Endemic characteristics of SARS-CoV-2 infection

The fourth year of the COVID-19 pandemic without decreasing trends in the global numbers of new daily cases, high numbers of circulating SARS-CoV-2 variants and re-infections together with pessimistic predictions for the Omicron wave duration force studies about the endemic stage of the disease. The global trends were illustrated with the use the accumulated numbers of laboratory-confirmed COVID-19 cases and deaths, the percentages of fully vaccinated people and boosters (additional vaccinations), and the results of calculation of the effective reproduction number provided by Johns Hopkins University. A new modified SIR model with re-infections was proposed and analyzed. The estimated parameters of equilibrium show that the global numbers of new daily cases will range between 300 thousand and one million, daily deaths—between one and 3.3 thousand.


Endemic characteristics of SARS-CoV-2 infection Igor Nesteruk
The fourth year of the COVID-19 pandemic without decreasing trends in the global numbers of new daily cases, high numbers of circulating SARS-CoV-2 variants and re-infections together with pessimistic predictions for the Omicron wave duration force studies about the endemic stage of the disease.The global trends were illustrated with the use the accumulated numbers of laboratoryconfirmed COVID-19 cases and deaths, the percentages of fully vaccinated people and boosters (additional vaccinations), and the results of calculation of the effective reproduction number provided by Johns Hopkins University.A new modified SIR model with re-infections was proposed and analyzed.The estimated parameters of equilibrium show that the global numbers of new daily cases will range between 300 thousand and one million, daily deaths-between one and 3.3 thousand.
The fourth year of the COVID-19 pandemic, the high numbers of circulating SARS-CoV-2 variants 1-3 and reinfected persons [4][5][6] , the lack of decreasing trends in the global numbers of new daily cases 7,8 , and the expected very long duration of the Omicron wave 9 make us think about the constant circulation of the pathogen, that is, about the endemic stage of the disease.To illustrate the global trends, we will use the accumulated numbers of laboratory-confirmed COVID-19 cases V j and deaths D j , the percentages of fully vaccinated people VC j and boosters BC j, and the results of calculation of the effective reproduction number R j 10 . Information corresponding to day t j (starting with January 22, 2020) is available in COVID-19 Data Repository by the Center for Systems Science and Engineering (CSSE) at Johns Hopkins University (JHU) 8 .We will use the version the JHU file updated on December 7, 2022 and the smoothed daily and annual characteristics 11 (see formulae (8), (9) and Table 1): The classical SIR-model [12][13][14][15][16][17] relates the numbers of susceptible S(t), infectious I(t) and removed persons R(t) over time t and can be successfully used for simulations of the first COVID-19 waves in individual countries and worldwide 11,18-20 .Nevertheless, this model yields only trivial equilibrium point I * * = R * * = 0 and cannot be used to estimate endemic characteristics.Numerous improvements of this model 9,11,[21][22][23][24] do not take into account re-infections.In this paper we will modify the classical SIR model by adding the re-infection terms, calculate a non-trivial equilibrium point, and estimate its stability and endemic characteristics of SARS-CoV-2 infection.

Results
The solid lines in Fig. 1 illustrate the vaccination levels VC j and BC j (green and yellow colors, respectively), R j values (magenta) and results of calculations of smoothed numbers of new daily cases DV i , deaths DD i , and the case fatality risk CFR i = DD i /DV i listed in Table 2 (blue, black and red, respectively).It can be seen that after October 2020, smoothed global numbers of new daily cases were never less than 300,000.After April 2020, the effective reproduction number 10 was close to the critical value 1.0 [25][26][27][28] .In 2022 we see the a sufficient drop in CFI figures.
To estimate the long term trends we calculated average values of DV, DD, CFI for every year of the pandemic with the use of accumulated characteristics on January 22 and December 31, 2020; January 1 and December 31, 2021; and January 1 and December 6, 2022.The results are listed in Table 1 and shown in Fig. 1 by dashed lines.
The DV values approximately doubled every year, the DD figure in 2021 was 1.8 times higher than in 2020.But in 2022 we see a sufficient decrease in mortality and much lower CFR values.This can be explained as a positive effect of vaccinations (the global VC values tend to some saturation in 2022, see green line), lower pathogenicity of the Omicron strain (which began to spread widely at the end of 2021 29,30 ), as well as the influence of natural immunity 24 .
To simulate the re-infections, we add terms ±δR to the first and last differential equations of the classical SIR- model [11][12][13][14][15][16][17] , which relates the numbers of susceptible S(t), infectious I(t) and removed persons R(t) over time t: (1) dS dt = −αSI + δR,  Taking into account (4), the eigenvalues of ( 6) at the equilibrium point can be calculated as follows: Eigenvalues 2,3 have negative real parts, but the presence of a zero eigenvalue 1 indicates that the system can approach no equilibrium with increasing time.Its asymptotic stability needs further investigations with the use of non-linear methods.
To estimate the endemic global number of new daily cases we can use the average DV value for 2022 (around 1 million), presented in Table 1.More optimistic prediction DV ≈ 300, 000 follows from the minimum of DV i values illustrated by the blue solid line and non-linear correlation approach 35 , which yields the best fitting curve for the 2022 dataset (shown by the dotted blue line).Thus, the global daily numbers of SARS-CoV-2 cases can vary from 300 thousand to one million and the daily numbers of deaths from 1.0 to 3.3 thousand (if we use the CFR value from the Table 1 corresponding to 2022).According to the recent WHO report "nearly 2.8 million new cases and over 13,000 deaths were reported in the week of 9-15 January 2023.In the last 28 days (19 December 2022-15 January 2023), nearly 13 million cases and almost 53,000 new deaths were reported globally" 7 .Calculating the average daily characteristics, we obtain approximately 400,000-464,000 new daily cases and 1857-1893 daily deaths.These figures are consistent with the given theoretical estimates of SARS-CoV-2 endemic characteristics.

Discussion
It is necessary to emphasize that the presented estimates refer to the laboratory confirmed cases only.Due to the high numbers of asymptomatic patients and insufficient testing level, the real number of SARS-CoV-2 cases is much higher [35][36][37] .To estimate the global visibility coefficient let us take the number of cases per million 72,524 registered worldwide as of August 1, 2022 8 .On the same day, the average value was 460,834 for European countries with high enough testing level (more than 3 tests per capita) 35 .It means that the real numbers of cases can be approximately 6 times higher than registered figures.In the future, reducing the testing level can lead to lower numbers of detected cases and deaths.Other restrictions on the accuracy of the given estimates may be related to the situation in mainland China, where the Zero-COVID policy 38 began to be lifted only after November 30, 2022 39 .Very high numbers of new daily cases in December 2022 reported by the media 40 do not look reliable 41 , nevertheless it is better to use the presented estimations excluding the Chinese datasets.
According to the presented estimations, the annual mortality caused by SARS-CoV-2 will range from 365 thousand to 1.2 million.Unfortunately, these figures are higher than in the case of seasonal influenza (between 294 and 518 thousand annual deaths in the period from 2002 to 2011 42 ).

Conclusions
Thus, the proposed modified SIR model with re-infections allowed us to make adequate estimations of the equilibrium (endemic) global daily numbers of SARS-CoV-2 cases and related deaths.In different countries and regions, these characteristics are different and need special investigations.To take into account the influence of other factors (e.g., vaccination and/or testing rates), more complicated mathematical models can be used.

Methods
Since numbers of new cases and deaths are random and characterized by some weekly periodicity, the use of the smoothed characteristics is recommended 11 : Some different smoothing procedure is used in 8 based on the values corresponding to the fixed and previous 6 days.To estimate the smoothed numbers of new daily cases DV i , deaths DD i , and the case fatality risk CFR i =DD i /DV i , the numerical derivatives of the smoothed values (8) are calculated as follows 11 : The results of calculations are listed in Table 2.
Human/animals involved.No humans or human data was used during this study.

(
− C i−1 , C j = V j ; D j

Table 1 .
Averaged characteristics of the COVID-19 pandemic in 2020, 2021 and 2022.Results of calculations are in [bold].

Table 2 .
Accumulated numbers of COVID-19 cases and deaths form JHU dataset 8 , smoothed daily values of DV i and DD i calculated with the use of (8), (9) and case fatality risk CFR i = DD i /DV i .