Modeling and Reviewing Study to the COVID-19 Epidemic in PR China

Background: COVID-19 epidemic has been widely spread all over the world. During it appears in China, Chinese government quickly put forward and implement prevention and control measures to keep its spread within limits. This study aims to investigate the impacts of the prevention and control measures in controlling COVID-19 epidemic in China, so as to give a clue to control its spread in the world. Methods: We establish a two-stage dynamics transmission model with "lockdown of Wuhan city" as the time line. The first stage is the SEIR derived model that considers the contagious of the exposed. It simulates the COVID-19 epidemic in Hubei Province before "lockdown of Wuhan city". The second stage is a novel transmission dynamics model named SEIRQH. It takes into account the influence on the COVID-19 epidemic from the series of measures such as travel restriction, contact tracing, centralized treatment, the asymptomatic infected patients, hospitalized patients and so on. It simulates the COVID-19 epidemic in China after "lockdown of Wuhan city". The least square method is used to estimate the parameters of SEIR derived model and the proposed SEIRQH model based on the collected epidemic data of COVID-19 from Hubei Province and the mainland of China. Results: The SEIR derived model fits the actual data in Hubei Province before "lockdown of Wuhan city". The basic reproduction number of COVID-19 epidemic in Hubei Province is 3.2035 before "lockdown of Wuhan city". The SEIRQH model fits the number of the hospitalized persons of COVID-19 in Hubei Province and the mainland of China perfectly. The control reproductive number are 0.11428 and 0.09796 in Hubei Province and the mainland of China, respectively. The prevention and control measures taken by Chinese government play the significant role against the COVID-19 spread in China. Conclusions: Our two-stage dynamics transmission model simulates the COVID-19 in China, especially our SEIRQH model fits the actual data very well. The prevention and control measures implemented by Chinese government are effective in preventing the wide spread of COVID-19 epidemic in China. These measures give the reference to World Health Organization and other countries in controlling COVID-19 epidemic.


Background
In December 2019, an atypical pneumonia caused by a new type of coronavirus appeared in Wuhan city, Hubei Province, China [1,2]. On January 20, 2020, this new type of coronavirus was named as 2019-nCoV (2019 novel coronavirus) by the WHO (World health organization) for the first time.
The atypical pneumonia caused by 2019-nCoV was subsequently named as  (Coronavirus disease 2019) on February 11, 2020. It has been reported that the natural host of 2019-nCoV may be the bat [3][4], but the intermediate host is still controversial [5]. Although the origin of 2019-nCoV is still disputed, the infectious disease COVID-19 caused by 2019-nCoV has spread globally and has become an infectious disease that threatens all of humankind. It is report by WHO that, as of August 13, 2020, the cumulative number of infected people around the world had exceeded 20 million and the number of deaths had reached to 742,878.
In the early and middle stages of the COVID-19 spread in China, there are many researchers who devoted themselves to study the trend of the COVID-19 epidemic using the differential equations of infectious disease transmission dynamics. The traditional transmission dynamic models comprise SI (Susceptible-Infectious), SIR (Susceptible-Infectious-Recovered), SEIR (Susceptible-Exposed-Infectious-Recovered) and its derived models. The SIR and SEIR models were widely adopted in the studies of COVID-19 epidemic at the early stage of COVID-19. The transmission mechanism of COVID-19 was not clear and the epidemic prevention and control measures were not explicitly published at that time. For example, Yang et al. [6] used the SIR model to predict the epidemic trend in Chongqing, and estimated that Chongqing would come to its inflection point on February 25. Fang et al. [7] used the SEIR model considering the incubation period to predict and analyze the specific time of inflection point of  in Wuhan under the condition of different incubation periods. Geng et al. [8] studied the impact of travel restriction measures on the trend of COVID-19 epidemic based on SEIR model.
To restraint the spread of COVID-19 epidemic in China, Chinese government took several policies, such as travel restriction, close contact tracing, centralized treatment, and so on. Therefore the traditional SIR and SEIR models are no longer suitable to simulate the trend of  China. Such that many derived models are proposed on the basis of the traditional SEIR model by adding some compartments, so as to investigate the impact of the prevention and control measures taken by Chinese government on the COVID-19 epidemic. For example, Tang et al. [9] proposed a SEIR derived model based on the epidemic prevention and control measures in China, which comprises some appropriate compartments such as quarantine, isolation and treatment. Their follow-up study [10] described the contact and diagnose rates as the time-dependent functions in their model, which effectively evaluated the trend of the COVID-19 epidemic in China under the gradually strengthening epidemic prevention and control measures. Cao et al. [11] established a SEIR derived model that considers the infectivity of latent patients and the measures of contact tracing. The SEI model established by Li et al. [12] studied the impact of unrecorded infected patients on the COVID-19 epidemic in China. They estimated that there were about 86% (95%CI: 82-90%) infected patients which had not been recorded before January 23, 2020, and about 79% recorded infected patients come from unrecorded infected patients. Chen et al. [13] proposed a BHRP (Bats-Hosts-Reservoir-People) transmission network model based on traditional SEIR model, and used its simplified model RP (Reservoir-People) to simulate the transmission mechanism of the early 2019-nCoV outbreak. They estimated that the R0 (Basic reproduction number) of COVID-19 was 3.58 in China. Wei et al. [14] established the SEIR +CAQ model based on the transmission  mechanism, infection spectrum and prevention and control measures of COVID-19, so as to provide   the basis for decision making about the development, prevention and control of COVID-19 epidemic. Li et al. [15] proposed a derived SEIR model to analyze the impact on the COVID-19 epidemic of the prevention measures and the people's compliance and the resumption of work and production. Wang et al. [16] proposed a SEIR derived network model by integrated multi-source data. They did a detailed analysis of when the resumption work and production should be done in

Infectious disease dynamic model
SEIR model is a widely used classical infectious disease transmission dynamics model. It assumes that all individuals are at the risk of being infected and divides populations into four compartments, which are susceptible (S), exposed (E), infected (I) and recovered (R). Compared to the SI and SIR model, SEIR model takes into account the incubation period properties of infectious diseases. It has evolved to several derived models [8,9,11] according to some factors such as whether the patients in incubation period will infect others, whether the cured patients will be infected again, and whether some prevention and control measures are taken. These derived models have been widely used in preventing, controlling, and predicting the trend of many major epidemics, such as MERS (Middle east respiratory syndrome) [17], HIV (Human immuno-deficiency virus) [18][19], SARS (Severe acute respiratory syndrome) [20][21], COVID-19 [7, 9-11, 15, 22-26], and so on.

The SEIR derived model considering the infectivity of latent patients
The SEIR model is a common SEIR derived model that considers the infectiousness of patients in incubation period, as shown in Fig. 2. This SEIR derived model partitions the populations into susceptible (S), exposed (E), infected (I), and recovered (R) parts. The individuals in incubation (exposed) and infected periods have got the same infective capability. Assume that the infectious probability is  , and the average contact number of individuals is c between the susceptible and the infected, and the contact factor is  between the exposed, relative to the infected, and susceptible individuals, then the dynamics differential equations of this SEIR derived model at time t are shown in formula (1). Fig. 2. The SEIR derived transmission dynamics model considering the infectivity of latent patients.
is the total number of individuals. Parameter  represents the transition probability from the exposed to the infected. Its value is the reciprocal of the incubation period, such as 1/ 5   in [16,27]. Parameters  and  are the recovery rate and death rate of the infected, respectively. In the early stage of COVID-19, due to the insufficient awareness and the poor prevention and control measures for the epidemic, we assume that the average contact number between the exposed and the susceptible individuals is the same as that between the infected and susceptible individuals, that is 1   . The specific definitions of each parameter in formula (1) are given in Table 1.

The SEIRQH model
With the increasing awareness of the transmission mechanism and clinical symptoms of COVID- Where, c represents the average contact number between the infected and the susceptible persons,  is the probability of infection. The parameter  is the contact factor between the asymptomatic infected, relative to the symptomatic infected, and the susceptible. The parameter  is the contact factor between the exposed, relative to the symptomatic infected, and the susceptible. The specific definitions of each parameter in formula (2) are given in Table 2.

Basic reproduction number
The basic reproduction number, R0 [28][29][30], refers to the number of secondary infections produced by an infected person through the entire duration of the infectious period when all individuals in a certain area have no immunity to infectious diseases and there is not external control in epidemiology. The R0 is used to measure the spread capability of an infectious disease. It is the quantitative base to judge the propagation and extinction of an infectious disease in the populations.
The control reproduction number, Rc [16,18], is the number of secondary infections by an infected individual during the entire infectious duration with the control and intervention existing.
During the process of transmission of an infectious disease, if R0 or Rc is greater than 1, then the number of infected people will increase exponentially, and the infectious disease will eventually spread throughout the whole populations. However, the infectious disease tends to disappear with the decreasing number of infected people when R0 or Rc is less than 1. COVID-19 is an infectious disease that has never appeared before. R0 and Rc are of great significance in studying its spread trend, prevention, and control.
The next generation matrix method is the one that calculates the basic reproduction number R0 by the spectral radius of the next generation matrix. The next generation matrix is obtained at the disease-free equilibrium point of the model [28][29]. Given an infectious disease model containing n compartments, this model can be divided into infected class with   reproduction number R0 is defined as the spectral radius of matrix 1  FV , shown in formula (3).
For SEIR derived model that considers the infectiousness of patients in incubation period, the 2 m  holds according to the biological significances of the infectious disease, that is, the infected class includes two compartments E and I. Therefore, the basic reproduction number R0 of SEIR derived model can be calculated by formula (4).
where, 0 For our proposed SEIRQH model, we can see that there are four compartments in the infected class. They are E, IA, IS and H, that is, 4 m  holds. Therefore, the control reproduction number Rc of our SEIRQH model can be calculated by formula (5).
Where,    S  S   I  I   I   I  I  I  I  I  I  I  I   I  I  I   H  I  I  I  H  I  I  I  H  I  I

Experimental results of SEIR derived model
Based on the COVID-19 epidemic data of Hubei Province from January 10, 2020 to January 22, 2020, the parameters of SEIR derived model are calculated by using the least square method. All the parameters estimated are shown in Table 1. Fig. 4(a) shows the fitting result of SEIR derived model to the number of infected cases in Hubei Province from January 10, 2020 to January 22, 2020. epidemic within 100 days after January 10, 2020, that is, from January 10, 2020 to April 19, 2020, based on the parameters shown in Table 1.

Effects of prevention and control measures on COVID-19
The  Table 2, respectively, by using the least squares algorithm and MATLAB simulation experiments. These two average contact numbers are both 12. We study the impact from the average contact number c on the trend of COVID-19 by changing its values. Fig. 6 shows the simulation results of the hospitalized case variations in Hubei Province ( Fig. 6(a)) and in the mainland of China (Fig. 6(b)) with the variations of parameter c within the 200 days after January 23, 2020, that is, from January 23, 2020 to August 10, 2020. The simulation results of peak time and the peak value of the hospitalized cases of COVID-19 are shown in Table 3.    Table 4.

SEIR derived model
It can be seen from the experimental results in Fig. 4(a) that the SEIR derived model fits the COVID-19 infection cases in Hubei Province from January 10, 2020 to January 22, 2020 to some extent. The simulation results in Fig. 4(b) show that the COVID-19 epidemic enters the outbreak phase in the late of January, and reaches its peak on Hubei Province from January 10, 2020 to April 30, 2020 in Fig. 1(a), it can be found that the SEIR derived model can simulate the trend of COVID-19 epidemic in Hubei Province after January 23, 2020. This fact demonstrate that the first stage of our two-stage transmission dynamics model is effective in predicting the COVID-19 epidemic trend in Hubei province after January 23, 2020.
Although the basic reproduction number R0 of the COVID-19 has not been known yet, its value in the early stage of COVID-19 outbreak in Hubei Province can be calculated by using the estimated parameters shown in Table 1 and the formula (4). It is 3.2035, slightly higher than the 1.4-2.5 released by the WHO [35]. But it is consistent with 2.3-3.58 published by Chen et al. [13], and is also consistent with 2.24-5.71 given by Zhao et al. in [36], and is also consistent with the 3.2 given by Huang et al. in [37] when they are considering the delay in treatment. It is lower than the 6.47 (95%CI: 5.71-7.23) given by Tang et al. [9].

SEIRQH model
It can be seen from the results in

Sensitivity analyses to the parameter c
It can be seen from Fig. 6 that the number of hospitalized cases in Hubei Province and in the mainland of China both increase with the increasing of average contact number. The peak time of the number of hospitalized cases comes earlier when the average contact number increases. That is to say, if the Chinese government reduces or fails to adopt strict control measures, such as home quarantine and travel restrictions, the COVID-19 epidemic may become aggravated, and the hospitalized cases may have got a higher peak value, and the epidemic inflection point will be earlier.
The experimental results in Table 3  China, but also minimize its harm to humankind.
Although the results in Fig. 6 and Table 3 show that the peak value of the hospitalized cases in Hubei Province and in the mainland of China will be reduced by taking more stricter prevention and control efforts, and the inflection point time will be delayed as well, that is, the peak time of the number of hospitalized cases in Hubei Province and the mainland of China comes later. But the stricter measures may cause unnecessary social panic. Therefore, the prevention and control measures need to be formulated according to the actual situation of COVID-19 epidemic, so as to prevent its widespread transmission and minimize its harm maximally.

Sensitivity analyses to the quarantined rate q
The experimental results in Fig. 7 show that the peak value of the hospitalized cases in Hubei Province and the mainland of China becomes larger as the value of q increases, and the inflection point of the hospitalized case number comes earlier. The number of hospitalized cases in Hubei Province and the mainland of China rises with the increasing q value. It can be concluded that people will be in the risk to be infected if they do not abide by the guidance from the government, because going out and gathering will increase the probability to contact the exposed, the infected, and the asymptomatic infected. This may cause the more severe COVID-19 epidemic than current.
It can also be seen from the results in Table 4  The theoretical analyses in Table 4 and Fig. 7 reveal that stricter measures of isolation at home and prohibition of gathering can further decrease the quarantine probability, so as to reduce the risk to be infected for susceptible persons. Therefore, citizens can do a favor against COVID-19 epidemic by abiding by the prevention and control measures.

Sensitivity analyses to the medical screening
It can be seen from Fig. 8(a) that the peak value of the hospitalized cases drops when the value of parameter 1  decreases. This will lead to more pressure on the later period of the epidemic and make the epidemic last more time. In other words, decreasing of the medical screening strength for the quarantined people will increase the later pressure against COVID-19 and will increase the risk of the epidemic spreading further and increasing the death numbers as well. Fig. 8(b) shows that the transition rate 2  has the similar impact on the trend of COVID-19 in Hubei Province as that of the rate 1  does. But the influence from 2  is bigger than that from 1  .
From the experimental results in Fig. 9, we can see that the impact of parameters 1  and 2  on the COVID-19 in the mainland of China is similar to that in Hubei Province. The transition rate