1. Introduction

2. Materials and Methods

3. Results

3.2. Remodeling

Soon after the omicron BA.1 outbreak, omicron BA.2 emerged [11]. It is essential to include it, too, as it has a larger basic reproduction number (R₀) than BA.1 while inducing strong protection against BA.1 and similar protection as BA.1 against Delta. We, therefore, present here a triple susceptible-infected-recovered-susceptible (SIRS) model [22,23] that accounts for an (exponentially decaying) time-dependent waning immunity and cross-variant immunization [20].

We define the following time-dependent variables:

• $f_D$, $f_O$ and $f_B$ are the fractions of actively infected populations in Delta, Omicron-BA.1, and Omicron-BA.2, respectively.
• $s_D$, $s_O$ and $s_B$are the effective fractions of susceptible populations to Delta, Omicron-BA.1, and Omicron-BA.2 infections, respectively, henceforth "susceptibilities". These variables present an average over the diverse immunity presented in the population, although, in the original SIR model, they simply present the fraction of population that is neither actively infected nor recovered.
• $r_D$, $r_O$ and $r_B$ are the fractions of recovered population from Delta, Omicron-BA.1, and Omicron-BA.2, respectively. The contribution of recovered individuals from previous outbreaks is accounted for in the initial conditions.

In addition, we use the following (time-independent) parameters:

• ${\tau }_D$, ${\tau }_O$ and ${\tau }_B$are the infection time-period of Delta, Omicron-BA.1, and Omicron-BA.2, respectively.
• $R_0^{\left(D\right)}$, $R_0^{\left(O\right)}$ and $R_0^{\left(B\right)}$are the basic reproduction numbers of Delta, Omicron-BA.1, and Omicron-BA.2, respectively.
• ${\tau }_{rD}$, ${\tau }_{rO}$, and ${\tau }_{rB}$ are the corresponding characteristic waning-immunity times, based on exponential decay of the immunity.

The model equations are as follows:

$$\frac{d}{dt}{f}_D=\frac{R_0^{\left(D\right)}}{{\tau }_D}{s}_D{f}_D-\frac{f_D}{{\tau }_D}$$

(1)

$$\frac{d}{dt}{f}_O=\frac{R_0^{\left(O\right)}}{{\tau }_O}{s}_O{f}_O-\frac{f_O}{{\tau }_O}$$

(2)

$$\frac{d}{dt}{f}_B=\frac{R_0^{\left(B\right)}}{{\tau }_B}{s}_B{f}_B-\frac{f_B}{{\tau }_B}$$

(3)

$$\frac{d}{dt}{r}_D=\frac{f_D}{{\tau }_D}-\frac{r_D}{{\tau }_{rD}}$$

(4)

$$\frac{d}{dt}{r}_O=\frac{f_O}{{\tau }_O}-\frac{r_O}{{\tau }_{rO}}$$

(5)

$$\frac{d}{dt}{r}_B=\frac{f_B}{{\tau }_B}-\frac{r_B}{{\tau }_{rB}}$$

(6)

where $s_D$, $s_O$ and $s_B$ take the following expressions, accounting for cross-variant immunization,

$$s_D=1-\left(f_D+r_D\right)-q_{DO}{r}_O-q_{DB}{r}_B$$

(7)

$$s_O=1-\left(f_O+r_O\right)-q_{OD}{r}_D-q_{OB}{r}_B$$

(8)

$$s_B=1-\left(f_B+r_B\right)-q_{BD}{r}_D-q_{BO}{r}_O$$

(9)

In Eqs. (1)-(3), the first term on the right-hand-side (RHS) of each equation represents an infection rate while the second term accounts for the recovery rate. In Eqs. (4)-(6), the first term on the RHS in each equation is the same recovery rate, while the second term is a waning-immunity rate.

In Eqs. (7)-(9), $q_{DO}$ represents the relative mean protection against Delta infection that a newly recovered individual from Omicron gained, and similarly $q_{OD}$ represents the relative mean protection against Omicron infection that a newly recovered individual from Delta gained; $q_{BO}, q_{OB}$, $q_{DB}$, and $q_{BD}$ have a corresponding meaning.

To minimize free-parameters, and due to the similarity between Omicron-BA.1 and Omicron-BA.2, we use $q_{DO}=q_{DB}$, $q_{OD}=q_{BD}$. Asymmetric cross-immunization, supported by antibody immunology studies [24,25], implies $q_{DO}<q_{OD}$, i.e., the protection against Omicron due to Delta past infection is higher than the protection against Delta due to Omicron past infection. However, these antibody immunity assays do not account for the full complexity of the immune system, such as the T-cell immunity, and so are an underestimate of the immunity against Delta due to Omicron past infection. Hence, the estimated ratio $\frac{q_{DO}}{q_{OD}}=0.25$ used in our previous publication [20], which stems from these immunity assays, is an underestimate, which we correct here to be $\frac{q_{DO}}{q_{OD}}=$0.6 to fit better the wastewater data. (In the SI of Yaniv et al (2022) [20]) we considered also $\frac{q_{DO}}{q_{OD}}=$0.5, closer to the ratio used here.)

To account for immunity gained from past pandemic waves and vaccination, the initial recovered fraction of Delta is set to a non-zero value, thereby determining the initial susceptibilities ($t=0$) associated with the three variants.

In addition, we use the following parameter values that fall within the acceptable range of known values to roughly fit the observed data:

• Basic reproduction numbers
  $R_0^{\left(D\right)}=3.7$
  $R_0^{\left(O\right)}=7.3$
  $R_0^{\left(B\right)}=1.5\times R_0^{\left(O\right)}=10.95$
• Infection periods
  ${\tau }_D=11$ (days)
  ${\tau }_O=7$ (days)
  ${\tau }_B=7$ (days)
• Characteristic waning-immunity times
  ${\tau }_{rD}=550$ (days)
  ${\tau }_{rO}{=\tau }_{rB}=400$ (days)
• Cross immunity probabilities
  $q_{OD}=0.66$, $q_{DO}=0.6\times q_{OD}=0.396$
  $q_{BD}=0.66$, $q_{DB}=0.6\times q_{BD}=0.396$
  $q_{OB}=0.8$, $q_{BO}=1.0875\times q_{OB}=0.87$

The following initial conditions are taken: $f_D\left(0\right)=6.641\times {10}^{-3}$, $f_0\left(0\right)=3.3375\times {10}^{-5}$, $r_D\left(0\right)=0.724$, $r_O\left(0\right)=r_B\left(0\right)=0$. The entrance of BA.2 is modelled as a (Dirac-delta function) source rate at day 34 with value ${10}^{-5}$ (${\mathrm{d}\mathrm{a}\mathrm{y}}^{-1}$), introduced on the RHS of Eq. (3).

Eqs. (1)-(9) form a set of non-linear differential equations and are therefore solved numerically. The results for the active infections ($f_D$, $f_O$ and $f_B$), scaled to the observed wastewater peak concentration of the Omicron-BA.1 wave, are shown in figure 2.

Figure 2. SARS-CoV-2 variants copy number from the model, scaled to the observed wastewater concentrations (Figure 1).

Figure 2. SARS-CoV-2 variants copy number from the model, scaled to the observed wastewater concentrations (Figure 1).

Comparing to the wastewater data (figure 1), the BA.1 wave is well described. The subsequent BA.2 wave, roughly 4+ months later, is consistent with the total wastewater viral count and MOH infection data [26]. Soon after BA.1 has peaked, there was a strong drop of Delta, consistent with it going below the detection threshold. This drop continues even after BA.1 dropped to low values, since the BA.2 wave keeps it down. In absolute terms, according to the model Delta is not eradicated completely, but it is kept extremely low.

The BA.2 wave keeps BA.1 dropping too, but not at a constant pace, and we can observe a “shoulder” of it roughly when BA.2 peaks, reminiscent of the “shoulder” in wastewater data (end March-early April). In the absence of BA.2, we would have had a second BA.1 wave coming due to the BA.1 waning immunity, and the “shoulder” is simply the signature left from this hypothetical wave before the BA.2 wave suppresses it. Note that, surprisingly, BA.1 is not detected in wastewater for the subsequent few weeks in April and emerges briefly at end-April (figure 1), which is clearly not predicted by the model.

In figures S2-S4, we perform sensitivity analysis to the variation of a few parameters: (i) $R_0^{\left(B\right)}$(figure S2), ranging from 30%-60% above $R_0^{\left(0\right)}$ [10], showing relatively weak sensitivity of all variants. (ii) $q_{DO}$ and $q_{DB}$ (figure S3), ranging from 40% to 70% of $q_{OD}$ and $q_{BD}$ (respectively), thus describing strong asymmetric cross-immunities between the pairs BA.1 and Delta and BA.2 and Delta. The analysis shows strong sensitivity of the faith of Delta but very weak sensitivity of the two other variants. (iii) $q_{BO}$ (figure S3), ranging from just 117.5% to 91% of $q_{OB}$, thus describing slight asymmetric cross-immunities between BA.1 and BA.2. The analysis shows strong effects on BA.1 and BA.2 and weaker effects on Delta. In particular, the timing of the BA.2 wave is very sensitive to this value, which in turn influences the appearance of the shoulder/peak in BA.1.

The presented model is based on homogeneous infection spreading model and cannot describe spatial (geographic) non-uniformity [27,28,29,30]. We suspect that the brief (single detection) emergence of BA.1 on end-April results from a late infection of one of the city neighborhoods. Unfortunately, the data for different neighborhoods is unavailable to us, so we cannot confirm this hypothesis. Nevertheless, this suggests that detection at multiple locations, even within the same city, could give insight into the spreading pattern of disease [31].

4. Discussion

References

1. WHO COVID-19 Weekly Epidemiological Update - Edition 144 2023.
2. Wang, H.; Paulson, K.R.; Pease, S.A.; Watson, S.; Comfort, H.; Zheng, P.; Aravkin, A.Y.; Bisignano, C.; Barber, R.M.; Alam, T.; et al. Estimating Excess Mortality Due to the COVID-19 Pandemic: A Systematic Analysis of COVID-19-Related Mortality, 2020–21. The Lancet 2022, 399, 1513–1536. [CrossRef]
3. Goswami, S.; Gupta Bakshi, U.; Bhattacharya, S. SARS-CoV-2 and Its Variants. In COVID-19 and SARS-CoV-2: The Science and Clinical Application of Conventional and Complementary Treatments; CRC Press, 2022; pp. 49–60 ISBN 978-1-00-317851-4.
4. Farheen, S.; Araf, Y.; Tang, Y.; Zheng, C. The Deltacron Conundrum: Its Origin and Potential Health Risks. J. Med. Virol. 2022, 94, 5096–5102. [CrossRef]
5. Tao, K.; Tzou, P.L.; Nouhin, J.; Gupta, R.K.; De Oliveira, T.; Kosakovsky Pond, S.L.; Fera, D.; Shafer, R.W. The Biological and Clinical Significance of Emerging SARS-CoV-2 Variants. Nat. Rev. Genet. 2021, 22, 757–773. [CrossRef]
6. Kannan, S.; Shaik Syed Ali, P.; Sheeza, A. Omicron (B.1.1.529) - Variant of Concern - Molecular Profile and Epidemiology: A Mini Review. Eur. Rev. Med. Pharmacol. Sci. 2021, 25, 8019–8022. [CrossRef]
7. Karim, S.S.A.; Karim, Q.A. Omicron SARS-CoV-2 Variant: A New Chapter in the COVID-19 Pandemic. The Lancet 2021, 398, 2126–2128. [CrossRef]
8. Singh, A.K.; Misra, A. Impact of COVID-19 and Comorbidities on Health and Economics: Focus on Developing Countries and India. Diabetes Metab. Syndr. 2020, 14, 1625–1630. [CrossRef]
9. Callaway, E. Heavily Mutated Omicron Variant Puts Scientists on Alert. Nature 2021, 600, 21–21. [CrossRef]
10. Chatterjee, S.; Bhattacharya, M.; Nag, S.; Dhama, K.; Chakraborty, C. A Detailed Overview of SARS-CoV-2 Omicron: Its Sub-Variants, Mutations and Pathophysiology, Clinical Characteristics, Immunological Landscape, Immune Escape, and Therapies. Viruses 2023, 15, 167. [CrossRef]
11. Kliker, L.; Zuckerman, N.; Atari, N.; Barda, N.; Gilboa, M.; Nemet, I.; Abd Elkader, B.; Fratty, I.S.; Jaber, H.; Mendelson, E.; et al. COVID-19 Vaccination and BA.1 Breakthrough Infection Induce Neutralising Antibodies Which Are Less Efficient against BA.4 and BA.5 Omicron Variants, Israel, March to June 2022. Eurosurveillance 2022, 27. [CrossRef]
12. WHO COVID-19 Weekly Epidemiological Update - Edition 115 2022.
13. Collivignarelli, M.C.; Collivignarelli, C.; Carnevale Miino, M.; Abbà, A.; Pedrazzani, R.; Bertanza, G. SARS-CoV-2 in Sewer Systems and Connected Facilities. Process Saf. Environ. Prot. 2020, 143, 196–203. [CrossRef]
14. Martin, J.; Klapsa, D.; Wilton, T.; Zambon, M.; Bentley, E.; Bujaki, E.; Fritzsche, M.; Mate, R.; Majumdar, M. Tracking SARS-CoV-2 in Sewage: Evidence of Changes in Virus Variant Predominance during COVID-19 Pandemic. Viruses 2020, 12, 1144. [CrossRef]
15. La Rosa, G.; Bonadonna, L.; Lucentini, L.; Kenmoe, S.; Suffredini, E. Coronavirus in Water Environments: Occurrence, Persistence and Concentration Methods - A Scoping Review. Water Res. 2020, 179, 115899. [CrossRef]
16. Westhaus, S.; Weber, F.-A.; Schiwy, S.; Linnemann, V.; Brinkmann, M.; Widera, M.; Greve, C.; Janke, A.; Hollert, H.; Wintgens, T.; et al. Detection of SARS-CoV-2 in Raw and Treated Wastewater in Germany – Suitability for COVID-19 Surveillance and Potential Transmission Risks. Sci. Total Environ. 2021, 751, 141750. [CrossRef]
17. Yaniv, K.; Ozer, E.; Lewis, Y.; Kushmaro, A. RT-QPCR Assays for SARS-CoV-2 Variants of Concern in Wastewater Reveals Compromised Vaccination-Induced Immunity. Water Res. 2021, 207, 117808. [CrossRef]
18. Yaniv, K.; Ozer, E.; Shagan, M.; Lakkakula, S.; Plotkin, N.; Bhandarkar, N.S.; Kushmaro, A. Direct RT-QPCR Assay for SARS-CoV-2 Variants of Concern (Alpha, B.1.1.7 and Beta, B.1.351) Detection and Quantification in Wastewater. Environ. Res. 2021, 201, 111653. [CrossRef]
19. Maida, C.M.; Tramuto, F.; Giammanco, G.M.; Palermo, R.; Priano, W.; De Grazia, S.; Purpari, G.; La Rosa, G.; Suffredini, E.; Lucentini, L.; et al. Wastewater-Based Epidemiology as a Tool to Detect SARS-CoV-2 Circulation at the Community Level: Findings from a One-Year Wastewater Investigation Conducted in Sicily, Italy. Pathogens 2023, 12, 748. [CrossRef]
20. Yaniv, K.; Ozer, E.; Shagan, M.; Paitan, Y.; Granek, R.; Kushmaro, A. Managing an Evolving Pandemic: Cryptic Circulation of the Delta Variant during the Omicron Rise. Sci. Total Environ. 2022, 836, 155599. [CrossRef]
21. Dreier, J.; Störmer, M.; Kleesiek, K. Use of Bacteriophage MS2 as an Internal Control in Viral Reverse Transcription-PCR Assays. J. Clin. Microbiol. 2005, 43, 4551–4557. [CrossRef]
22. Nill, F. Endemic Oscillations for SARS-CoV-2 Omicron -- A SIRS Model Analysis. Chaos Solitons Fractals 2023, 173, 113678. [CrossRef]
23. Goncalves, S.; Abramson, G.; Gomes, M.F.C. Oscillations in SIRS Model with Distributed Delays. 2009. [CrossRef]
24. Laurie, M.T.; Liu, J.; Sunshine, S.; Peng, J.; Black, D.; Mitchell, A.M.; Mann, S.A.; Pilarowski, G.; Zorn, K.C.; Rubio, L.; et al. SARS-CoV-2 Variant Exposures Elicit Antibody Responses With Differential Cross-Neutralization of Established and Emerging Strains Including Delta and Omicron. J. Infect. Dis. 2022, 225, 1909–1914. [CrossRef]
25. Suryawanshi, R.K.; Chen, I.P.; Ma, T.; Syed, A.M.; Brazer, N.; Saldhi, P.; Simoneau, C.R.; Ciling, A.; Khalid, M.M.; Sreekumar, B.; et al. Limited Cross-Variant Immunity after Infection with the SARS-CoV-2 Omicron Variant Without Vaccination; Infectious Diseases (except HIV/AIDS), 2022;
26. Data Resources - COVID-19 Data Tracker - IsraeliMinistryofHealthDashboard Available online: https://datadashboard.health.gov.il/portal/dashboard/corona.
27. Tsori, Y.; Granek, R. Epidemiological Model for the Inhomogeneous Spatial Spreading of COVID-19 and Other Diseases. PLOS ONE 2021, 16, e0246056. [CrossRef]
28. Tsori, Y.; Granek, R. Spatio-Temporal Spread of COVID-19: Comparison of the Inhomogeneous SEPIR Model and Data from South Carolina. PLOS ONE 2022, 17, e0268995. [CrossRef]
29. Dos Santos, J.P.C.; Monteiro, E.; Ferreira, J.C.; Teixeira Lemes, N.H.; Rodrigues, D.S. Well Posedness and Qualitative Analysis of a SEIR Model with Spatial Diffusion&nbsp; for COVID-19 Spread. SSRN Electron. J. 2022. [CrossRef]
30. La Torre, D.; Liuzzi, D.; Marsiglio, S. Geographical Heterogeneities and Externalities in an Epidemiological-macroeconomic Framework. J. Public Econ. Theory 2022, 24, 1154–1181. [CrossRef]
31. Yaniv, K.; Shagan, M.; Lewis, Y.E.; Kramarsky-Winter, E.; Weil, M.; Indenbaum, V.; Elul, M.; Erster, O.; Brown, A.S.; Mendelson, E.; et al. City-Level SARS-CoV-2 Sewage Surveillance. Chemosphere 2021, 283, 131194. [CrossRef]