Introduction

The fourth year of the COVID-19 pandemic, the high numbers of circulating SARS-CoV-2 variants1,2,3 and re-infected persons4,5,6, the lack of decreasing trends in the global numbers of new daily cases7,8, and the expected very long duration of the Omicron wave9 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 Vj and deaths Dj, the percentages of fully vaccinated people VCj and boosters BCj, and the results of calculation of the effective reproduction number Rj10. Information corresponding to day tj (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 characteristics11 (see formulae (8), (9) and Table 1):

Table 1 Averaged characteristics of the COVID-19 pandemic in 2020, 2021 and 2022.
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The classical SIR-model12,13,14,15,16,17 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 worldwide11,18,19,20. Nevertheless, this model yields only trivial equilibrium point \(I_{**} = R_{**} = 0\) and cannot be used to estimate endemic characteristics. Numerous improvements of this model9,11,21,22,23,24 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 VCj and BCj (green and yellow colors, respectively), Rj values (magenta) and results of calculations of smoothed numbers of new daily cases DVi, deaths DDi, and the case fatality risk CFRi = DDi/DVi 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 number10 was close to the critical value 1.025,26,27,28. In 2022 we see the a sufficient drop in CFI figures.

Figure 1
figure 1

Global characteristics of the COVID-19 pandemic. Green, yellow and magenta lines respectively show VCj, BCj, Rj values listed in8. Solid blue, black and red represent calculations of DVi, DDi and CFRi, respectively (Eqs. (8) and (9)). Corresponding dashed lines represent the average values for different years listed in Table 1. The dotted blue line shows the results of non-linear correlation (Eq. 7).

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Table 2 Accumulated numbers of COVID-19 cases and deaths form JHU dataset8, smoothed daily values of DVi and DDi calculated with the use of (8), (9) and case fatality risk CFRi = DDi /DVi.
Full size table

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 202129,30), as well as the influence of natural immunity24.

To simulate the re-infections, we add terms \(\pm \delta R\) to the first and last differential equations of the classical SIR-model11,12,13,14,15,16,17, which relates the numbers of susceptible S(t), infectious I(t) and removed persons R(t) over time t:

$$\frac{dS}{{dt}} = - \alpha_{{}} SI + \delta R,$$
(1)
$$\frac{dI}{{dt}} = \alpha SI - \rho_{{}} I,$$
(2)
$$\frac{dR}{{dt}} = \rho_{{}} I - \delta R.$$
(3)

Now compartment S(t), which includes the persons who are sensitive to the pathogen, is increasing with rate \(\delta R\) due to the persons who have lost their immunity and can be re-infected. The same term with opposite sign appears in Eq. (3).

Parameters \(\alpha\) (primal infection rate), \(\rho_{{}}\) (removal rate) and \(\delta\) (re-infection rate) can be supposed to be constant for every epidemic wave. Corresponding generalized SIR model, initial conditions and parameter identification procedures (at \(\delta\) = 0) were successfully used in9,11,31 to simulate and predict different waves of COVID-19 pandemic in different countries and worldwide.

The set of differential Eqs. (1)–(3) has the non-trivial equilibrium point32,33,34 \((S_{*} ,I_{*} ,R_{*} )\) corresponding to zero values of the derivatives:

$$S_{*} = \rho /\alpha ,\quad I_{*} = \delta R_{*} /\rho ,\quad R_{*} = \rho I_{*} /\delta .$$
(4)

At the equilibrium, the daily number of new cases DV* must be equal to the daily number of new infections; to the sum of immunized, isolated and dead persons; and to the number of persons who have lost their immunity and moved from compartment R to S, i.e.:

$$DV_{*} = \alpha S_{*} I_{*} = \rho I_{*} = \delta R_{*}$$
(5)

The Jacobian matrix32,33,34 of set (1)–(3) can be written as follows:

$$J = \left\| \begin{gathered} - \alpha I\quad - \alpha S\quad \delta \hfill \\ \alpha I\;\;\;\alpha S - \rho \quad 0 \hfill \\ 0\quad \quad \;\rho \quad \; - \delta \hfill \\ \end{gathered} \right\|$$
(6)

Taking into account (4), the eigenvalues of (6) at the equilibrium point can be calculated as follows:

$$\lambda_{1} = 0,\lambda_{2,3} = \frac{{ - (\alpha I_{*} + \delta ) \pm \sqrt {(\alpha I_{*} + \delta )^{2} - 4\alpha I_{*} (\rho + \delta )} }}{2}$$

Eigenvalues \(\lambda_{2,3}\) have negative real parts, but the presence of a zero eigenvalue \(\lambda_{1}\) 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 \(DV \approx 300,000\) follows from the minimum of DVi values illustrated by the blue solid line and non-linear correlation approach35, which yields the best fitting curve

$$DC \approx \frac{{1.2484 \cdot 10^{8} }}{{(t - 674.5)^{0.9684} }},\quad t > 710$$
(7)

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 higher35,36,37. To estimate the global visibility coefficient let us take the number of cases per million 72,524 registered worldwide as of August 1, 20228. 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 policy38 began to be lifted only after November 30, 202239. Very high numbers of new daily cases in December 2022 reported by the media40 do not look reliable41, 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 201142). 

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 recommended11:

$$\overline{C}_{i} = \frac{1}{7}\sum\limits_{j = i - 3}^{j = i + 3} {C_{j} } ,\quad C_{j} = V_{j} ;D_{j}$$
(8)

Some different smoothing procedure is used in8 based on the values corresponding to the fixed and previous 6 days. To estimate the smoothed numbers of new daily cases DVi, deaths DDi, and the case fatality risk CFRi =DDi/DVi, the numerical derivatives of the smoothed values (8) are calculated as follows11:

$$\left. {DC_{i} \equiv \frac{{d\overline{C} }}{dt}} \right|_{{t = t_{i} }} \approx \frac{1}{2}\left( {\overline{C}_{i + 1} - \overline{C}_{i - 1} } \right),\quad C_{j} = V_{j} ;\,D_{j}$$
(9)

The results of calculations are listed in Table 2.

Human/animals involved

No humans or human data was used during this study.