key: cord-0921901-yuptabvi authors: Ikeda, Y.; Sasaki, K.; Nakano, T. title: A new compartment model of COVID-19 transmission: The broken-link model date: 2022-03-07 journal: nan DOI: 10.1101/2022.03.04.22271940 sha: 5b438f5555f8d7c6565c3a0d6cdbfd36502a5fa0 doc_id: 921901 cord_uid: yuptabvi We propose a new compartment model of COVID-19 spread, the broken-link model, which includes the effect from unconnected infectious links of the transmission. The traditional SIR-type epidemic models are widely used to analyze the spread status, and the models show the exponential growth of the number of infected people. However, even in the early stage of the spread, it is proven by the actual data that the exponential growth did not occur all over the world. We consider this is caused by the suppression of secondary and higher transmissions of COVID-19. We find that the proposed broken-link model quantitatively describes the mechanism of this suppression and is consistent with the actual data. Since the first patient of novel coronavirus infectious disease (COVID-19) was 20 reported in Wuhan, China, COVID-19 has spread all over the world. In order to save the 21 lives from the threat of COVID-19 and maintain social activities from the viewpoint of 22 economy, it is vital to ascertain accurately the status of the spread. 23 The SIR (susceptible-infected-removed) model and its family such as the SEIR 24 (susceptible-exposed-infected-removed) model have been widely used compartment 25 models trying to describe the projection of COVID-19 spread. The SIR model was first 26 applied to the plague in the island of Bombay over the period Dec. 1905 to July 1906 [1] . 27 The first order coupling between susceptible and infected people was assumed, and 28 such treatment was justified for the plague mediated by carrier rats which form a mean 29 field of the plague, and thus the susceptible people have an equal probability of being 30 infected. Indeed, the calculated epidemic curve during the period of epidemic roughly 31 agreed with the reported numbers. One of typical features of the SIR-type models is that 32 the models predict the exponential growth of the number of infected and removed 33 people for the early stage of the spread [2] . 34 Meanwhile, the indicator of the spread rate, what is called the K-value, defined by 35 ( ) = 1 − ( − 7)⁄ ( ) with ( ) being the cumulative number of confirmed cases at 36 day from a reference date, exhibits nonexponential growth of ( ) even in the early 37 stage of the spread but exhibits approximate linear decrease of the K-value transition 38 universally in many countries [3] . The linearly decreasing behavior of the K-value 39 transition was well reproduced by the phenomenologically developed constant 40 attenuation model [3] , where ( ) is expressed as ( ) = (0) exp( ( ) ), and ( ) is 41 defined by the geometric progression, ( ) = exp[−(1 − )] ( − 1) with a constant 42 attenuation factor . Based on the constant attenuation model, it was found that ( ) 43 follows the Gompertz curve [4] [5] [6] . In this paper, we propose a new compartment model, the broken-link model, in order 45 to microscopically understand why the COVID-19 transmission follows the Gompertz 46 curve. The model is naturally derived from the observation of suppression of COVID-19 47 transmission in the secondary cases generated by the primary ones [7] . We also apply 48 the model to the epidemic surges generated by Delta ( ) and Omicron ( ) variants in 49 Japan, South Africa, Unites States, France and Denmark. To derive the broken-link model, we start with the SIR model. In the SIR model, we 52 partition the total population into three compartments: susceptible, infected and 53 removed individuals, and represent the numbers of three compartments at time by 54 ( ), ( ) and ( ). The SIR model is then described as coupled ordinary differential 55 equations (ODEs), where and are contact and removal rates of infections, respectively. The basic 57 reproduction number is denoted by ! = ⁄ with being the total population 58 number. When the cumulative number of infected persons ( ) is much less than the 59 total population, ( ) can be approximated by . Then one finds the exponential growth 60 of ( ) and ( ) which cannot be inevitable unless the contact and removal rates are 61 assumed to be constant in the period of epidemic. One of good indicators to find out the behavior of COVID-19 transmission is the K-63 value. The analysis using the K-value has revealed that the cumulative number of 64 confirmed cases ( ) follows the Gompertz curve even in the early stage of the spread, 65 where the herd immunity has not been achieved at all. A natural reason is that COVID-66 19 spread through the contact and/or local droplet processes. As reported in Ref. [7] , the 67 secondary transmission generated from the primary cases in non-close environments is 68 highly suppressed. 69 We model the suppression of the secondary and higher transmission in terms of 70 compartment models. According to Ref. [7] , all the transmission links are not connected 71 to the next generation. When the links are connected through the probability , in other 72 words through the broken-link probability (1 − ), the subsequent transmissions are not 73 generated. Therefore, we cut these contributions from a transmission tree as shown in 74 Figure 1 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Now, we formulate the broken-link model. In addition to the , and compartments, 83 it is natural to introduce the ′ (temporary removed) compartment due to unconnected 84 transmission links as shown in Figure 1 (b). The time evolutions of ( ), ( ), and ( ) 85 are respectively expressed by the following coupled ODEs: where ! in Eq. (2) The analytic solutions of ( ) and ( ) are found as with = − ln in Eqs. (3) and (4), and $ = (0) exp( ! ) which represents the 94 cumulative number of infected people in each infection wave generated by a 95 coronavirus with the basic reproduction number ! . In Eq. (6), we can see that the 96 cumulative number ( ) satisfies the Gompertz curve. It also turns out that the 97 probability is equivalent to the constant attenuation factor [3] , so that the 98 phenomenologically introduced constant attenuation is consistent with the suppression 99 of transmissions due to the unconnected transmission links. is close to one. Therefore, even though only the small regional difference in the 106 probability is obtained from the actual data, the orders of magnitudes of the confirmed 107 cases can be largely different. Engineering (CSSE) at Johns Hopkins University [8] . In the analysis, the bump structure 114 appears in daily confirmed cases as a counterpart of the Gompertz function in the 115 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. Table 1 . (which was not certified by peer review) 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 7, 2022. ; https://doi.org/10.1101/2022.03.04.22271940 doi: medRxiv preprint nationwide. As seen in Figure 3 , the decomposition of the surge into two partial waves 143 was justified by the trend of the K-value. 144 The fit parameters of the Gompertz curves for the surge in Japan are summarized 145 in Table 2 . The broken-link probability (1 − ) is smaller than that in the surge, which 146 implies that the subsidence of the surge gets slow comparing to the surge. The 147 cumulative number of confirmed cases in Japan is predicted to be about 5 times larger 148 than that of the case. In this subsection, we survey the status of the and surges in the other countries. variant was reported to spread, we were able to confirm the existence of three partial 165 waves in the surge from November 2021 to mid-February 2022 in Figure 5 (d). Again, we 166 easily see that the number of daily cases at peak was getting larger and larger than each 167 prior wave. The results of the fit parameters are summarized in Table 3 . All rights reserved. No reuse allowed without permission. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. As a new compartment model of COVID-19, we have proposed the broken-link 178 model, where the suppression of secondary and higher transmissions is taken into 179 account. The model predicts the Gompertz curve for the cumulative number of 180 confirmed cases, which is consistent with the observations shown in Figures 2 to 4 . In 181 the model, the shape of epicurves is controlled by the probability , and the magnitude 182 is proportional to exp( ! ) in which the basic reproduction number ! is obtained as 183 for ≅ 1 with a constant . Therefore, the small regional 184 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 7, 2022. ; https://doi.org/10.1101/2022.03.04.22271940 doi: medRxiv preprint difference of the probability observed in Section 3 is enhanced in the numbers of daily 185 and total confirmed cases. Shown in Figure 5 is the predicted dependence of the 186 number of the daily cases at peak in the model. From Ref. [3] , the mean value of the probability in Japan was found to be = 0.92 194 (8% for the broken-link probability), which is also consistent with the 1 st partial waves in 195 the surge as shown in Table 1 . On the other hand, for example in France, the broken-196 link probability was approximately 30% smaller as shown in Table 3 . This difference 197 gives about 12 and 17 times larger in the number of daily cases at a peak and the 198 cumulative number than those in Japan, respectively. The regional difference would be 199 attributed to the immune response to coronaviruses [9] . Indeed, due to double 200 vaccination, about 20% and 30% increases in the broken-link probability were observed 201 for the 2 nd and 3 rd partial waves in Japan, respectively. Thanks to the double 202 vaccination, the 2 nd and 3 rd partial wave were suppressed to about 40% and 30%, 203 respectively. 204 It is notable that the onset of epidemic surges or even partial waves has 205 synchronized the appearance of new variants of coronaviruses in country to country. In It is also important to investigate the situation in other countries with respect to the 215 genomic surveillance. In South Africa shown in Figure 5 were caused by the and variants, respectively. This fact was able to be confirmed by 221 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 7, 2022. ; https://doi.org/10.1101/2022.03.04.22271940 doi: medRxiv preprint the results of ! given in Table 3 , because the ! s in the 1 st and 2 nd waves are consistent 222 with the typical values for the [12] and [13] variants, respectively. The situation is slightly complicate in Denmark shown in Figure 5 (d). The 1 st wave 224 was generated by the variant and the others were caused by the variant. The 225 genomic surveillance report from the outbreak.info [14] indicated that BA.2 cases emerged 226 from mid-December 2021 and became dominant at the end of January 2022. According 227 to the report [14] , we find two points that the 2 nd and 3 rd waves were caused respectively 228 by BA.1 and BA.2, and BA.2 has enough strong infectivity to generate a new wave in 229 daily confirmed cases. We proposed a new compartment model of COVID-19 spread, the broken-link 233 model, which includes the effect from unconnected infectious links of the transmission. 234 The model took into account the suppression of secondary and higher transmissions of 235 COVID-19. The cumulative number of confirmed cases ( ) in the model satisfies the 236 Gompertz curve whose parameters are characterized as the cumulative number of 237 infected people $ , the basic reproduction number ! and the connection probability of 238 transmission links , which was defined as the attenuation factor in the previous paper 239 [3] . The model applied to the actual data for epidemic surges of coronaviruses in Japan, 242 South Africa, Unites States, France and Denmark. From these results with the detailed 243 genomic surveillance, we found that the onset of a partial wave has synchronized the 244 appearance of new variants of coronaviruses and a scale of total infected people is 245 closely related to the probability . The typical value of in Japan evaluated in this 246 study is smaller than those in European countries for the surge, but it gets close to 247 European ones for the surge. A Contribution to the Mathematical Theory of Epidemics Qualitative analyses of communicable disease models Novel Indicator to Ascertain the Status and Trend of COVID-19 Spread: Modeling Study The K indicator epidemic model follows the Gompertz curve Universality in COVID-19 spread in view of the Gompertz function. Progress of 275 Theoretical and Experimental Physics Predicting the trajectory of any COVID19 epidemic from the best straight line. medRxiv, 277 Preprint posted online Closed environments facilitate secondary transmission of coronavirus disease 2019 (COVID-19). medRxiv. 280 Preprint posted online An interactive web-based dashboard to track COVID-19 in real time Identification of TCR repertoires in functionally competent cytotoxic T cells cross-284 reactive to SARS-CoV-2 Ministry of Health, Lavour and Welfare (MHLW) of Japan; Situation report for COVID-19 A dynamic nomenclature 288 proposal for SARS-CoV-2 lineages to assist genomic epidemiology The reproductive number of the delta variant of SARS-CoV-2 is far higher compared to the ancestral SARS-292 CoV-2 virus Omicron variant and booster COVID-19 vaccines We thank Prof. Yoshiharu Matsuura, Prof. Fumio Ohtake and all the member 261 of Division of Scientific Information and Public Policy (SiPP) at Center for Infectious Disease 262 Education and Research (CiDER) Osaka University for useful discussions and comments. We also 263 thank Mr. Toru Ohmuta for the comments.