Data-driven analysis of the simulations of the spread of COVID-19 under different interventions of China

ABSTRACT Since February 2020, COVID-19 has spread rapidly to more than 200 countries in the world. During the pandemic, local governments in China have implemented different interventions to efficiently control the spread of the epidemic. Characterizing transmission of COVID-19 under some typical interventions is essential to help countries develop appropriate interventions. Based on the pre-symptomatic transmission patterns of COVID-19, we established a novel compartmental model: Susceptible-Infectious-Confirmed-Removed (SICR) model, which allowed the effective reproduction number to change over time, thus the effects of policies could be reasonably estimated. Using the epidemic data of Wuhan, Wenzhou, and Shenzhen, we migrated the corresponding estimated policy modes to South Korea, Italy, and the United States and simulated the potential outcomes for these countries when they adopted similar policy strategies to China. We found that the mild interventions implemented in Shenzhen were effective in controlling the epidemic in the early stage, while more stringent policies which were implemented in Wuhan and Wenzhou were necessary if the epidemic became severe and needed to be controlled in a short time.


Introduction
Since December 2019, the report of COVID-19 detected was sent to the World Health Organization (WHO) by the Chinese government [23]. China issued a series of intervention policies to reduce the transmission of COVID-19, such as suspending public transportations in areas of the severe outbreak (e.g. Wuhan), cancelling public gatherings, and delaying the reopening of enterprises. These endeavours have led to the significant suppression of the epidemic of COVID-19. There were no domestically newly reported confirmed cases for the first time in China on 18 March 2020 [25]. The local and national responses of China have made remarkable contributions to the prevention and control of  In this article, we selected Wuhan, Wenzhou, and Shenzhen cities as the typical representatives of the policy interventions of China. Wuhan was the most severe epidemic city of COVID-19 outbreak in China. It suspended public transportation, cancelled all outbound trains and flights [21], and started to construct a specialist emergency hospital (i.e. Huoshenshan Hospital) for patients on 23 January 2020. Wenzhou was the city with the largest number of cumulative confirmed cases in Zhejiang province and implemented comparatively stringent policy interventions for the control of COVID-19. It announced that every household might send one person every 2 days outside for necessity purchases [22]. Shenzhen was the city of the confirmed cases initially reported in Guangdong province, and the intervention policy was relatively moderate due to its special economic status in China and management level. It announced that all the entertainment places were suspended from opening to the public on 24 January 2020 [18], and the residential units with confirmed cases were forced to implement 'hard quarantine' for 14 days. All details about the interventions implemented in these cities were given in Figure 1.
Nevertheless, these policy interventions were implemented at the expense of the economic loss. Taking the individual consumption industry as an example, the amount of total retail sales of consumer goods decreased by about 20.5%, with a sharp drop of catering services by 43.1% from January to February 2020 compared to the same duration of 2019 according to the National Bureau of Statistics in China [14]. Consequently, policymakers needed to anticipate the likely outcomes of interventions in terms of their epidemic situations.
The epidemic has rapidly spread to more than 200 countries around the world. WHO declared that COVID-19 became a pandemic on 11 March 2020 [24]. As an epitome of the pandemic, the numbers of cumulative confirmed cases and deaths on 31 May 2020 were 11,468 and 270 in South Korea, 232,664 and 33,340 in Italy, and 1,819,788 and 105,634 in the United States, respectively [1]. Some details about the spread of COVID-19 in South Korea, Italy, and the United States are given in COVID-19 prevalence in Supplemental material. In this situation, the control measures of China, which have significantly suppressed the spread of COVID-19, are worth learning for other countries confronting with coronavirus. Therefore, it would be instructive to simulate the potential outcomes for other countries when they adopted similar policy strategies to China.
In order to simulate the likely outcomes for other countries under different policy patterns, an epidemic model considering the transmission mode of COVID-19 and the effects of policy interventions is needed. Since the transmission of COVID-19 could be caused by the individuals infected with the virus before significant symptoms developed [8], the Susceptible-Infectious-Removed (SIR) model [4,6] is inappropriate for evaluating the spread of COVID-19 due to the lacking assumption of the incubation period. In this paper, we propose a novel epidemic model: Susceptible-Infectious-Confirmed-Removed (SICR) model, which considers the pre-symptomatic transmission of COVID-19 by an appropriate compartment assumption. In the SICR model, the numbers of infected and infectious individuals without isolation are the key ingredients for evaluating the spread, which are assumed to be unobserved and estimated by the latent dynamic of the SICR model. We also construct a suitable contact function among the SICR model to Taxi services were suspended.
All public transportation was suspended from 10 a.m. onwards; All outbound trains were halted; The construction for Huoshenshan Hospital began.
The construction for Leishenshan Hospital began.
The first shelter hospital was put into use.
The closure management was implemented to all residential areas.

Wuhan
Inter-provincial and municipal bus services were suspended.
Every household might send one person every two days outside for purchases.
All the entertainment places were suspended from opening to the public.
All the individuals returning from Hubei needed to be quarantined at home or other suitable facilities for 14 days.
The residential units with confirmed cases were forced to implement "hard quarantine" for 14 days. Checkpoints were set up in all entrances of residential areas.
Measures were rolled out to support companies to resume businesses.
Community property service centers should register the information of visitors and cooperate with quarantine observations for close contacts of the confirmed cases. capture the policy pattern duration of the outbreak period and use the estimated timevarying reproduction number [9,17] to reflect the effects of interventions implemented in Wuhan, Wenzhou, and Shenzhen. After the estimation of the cities of China, we migrate the corresponding policy intensity by parameters transition to simulate the likely outcomes if South Korea, Italy, and the United States adopt these interventions, respectively, so the lessons can be learned from the key decisions made in the representative cities of China. The rest of this paper is organized as follows. In Section 2, we introduce the construction of our proposed model and further show the mode migration approaches in policy pattern assessment. In Section 3, we present the simulation studies for the spread of COVID-19 in South Korea, Italy, and the United States under different interventions of China by the SICR model. Finally, we end up with conclusion in Section 4.

Data sources
The epidemic data were extracted for Wuhan, Wenzhou, and Shenzhen from the Chinese Center for Disease Control and Prevention (China CDC) [15]. We used the reported numbers of cumulative confirmed cases, cumulative recovery cases, and total deaths from January to February 12 in Wuhan, from January 21 to February 9 in Wenzhou, and from January 19 to February 6 in Shenzhen, respectively. Similarly, the reported numbers of cumulative confirmed cases, cumulative recovery cases, and total deaths for South Korea, Italy, and the United States were downloaded through 23 April 2020 from 'nCov2019' package [1]. The data and R files supporting the conclusions of this study are available at https://github.com/tingT0929/The-simulations-of-the-spread-of-COVID-19under-different-interventions-.

The epidemic SICR model
To consider the pre-symptomatic transmission patterns of COVID-19 [20], we assume an additional state compared to the standard SIR model, and introduce the SICR model with four compartments: susceptible class (S), infectious class without isolation (I), confirmed class (C), and removed class (R). We assume that: Based on the above assumptions, the transfer diagram among compartments S, I, C, R are given in Figure 2: In the SICR model, the increase of the number of infections only happens through the contact between S and I, similar to the SIR model [4,6], we can define the corresponding dynamic equations of four compartments as follows: where N represents the total number population. For the simplicity, the disturbance caused by the new-born and death is not considered in the dynamic equation.
are the numbers of corresponding compartments at time t. α denotes the product of the transmissibility of infectious disease and the average contact numbers per person during the outbreak, indicating that the α S(t) N represents the second attack numbers of an infection per unit time. β and γ are the average lengths of retention time of a person for compartments I and C, respectively, where the length of β is assumed to be dominated by the mean length of the incubation period.
The basic reproduction number R 0 , which is the expected number of secondary cases produced by a single (typical) infection, is defined as the spectral radius of the next generation operator [9,17] of the SICR model: It is reasonable to assume the time-varying transmission patterns in the SICR model by considering the policy interventions. We construct a logistic function to simulate the decreasing trend of α: where c > 0. Notice that: If we chose c as then Let ε = 0.01, which implies that α 0 ≈ α α 0 ,d 1 ,d 2 (t) for t ≤ d 1 , reflecting the contact pattern until d th 1 day when the control measures start to be effective, and d 2 represents the duration of the decreasing process when the contact is to nearly vanish. Specifically, the smaller values of d 1 and d 2 , the earlier effectiveness and the stronger intensity of interventions were implemented. The graph of time-varying α is given in Figure 3. By the setting above, assuming S(0) ≈ N for the first wave of infection, the corresponding time-varying reproduction number R t was calculated as For the simplicity, let To extract the parameters of the SICR model, we utilize the negative binomial mass as the loss function to fit the observed daily increasing numbers of cumulative confirmed cases (C) and the removed cases (R), which is feasible to capture the over-dispersion of the counting process [13]. Let := (α 0 , d 1 , d 2 , β, γ ) and (S 1 , I 1 , C 1 , R 1 ) be the initial numbers of each compartment. C t and R t denote the observed daily increasing numbers of cumulative confirmed cases (C) and the removed cases (R) at time t, respectively. Moreover, C fit t is the fitting value of C t given the initial condition (S 1 , I 1 , C 1 , R 1 ) and by Equation (1). Similarly, R fit t is the fitting values of R t . The likelihood of the observed data is given as where T is the time length of the observed data. NegB(· ; μ, ν) denotes the negative binomial mass with mean μ and size ν, which is where p(μ, ν) := ν/(ν + μ). The size parameters ν 1 and ν 2 are introduced to evaluate the overdispersion of C t and R t , i.e.
The fitting values of C fit t and R fit t could be efficiently calculated by the R package desolve [19].
Here, we focus on three typical interventions of Wuhan, Wenzhou, and Shenzhen: the first lockdown in Wuhan (January 23) and Wenzhou (February 1), and the suspense of the places of entertainment for Shenzhen (January 24). In particular, we assume the lockdown policies of Wuhan were completely effective after 3 days of implementation (January 26) [7]. Therefore, we chose d 1 as 11, 11, 5 for Wuhan, Wenzhou, and Shenzhen, respectively, according to their starting dates of the observed data in Figure 1 (January 15, January 21, and January 19, for Wuhan, Wenzhou, and Shenzhen, respectively).
For the parameter β, we chose N [0,30] (5.1, 0.325) as the prior distribution for Wenzhou and Shenzhen, in terms of study about the incubation period of COVID-19 [12], where N [a,b] (μ, σ ) is the truncated normal mass with mean μ and standard error σ valued in [a, b]. In Wuhan, since there is a significant waiting time of the testing for the confirmation of infections (averagely 11 days [26]), we assume the β before February 5 (denoted by β 1 ) and the β after February 5 (denoted by β 2 ) satisfy: β 2 ≤ β 1 , where the change point is chosen based on the timing of the first shelter hospital was put into use in Wuhan (February 5) and the reported date for the acceleration of diagnosis [26]. Similarly, we chose N [0,30] (16.1, 0.325) and N [0,30] (5.1, 0.325) as the prior distributions of β 1 and β 2 in Wuhan. We abuse of notation β to represent (β 1 , β 2 ) for Wuhan.
Once the prior distribution of : π( ) is given, we use the posterior distribution of : π( |C 1 , R 1 , d 1 ) for evaluating the dynamics of the epidemic process, which is calculated as Since π( |d 1 , C 1 , R 1 ) has no closed form, combining with the prior distribution of , we use Gibbs sampler embedded random walk Metropolis-Hastings steps [2,3] to sample the posterior distribution. We randomly chose different initial values and generate the Markov chain Monte Carlo (MCMC) samples of the posterior distribution parallel by R package snowfall [10] as the burn-in procedure. After the convergence of these MCMC chains, we collect at least 4000 samples from the post-burn-in period of different chains to approximate π( |C 1 , R 1 , d 1 ). Besides, we could use the posterior distribution of the fitting numbers for the prediction of the epidemic. Let Z t = K t (Z, ) be the dynamic numbers of compartments at time t by Equation (1), given the initial condition Z := (S 1 , H 1 , C 1 , R 1 ) and parameters . The posterior distribution of Z t could be calculated as hence the posterior samples of Z t could be achieved by the k th posterior samples. The point estimation of the parameters and predicted numbers are presented as the median of the posterior distribution, while 95% credible intervals are constructed with 2.5% and 97.5% quantiles.

The mode migration
For simulating the epidemic trends of other countries, we applied the SICR model where the R t is time invariant, i.e. we replace the functional parameter α α 0 ,d 1 ,d 2 (t) by a single parameter α within the estimation procedure of South Korea, Italy and the United States, respectively. Moreover, we migrate the estimated intensity of policy intervention of Wuhan, Wenzhou, and Shenzhen to South Korea, Italy, and the United States after M days' original outbreak from the corresponding starting dates, respectively. In doing so, we specify the time-varying α α 0 ,d 1 ,d 2 (t) for South Korea, Italy, and the United States, where α 0 is chosen as the corresponding posterior samples of α, d 1 is chosen as M and d 2 is chosen as the posterior samples of d 2 of Wuhan, Wenzhou, and Shenzhen, respectively. Taking M as 3 to denote the short-term transmission, we then apply the estimated policy intensity of China and simulate the hypothesis epidemic trends of South Korea, Italy, and the United States by Equation (3), respectively. Similarly, the point estimates were presented as the median of the posterior distribution while 95% credible intervals were constructed with 2.5% and 97.5% quantiles.

The estimation of policy patterns in China
To describe the effects of interventions in different cities of China, we used the epidemic data from January 15 to February 12 in Wuhan, from January 21 to February 9 in Wenzhou, and from January 19 to February 6 in Shenzhen, to estimate parameters of the SICR model. The mainly estimated parameters and the trend of R t for Wuhan, Wenzhou and Shenzhen were presented in Table 1 and Figure 4. We found that the initial R t for Wuhan was the largest among the three cities, while Shenzhen was the city where the reproduction number started to decrease earliest, followed by Wuhan and Wenzhou. The decreasing speed of R t in Wenzhou was faster than that in Wuhan, followed by Shenzhen.

The migration of policy patterns
The simulation of COVID-19 trend without policy intervention and under three policies of China in South Korea, Italy, and the United States is given in Figures 5-7. The

Discussion
In SICR model, the estimated R t could reasonably reflect the patterns of policy interventions. From Figure 4, the different decreasing speeds of R t imply different intensities of policy interventions. The intensity of policies for Wenzhou was the strongest, as opposite to be the weakest in Shenzhen. Comparing with details in Figure 1, Shenzhen issued relatively mild policies in the early period of outbreak, such as suspending entertainment places and tracking on the close contacts of confirmed cases, unlike the stringent intervention policies in Wenzhou, for example, block both long-distance and short-distance social contacts. Indeed, mild strategy would cause less economic damage, oppositely, stringent strategy was used for the quick control of the infectious disease.
For South Korea, Italy and the United States, if interventions were not implemented, it can be expected that how tremendously large populations would be infected in these countries. Thus the interventions are critical for the control of the fast spread of infectious disease. Based upon our simulation, the actual cumulative confirmed cases of COVID-19 in South Korea from February 26 to March 9 were quite similar to Shenzhen pattern, whose intensity was relatively mild compared with both Wuhan and Wenzhou patterns. According to the report [16], South Korea has implemented similar interventions to Shenzhen pattern, where it implemented policies closely following and isolating the close contact individuals of confirmed cases to reduce the transmission of COVID-19. This indicated that the mitigation policies like Shenzhen pattern implemented at an early stage of the epidemic could effectively curb the outbreak of infectious diseases.
On the other hand, note that the expected number of cumulative confirmed cases under the interventions of Wenzhou for Italy would be the smallest. Comparing the actual situations of Italy, the cumulative confirmed cases were 187,327 on 23 April 2020, which were 38.6% higher than the simulation of COVID-19 under interventions of Wenzhou. If taking such highly stringent interventions, the epidemic situation could be mitigated where the expected number of cumulative confirmed cases would be 135,308 on May 31, decreasing by 14.2% and 16.8% compared to the implementation of policies of Wuhan and Shenzhen, respectively. It showed that containment policies like Wenzhou pattern were effective for quick control of the magnitude of the outbreak of infectious disease.
In the United States, the cumulative confirmed cases were 849,094 on 23 April 2020 and were 160.8% more than the simulation of COVID-19 under interventions of Wenzhou, which implied that the implementations of interventions of Wenzhou may significantly decrease the magnitude of the outbreak of COVID-19 for the United States.

Conclusion
In this paper, we proposed a novel compartmental epidemic model: SICR model, which took the pre-symptomatic transmission of COIVD-19 into account and could estimate the policy patterns according to the real data. Subsequently, we can get reasonable assessments for the corresponding policy patterns through the mode migration between countries.
As a typical example, we used the epidemic data of three representative cities in China (Wuhan, Wenzhou, and Shenzhen) and migrated the estimated policy modes to South Korea, Italy, and the United States, respectively. Our simulations indicated that the mild interventions were difficult to control the magnitude of the outbreak of infectious diseases when the number of infected and infectious without isolation was quite large, but it would be effective to implement at the early stage of the epidemic. Once the epidemic became severe, the highly stringent interventions were needed to be implemented for the control of the epidemic. Although it still takes time to make a reasonable assessment of the patterns and effects of these policies, the transmission patterns of COVID-19 under some typical intervention measures could be learned from them.