Modelling the effect of lockdown on COVID-19 pandemic in 22 countries and cities

Backgrounds: COVID-19 is currently spreading around the world, and the cumulative number of cases worldwide exceeded 5 million on 23 May 2020 (10:00 GMT+2). At present, many countries or cities have implemented lockdown measures. This study evaluated the inhibitory effect of lockdown measures on the pandemic by the use of lockdown or similar lockdown in 22 countries or cities. Methods: An SEIQR epidemiological model was developed to capture the transmission dynamics of COVID-19. With the data related to COVID-19 from 22 countries or cities, the optimal parameters of the model were estimated, respectively. Results: The average basic reproduction numbers of 22 countries or cities were between 1.5286-3.8067. And Russia Federation, Spain, Italy, France, Germany, the United Kingdom, Singapore, the United States of New York and the United States of New Jersey were hardest hit by COVID-19. Conclusion: Although the pandemic has not been fundamentally controlled for a short time after lockdown, lockdown was proved to be an extremely effective control measure, which significantly scaled the number of patients down, thereby reduced the harmfulness of the


control.
To prevent the virus from spreading, the Chinese government put up the "lockdown" measure in Wuhan on 23 January 2020, which was a new approach curbing infectious diseases [3]. Other countries have followed suit. "Lockdown'' means home quarantine, sealing off the city from all outside for the interruption of the disease transmission. So the schools and businesses are closed, the public should observe home quarantine, keep social distance, stop gathering together and even must insist on wear a mask when they go out for essential activities during the lockdown period [4,5].
In this contribution we developed an SEIQR (susceptible-exposed-infected but not hospitalized-infectious and isolated-recovered) epidemic model to capture the transmission dynamics of COVID-19. In details, based on the cumulative case data and permanent population data of 22 countries or cities, considering the time of lockdown and lifting the lockdown, and with the help of the global optimization algorithm, the parameters of the model were obtained, then the actual confirmed case data were fitted. Further, we considered the trend and scale of the epidemic if the countries or cities were not in lockdown. Our results showed that lockdown measures significantly reduced the number of confirmed cases. In 22 countries or cities, before the lockdown, the basic reproduction numbers were between 1.3884-4.1926; after the lockdown, the basic reproduction numbers were between 1.2141-3.7829; the average basic reproduction numbers were between 1.5286-3.8067. The number of basic reproduction numbers in all countries or cities had decreased sharply after lockdown. Although the epidemic has not been fundamentally controlled for a short time after lockdown in Russia Federation, Spain, Italy, France, Germany, the United Kingdom, Singapore, the United States of New York and the United States of New Jersey owing to their high severity of the epidemics, lockdown remains an extremely effective control strategy that significantly reduces the number of patients and the harmfulness of the pandemic.

Data
This study simulated the cumulative numbers of cases in 22 countries or cities, which were collected and sorted out from four official reports: (1) WHO (World Health Organization) [2] reports the total confirmed COVID-19 daily new cases worldwide, so the data of the 16 countries Department of Health of Italy [35] reports the daily total case number. The data of Wuhan city that we select isn't over until 1 March 2020 (24:00 GMT+8) while the data of other countries and cities isn't over until 19 April 2020 (10:00 GMT+2). The population data of 22 countries or cities researched above are from World Population in 2020 [36] .
Lockdown suppressed the epidemic in the countries with serious epidemics. As we all know, Wuhan, China is the first city to adopt the strict lockdown measures, followed by other cities in China. It was worth mentioning that South Korea adopted a similar lockdown approach, although South Korea failed to implement strict measures to lockdown the city, such as Daegu and Gyeongsangbuk-do, which were seriously affected by the epidemic. The time of lockdown, time of remove the lockdown and the specific measures of lockdown in 22 countries or cities were listed in Supplementary Materials (see Table 3 for details).

The COVID-19 Model
In what follows, the impact of lockdown on the COVID-19 epidemics in the 22 countries or cities was investigated by formulating an SEIQR epidemic model. The total population related to COVID-19, N, was divided into 5 epidemiological subgroups: susceptible, S; exposed, E; infectious but not hospitalized (maybe quarantine at home), I; infectious and isolated Q and recovered, R individuals, thus N=S+E+I+Q+R. Meanwhile,``M '' represents the number of the cumulative infectious and isolated individuals,``C '' denotes the number of the individuals who are identified not to be patients with COVID-19 and removed from the system. Note that the suspected cases during the COVID-19 pandemic may be influenza patients from current influenza season since their symptoms are similar to those of COVID-19 patients. Consequently, the following modelling assumptions were made based on some biological significance: (1) Natural birth and death were ignored since we focused on the short-term transmission dynamics; (2) The individuals in incubation period have the potential to transmit the virus. In addition, infectious and quarantined individuals Q still also have a certain probability to transmit the virus to medical workers. Therefore, the exposed (E) and the infectious and quarantined (Q) were considered infectious, with infectivity reduction factors p and q, respectively; (4) It was assumed that the transmission rate (  ) will decrease and the proportion of confirmed cases ( 1   ) will increase after the lockdown. It was realistic to suppose that more people stay in a safer isolation place on the lockdown day, resulting in a decrease in the number of susceptible people.
These considerations above yielded the schematic flow diagram illustrating the transmission dynamics of the COVID-19 in Fig 1, and then the model was described by the following system of ordinary differential equations: .
The biological meanings and acceptable ranges of all parameters of model (1) were demonstrated in Table 1. The basic reproduction number (R0) represents the number of infected during the initial patient's infectious (not sick) period. This threshold value may determine whether a disease will die out (if R0< 1) or become epidemic (if R0> 1). As far as the epidemic models with complex dynamics, R0< 1 is not only the condition guaranteeing that the disease is extinct, but also the smaller the better. Following Van : .
Here, R01, R02 and R03 represent the average numbers of the infected individuals by a single exposed individual E, infectious but not hospitalized individual I or infectious and quarantined individual Q in a fully susceptible population, respectively. This also suggests that three transmission ways of COVID-19 contribute to the basic reproduction number R0.

Parameter Estimation
The average incubation period ( 1  ) of COVID-19 is 5.1 days [4], so transition rate of exposed individuals E read  =0.1961. We consulted the values of p=0.1 and q=0.38 by Chowell et al. [38], disease-induced death rate d=1.7826  10 -5 from Tang et al. [15], then the appropriate range of p, q, and d could be obtained. Considering that the disease course (1  ) is more than 10 days, and the time requiring to detect a suspected patient ( 1  ) is 5 to 15 days, we thus determined the ranges of parameters  and  , respectively. The lower and upper limits of other parameters and initial values of model (1) were shown in Table 1. By application of model (1), we fitted the numbers of the cumulative confirmed cases (M(t)) to estimate the parameters and initial values through calculating the objective function as follows [39,40]    (Table 4).

Results and Discussion
Detailed results could be found in the second, third and fourth columns of Table 2.
At the same time, we defined the growth ratio of cumulative confirmed cases before the lockdown, indicated the cumulative number of confirmed cases on the day before locking down the country (or city) j and on the fourth day before locking down the country (or city) j, respectively. Detailed results were listed in the last column of Table 2.
In the 22 countries or cities, before the lockdown, the basic reproduction numbers were between 1.3884-4.1926; after the lockdown, the basic reproduction numbers were between 1.2141-3.7829; the average basic reproduction numbers were between 1.5286-3.8067. The results were in good agreement with many existing results [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32]. The basic reproduction numbers in the 22 countries or cities decreased significantly after the lockdown, which suggested that lockdown measures indeed suppressed the epidemics. The basic reproduction numbers after the lockdown were greater than 1, so the epidemic were not fundamentally controlled for a short time after the lockdown (except for Wuhan until 1 March 2020 and other countries or cities until 19 April 2020). At the same time, we also calculated the growth ratio of confirmed cases in the five days before lockdown. This value intuitively reflects the increase rate in cases before the lockdown. Judging from the values of basic reproduction numbers, Russia Federation, Spain, Italy, France, Germany, the United Kingdom, Singapore, New York and New Jersey were very severely affected areas. This conclusion was also consistent with the case data reported by WHO [2].
Combining basic reproduction numbers, lockdown time and case growth ratio, Denmark, Egypt and Singapore adopted lockdown measures earlier when the outbreaks were relatively less severe.

Conclusion
According to the epidemiological characteristics of COVID-19, an SEIQR model was established. Considering the lockdown time and the number of confirmed cases in 22 countries or cities, we simulated the epidemiological parameters of each country or city, and made a detailed comparative analysis on whether to implement the lockdown measures (as shown in Fig. 2). A significant reduction in basic reproduction number (as shown in Table 2) and the number of cases reported by 22 countries or cities is significantly lower than that without lockdown measures (Fig.   2) show that lockdown is an extremely effective control measure.
Currently, COVID-19 still has been rampant around the world, threatening people's health and affecting people's normal lives. Many countries and territories have been carrying out strict measures to lock down the cities and even country. Obviously, timely lockdown could limit the epidemics to the smaller areas and avoid large-scale outbreaks, and thus lockdown is such an extremely effective control measure that can significantly reduce the number of patients and the harmfulness of the epidemic. In the face of the persistent spread of the epidemic, it is necessary for us to consider the measures of lockdown the city in time, which also brings inspiration for similar sudden infectious diseases in the future.

Acknowledgments
We would like to thank anonymous reviewers for very helpful suggestions which improved greatly this manuscript.

Availability of data and materials
All data are publicly available.

Ethics approval and consent to participate
Since no individual patient's data was collected, the ethical approval or individual consent was not applicable.

Consent for publication
Not applicable.  (1). Table 2 -The basic reproduction numbers and growth ratios of cumulative confirmed cases.

Figure Captions
Germany The government issued a nationwide curfew to keep residents staying at home except some certain activities.
People could hitchhike to work. They were able to exercise themselves and made purchases. However more than two people were not allowed unless they came from the same family.