Characteristics of the Covid-19 Pandemic in the United States, 2020

The novel SARS-2 coronavirus spread from China to most of the rest of the world in the year 2020. The virus spread to the U.S. from travelers to China in January, and the earliest signifi cant spread into public communities occurred in March. The virus spread to Europe and from there spread through travelers to New York. It is documented that by this time the virus had mutated to a different form, having a spike protein with an aspartate replaced by a glycine amino acid. This conferred an increased infectivity [1]. In most countries of the world that ordered a National policy of social distancing with mask wearing in public caused the cases to decline and a virtual end to the pandemic, such as fi rst in South Korea [2], and later by countries in Europe after experiencing a huge increase in cases and deaths. California and Washington ordered Statewide lockdowns initially in March. Around Memorial Day these restrictions were lifted with a large return to work allowance. Unlike the biphasic curve that occurred in those countries with widespread ordered restrictions, the U.S. exhibited a resurgence in cases and deaths through July 2020. An analysis of these trends is made here in hopes of assisting the US. with the goal of halting the pandemic.


Introduction
The novel SARS-2 coronavirus spread from China to most of the rest of the world in the year 2020. The virus spread to the U.S. from travelers to China in January, and the earliest signifi cant spread into public communities occurred in March.
The virus spread to Europe and from there spread through travelers to New York. It is documented that by this time the virus had mutated to a different form, having a spike protein with an aspartate replaced by a glycine amino acid. This conferred an increased infectivity [1]. In most countries of the world that ordered a National policy of social distancing with mask wearing in public caused the cases to decline and a virtual end to the pandemic, such as fi rst in South Korea [2], and later by countries in Europe after experiencing a huge increase in cases and deaths. California and Washington ordered Statewide lockdowns initially in March. Around Memorial Day these restrictions were lifted with a large return to work allowance.
Unlike the biphasic curve that occurred in those countries with widespread ordered restrictions, the U.S. exhibited a resurgence in cases and deaths through July 2020. An analysis of these trends is made here in hopes of assisting the US. with the goal of halting the pandemic.
following principles of the Calculus to see if changing trends could be mathematically described. First derivatives that determine the slopes at any given date, and second derivatives that determine the curvature or cupping of the curves at any particular date were estimated visually. Modifi ed sinusoidal mathematic functions appear to reasonably model the real data. Figure 1 shows the daily number of confi rmed Covid-19 cases in the U.S. as a function of time from January to July, 2020 [3].       Figure 8 indicates the percent of cases in the population of Los Angeles, New York City, the U.S., and the world [2].

Discussion
Analyzing curves with tortuous courses is in the realm of the Calculus. The fi rst derivative of any function represents and computes the slope of the curve at any particular position.
The second derivative of a function generates a formula for and represents the curvature or cupping of the curve, which is the slope of the slope for the curve. The second derivative of a curve generating a valley is positive because the downward slope becomes less negative with time, approaches a slope of zero at the bottom of a well shape, and begins to rise. A curve with a negative second derivative has a negative curvature or cupping such as at the top of a hill where the upward slope begins to decrease in steepness, levels off at the top of the hill shape, and begins a downward negative slope [4].
Notice that in all the curves describing incidence of confi rmed Covid-19 cases in this pandemic in the U.S. that in an initial phase of an upward slope in number of cases and deaths is eventually slowed to a point of leveling off on the top       [5] in Figure 9 from March through July, 2020.trace such a curve. This is because the downward cupping with a negative second derivative stops the hill from proceeding higher and brings the curve down while a positive later cupping second derivative causes the downward trend to level off and begin to increase again. The downward cupping followed by upward cupping occurred at specifi c dates and help to understand the nature of the infectious process in the country.
The initial pandemic was exemplifi ed by a steep rise in number of cases with time in March. This is accompanied with a drop in number of employed people working, as one would expect. This may be due to the virus causing illness with more people staying home from work, coupled with healthy persons staying home to avoid the infection, and any government orders to stay at home. This was followed by a slowing of the rise in cases until a peak was reached and then a drop in daily new cases occurred. This was followed by an increase in employment probably due to recovery from illness, relaxing of stay at home restrictions and requests to open up and to return to work from various offi cials. This again led to an increase in number of cases perhaps due to infection at work especially in work places where closeness or direct contact among workers is necessary. This was followed in time with increased numbers of cases again. Death incidence typically followed the incidence curve by a few weeks since this is the time required for the virus to replicate and spread in an infected victim.
The two strains or mutants of the virus appear to cause differing rates of infection and different lethality. Notice the percent of cases that occurred in Los Angeles and New York are widely different, as are the deaths or lethality rates in the two cities.
The early widespread closure of public schools in the U.S., like elsewhere in the world, has undoubtedly helped mitigate the pandemic. Many reports have indicated that children are infected at a far lower incidence rate than adults and that the lethality or virulence of the virus in children may be lower as well. However it must be emphasized that obviously the spread of the virus through children would be much reduced when social contacts children have while not attending school would be much reduced compared to adults or compared to children in school. As long as a pandemic exists without a cure and a signifi cant lethality (2 % in CA and 6-10% in the NorthEastern U.S.) and a lethality of generally 10% in the elderly, it is prudent to have off-campus, public schooling consistent with good home schooling practices. In some cases facilities might permit distanced on-campus schooling with students mature enough to consistently be mask-protected.
Los Angeles and New York City appear to have been infected with two different mutants of the novel coronavirus. The East Coast was infected from travelers from Europe with the G virus, while the West Coast was directly infected from travelers from China. The New York City experience thus differed from Los Angeles, where the infectivity of the G form is greeter. NYC in its 5 boroughs or counties had 204,569 cases as of July 23 with 30,187 deaths. This is a death rate or virulence or lethality of 14.8%. Los Angeles at this date had 165,134 confi rmed cases with 4,215 deaths, a lethality or virulence of 2.6%..This supports the notion that two forms of the virus infected these locations since the relative proportion of the population that are elderly and more susceptible is not widely different between the two cites, and the capability of the health care systems to treat the condition are also not considered largely different. The difference also however may be somewhat related to the fact that   These observations are consistent with data for the U.S. and for the world as well, as shown in Figure 11 over the updated more lengthy range from March to September, 2020.

Viral infection rates affect social behavior, and social
behavior also directly modifi es viral infection rates. In general, when cases are on the rise, it is natural to halt normal employment arrangements, by working at home and to change other social activities. This is not only due to increased sickness but largely also is an attempt to avoid becoming infected. This is followed by a downward trend in number of cases which is then followed by an opening of facilities again. This is followed by an upward trend in new infections again, which again leads to more cautious public activity, and so on. As Covid cases rise at a rapid rate (time t = 0), employment begins to decline from a maximum. This mitigates the transmission of the virus, and the rate of increase in active cases of infection begins to slow, and at t = 100 reaches a peak and begins to decline. This decrease in the rate of active infections causes the rate of employment to become less negative so that employment numbers decrease more slowly until employment is at a minimum at t = 200. This is followed by continued declines in cases of infection which decrease at a less rapid rate until reaching a minimum at t = 300. Employment rises until 360 which causes the steepest rate of increase in infectious cases. This causes the cycle to continue again where increasing cases cause a decrease in the rate of employment again, and so on. The proper way to halt this endless cyclical behavior is to regulate employment so that very safe practices are followed, particularly when employees work in enclosed or relatively close spaces. This would also apply to activities where people congregate closely without proper precautions. Droplets in breath travel substantial distances from merely casual talking. Work involving exertion causes  droplets to spread much further. Figure 12 shows a theoretic curve of deaths due to the infections described in Figure 13, where an arbitrary 10% lethality was employed at an average 14 days after infection. The link between public exposure in unregulated workplaces, or unprotected gatherings, with lethal consequences is expected.
Recent data confi rms that this sinusoidal cyclical behavior continues repeatedly in the actual pandemic. This is shown for a longer extended time period recently published for San Diego County through September, 2020 (see Figure 14). Two successive sinusoidal waves are clearly seen in these data of confi rmed cases averaged in 14 day time intervals.. Notice that the second sine wave is of lower amplitude than the fi rst. This suggests that the public became more engaged in following protective behaviors. A greater percent of the population may be wearing masks and social distancing as employees in the workplace and as citizens in public and social activities.
In a previous study [2] conditions were determined for the chronic inhalation of heated air or steam as a possible mitigant for those infected or exposed to this novel virus. The virus is known to be rendered non-infective after a 20 minute exposure to ~154 o F (~68 o C), which is possible to achieve with a boiled water or heated air without undo discomfort. Although clinical direct studies have not yet been done, some evidence indicates that steam treatment of those with the fl u experience signifi cantly decreased convalescent time, and this seems a prudent precaution immediately after one is suspected of being exposed to the novel coronavirus, before initial symptoms appear several days after exposure. Recent studies indicate that remdesivir, the RNA polymerase virus replication inhibitor, and dexamethasone, which impairs infl ammation, are clinically effective in shortening recovery times for victims of Covid-19.
Finally, recent studies indicate that the fl avenoid dietary supplement quercetin (together with vitamin C) is effective at blocking virus insertion into cells and subsequent replication and has been recommended for Covid-19 clinical trials [6].

Recommendations
These data indicate that when at work or when close to people in public, it is necessary to minimize infection spread by wearing masks and practicing social distancing to prevent signifi cant spread of exhaled breath droplets, wearing of goggles in places where very close contact is unavoidable or in an indoor compact spaces, washing hands often, and bathing and laundering clothes after returning home from work in some cases. The degree to which these measures are followed determines the shape of the incidence (or infectivity) and the death (or lethality) curves for any country. As proof, notice Figure 14 which shows a fi rst wave and then a second wave of infection incidence in San Diego County from March to June corresponding to health warnings which were later followed by relaxation of warnings or a re-opening, and then again from July to September corresponding again to precaution warnings followed by relaxation of warnings. The difference in amplitude of the two waves is undoubtedly due to the fact that a larger segment of the population has become aware of the need to practice precautionary measures. It is hoped that as information is provided across the country, that the U.S. population sinusoidal curves will also become more attenuated than has been found thus far for the successive waves shown in Figures 1 and 11. Attenuating these curves is essential to control the pandemic and to approach our former way of life.  Upper curve for number of people gainfully employed is given by y1 = cos(6t) + 3 where 3 could represent 3 million, or 3 thousand, etc. people in a particular population, and t represents time in days. The lower curve for active cases of Covid infection is given by y2 = sin(6t). Each hump occurs over a 30 day period of time, approximating the time-dependence of curves seen in the U.S. The height of the cases curve is arbitrary and may refer to a particular unit in a given population. Figure 14: Average positive Covid-19 tests plotted as 14 day averages for San Diego County, CA from March through September 13. As incidence of infection increases, precautions are taken to cause daily incidence to lessen. As incidence lessens, precautions are relaxed and infection incidence increases again. This cycle occurred from March to June and then again from July to September. For example, many countries widely followed mask wearing and distancing protocols which prevented sinusoidal curves of lethal cases. Only 300 total deaths occurred in South Korea for their entire pandemic remaining as a bell-shaped curve.

Acknowledgments
The people of the State of New York are honored and thanked for the tenacious work in battling and eventually controlling the Covid-19 pandemic there. This confi rms that although a virus can dramatically affl ict a population, people can greatly control a viral pandemic even in the absence of a vaccine or other cure.

Note added in proof
As of September, 2020, data from The Johns Hopkins Coronavirus Resource Center refl ect a 3.5 fold higher lethality for the Covid-19 pandemic in New York than in California, due presumably to a more virulent virus strain (assuming similar demographics and access to health care). Los Angeles County at this date has a total 6,401 deaths among 262,133 cases, a lethality of 2.4%. Queens County now has a total 6,017 deaths among 71,063 cases at 8.5% lethality, though fortunately now the pandemic has been effectively controlled in NY. Some public offi cials argue that the U.S. has turned a corner for the Coronavirus pandemic now, but this is clearly not the case. With total cases in excess of 7 million, and total deaths in excess of 200,000 in the U.S., over 40,000 new confi rmed cases and approximately 800 new Covid-19 confi rmed deaths continue to occur daily in the country [7]. The U.S. has less than 5 percent of the world's population but more than 20% of the nearly 1 million reported Covid-19 deaths worldwide (San Diego Union, September 23, 2020, 'U.S. virus death toll tops 200,000'), with a per capita death rate that is 6 th in the world, behind Peru, Bolivia, Chile, Spain, and Brazil.