key: cord-1010606-qp3vfwhq
authors: Piantham, C.; Ito, K.
title: Predicting the time course of replacements of SARS-CoV-2 variants using relative reproduction numbers
date: 2022-03-31
journal: nan
DOI: 10.1101/2022.03.30.22273218
sha: eb74943c8fcba555e7f0d9cf73e782e47b2f46f5
doc_id: 1010606
cord_uid: qp3vfwhq

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has continuously evolved since its introduction to the human population in 2019. Natural selection selects variants with higher effective reproduction numbers, increasing the overall transmissibility of the circulating viruses. In order to establish effective control measures for a new variant, it is crucial to know its transmissibility and replacement time course in early phases of the variant replacement. In this paper, we conduct retrospective prediction tests of the variant replacement from Alpha to Delta in England. Our method firstly estimated the relative reproduction number, the ratio of the reproduction number of a variant to that of another, from partial observations up to a given time point. Secondly, the replacement time course after the time point was predicted based on the estimates of relative reproduction number. Thirdly, the estimated relative reproduction number and the predicted time course were evaluated by being compared to those estimated using the entire observations. We found that it is possible to estimate the relative reproduction number of Delta with respect to Alpha when the frequency of Delta was more than or equal to 0.25. Using these relative reproduction numbers, predictions targeting on 1st June 2021, the date when the frequency of Delta reached 0.90, had maximum absolute prediction errors of 0.015 for frequencies of Delta and 0.067 for the average relative reproduction number of circulating viruses with respect to Alpha. These results suggest that our method allows us to predict the time course of variant replacement in future from partial datasets observed in early phases of variant replacement.

Since its first emergence in the human population in 2019, the severe acute respiratory syndrome 31 coronavirus 2 (SARS-CoV-2) has been generating new variants. Natural selection selects new 32 variants that has higher effective reproduction numbers than other circulating variants. As a 33 result, the average transmissibility in the viral population increases over time [1] . As of date, 

(1) 120

(2) 121

Since the effective reproduction number of variant is times higher than that of variant , the 122 effective reproduction number of variant at time is given by 123

(3) 124

Assuming that the viral population at time comprises of only variants and , the frequency of 125 variant at calendar time , # ( ), can be calculated as 126

(4) 127

We assume that the numbers of new infections do not vary greatly for 20 days, i.e. 128

for > " . Using this approximation with Equations (1), (2), and (3), we can rewrite Equation 130

(4) using # ( − ) for 1 ≤ ≤ 20 as 131

The average relative reproduction number of circulating viruses at time w.r.t. variant is given 133 by 134

. The beta-binomial distribution becomes the binomial 141 distribution when = ∞. The following equation gives the likelihood function of parameters , 142 # ( # ), and for observing * ( ) and # ( ) sequences of variant and at calendar time : 143 predicted by substituting and # ( # ) in Equations (6) and (7), respectively. 166 . 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 31, 2022. ;

Estimation of relative reproduction number from entire observations 168 Parameters of the model were estimated using the entire observations from 18 th March to 4 th July 169 2021 in England (Table 1) Estimation of relative reproduction number from partial observations 182 Table 3 shows the parameters of our model estimated using the partial data collected no later 183 than each maximum likelihood date in Table 2 . The maximum likelihood estimate of using 184 observations of the entire period in the Alpha-Delta replacement was 1.67 (Table 1) (Table 3) . 196 Prediction of variant frequency and average relative reproduction 197 number 198 We conducted retrospective prediction tests on the future frequency of Delta and the average 199 relative reproduction number of circulating viruses w.r.t. Alpha using model parameters in Table  200 3, which were estimated from partial observations. Figure 2 shows predicted trajectories of the 201 Alpha-Delta replacement using partial observations up to different time points in Table 2 (Table 2) . We evaluated 208 the accuracy of predictions by analyzing predictions targeted on these dates ( Figure 3 ). As the 209 relative reproduction numbers were overestimated when predictions were made before 210 frequencies of Delta reached 0.25, the frequencies of Delta on the target dates were also 211 overestimated in these predictions (Figure 3a (Table 5 ). Predictions made when frequencies of Delta were greater than 235 or equal to 0.25 have significantly smaller prediction errors than those made when frequencies of 236 . 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) Table S2 ). This means that the observed variance was larger than the variance of 265 the binomial distribution. The additional variance to the binomial distribution might be attributed 266 . 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. 

Acknowledgement 366

We gratefully acknowledge the laboratories responsible for obtaining the specimens and the 367 laboratories where genetic sequence data were generated and shared via the GISAID Initiative, 368 on which this research is based. The information on originating laboratories, submitting 369 laboratories, and authors of SARS-CoV-2 sequence data can be found in Supplementary Table  370 S1. We thank Brad Suchoski, Heidi Gurung, and Prasith Baccam from IEM, Inc. in the United 371 Education, Culture, Sports, Science, and Technology, Japan. The funders had no role in the study 377 design, data collection and analysis, decision to publish, or preparation of the manuscript. 378

Author contributions . 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. . 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. 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) . 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 31, 2022. ; https://doi.org/10.1101/2022.03.30.22273218 doi: medRxiv preprint

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