Effects of behavioral restrictions on COVID-19 spread

Several measures including behavioral restrictions for individuals have been taken to 1 control the spread of COVID-19 all over the world. The aim of these measures is to prevent infected 2 persons from contacting with susceptible persons. Since the behavioral restrictions for all citizens, 3 such as the city-wide lockdown, are directly linked to stagnation of economic activities, the assessment 4 of such measures is crucial. In order to evaluate the effects of behavioral restrictions, we employ 5 the broken-link model to compare the situation of COVID-19 in Shanghai where the lockdown 6 was implemented from March to June 2022 with it in Taiwan where a spread of COVID-19 was 7 known to be well controlled so far. The result shows that the small link-connection probability is 8 achieved by substantial isolation of infected person including the lockdown measures. Although 9 the strict measures for behavioral restrictions are effective to reduce the total infected people, the 10 daily confirmed cases follow the curve which is evaluated by the broken-link model. This result is 11 considered as unavoidable infections for population. 12

COVID-19; epidemic model; broken-link model; lockdown measure; Omicron variant 13 1. Introduction 14 A novel coronavirus occurred in Wuhan in 2019 [1,2] was spread over the world. More 15 than two years later, the virus is still mutating and causing infections around the world, 16 with a cumulative total of 550 million infections and 6.3 million deaths. 17 Several measures have been taken to suppress the spread of COVID-19 all over the 18 world. Basic idea is to prevent infected individuals from contacting with susceptible people. 19 There are roughly two ways which are monitoring method and/or lockdown method to 20 restrict citizen’s activity. 21 The monitoring method is a primitive but efficient strategy to pick up the person 22 who is required a treatment or isolation. In this method, thorough contact tracing and 23 quarantine of infected people are made as needed by collecting positional information of 24 infected individuals. However, due to invasions of privacy, this method has been applied 25 by limited countries, such as China [3,4], Taiwan [5–7] and South Korea [8,9]. 26 The strongest measure to control a spread of infectious diseases is considered to be 27 the (local) lockdown which forces one not to contact with the others in the community 28 regardless of one’s health condition. The lockdown method has been implemented in many 29 countries and is believed to have reduced the size of the spread of infectious diseases and 30 avoided the collapse of the medical system caused by the rapid increase in the number of 31 infected patients. However, it is difficult to sustain the lockdown on a long-term since it 32 has caused significant damage to people’s daily lives and economic activities. 33 China is still trying to achieve "zero COVID" by the monitoring method with the 34 help of Information Technologies. Indeed, it is credited with suppressing the numbers of 35 COVID-19 patients from 2020 to 2021 after the outbreak in Wuhan, reducing the cases to a 36 significantly lower level. However, after two years from outbreak in Wuhan, the situation 37 has changed by emergence of Omicron variant which has a strong infectivity via not only 38 droplet but also aerosol transmission and can easily breaks through immunity gained by 39 Version August 8, 2022 submitted to Journal Not Specified https://www.mdpi.com/journal/notspecified All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for thisthis version posted August 9, 2022. ; https://doi.org/10.1101/2022.08.08.22278490doi: medRxiv preprint NOTE: This preprint reports new research that has not been certified by peer review and should not be used to guide clinical practice. Version August 8, 2022 submitted to Journal Not Specified 2 of 8 vaccination. Due to the surge of COVID-19 by Omicron variants, massive outbreaks have 40 occurred in many parts of the world, and the lockdown method has been made to subside 41 it in China. 42 There have been some evaluations of the extent to which the local lockdown or behav- 43 ioral restrictions of infected persons have reduced the spread of infectious diseases [10–13]. 44 This is because the counterfactual assumption based on the mean field SIR model over 45 the population in which the cases continue to increase by the basic reproduction number, 46 R0, for every infected persons in the infection period till the hard immunity is achieved. 47 Consequently it would lead to the conclusion that any measures are always effective if the 48 daily confirmed cases tend to be reduced if it is not considering the transmission networks 49 between individuals. The goal of this paper is to avoid such problems and to estimate the 50 effects of the measures to restrict social activity by using the broken-link model. 51 2. Methods 52 We briefly review the broken-link model [ 14,15] which is proposed as a new com- 53 partment model with the consideration of unconnected inflectional transmission link. The 54 cumulative number of cases in this model is described by the Gompertz function with three 55 parameters as the cumulative number of infected people N∞, the connection probability 56 of transmission links k, and the basic reproduction number R0 = −a/ lnk with constant a. 57 These parameters are determined by fitting procedure to the reported data. 58 In SIR model which is traditional compartment model first developed by Kermack 59 and McKendrick in 1927 [16], the basic reproduction number R0 is the averaged number of 60 cases directly generated by one case in populations where all individuals are susceptible to 61 infections. The R0 is composed by several factors including the duration of infectivity of 62 affected individual, the infectiousness of the virus, and the contact frequency of infected 63 people in the population, so that it is determined on the regional basis. The R0 in the 64 broken-link model is defined in the similar sense and can be compared with that in SIR 65 model. 66 The surge of infectious disease analysed the broken-link model always subsides 67 without any measure because the infected people is quarantined by oneself if PCR testing 68

is positive either symptomatic or asymptomatic, which leads to the k less than 1. 69 The small k is favorable for suppression of infectious diseases and is achieved by a strong 70 quarantine system under the monitoring method. 71 In this paper, the effects of behavioral restriction including lockdown are evaluated by 72 comparing the sizes and link connection probabilities in two surges in Taiwan where the 73 COVID-19 was well controlled without lockdown and one surge in Shanghai where the 74 lockdown was implemented for suppressing the Omicron surge. 75 3. Results 76 3.1. The Alpha surge in Taiwan 77 From the beginning of May to the mid July 2021, the surge of COVID-19 by Alpha 78 variant of the novel coronavirus occurred in Taiwan. Though measures such as school 79 blockages were taken, the lockdown measure was not taken. But “zero COVID” policy 80 with redthe monitoring method took place in this period. As a result of Alpha surge, about 81 13 thousands people were infected in Taiwan. 82 The epicurve whose numbers are reported by the Center for Systems Science and 83 Engineering (CSSE) at Johns Hopkins University [ 17] is fitted by reda single Gompertz 84 curve called a wave. Both the numbers of the daily confirmed cases shown in Fig. 1 (a) and 85 the K-values in Fig. 1 (b) are well reproduced with the fit parameters listed in Table1. It 86 is interesting to compare the k in the Alpha surge with it in the first surge in Taiwan from 87 20th March to 10th April 2020. The link connection probability in the Alpha surge is larger 88 than k ≃ 0.85 (K′ ≃ 0.0524) in the first surge [14]. 89 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for thisthis version posted August 9, 2022. ; https://doi.org/10.1101/2022.08.08.22278490doi: medRxiv preprint Version August 8, 2022 submitted to Journal Not Specified 3 of 8 This leads that, under the same policy, the k is more or less the same in magnitude. 90 Effects of changing from k = 0.85 to k = 0.90 is about 5 times enhancement in the number 91 of daily confirmed cases at a perk. 92 (a) (b) Figure 1. The logarithmic plot of the epicurve for the Alpha surge of COVID-19 in Taiwan from May to July 2021. (a) The he daily confirmed cases and fit results (solid curve). The fit is performed with a single partial wave denoted by dashed lines. (b) The observed data and fit result of the K-value. The red bands stand for systematic errors for the choice of fit range of the partial wave. Table 1. The parameters of Gompertz curves for the Alpha surge in Shanghai. The N∞, R0 and k are the cumulative number of infected people, the basic reproduction number and connected probability of transmission links, respectively. The “Shift” stands for the onset of Gompertz curves from the

date (5th May 2021). partial wave N∞ R0 k Shift 1st wave 14k 4.92 0.895 7.0 3.2. The Omicron surge in Taiwan 93 For the period of the Omicron surge in Taiwan, they changed the policy of "zero 94 COVID" to "pandemic management" policy [18–20]. After the mitigation of measures, the 95 daily confirmed cases markedly increased by proliferation of BA.1.1 and BA.2 [21] in terms 96 of PANGO (Phylogenetic Assignment of Named Global Outbreak) Lineages [22]. As shown 97 in Fig. 2, the Omicron surge in Taiwan is decomposed into two waves whose onsets are 98 synchronized to the emergence of new variants of coronavirus. From Tab. 2, the size of the 99 first wave is quite small comparing with the size of second wave. Thus we mainly discuss 100 the second wave though the first wave is indispensable to reproduce the epicurve. 101 It is notable that the ks for both 1st and 2nd wave are much larger than the previous 102 waves in Taiwan because of the mitigated policy. The relaxations of measures are evaluated 103 as the change of k from that in the Alpha surge which is about 0.05 leads to about 15 times 104 enlargement of the number of daily confirmed case at a peak from Fig. 5 in Ref. [15]. It is 105 concluded that "zero COVID" policy without lockdown sufficiently reduces the value of 106 the probability k, when we compare the values for the Alpha surge. Also the value of k in 107 the Omicron surge is comparable with that in Japan [14,15], where the monitoring method 108 was not applied. 109 It is also worth noting that Taiwan did not change any major policy during the period 110 of the Omicron surge. This fact is consistent with the natural suppression of infectious 111 diseases due to voluntary behavioral changes assumed in the broken-link model. 112 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for thisthis version posted August 9, 2022. ; https://doi.org/10.1101/2022.08.08.22278490doi: medRxiv preprint Version August 8, 2022 submitted to Journal Not Specified 4 of 8 (a) (b) Figure 2. The logarithmic plot of the epicurve for the Omicron surge of COVID-19 in Taiwan from April to July 2022. (a) The daily confirmed cases and fit results (solid curve). The fit is performed with two partial waves denoted by dashed lines. (b) The observed data and fit results of the K-value. The red bands stand for systematic errors for the choice of fit range of the first and second partial wave. Table 2. The parameters of Gompertz curves in the Omicron surge in Taiwan. The definition of the parameters is the same as in Table 1, but the reference date is 3rd April 2022. partial wave N∞ R0 k Shift 1st wave 110k 7.00 0.943 -4.7 2nd wave 4126k 10.63 0.946 10.5 3.3. The Omicron surge in Shanghai 113 In Shanghai, the Omicron surge occurred in early March is analyzed based on the 114 broken-link model. In order to control the surge of COVID-19, Shanghai commenced the 115 city-wide lockdown in late March, dividing the city into east and west districts (from 116 28th March for the east district and from 1st April for the west district). In each of the 117 east and west districts, all residents were given two PCR testings at the beginning of 118 lockdown. While those who tested positive were sent to a medical facility, those who had 119 close contacts with a tested positive person were placed in an isolation facility. This is 120 called the half-lockdown [23]. 121 Subsequently, the full-lockdown [24–26] was commenced on 11th April, in which the 122 entire city was managed in three areas as follows, 123 1. "blockade area" : The area was closed and one was not allowed to leave the house. An 124 attendant visit them to supply foods or if necessary. 125 2. "controlled area" : Walks, etc. within a small area were allowed, but it was prohibited 126 to gather or leave one’s area. 127 3. "precautionary area" : One could do anything only within the administrative ward. 128 These measures were important to see how effective severe behavioral restrictions on 129 people had been against the spread of infectious diseases. 130 Fig. 3 (a) shows the daily confirmed cases including both symptomatic and asymp- 131 tomatic cases in Shanghai from mid March to June 2022 and the corresponding K-values 132 are given in Fig. 3 (b). These curves are plotted with the data reported by the China-CDC 133 (Chinese Center for Disease Control and Prevention) [27]. It is worth mentioning that the 134 epicurve has a peak just after the commencement of full-lockdown measures. 135 This fact does not lead the conclusion that only the full-lockdown is meaningful to 136 reduce the daily cases even the half-lockdown is comparatively strong measure. 137 The broken-link model revealed that the epicurve was consisted of two partial waves. 138 As shown in Fig. 3 (a), the magnitude of 1st wave was much smaller than that of 2nd 139 wave. The curve by the broken-link model nicely reproduce the epicurve and K-values 140 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for thisthis version posted August 9, 2022. ; https://doi.org/10.1101/2022.08.08.22278490doi: medRxiv preprint Version August 8, 2022 submitted to Journal Not Specified 5 of 8 in Shanghai. The 1st and 2nd wave are considered to be caused by BA.2 and by BA.2.2, 141 respectively. The parameters of each partial wave are given in Table3. 142 (a) (b) Figure 3. The logarithmic plot of the epicurve for the Omicron surge of COVID-19 in Shanghai from March to June 2022. (a) The daily confirmed cases and fit results (solid curve). The fit is performed with two partial waves denoted by dashed lines. (b) The observed data and fit results of the K-value. The red bands stand for systematic errors for the choice of fit range of the first and second partial wave. Table 3. The parameters of the Gompertz curves in the Omicron surge in Shanghai. The definition of the parameters is the same as in Table 1, but the reference date is 18th March 2022. partial wave N∞ R0 k Shift 1st wave 98k 6.89 0.865 -0.1 2nd wave 565k 8.64 0.891 8.0 4. Discussion 143 We analyze the daily confirmed cases for the Alpha and the Omicron surge in Taiwan 144 and the Omicron surge in Shanghai by using the broken-link model. These data are nicely 145 described by the combination of partial wave components. 146 In Taiwan, though the lockdown method was not taken for both the Alpha and the 147 Omicron surge periods, they changed from "zero COVID" policy [5] in period of the Alpha 148 surge to the "pandemic management" policy [ 18–20] in period of the Omicron surge by 149 mitigating the measures. The drastic policy change in Taiwan decreased the broken-link 150 probability 1 − k, which cause about 15 times larger number of daily confirmed cases at a 151 peak position. 152 We also find that the k is similar in magnitude under the same policy. This fact is 153 helpful to predict the size of infectious diseases by the broken-link model. 154 Next, we discuss the Omicron surge in Shanghai. As shown in Fig. 3, the broken-link 155 model nicely reproduces both daily confirmed cases and the K-values. From Table 3, both 156 of these two waves have large basic reproduction numbers as R0 = 6.89 and 8.64, while 157 both of ks are at most 0.89 which is quite smaller comparing with the other countries under 158 the Omicron surge [15]. In spite of large R0s in both waves, the cumulative cases in period 159 of the Omicron surge is only up to 2.4% of population because of small k. The k depends 160 on a level of behavioral restrictions, political measures, immunity of human, etc. Of cause, 161 it also depends on the strictness of lockdown measures but is considered to be far less 162 sensitive under the stringent monitoring the activity of people under quarantine or isolation 163 because whole epicurve can be reproduced with single k including before and after the 164 commencement of lockdown. We rather have to take care of the risk on the commencement 165 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for thisthis version posted August 9, 2022. ; https://doi.org/10.1101/2022.08.08.22278490doi: medRxiv preprint Version August 8, 2022 submitted to Journal Not Specified 6 of 8 of lockdown since residents are converged for hoarding commodities against it. Such 166 behavior of the population possibly enhances COVID-19 patients. 167 As a counterfactual hypothesis, we consider the 2nd partial wave in the Omicron 168 surge in Shanghai by changing solely the link-connected probability to k = 0.920 which is 169 the typical value in Japan [14,15]. The result of a counterfactual hypothesis shown in Fig. 4 170 indicates that the maximum number of infected persons per day is 23 times larger and the 171 size of infected people is 30 times larger than the actual numbers. This can be considered 172 as the effect of stringent behavioral restrictions against a COVID-19 spread. 173 Figure 4. The logarithmic plot of the counterfactual hypothesis curve in the 2nd wave in Shanghai by changing the k from 0.891 to 0.920 which is typical value in Japan [14,15]. The red bands stand for the fluctuations with the ±0.05 changes in k. It is interesting to compare the results of the Omicron surge in Taiwan as the mitigated 174 behavioral restrictions and in Shanghai as the stringent behavioral restrictions with the 175 similar population size. The size of cumulative cases and the link connection probability 176 in Taiwan is 6 times and 5% larger than that in Shanghai, respectively. This difference is 177 attributed to the mitigation behavioral restriction policy to recover the economical activity 178 to the normal. Therefore, we have to think how to save the citizen’s live and economy from 179 infectious diseases with the considerations of the effectiveness of behavioral restriction 180 measures and the virulence of causative viruses. 181 5. Conclusions 182 We found that the link connection probability k defined in the broken-link model 183 which mainly controls the size of cumulative cases is insensitive to lockdown measures if 184 the activity of people under isolation or quarantine is strictly monitored. Although there 185 is discussion that the behavioral monitoring of person may be an invasion of privacy, the 186 stringent monitoring method are effective to suppressing the sarge of infectious disease. 187 On the other hand, we found that the link connection probability k is not to be very small 188 even if the strict lockdown is taken. It indicates that there are unavoidable infections 189 or transmission from one’s intimate neighbors if infectious disease with high infectivity 190 intrude one’s community. 191 Author Contributions: All authors contributed to the interpretation of the results obtained in this 192 study and the final manuscript. 193 Funding: Not applicable 194 Institutional Review Board Statement: Not applicable 195 Informed Consent Statement: Not applicable 196 Data Availability Statement: Not applicable 197 Acknowledgments: We thank all the member of Division of Scientific Information and Public Policy 198 (SiPP) at Center for Infectious Disease Education and Research (CiDER) Osaka University for useful 199 All rights reserved. No reuse allowed without permission. perpetuity. preprint (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in The copyright holder for thisthis version posted August 9, 2022. ; https://doi.org/10.1101/2022.08.08.22278490doi: medRxiv preprint Version August 8, 2022 submitted to Journal Not Specified 7 of 8 discussions. This research was supported by The Nippon Foundation - Osaka University Project for 200 Infectious Disease Prevention. 201 Conflicts of Interest: None declared. 202

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