosen for the analysis were for selected states and nations, for which we believed the available \npublished data to be reliable. We avoided nations suspected of manipulating data for political reasons \n(e.g., Russia, China). We also excluded others for which the results were similar to those of included \ncountries.  Iceland, for example, suppressed the pandemic at its inception by methods like those employed \nin South Korea and New Zealand and by taking advantage of its relative isolation.  \nConclusions \nThe most important conclusions to be drawn from these data are as follows. \nMost COVID-19 infections are transmitted from people who are not aware of being infectious. \nIn the U.S. and western Europe all attempts to arrest the pandemic at its inception by behavior \nrestrictions failed.  The lowest fatalities were in Sweden, where the government focused on \ninforming the public rather than imposing regulations. \nAntivirus survey tests indicate that, on average one person died for each 169 persons infected, \ngiving a probability of death of 0.6%.  Less than half of those with positive antivirus tests had \nbeen counted as “cases”. \nThe apparent asymptote in deaths in the summer 2020 was the result of attaining herd immunity \nin the about 20% of the population that defied restrictions, or it was the first phase of long-term \noscillations.  \nThe second wave experienced at the end of 2020 was the manifestation of long-term oscillations \ncaused by the population reacting to information on the pandemic, information that is \nsignificantly delayed from the infecting events.   \nUnless a sufficient fraction of the population is vaccinated and a virus variant does not defeat the \nvaccine, we predict another serious infection wave in September 2021.   \nThe restrictions imposed in some states, but apparently not obeyed by a minority group, thus had the \nfollowing effects: \n● They protected the medical facilities from overload. \n● They prolonged the shutdown of economic activity.  \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n18 \n \n● They enhanced “cabin fever”. \n● They failed to arrest the pandemic. \nThese conclusions lead to the question, “What would have resulted if the U.S. and state governments and \nacted differently in 2020-2021?”.  Speculations are presented in Appendix B. \nAccording to our model a virus mutation that increases the probability of infection by a factor of 2 will \nconvert the current death rate, which appears to be acceptable to the U.S. population, into an exponential \ngrowth with an exponentiation time of about 10 days.  Since it takes about 25 days for the infection rate to \nappear in the death rate, by the time the renewed growth is apparent the infection rate will already have \nincreased by a factor of about 12. \nAcknowledgment \nThe author gratefully acknowledges the enormous contribution of Dr. Neal J. Carron for his very careful \nreview, checking the mathematics, and suggesting clarifying improvements. \n \nAppendix A: PANDEMIC MODEL \nMathematical Model \nA model of the COVID-19 pandemic is based on the following assumptions and definitions: \n• The infection is spread primarily by an encounter between an infected and an uninfected person. \n• The newly infected person becomes infectious after a period Tdelay, typically 3 days [5]. \n• The infected person is infectious for a period Tinf before he/she is removed from the interacting \ngeneral population by quarantine or hospitalization. We assume that the person is removed \nwithin one day after the onset of symptoms. Symptoms appear approximately 5 days after \ninfection, so Tinf typically lasts for 3 days. [5] \n• The fraction of the population that has acquired the infection at time t is f(t). \n• The rate at which the average person encounters others is µ. µ is 0 if an individual never interacts \nclosely with another person; it may be several per day or hundreds per week if he/she is \nengaging in normal life. \n• The probability that an encounter will transfer the infection from an infectious person to an \nuninfected person is P. \nµTinf is the number of persons encountered by an infectious person, and Ro = µPTinf, the infection \nmultiplier, is the average number of people that each infectious person infects during his/her infectious \nperiod.  \nTherefore, at the beginning of the pandemic, when f is near zero, Ro others will be infected by each \ninfected person. It is obvious that the pandemic will expand or shrink according to whether Ro is greater or \nless than 1.0: if Ro is greater than 1.0, the rate of increase of infections, df/dt, will grow until f becomes \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n19 \n \nlarge enough to reduce the effective multiplication rate below 1.0 (because encounters with previously \ninfected persons are assumed not to create new infections). \nThus, Ro = µPTinf is a critical parameter, the infection multiplier.  It consists of three independent factors, \neach of which is under people’s control, in principle. \nConsequently, before developing any equations, recommended ameliorative actions are obvious: \n• Decrease Tinf: isolate infected people as soon as possible by testing and quarantine or \nhospitalization.  \n• Decrease µ: practice social distancing. \n• Decrease P: wear masks and maintain separation. \nThe factors that make COVID-19 particularly dangerous are the following:  \n• The virus is highly contagious, so that P is large unless our behavior is modified. \n• There is a period of two or three days after Tdelay during which an infected person is infectious but \nnot symptomatic; that is, he/she is not aware of being infectious. \nDevelopment of Controlling Equation \nIf N0 is the total initial population, the rate at which the number of infected persons f(t)No increases is \nequal to the product of four factors: \nThe number of infectious persons, f(t)N0 \nThe rate at which each person encounters another person, µ, \nThe probability that the other person is not already infected or immune (1-f),  \nThe probability that an encounter transfers an infection P,  \nFirst, suppose that once a person is infected, he or she remains infectious forever and that Tdelay = 0.  Then, \nthe number of infectious persons would be the same as the number infected fNo and that number would \nincrease according to  \n \no\n0\n() ( )( )(1 )d f N f N Pµ fdt =−  (1) \nWhere fN0 is the number of infected persons, µP is the number of encounters per unit time that can \ntransfer the infection, and (1-f) is the probability that the other person in the encounter is not immune due \nto a previous infection. \nHowever, for COVID-19 a person is not infectious until Tdelay after his/her infection, and only for a time \nTinf after that.  Thus, the fraction of infectious persons is not f(t) but \n \ninf inf( ) ( ) ( ) delay delayf t f t T f t T T= − − − −  (2)  \nthat is, those who were infected between (t - Tdelay- Tinf) and (t - Tdelay), which should replace the first f in \nEq. 1.  Thus, we obtain: \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n20 \n \n \no\ninf o\n() ( )( μ )(1 )d f N f N P fdt =−   (2) \nas the controlling equation.  \nfinf can be approximated by  \n( )\n2 2 2\ninf\n1/ 2 /   2\ninf inf delay inff T df dt T T T d f dt − +\n , with f evaluated at t, \nleading to \n \n2\n2\ninf 0\n21 1( 2 ) (1 ) []\ndelay\nd f df\ndt T T dt R f− +−  . (3) \nAs expected, if the actual infections per infectious person \n0(1 )Rf − falls to 1.0, the second derivative of f \nbecomes zero and the rate of new infections df/dt becomes constant. Thereafter, df/dt decreases toward \nzero as f continues to increase. \nIn the early stages, when f <<1, f increases exponentially with an e-folding time \ndelT that satisfies the \ntranscendental equation** \n \n0 infexp( 1/ )[1 exp( / )] 1 delayR T T   − − − = . (4) \nBecause we do not know how to solve Eq. (2) or (3), we performed numerical calculations using a spread \nsheet††. Typical values reported by experts are that symptoms appear approximately 5 d after infection \nand that the person is infectious approximately 2 d prior to presenting symptoms [5]. If we assume that a \nprudent person will enter quarantine within 1 d after developing symptoms, then Tinf = 3 d and  \nTdelay = 3 d.  \nNumerical Solutions \nFigure A1 presents the evolution of f in time units of Tdelay for different values of R0 and Tinf = Tdelay. The \ntime scale was determined by Tdelay.  If R0 = 1.0, the infected fraction‡‡ remains nearly constant and \neventually decreases very slowly. \n \n \n** This is derived by assuming exponential growth for \n0 exp( / ) delayf f t T =  and approximating \n(1 ) 1f− . \n†† The limiting case of Eq. 1 is readily solved by letting f = 1/g, yielding an easily solved linear equation for g: find \n0\nμ\n00\n() (1 ) Pt\nfft f f e −= +−\n, where fo is a small initial seed value for f.  \n‡‡ We used an arbitrary seed value of \n8\n0 1*10f −=  . \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n21 \n \nFigure A1. Development of cumulative infected fraction  \nStarting from the seed fraction, the infected fraction increases approximately exponentially with time, \n0 exp( / ) delayf f t T =\n until f approaches unity, that is, until a significant fraction of the population was \ninfected and recovered or died, and is no longer susceptible to infection.  The exponentiation time, κTdelay \ndecreases with increasing R0.   The most useful form of this relationship is its inverse, the dependence of \n1/κ, the reciprocal of the exponentiation time measured in units of Tdelay, as shown in Figure A2 \n \nFigure A2. Dependence of reciprocal of the exponentiation time on R0  \n1.0E-08\n1.0E-07\n1.0E-06\n1.0E-05\n1.0E-04\n1.0E-03\n1.0E-02\n1.0E-01\n1.0E+00\n0 5 10 15 20 25 30 35 40 45 50\nf\nTime (Units of Tdelay)\nCumulative Infected Fraction\nRo = 2.0 Ro = 4.0 Ro = 7.0 Ro = 10\n-3.0\n-2.0\n-1.0\n0.0\n1.0\n2.0\n3.0\n4.0\n5.0\n-2.0 -1.0 0.0 1.0 2.0 3.0 4.0\n1/κ\nln(Ro)\nDependence of Exponentiation Time on Ro \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n22 \n \nThe logarithm of R0 is almost - not exactly - proportional to the reciprocal of the exponentiation time, i.e., \n( )01/  /1.5.ln R \n \nIf R0 remains constant, the reciprocal of the exponentiation time, 1/κ, decreases as f approaches unity, as \nshown in Figure A3.  \n \nFigure A3. Dependence of reciprocal exponentiation time on f \nActually, people change their behavior during the course of a pandemic, so high initial values of R0 are \nusually substantially reduced before half the population has been infected.  Thus, the histories of actual \nreciprocal exponentiation times will fall more steeply than shown in Figure A3 as people respond to \nalarming data. \nThe increment in f during each unit of time is proportional to the rate of infection, but it is proportional to \nthe rate of deaths at a later time, t + Tdie.  It is shown as the increment during each Tdelay period in Figure \nA4. The number of daily deaths can be derived from these curves by multiplying the ordinate by the \npopulation and the fraction of the infections that result in death and dividing the result by Tdelay.  It is \nremarkable that for Tdelay = 3 d, more than 3% of the population can be infected per day at the peak. This \nis an implication of exponential growth.  In practice, people will become frightened and moderate their \nbehavior, decreasing R0 before it reaches this peak.  \nThese curves are approximately exponential, upward when f <<1 and downward when (1-f)<<1. The peak \nrates were reached, that is, herd immunity overcame the effect of R0, when the infected fraction values \nreached values ranging from 0.43 for R0 = 2.0 to 0.70 for R0 = 10.  \n \n0\n0.5\n1\n1.5\n2\n2.5\n-0.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20\n1/κ\nf\nReciprocal Exponentiation Time\nRo = 2.0\nRo = 4.0\nRo = 10\nRo = 30\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n23 \n \n \nFigure A4. Fraction of population infected during each Tdelay interval \nHybrid Population \nThe foregoing calculations show that the rate of pandemic growth is determined by the value of R0, which \nis the product of three factors: the frequency of interactions between people, the efficiency with which the \ninfection is transferred, and the average time during which an infected person is infectious.  \nCalculation result presented so far assumed homogeneous populations, that is, they assume that everyone \nfollows similar isolation guidelines.  However, experience in the U.S. suggests otherwise. We can model \nthis situation by postulating two sub-populations: \nCompliers: those that take the pandemic seriously and comply with social restriction guidelines. \nDeniers: those that deny the seriousness of the pandemic and defy restrictions. \nTo investigate this situation, we need to divide equation (3) into separate equations for each sub-\npopulation.  Let α1 and α2 be the fractions of the total population, N0, represented by the two groups \n(compliers and deniers), f1(t) and f2(t) their cumulative infected fractions and f1inf and f2inf their respective \ninfectious fractions.  P is the probability of transferring the infection from an infectious to a non-infected \nperson during an interaction. (We assume the same value for both populations, even though deniers also \ntend to eschew face masks).  \n11 12 22,,    are the rates at which members of each population would \ninteract with others if all others were of the same group.  Thus, the rate at which a member of group 1 \ninteracts with members of group 2 is \n2 12 .  \nEquation (3) then becomes: \n1.0E-08\n1.0E-07\n1.0E-06\n1.0E-05\n1.0E-04\n1.0E-03\n1.0E-02\n1.0E-01\n1.0E+00\n0 5 10 15 20 25 30 35 40 45 50\nTdelay*df/dt\nTime (units of Tdelay)\nFraction of Population Newly Infected per Tdelay\nRo = 2.0\nRo = 4.0\nRo = 7.0\nRo = 10\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n24 \n \n1 1 o\n1 1inf o 1 11 1 2 2inf o 1 12 1\n( α) (α )(α μ )(1 ) (α )(α μ )(1 )d f N f N P f f N P fdt = − + −\n     (5) \n( ) ( ) ( ) ( )( )2 2 o\n1 1inf o 2 12 2 2 2inf o 2 22 2\n( α) α α μ (1 ) α α μ 1d f N f N P f f N P fdt = − + −\n \nThe fraction of the total population that is infected is:  \n \n1 1 2 2( ) ( ) ( )f t f t f t=+ .  \nThe effects of the different behaviors by the two groups are contained in the µ values.  \nWe investigated the effect of a particular population composition: the compliers, who comprise most of \nthe population, maintain R0 = 1.0, whereas the deniers continue nearly normal interactions, but only with \nothers in their own group.  Their value of R0 is adjusted so that the average of the total population is \nmaintained at R0 = 2.0.  The complier group on its own would maintain the infection at a constant level \nsince 1.0 person is infected by each infectious person.  The results are displayed in Figure A5. \n \nFigure A5. Effect of  deniers on population infection with a population average R0 = 2.0                                                                                                     \nThe heavy solid blue curve represents the fraction of the total population that becomes infected if 20% of \nthe population maintains R0 =6.0, while for 80% of the population R0 =1.0.  The other solid curves \nrepresent different denier fractions, each with its R0 adjusted to make the population average R0 =2.0.   \n1.0E-03\n1.0E-02\n1.0E-01\n1.0E+00\n0 50 100 150 200\nf\nTime (units of Tdelay)\nCumulative Infected Fraction for Hybrid Population\n5%/95% Hybrid Population\n10%/90% Hybrid Population\n20%/80% Hybrid Population\nf-20% Ro = 6\nf-80% Ro =1.0\n30%/70% Hybrid Population\nUniform Population Ro = 2.0\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n25 \n \nThe development of the 80%/20% pandemic is slower than the uniform R0 = 2.0 case, and its asymptotic \nvalue of infections is 39% instead of 80% for the uniform population.  The short-dashed curve represents \nthe 80% complier population, almost all of whom would not be infected in the absence of deniers.  The \nexponentiation rates for the two populations are identical at 0.21 Tdelay-1 since the infection growth is \ncontrolled by the denier population.  The exponentiation rate is characteristic of a population R0 = 1.36, a \nvalue slightly higher than the denier population R0 times its population fraction.   \nThe effect of deniers on the limiting infected fraction and the exponentiation rate, all for populations with \nan average R0 =2.0, are shown in Fig. A6.   \n   \nFigure A6.  Asymptotic infection fraction, exponentiation rates and 1/R02 required to maintain average R0 = 2 \nEffect of Community Response \nThe foregoing calculations have assumed that R0 of the population, or parts of the population, is fixed in \ntime, even though it is largely determined by the popular behavior.  More realistically, R0 is influenced by \nthe information received by the population, particularly when the data are alarming.  The predominant \ninformation is provided by official government communications and favorite news sources, which \nemphasize the most dramatic occurrences.  During the COVID-19 pandemic in 2020-2021 the principal \ndata available to the U.S public were the number of positive virus tests, estimates of the number of \ninfections, the reported number of deaths attributed to the virus, and impending crises in hospital staffing, \nICU beds and respirators.  Data on death rates were the most reliable, but they lagged the infection rate by \nabout a month.  Even the hospitalizations occurred a week or two after infection.   \nA model of community response must deal with this delay.  Our choice was to incorporate an adjustment \nto R0 when the rate of change in f evaluated at 5*Tdelay previously exceeds a response threshold of  \n.001 Tdelay-1: \n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\nLimiting f,  1/κ, 1/Ro2\nDenier Fraction\nDependence on Denier Fraction\nLimit\n1/Kappa\n1/Ro2\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n26 \n \nFor Tdelaydf/dt < .001 R0 remains at its base value, or is increased back toward its base value by \na factor of 1.1 per Tdelay \nFor Tdelaydf/dt > .001 R0 is decreased by a factor of 1.1 per Tdelay until it reaches a value of 0.5 \nThe resulting f and Tdelaydf/dt for R0 = 2.0 are compared to the corresponding values without response in \nFig. A7.   \n \nFigure A7. Effect of population response on pandemic history for R0 = 2.0 \nThis oscillatory behavior is typical of a control loop with delayed feedback: the output overshoots in both \nthe upward and downward directions.  The level at which the first plateau in f is reached is approximately \nproportional to the df/dTdelay threshold.  The oscillation period is shortened by responding more quickly, \ne.g., dividing or multiplying R0 by 1.2 rather than 1.1 per Tdelay. \nEffect of Vaccination \nThe foregoing calculations demonstrated how the development of the pandemic depended on the value of \nthe effective infection multiplier \n0(1 )Rf − .  Introducing an effective vaccine in Dec. 2020 added a new \nfactor, fvacc, the fraction of the population that is immunized by vaccination.  Since vaccines are \nadministered irrespectively of prior infection, the appropriate means to incorporate vaccination into the \nmodel is to replace the factor (1-f) in equations (3) and (6) by (1 - f) (1-fvacc).   \n \nAppendix B: SPECULATIONS on ALTERNATE APPROACHES \n1.E-08\n1.E-07\n1.E-06\n1.E-05\n1.E-04\n1.E-03\n1.E-02\n1.E-01\n1.E+00\n0 50 100 150 200\nf or  Tdelay*df/dt\nt/Tdelay\nEffect of Response on Pandemic History   (Ro = 2.0)\nf; Ro = 2.0\nTdelay*df/dt; Ro = 2.0\nf with response\nTdelay*df/dt w.\nresponse\n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n27 \n \nIntroduction \nGiven our model, key parameters and some indications of the population’s response, we can speculate \nabout the outcomes of other approaches to controlling the pandemic.  Lack of preparation denied the U.S. \nthe preferred approach, which was demonstrated by S. Korea: contain the pandemic at its inception.  \nOtherwise, we can speculate about the result if the national and state governments had done nothing, \nimposed restrictions on travelers from Europe as well as China, or if state and local government \nrestrictions had been more or less severe, and/or imposed earlier or later.   \nWe commend the federal government for encouraging the development and deployment of vaccines in \n2020.  However, it had minimal impact on the pandemic development through May 2021 under any of \nthese alternatives.  \nWe offer a general recommendation for federal action under the threat of a similar pandemic.  It should \nadvise, recommend strongly, and/or insist that we wear a mask whenever we expect to come near anyone \nnot in our own family enclave and possibly not vaccinated.  If not, we should at least turn our head \nsideways when we talk with others.  Since the primary means by which the infection is spread is by \nperson-to-person contact via transfer of exhaled air into a recipient’s lungs, minimizing the transfer of \nbreath from one to the other reduces R0 by a significant factor.  Since there is a delay between infection \nand onset of symptoms, decreasing R0 is particularly important during the unseen inception phase.   \nNo Action \nThe result of taking no action on eventual deaths is clear if we assume, optimistically, that the peak \ninfection rates do not stress the medical system sufficiently to increase the probability of dying.  Then the \npandemic would have propagated throughout the U.S. until herd immunity was reached when about 80% \nof the population had become immune.  The result would have been 0.80 x 331 million /169 = 1.6 million \ndeaths in the U.S.  Any overstress in medical care would increase this number.  Despite the record-setting \npace of vaccine development and deployment, it would have arrived too late to prevent many of these \ndeaths.   \nControl Travelers from Europe and Asia  \nThe President curtailed travel from China on Jan. 31, 2020; in March 2020 he imposed a similar ban on \ntravelers from UK and Ireland.  Technically, they applied only to foreign nationals, although immigration \nauthorities encouraged citizens to quarantine themselves for two weeks.  Even though the earliest \ninfections were detected in Washington state, the pandemic remained relatively mild in west coast states \nthroughout its first wave.  On March 19, 2020, Governor Newsom of California and on March 22 \nGovernor Cuomo of New York issued orders intended to restrict dense gatherings.  At those times \nCalifornia had suffered 23 deaths and New York had accumulated 157.  Eventually, the first wave peaked \nin California at 3.5 dpM/day in August; New York reached 38 dpM/day in mid-April.  \nA key factor in the COVID-19 development is the incredibly fast buildup of infections under normal-\nbehavior conditions.  As illustrated under Discussion, in New York state more than a million persons had \nbeen infected by the time the first person died.  At this point any additional imported infectious persons \nare negligible compared to domestically induced infections.  \nTherefore, import restrictions can serve to limit the influx of infections to the degree that available \ntest/track/quarantine resources can maintain control, or delay the onset.  But they will be effective only if \napplied long before a significant number of deaths occur.  Therefore, the COVID-19 pandemic on the \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n28 \n \nU.S. west coast was delayed by restricting travel from China, but a restriction on travel from Europe \nwould have been effective only if it were applied before Feb. 18, 2020. \nMore or Less Social Restrictions at State/Local Level \nUnwittingly the U.S. performed an experiment to evaluate different approaches to controlling the impact \nof the COVID-19 pandemic because the political controversy encouraged “Blue” states to be more \naggressive than “Red” states.  The resulting deaths is a measure of the relative value of the approaches.  \nUnfortunately, data are not yet available on another important measure: the relative impact on the \nstates’economies.  There is no doubt that it was severe and that pre-emptive restrictions made it more so. \nWe compare the pandemic history in three states for insight: California and New York as solidly “Blue” \nstates and Florida as a representative “Red” state.  The final death tolls are 1570 dpM for California, 2115 \ndpM for New York and 1710 dpM for Florida.  The first-wave death tolls were 109 dpM for California, \n1189 for New York and 114 for Florida.  In New York the first wave accounted for 56% of the eventual \ndeaths; in California and Florida it accounted for about 7%. \nIn California Gov. Newsom imposed in March 2020 the earliest restrictions on large gatherings.  In New \nYork Gov. Cuomo followed suit, but the death rate had already soared.  In Florida, Gov. DeSantis also \nrestricted activities to essential services.  During the summer and early fall of 2020, the ongoing relatively \nbenign death rate caused Gov. Newsom and Gov. Cuomo to adjust restrictions, but in Sept. 2020 Gov. \nDeSantis nullified most restrictions imposed by local authorities.  The resulting Dec./Jan. peak enabled \nCalifornia to reach death levels comparable to New York.  \nWe must rate all three approaches as failures.  The motivations were worthy: protect lives and protect the \nstate’s economy.  Neither goal was achieved.  The only conclusions consistent with the data are: \nPre-emptive restrictions are more effective than those applied when the death rate is already \nsignificant. \nThe population tires of long-term restrictions and relaxes them, even while the death rate remains \nsignificant. \nInstead, we study the two states that were relatively successful in controlling the pandemic: Hawaii and \nAlaska, one “Blue” and one “Red”.  Their final deaths were comparable at 353 dpM and 495 dpM, \nrespectively.  Their remoteness enabled them to control ingress of infected persons, but the initial deaths \nin Alaska coincided with those in California; in Hawaii they occurred 9 days later.  As we’ve argued \nabove, at this stage home-grown infections outnumber imported ones.  Alaska has the advantage of the \nlowest state population density at 1.3 persons/sq. mile, but Hawaii’s at 219 persons/sq. mile is comparable \nto California at 246 persons/sq. mile.  The state governments provided guidance but did not impose \nsevere restrictions other than controlling ingress.   \nThus, we are forced to speculate further to explain the data.  Without direct evidence to support it, we \noffer the following hypothesis: \nThe remoteness of Hawaii, Alaska and New Zealand promotes in their citizens a sense of pride \nthat induces them to exercise extra care to protect their state from pandemic infection.  Restricting \ningress from elsewhere is one manifestation of that pride.  Conscientiously following guidance \nfor social separation is another. \n  \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint \n\n29 \n \n \nReferences \n \n[1]   https://www.cdc.gov/coronavirus/2019-ncov/cases-updates/geographic-seroprevalence-surveys.html    \n[2]  https://www.startribune.com/sturgis-rider-from-minnesota-dies-of-covid-19/572296082/   \n[3]  https://en.wikipedia.org/wiki/COVID-19_pandemic_in_New_York_City   \n[4]  www.cnn.com/politics/live-news/fauci-senate-hearing-09-23-20/index.html   \n[5]https://medical.mit.edu/covid-19-updates/2020/07/how-long-symptom-onset-person-contagious   \n . CC-BY 4.0 International licenseIt is made available under a \n is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)\nThe copyright holder for this preprint this version posted July 15, 2021. ; https://doi.org/10.1101/2021.07.12.21260326doi: medRxiv preprint","source_license":"CC-BY-4.0","license_restricted":false}