key: cord-0311070-2gaqk4h6
authors: Bonasera, A.; zheng, h.
title: Controlling the worldwide chaotic spreading of COVID-19 through vaccinations
date: 2021-12-27
journal: nan
DOI: 10.1101/2021.12.23.21268184
sha: 8b0c7cd54d4245a8a9a835b45e7b20dfb64b393c
doc_id: 311070
cord_uid: 2gaqk4h6

The striking differences and similarities between the Spanish-flu of 1918 and the Coronavirus disease of 2019 (COVID-19) are analyzed. Progress in medicine and technology and in particular the availability of vaccines has decreased the death probability from about 2% of the affected for the Spanish-flu, to about 10-5 in the UK and 10-3 in Italy, USA, Canada, San Marino and other countries for COVID-19. The logistic map reproduces most features of the disease and may be of guidance for predictions and future steps to be taken in order to contrast the virus. We estimate 6.4 107 deaths worldwide without the vaccines, this value decreases to 2.4 107 with the current vaccination rate. In August 2021, the number of deceased worldwide was 4.4 106. To reduce the fatalities further, it is imperative to increase the vaccination rate worldwide to at least 120 millions/day.

powerful approach to quantify the 'degree of chaoticity' is to calculate an ensemble of N pairs of trajectories, separated initially by a very small distance d 0 . We introduce the mean distance between them at the iteration n as:

where

The initial starting points x (i) 0 are chosen from a uniform distribution spanning the defining interval of the map. For fully chaotic maps the average distance d n after n-iteration may be expressed by the relation

where the Lyapunov exponent λ > 0 indicates that nearby trajectories diverge to a finite (since the phase space is finite) value d ∞ (< 1). In the ergodic limit r = 4, λ = 2 which can be considered as the 'highest chaoticity' which can be reached. In the same limit [4] d ∞ = 4 π 2 , the largest average distance between nearby trajectories. Notice that even in the chaotic regime there are values of the control parameter where λ < 0, i.e., the map is not chaotic. We may argue that for the S1918 pandemic one of such values was accidentally hit and the epidemic disappeared suddenly. Notice that in such cases eq. (4) gives d ∞ = d 0 → 0 [4] .

In refs. [5] [6] [7] [8] we discussed the fact that the cumulative probability to be infected by COVID-19 (number of cases divided by number of tests) follows the same eq. (4) with the iteration n substituted by time t, thus if we know the Lyapunov exponents for the data and the map we can easily make a connection between time and iteration i.e., t = λ Λ n.

Under this assumption it follows that the maximum probability to become infected is 4 π 2 . In 1918 the world population was about 2 billion which means that about 0.8 billions got infected by the S1918 in the period 1918-20. As we will discuss below about 2% of the infected died which gives 16 millions deaths, a number comparable to the casualties of WWI. Extending these estimates to 2020, the world population is about a factor 4 higher, which increases the estimates above by the same factor. However, in November 2021, 7.3 billion vaccine doses have been administrated which means that about 3.65 billion people have been fully vaccinated. Since the mortality rate is about 2% of the positive cases, then we expect the mortality rate (mostly non vaccinated people) to be given by (8 − 3.65) × 2% × 4/π 2 = 35 millions.

Also by the end of the year, people who received the first vaccine dose, will get the second thus reducing the maximum estimate to 18 millions. Optimistically 75% of the world population may be vaccinated (excluding children age less than 12 for whom vaccination is not possible yet, but more recently the use of some vaccines have been extended to . CC-BY 4.0 International license It is made available under a perpetuity.

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The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint children aged 5 or older). This gives a mortality rate 0.25 × 8 × 10 9 × 2% × 4/π 2 = 1.6 × 10 7 , currently, the number of recorded deaths is 5.1 million. Our estimate above assumes that the spreading is fully chaotic and as we will show below this is not the case. If we take Italy as an example, the average positive probability is about a factor 4 below the ergodic limit of the logistic map. These are in any case very important numbers thus the urgency to increase the number of vaccines worldwide, the faster we do the more we protect children and people with medical conditions and that cannot be vaccinated. We would like to stress that the extension of vaccinations to the youngest population may be necessary since there is some part of the population refusing to get vaccinated denying the clear success vaccines have obtained so far. But this is not enough especially if new variants will appear. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint In the figure 1 (top panel) we plot the cumulative number of cases (positive and deceased) as function of time starting from January 1, 2020 for Italy (top-left panel) and the UK (top-right panel) [10, 11] . In order to directly compare the two countries we have divided the number of cases by their population (60,177,000 in Italy and 68, 207, 116 in the UK). The two cases are strikingly similar and we have observed no major differences to other countries, see also supplemental material. After the first rapid increase there is a plateau, which roughly starts at the end of the lockdowns and the beginning of the summer season. In the fall of 2020 the cases increase rapidly again forcing new lockdowns to slow them down and fortunately in December 2020 vaccinations started. The final increase in the UK is due to the Delta variant of the disease. The number of deceased is roughly a factor 50 smaller than the number of positives and no increase is seen at later times thanks to vaccinations. If we treat these data as representative of cases worldwide we can estimate a total number of positive to the virus in 790 millions (10% of the current world population) and a total number of deceased in 16 millions (0.2% of the current world population), in November 2021 those values are 253 millions and 5.1 millions deaths respectively. If this trend is followed, i.e., no increase in the number of vaccines and no other more deadly variant, we can estimate the maximum total duration of the disease in 5 years, two more than the S1918. These values are of course not acceptable and a worldwide effort is needed, especially vaccinations. We notice in passing that some countries, notably S. Korea and others, have adopted strict distancing, tracking and wearing mask measures, which successfully stopped the spreading, see supplemental material.

Other countries have not been able to follow these examples because of unrest from part of the population thus leaving vaccinations and new medicines to contrast the spreading. Unfortunately, in the same countries there has been some resistance to vaccination as well but this may be overcome through compulsory vaccinations or the natural spreading of the virus from vaccinated to unvaccinated ones. In the latter case, the weak part of the population, which cannot be vaccinated, may pay a high price unwillingly. A fit to the top part of the figure 1 using eq. (4) gives about 7% and 0.2% total number of positives and deceased of the population respectively. These values are confirmed by many other countries, see supplements, with the notable exception of S. Korea reducing to 0.6% and 0.004% respectively.

The approach and methods used by S. Korea and other Asiatic countries should be part of the civic study curricula of students starting in elementary schools worldwide to avoid the drama for the next pandemic, which could occur in the next 10 or 100 years.

The estimates above could be misleading because, especially at the very beginning of the disease, only part of the population was tested. Since the number of tests each day are known we can define probabilities as the number of cases divided the number of tests [5] [6] [7] [8] . is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint

The dashed lines in the bottom panels are given by the logistic map value 4 π 2 and in no cases it is reached apart Norway, see fig. S1 , at the very beginning of the pandemic. In particular the rapid increase at the beginning reaches a maximum value on March 14, 2020 when the lockdown was imposed in Italy and April 5, 2020 when the UK prime minister was admitted at the hospital in serious conditions having contracted COVID-19. As we noticed before the deceased probability follows the same trend and it is about a factor 50 lower than the positives, shifted in time of about one week. Notice that the 'population' for the deceased is given by the positives, thus we can define another probability as the number of deceased divided by the number of positives. This quantity is plotted in the bottom part of figure 1 (blue symbols) as function of time and it is very close to the probability to be positive to COVID-19 as expected. In particular, after a first rapid increase at the beginning it plateaus at about 10% and jumps down to 2-3% again in December 2020 when the vaccination campaigns started, clearly demonstrating their efficacity. We stress that in the S1918 case, if the disease was completely out of control, then the number of deaths may be a factor of 5 higher than estimated above, i.e., of the order of 80 millions. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint In order to compare to the logistic map, we plot in figure 2 the positive probabilities in two different periods for Italy (top panel) and UK (bottom panel). The logistic map follows rather well (red symbols) the data and the values of the Lyapunov and d ∞ can be extracted for all cases. From ref. [4] we know that d ∞ can be parametrized as

as function of r with ν/2 = ln α ln δ , α = 2.502807, δ = 4.6692016 and r ∞ = 3.569946. And similarly for the Lyapunov

where β = ln 2 ln δ . The values c 1 and c 2 can be fixed to their analytical values in the ergodic limit i.e., r=4.

From the number of positives we know the value of the asymptotic probability which plays the role of d ∞ and it is the same as for the logistic map, thus we can extract the corresponding value of r from eq. (6) and, knowing r we obtain the Lyapunov exponent for the map eq. (7). From the ratio of the Lyapunov exponent for the map and the data we can make the transformation from the number of iterations to time, eq. (5). In is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint Table 1 . The r and λ calculated from Eqs. (6) and (7) for Fig. 1 is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint 

We have discussed in the introduction the connection between the control parameter r of the logistic map with λ and d ∞ . In order to have r < r c we can decrease the value of d ∞ through lockdowns, masks, tracing and vaccinations. In particular we can write:

where, d 1 ∞ is the current value of the probability while d 0 ∞ is the same quantity but exactly in the same period of the year before. χ is the controlling factor which we assume equal to

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The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint thus another reason to get vaccinated as soon as possible. Also vaccinated people may be justified not wearing masks in presence of unvaccinated by choice, this may spread the virus faster when it is harmless to vaccinated and there are medicines which can save the life of the unvaccinated, keeping in mind that 2-3% of the positives will die, see figure 1. Future, more deadly variants may develop thus the importance to stop the epidemic as quickly as possible. In figure 3 we plot the probabilities as function of the vaccination ratio χ. The decrease is very encouraging but it maybe misleading since could be due to lockdowns or simply to a major awareness from the population. A more suitable quantity may be given by the ratio of probabilities or cases in two different periods, see eq. (8).

From eq. (8) we can define the probability ratios for the year 2021 divided the same period in 2020. Since, in Italy for instance, test data was taken starting Feb 24, 2020 that is the first day we can build the ratios (for the UK, US and Russia are Mar 31, 2020, Mar 1, 2020 and Mar 4, 2020 respectively). This ratio is plotted in figure 4 as function of the parameter χ defined in eq. (9). The divergence for low vaccination rates is due to the low statistics at the very beginning of the pandemics when the virus was starting to diffuse among the people. From the expansion, eq. (8), we should get an initial decrease followed by higher order corrections that could be positive or negative. If 75% of the population is vaccinated, we expect the number of cases to decrease of a factor at least 4 as respect to the year before. The figure displays a fast decrease at the beginning of the vaccinations followed by a later increase due most probably to the Delta variant and the release of strong restrictions and lockdowns. The fact that the ratio for positives is about 1 for large χ values indicates that herd immunity cannot be reached. On the other hand the ratio for deceased is about 20% for the US and Italy and 4% for the UK indicating that vaccines are working in preventing deaths but with different efficacities. As we noticed above, the number of deaths should be compared to the number of positives and not to the number of tests, thus in the same figure we include this ratio with blue symbols. Now the results are comparable to the deceased probabilities as expected with the exception of Russia. The important lesson to take from this plot is that the vaccines are working and in particular the approach used in the UK seems to be the most effective.

A striking difference is given by Russia, which follows the same trend as other countries but the ratios are larger than one and the deceased are above the positives, figure 4 . This suggests that an important epidemic wave is underway in Russia more important than the previous year together with the resistance of people to get vaccinated. We had noticed in previous works [7] that in the year 2020 while the other countries were "under siege" from the virus, Russia was relatively free from it. The general behavior as function of vaccination confirms the quality of the vaccine used, in any case we can test this by analyzing cases from the Republic of San Marino (RSM), which adopted the same vaccine as Russia. The figure 4 shows an increase in the ratio with increasing vaccination rate. We have attempted fits using eq. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint confirming once more that we are nowhere near herd-immunity. This is not going in the direction of herd immunity but it could be an artifact due to the different number of tests in the different periods. To explore this feature we will discuss below the ratios of the number of cases in the two periods. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint we believe it is due to the different ways of counting the deceased. A careful investigation should be performed. Other vaccines are used worldwide such as Sinovac, Novavax. However we have not found any country using these vaccines and providing complete data. In the supplement we discuss some partial results from the incomplete data available from Mongolia (not the ratios), which seem to confirm the quality of these vaccines as compared to the others. As we discussed above, the divergence seen at low χ is due to low statistics data at the beginning of the pandemics, i.e., early 2020. In order to compare different responses on the same scale we define the ratio of the period 2021/(2020+2021). In this case if the number of deaths in the period 2021 is much larger than the corresponding period in 2020, the ratio converges to 1. This is seen in figure 5 (bottom panel) for very low χ and it is due to statistics at the beginning of the pandemics. If the vaccines are not working, because of a new variant or loss of efficacity with time, then the ratio should converge to 0.5, dashed line in the figure 5 (bottom panel) . The latter is observed for Italy, USA and Russia (larger than 0.5). Canada seems to be approaching the 0.5 value while the UK and RSM are well is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint below it meaning that the vaccines used in those countries are still effective. We notice the increase of the ratio for the UK at larger vaccination rates, corresponding to later times, which may be due to the choice of the UK government to use other vaccines instead of the AstraZeneca (AZ) for people below 40 years of age. This was due to the claim that the AZ vaccines may provoke blood clots. This claim was largely amplified by the press leading to many countries to halt or slow down the use of this particular vaccines. In the UK, 49 blood clots cases have been reported after 28.5 million AZ vaccines have been administrated, out of about 49 million vaccines administrated in the same period (Pfizer is the other one). This is about 2 cases per million people which is about double the value observed normally in the population, thus a negligible factor. Nevertheless many countries opposed it or slowed down its use (Germany for instance). Clearly the economical factor is huge, recall that the AZ vaccine was led by the university of Oxford-UK with the intent of creating a vaccines readily available to all for a low price, about $2, i.e., an order of magnitude less than other more 'popular' vaccines such as Pfizer and Moderna, see table 2. Our results show a quite different scenario about the vaccines efficacity, thus while rich countries should continue their economical games, 'third world' countries can confidently rely on the AZ vaccine. One question remains is why prices are so much different. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint 

In this work we have explored the similarities and differences of COVID-19 with the Spanish flu of 1918 using the logistic map as guidance. We have made some prediction on the number of cases and estimated the progress reached through vaccinations. We have shown that herd immunity cannot be reached because of the Delta variant. However, the death probability has been largely reduced and it affects almost exclusively the unvaccinated. A comparison of . CC-BY 4.0 International license It is made available under a perpetuity.

is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The most important action in those cases is information. People that gets positive to Covid have 1 probability out of 50 to die. Since about 70-80% of the positives are vaccinated and their probability to die is close to zero, it means that the unvaccinated positive to Covid have probability 1 out of 10-15 to die. Furthermore, given the high probabilities to be positive to the virus (about 20-30% of the tested), sooner or later we may become positive and it is better to be ready, i.e. vaccinated, to reduce the risk practically to zero to die. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 27, 2021. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint

The copyright holder for this this version posted December 27, 2021. ; https://doi.org/10.1101/2021.12.23.21268184 doi: medRxiv preprint

Fractals and chaos in geology and geophysics

Deterministic Chaos (VCH

DENA0003841 (CENTAUR)

Note added in proof: extending the analysis of this paper to December 21, 2021 did not show any relevant changes due to the Omicron variant

designed the research and drafted the work, H.Z. collected the data, conducted the data analysis and revised the draft

This work was supported partly by the National Natural Science Foundation of China (Grant Nos. 11905120 and

The authors declare no competing interests.

In this supplement we discuss different countries, which adopted different strategies to contrast COVID-19. Sweden was one of the few countries letting the disease to spread freely and as a results had the largest number of fatalities as compared to nearby countries like Norway and Denmark, see figure S1. When divided by the population, their death