key: cord-1052476-ii0ceksc
authors: Ito, K.; Piantham, C.; Nishiura, H.
title: Estimating relative generation times and relative reproduction numbers of Omicron BA.1 and BA.2 with respect to Delta in Denmark
date: 2022-03-04
journal: nan
DOI: 10.1101/2022.03.02.22271767
sha: 8dc770573e1609e5bca348f0d922fe9850410fee
doc_id: 1052476
cord_uid: ii0ceksc

The Omicron variant is the most transmissible variant of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) we had so far. The BA.1 and BA.2 sublineages of Omicron are circulating in Europe and it is urgent to evaluate the transmissibility of these sub-lineages. Using a mathematical model describing trajectories of variant frequencies that assumes a constant ratio in generation times and a constant ratio in effective reproduction numbers among variants, trajectories of variant frequencies in Denmark from November 22, 2021 to February 26, 2022 were analyzed. We found that the generation times of Omicron BA.1 and BA.2 are 0.60 (95%CI: 0.59-0.62) and 0.51 (95%CI: 0.50-0.52) of the length of that of Delta, respectively. We also found that the effective reproduction number of Omicron BA.1 is 1.99 (95% CI: 1.98-2.02) times and that of Omicron BA.2 is 2.51 (95% CI: 2.48-2.55) times larger than the effective reproduction number of Delta. The generation times of Omicron BA.2 is 0.85 (95% CI:0.84-0.86) the length of that of BA.1 and that the effective reproduction number of Omicron BA.2 is 1.26 (95% CI:1.25-1.26) times larger than that of Omicron BA.1. These estimates on the ratio of generation times and the ratio of effective reproduction numbers has epidemiologically important implications. The duration of quarantine for people who contacted with an Omicron BA.1 and BA.2 patient can be reduced to 60% and 51% of that for Delta, respectively. The control measures against Omicron BA.1 and BA.2 need to reduce contacts between infectious and susceptible people respectively by 50% (95% CI: 49-50%) and 60% (95% CI: 60-61%) compared to that against Delta to achieve the same effect on their control.

The Omicron variant is the most transmissible variant of the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) we had so far. The variant was designated as a variant of concern by the World Health Organization (WHO) on November 26, 2021 (World Health Organization, 2021 . As of February 7, 2022, infections by Omicron were reported to WHO from official sources of 159 countries (World Health Organization, 2022) . The rapid replacement of Delta by Omicron followed by a steep rise in SARS-CoV-2 infections has been observed after the introduction of Omicron in many countries, indicating that Omicron has considerably higher transmissibility than Delta. Furthermore, the BA.2 sublineage of Omicron is replacing the BA.1 sublineage in Denmark. Although attenuated disease severity of Omicron was reported (Halfmann et al., 2022; Shuai et al., 2022) , there is an urgent need to evaluate the transmissibility of these Omicron sub-lineages.

Transmissions of a new variant should be characterized by two factors associated with the transmission of the virus. The first factor is effective reproduction number ( ! ), which measures how many secondary cases are generated by a single primary case at time . The second factor is the generation time distribution ( ), which describes how secondary infections are distributed as a function of time since the infection of primary case (Fraser, 2007) . Alex Selby has estimated the generation time of Omicron using case counts in England in 2021 (Selby, 2022) . In a logistic regression assuming that the generation time of Omicron is the same of that of Delta, the log odds of Omicron and Delta frequencies should be on a straight-line relationship. However, the case counts in England showed a bend in the log odds of observations that deviated from its theoretical straight line. Using this bend, Selby estimated the ratio of the mean generation time of Omicron to the mean generation time of Delta to be 0.46 (95% confidence intervals (CI): 0.38-0.61).

Laung et al. have developed a mathematical model describing trajectories of variant frequencies that assumes constant ratios of effective reproduction numbers among variants (Leung, Lipsitch, Yuen, & Wu, 2017; Leung, Shum, Leung, Lam, & Wu, 2021) . This method needs daily numbers of infections of each variant. In our previous paper we have proposed an approximated form of the model which is free from using daily numbers of infections (Kimihito Ito, Chayada Piantham, & Hiroshi Nishiura, 2021 ). Using the model, we have analyzed nucleotide sequences of SARS-CoV-2 collected in Denmark (K. . This model, however, assumed that generation times of variants followed the same probability distribution and that effective reproduction numbers of Omicron BA.1 and BA.2 were the same. In this paper, we propose a new model in which the generation times of variants may be different from each other. Analyzing latest variant count data observed in Denmark using the new model, we compare generation times and effective reproduction numbers of Omicron with those of Delta and compare effective reproduction numbers of Omicron BA.1 and Omicron BA.2 with respect to (w.r.t.) Delta.

Consider a situation where viruses of variant are circulating in the population at the beginning of the target period of analysis and variant " and # are newly introduced to the same population at calendar times $ ! and $ " , respectively. Let % ( ), $ ! ( ), and $ " ( ) be frequencies of variant , " , and # in the viral population at calendar time , respectively.

Since there is no infection of " before its introduction, $ ! ( ) = 0 for a calendar time < $ " , and the same is true for # for < $ # .

The dynamics on the frequencies of infections by , " , and # is determined by the generation times and effective reproduction numbers of infections of the two variants. We assume that the generation time (GT) of infections by " and # are " and # times longer than that of , respectively. We call the value of & the relative generation time of & with respect to . Let % ( ), $ ! ( ), and $ " ( ) be probability density functions of generation time distributions of infections by , " , and # , respectively. The assumption on generation times can be described by the following equation:

(1)

for any generation time ≥ 0. To deal with the calendar time system, we truncate the generation time distributions at = 1 and = and discretize it as follows:

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Let ( ) be the total number of new infections by either , " , or # at calendar time , and let % ( ), $ ( ), and $ ( ) be effective reproduction numbers of variants , " , and # at calendar time , respectively. From the definition of the instantaneous reproduction number (Fraser, 2007) , % ( ), $ ! ( ), and $ " ( )can be written as follows:

We assume that a patient infected by " and # generates respectively " and # times as many secondary transmissions as those of a patient infected by regardless of . Using effective reproduction numbers of , " , and # , this assumption can be described by the following equation:

Under the two epidemiological assumptions described by Equations (1) and (6), the frequency of variant & in the viral population at time is now modeled as . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

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To allow statistical estimation of & and & without knowing ( ), we assume that the number of new infections does not greatly vary within a single generation of transmission from time

for each calendar time . Then we obtain our main formula:

Note that Equation (8) is used just for approximating Equation (7) with Equation (9) for each calendar time . We do not assume that ( ) is constant during the entire period of the analysis, although a chain of the assumptions leads such a consequence.

Let % ( ), $ ! ( ), and $ " ( ) be the number of variants , " , and # observed at calendar time . Assuming that variants , " , and # is sampled following a multinomial distribution

and & and the initial variant frequencies of $ ! R $ ! S and $ " ( $ " ), can be estimated by maximizing the likelihood function of the multinomial distributions, which is described as follows:

As of February 28, 2022, the earliest Danish Omicron sequences registered in the GISAID database (Shu & McCauley, 2017) were sampled on November 22, 2021. We downloaded a . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) Hart et al. reported that the distribution of the generation time of the Delta variant had a mean of 4.7 days with a standard deviation of 3.3 days by analyzing household transmission data from UK Health Security Agency (Hart et al., 2021) . Based on the mean and the standard deviation, the generation times of infections by Delta were assumed to follow the gamma distribution with a shape parameter of 2.03 and a scale parameter of 2.32. We used the probability density function of this gamma distribution as % ( ).

We estimated the parameters of our model from the dataset using three assumptions: The One-GT model, the Two-GT model, and the Three-GT model. The Three-GT model assumes that the GT distribution of Omicron BA.1 and BA.2 may be different from Delta in a manner described by Equation (1) and that the reproduction number of Omicron BA.1 and BA.2 may be different in a manner described by Equation (6). More precisely, " may be different from # and " may be different from # . The Two-GT model assumes that BA.1 and BA.2 shares the same generation time distribution. In other words, " must be equal to # and " may be different from # . The One-GT model assumes that the GT distributions of Omicron BA.1 and BA.2 infections are the same as that of Delta, i.e. " = # = 1, and that the reproduction number of Omicron may be different in a manner described by Equation (6). Note that " = .

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The copyright holder for this preprint this version posted March 4, 2022. ; https://doi.org/10.1101/2022.03.02.22271767 doi: medRxiv preprint # in the Three-GT model implies the Two-GT model and that " = # = 1 in the Three-GT model implies the One-GT model. The 95% CIs are calculated using the profile likelihood method (Pawitan, 2013) . The maximum likelihood estimation and the 95% CI calculation were done using daily count data from November 22, 2021 to February 22, 2022. We used the augmented Lagrangian algorithm implemented in the NLopt module of the Julia language (Conn, Gould, & Toint, 1991) for the maximum likelihood estimation and the calculation of 95% CIs of parameters. Models were compared using the Akaike information criterion (AIC) (Akaike, 1974) .

The relative generation time of Omicron w.r.t. Delta, & , and the relative reproduction number & , were estimated from the daily count of Delta and Omicron BA.1 and BA.2 in Denmark ( Table 1 ). The Three-GT model estimated " and # to be 0.60 (95%CI: 0.59-0.62) and 0.51 (95%CI: 0.50-0.52) and " and # to be 1.99 (95% CI: 1.98-2.02) and 2.51 (95% CI: 2.48-2.55), respectively. The Two-GT model estimated " (= # ) to be 0.63 (95%CI: 0.61-0.67), " and # to be 2.03 (95%CI: 1.99-2.06) and 2.63 (95%CI: 2.57-2.72), respectively. The One-GT model estimated " and # to be 2.40 (95%CI: 2.38-2.42) and 3.66 (95%CI: 3.63-3.69, respectively. 

2.2 (1.9-2.4) ×10 -4 5.8 (5.4-6.3) ×10 -4 5.5 (5.1-6.1) ×10 -4

The maximum log likelihood of the Three-GT model was -651.4 with an AIC of 1320.8 and the Two-GT model was -670.4 with an AIC of 1356.8 while the maximum log likelihood of the One-GT model was -797.6 with an AIC of 1356.8 ( Table 2) . The values of AIC suggested that the selective advantage of Omicron over Delta can be decomposed into . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 4, 2022. . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 4, 2022. ; . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted March 4, 2022. ; B . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) contacts between infectious and susceptible people by 50% (95% CI: 49-50%) compared to that against Delta to achieve the same effect on their control. In the same manner, control measures against Omicron BA.2 need to reduce contacts by 60% (95% CI: 60-61%).

Second, generation times of Omicron BA.1 and BA.2 are respectivery 0.60 (95%CI: 0.59-0.62) and 0.51 (95%CI: 0.50-0.52) of the length of that of Delta. This suggests that the time needed for quarantine of people contacted with an Omicron patient may be reduced to 60% (95%CI: 48-53%) of time needed for people contacted with a Delta patient. The control of infections by Omicron needs to consider trade-off between the effort to prevent virus carriers to contact with others and the effort to encourage non-carrier essential workers to continue working. The estimates obtained by this study gives important evidence to establish a separated control measure for Omicron.

In our previous paper, we have estimated the relative reproduction number of Omicron w.r.t. Delta to be 3.19 (95%CI: 2.82-3.61) using variant frequencies observed in Denmark from November 1, 2021 to December 9, 2021 (K. . The discrepancy in the relative reproduction numbers from this study is attributed to the following reason. assumed the same generation time for Omicron and Delta. In fact, the estimate using the One-GT model estimated relative reproduction numbers of 2.40 (2.38-2.42) for BA.1 and 3.66

(3.63-3.69) for the BA.2 w.r.t Delta. Both BA.1 and BA.2 were circulating in Denmark in December 2021, and the relative reproduction number estimated by our previous study is consistent with results of this study.

The estimations of this study completely depend on the variant counts based on the nucleotide sequences submitted to the GISAID database from Denmark. Most sequences from Demark were submitted by member laboratories of Danish Covid-19 Genome Consortium, which analyzed positive specimen from hospitals (Danish Covid-19 Genome Consortium, 2021). As of February 24, the sequencing rate from the 47th week of 2021 to the 7th week of 2022 is 7.2% (157,049 / 2,156,159). Given such a high sequencing rate, the . CC-BY-NC-ND 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted March 4, 2022. extent of sequencing bias may be minimal. Evaluation of sequences submitted from other countries may confirm the results of present study.

the author/funder, who has granted medRxiv a license to display the preprint in perpetuity

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We gratefully acknowledge the laboratories responsible for obtaining the specimens and the laboratories where genetic sequence data were generated and shared via the GISAID Initiative, on which this research is based. The information on originating laboratories, submitting laboratories, and authors of SARS-CoV-2 sequence data can be found in Supplementary Table 1 

Supplementary Table 1 contains all data needed to reproduce the result of this study.

We declare that there is no conflict of interest.

KI and HI design the study. KI and CP collected data and conducted estimation. HN addressed public health implications. KI, CP, HN wrote the paper.