# The Impact of Lockdown, Patient Classification, and the Large-Scale Case Screening on the Spread of the Coronavirus Disease 2019 (COVID-19) in Hubei

Authors: Shengtao Wang; Yan Li; Ximei Wang; Yuanyuan Zhang; Yiyi Yuan; Yong Li
Journal: BioMed Research International
 (2022)

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## Abstract

The coronavirus disease (COVID-19) which emerged in Wuhan, China, in December 2019, is widely controlled now in China. However, the global epidemic is still severe. To study and comment on Hubei’s approaches for responding to the disease, the paper considered some factors such as suspected cases (part of them are influenza patients or common pneumonia patients, etc.), quarantine, patient classification (three types), clinically diagnosed cases, and lockdown of Wuhan and Hubei. After that, the paper established anSELIHR model based on the surveillance data of Hubei published by the Hubei Health Commission from 10 January 2020 to 30 April 2020 and used the fminsearch optimization method to estimate the optimal parameters of the model. We obtained the basic reproduction number R0=3.1571 from 10 to 22 January. R0 was calculated as 2.0471 from 23 to 27 January. From 28 January to 30 April, R0=1.5014. Through analysis, it is not hard to find that the patients without classification during the period of confirmed cases will result in the cumulative number of cases in Hubei to increase. In addition, regarding the lockdown measures implemented by Hubei during the epidemic, our simulations also show that if the lockdown time of either Hubei or Wuhan is advanced, it will effectively curb the spread of the epidemic. If the lockdown measures are not taken, the total cumulative number of cases will increase substantially. From the results of the study, it can be concluded that the lockdown, patient classification, and the large-scale case screening are essential to slow the spread of COVID-19, which can provide references for other countries or regions.


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## Body

## 1. Introduction

In December 2019, a novel coronavirus emerged in Wuhan, Hubei province. Then the World Health Organization (WHO) formally named the novel coronavirus COVID-19 on 11 February 2020 [1]. Coincided with transportation during the Spring Festival (called Chunyun in China), China took strict prevention measures to control the spread of the outbreak like closing all premises, imposing certain restrictions on the passage of the provinces, and paying attention to personal hygiene and other actions. On 23 January 2020, the authorities in Wuhan announced the lockdown of the city and closed all expressways leaving Wuhan [2]. After that, 12 cities and an autonomous prefecture took the same measures, and all people in Hubei province were advised to quarantine at home. When the situation got better, the traffic control in Hubei province was orderly cancelled except for Wuhan on 25 March. People can only rely on the Hubei health green code to other places smoothly [2]. After 76 days of continuous lockdown, the government in Wuhan gradually lifted its lockdown and people returned to their normal life. On 16 April 2020, the number of new cases in Hubei province decreased to 0 [2, 3]. On 26 April 2020, all cases of COVID-19 hospitalization in Wuhan city were cleared [2, 4]. On 11 May 2020, Wuhan conducted a 10-day city-wide nucleic acid detection of all its residents to determine the current epidemic situation [5].COVID-19 spreads mainly through respiratory droplets, contact, aerosols, feces-mouth, and other forms [6]. The population can be easily infected in direct or indirect ways when they are exposed to long-term closed environments. The source of infection found so far is mainly in patients infected with COVID-19 and patients with asymptomatic infections. Based on current epidemiological investigations and findings, the incubation lasts 1 to 14 days theoretically, but usually 3 to 7 days in reality [6].Patients who were infected with COVID-19 had main symptoms like fever, dry cough, and fatigue, and a small number of patients were accomplished with nasal congestion, runny nose, sore throat, myalgia, diarrhea, and other symptoms [6]. During the treatment of confirmed cases, the cases were divided into three types [7]: mild, severe, and critical. For the three cases, there were different clinical classification standards. Patients who had light clinical symptoms were mild, and their images did not show signs of pneumonia. Severe patients appeared with obvious respiratory distress with drowsiness, food refusal, and other symptoms. Critical cases referred to patients who appeared with respiratory failure, shock, and other serious life-threatening characteristics. However, it is still unknown whether such classification measures can have an impact on the suppression of the epidemic. Therefore, what impact it will bring is one of the main purposes of this research.It is well known that the basic reproduction number can measure the level of the epidemic, and the magnitude ofR0 allows one to determine the amount of effort which is necessary either to prevent an epidemic or to eliminate infection from a population [8], so it is crucial to estimate R0 for a given disease in a particular population. Until now, many researchers have paid attention to this outbreak and have some results [9–46] about the basic reproduction numbers in Wuhan, Hubei, or China which are summarized in Table 1.Table 1
The results ofR0 in Wuhan, Hubei or China.
AreaResearch timeValue (95% or 90% CI)ResourcesChinaJan. 22, 2020-Mar. 30, 20200.454Collins and Duffy [9]WuhanFeb. 2, 2020-Feb. 11, 20200.84 (0.81-0.88)Mizumoto et al. [11]WuhanJan. 4, 2020-Mar. 9, 20200.945Ndaïrou et al. [29]ChinaJan. 24, 2020-Feb. 8, 20200.99 (0.76-1.33)Li et al. [30]WuhanFeb. 4, 2020-Feb. 12, 20201.08Zhang et al. [21]WuhanJan. 23, 2020-Feb. 1, 20201.3065 (0.5273-2.0858)Xue et al. [31]WuhanJan. 21, 2020-Feb. 8, 20201.32 (1.16-1.48)Du et al. [32]WuhanJan. 15, 2020-Feb. 4, 20201.3469, 2.8349Sun et al. [16]ChinaJan. 24, 2020-Feb. 3, 20201.36 (1.14-1.63)Li et al. [30]ChinaJan. 18, 20201.40-2.50WHO [33]WuhanJan. 23, 2020-Feb. 3, 20201.55Zhang et al. [21]WuhanJan. 24, 2020-Feb. 11, 20201.7549Pang et al. [17]WuhanJan. 25, 20202.00-3.00Abbott et al. [27]ChinaJan. 24, 20202.10 (2.00-2.20)Jung et al. [12]China and overseasJan. 18, 20202.20 (90% HDI 1.40-3.80)Riou and Althaus. [13]WuhanBefore Jan. 22, 20202.20 (1.40-3.90)Li et al. [14]WuhanJan. 10, 2020-Jan. 24, 20202.24 (1.96-2.55)Zhao et al. [15]WuhanJan. 10, 2020-Jan. 23, 20202.38 (2.03-2.77)Li et al. [30]WuhanDec. 10, 2019-Jan. 21, 20202.42Hu et al. [10]WuhanJan. 1, 2020-Jan. 15, 20202.56 (2.49-2.63)Zhao et al. [34]WuhanJan. 23, 20202.57 (90% CI 2.37-2.78)Chinazzi et al. [18]ChinaJan. 20, 2020-Feb. 11, 20202.68Liu et al. [35]WuhanDec. 31, 2019-Jan. 25, 20202.68 (2.47-2.86)Wu et al. [36]WuhanJan. 18, 2020-Feb. 13, 20202.70Guo et al. [37]WuhanJan. 23, 2020-Mar. 6, 20202.71Wang et al. [26]WuhanDec. 22, 2019-Mar. 15, 20202.80Musa et al. [38]ChinaJan. 25, 20202.80-3.30Zhou et al. [39]WuhanDec. 12, 2019-Feb. 22, 20203.04Huang et al. [40]WuhanJan. 1, 2020-Jan. 22, 20203.11 (2.39-4.13)Read et al. [28]ChinaDec. 31, 2019-Jan. 23, 20203.15 (3.04-3.26)Tian et al. [19]ChinaJan. 24, 20203.20 (2.70-3.70)Jung et al. [12]WuhanJan. 24, 2020-Feb. 5, 20203.30 (2.66-3.95)Ma et al. [41]China (excluding Hubei province)Jan. 20, 2020-Mar. 3, 20203.36 (3.20-3.64)Wan et al. [20]WuhanJan. 11, 2020-Jan. 23, 20203.4074 (2.9959-3.8188)Xue et al. [31]Wuhan2019-20203.49 (3.39-3.62)Mizumoto et al. [11]WuhanDec. 7, 2019-Jan. 1, 20203.58Chen et al. [42]WuhanJan. 10, 2020-Jan. 24, 20203.58 (2.89-4.39)Zhao et al. [15]WuhanDec. 8, 2019-Jan. 22, 20203.6Zhang et al. [21]WuhanJan. 26, 2020-Feb. 9, 20203.66Wang et al. [43]HubeiJan. 27, 2020-Feb. 11, 20203.7732Li et al. [22]WuhanJan. 10, 2020-Jan. 30, 20204.30Song et al. [44]WuhanDec. 31, 2019-Jan. 23, 20204.6355Pang et al. [17]HubeiJan. 11, 2020-Jan. 22, 20205.6015Li et al. [22]HubeiJan. 23, 2020-Feb. 19, 20205.6870, 2.2426, 1.0560Jia et al. [23]WuhanJan. 15, 2020-Jan. 30, 20205.70 (3.80-8.90)Sanche et al. [45]ChinaJan. 10, 2020-Feb. 4, 20205.78 (5.71-5.89)Wang et al. [46]ChinaJan. 10, 2020-Jan. 22, 20206.47 (5.71-7.23)Tang et al. [24]ChinaJan. 23, 2020-Jan. 26, 20206.6037Li et al. [22]WuhanJan. 4, 2020-Jan. 23, 20207.53Song et al. [25]Since the outbreak of the epidemic, scholars at home and abroad have done a lot of work and breakthroughs. Mizumoto et al. [11], Jung et al. [12], Riou et al. [13], Li et al. [14], and Zhao et al. [15] made a certain assessment of the transmission potential and prevalence of COVID-19 at the beginning of the outbreak and believed that COVID-19 had a risk of global spread. Anderson et al. [47] argued that contact tracing can help contain the spread of COVID-19 but is still needed to control imported cases. Some papers [9, 16–25] have studied travel bans or isolation measures such as lockdown, among which Collins and Duffy [9] and Pang et al. [17] have discussed the effects of multiple control measures, and their studies suggested that in a series of measures, isolation plays the most important role. Zhang et al. [21], Sun et al. [16], and Wang et al. [26] also considered the impact of medical resource capacity on the spread of the epidemic. In addition, similar to the patient classification in Hubei, the Indonesian government also classified COVID-19 patients into three types, people under monitoring (ODP), patients under surveillance (PDP), and confirmed patients. Artiono et al. [48] included the three types in the established model to study. However, there are still few papers studying patient classification combined with mathematical models.As of 16 April, the cumulative reported cases of COVID-19 in Hubei province were 68,128, including 50,333 cases in Wuhan, and there were 4,512 people dying of the disease (3,869 in Wuhan) [49]. The number of patients in Hubei province was cleared at the end of April 2020 [4]. In the following nearly two years, there has never been a large-scale outbreak in Hubei, and only a few sporadic cases have occurred until now. Compared with the global epidemic, undoubtedly, the prevention and control measures taken by Hubei have achieved great success. Considering the actual epidemic situation and various response measures taken by the government in Hubei, a dynamic model has been established in this paper to study the inhibitory effect of some kinds of control factors: whether to lock down the city, timely lockdown, patient classification, and large-scale case screening. Through carefully studying the successful case of epidemic prevention in Hubei, the purpose of this paper is to provide some references for the prevention and control of the COVID-19 epidemic in other countries or regions with similar characteristics.This paper is divided into five sections: Introduction opens with the origin, epidemiology, diagnosis of COVID-19, and the situation in Hubei province; Method develops the study by introducing the method of the study, with which we build a model of COVID-19 based on the real situation and construct the differential equations. We collect the surveillance data and choose the fminsearch optimization method to estimate parameters. The results of the model-based estimates, basic reproduction numbers, and sensitivity analysis are provided in Results. And finally, in Discussion and Conclusion, we discuss the results and give the conclusion of the study.


## 2. Method

### 2.1. The COVID-19 Model

With the outbreak of COVID-19 in China, the decisive measures the Chinese government has taken effectively restrict the movement of people across provinces, especially in Hubei province [23]. On this account, we propose a compartmental dynamics model, and we divide COVID-19-related populations into nine epidemiological subgroups: susceptible (S); exposed (E); suspected and undetected (L); removed, such as influenza patients or common pneumonia patients (C); clinically diagnosed cases (I); mild patients (H1); severe patients (H2); critical patients (H3); and recovered (R). The total population size is denoted by N=S+E+L+I+H+R (H here is the total number of confirmed cases, H=H1+H2+H3). In order to make the model more suitable for the local situation in Hubei province, we put forward the following conditions:
(1)
Since we have only studied for a relatively short period of time, we ignore natural births and deaths, but we do not ignore deaths from COVID-19(2)
Asymptomatic transmission is a form of COVID-19 disease transmission [7]. For example, a person who contracts COVID-19 from someone with few or without symptoms may still be infected [50]. If we put the above examples into the model: exposed (E), suspected and undetected (L), clinically diagnosed cases (I), and confirmed cases (H), they might lead susceptible patients (S) to infect COVID-19 with different infection probabilities(3)
Novel coronavirus nucleic acid test-negative patients and a small number of COVID-19 self-healing patients from suspected and undetected (L) will still have a certain possibility of contacting COVID-19, so some people in this group will become susceptible (S)(4)
The city of Wuhan in Hubei province was locked down on 23 January 2020, and other areas in Hubei province were also locked down one after another. Xiangyang city, the last city in Hubei province, was locked down on 27 January 2020. Considering that the symptoms of COVID-19 are very similar to those of influenza patients or common pneumonia patients, etc. In addition, some suspected and undetected (L) are actually just influenza patients or common pneumonia patients, etc. So, we introduce (βi,i=1,2) the following. The β1 represents the transmission rate of COVID-19, and β2 represents the transmission rate of influenza or common pneumonia, etc.(1)βi=βia,10−22January2020,Wuhanwasnotlockeddown,βib,23−27January2020,thecitiesinHubeihavebeenlockeddown,βic,28January‐30April2020,afterHubeiwaslockeddown(5)
The probability of medical personnel contracting COVID-19 (k, q) before and after the lockdown in Hubei is different. The risk before the lockdown should be higher than that after it was taken, which is due to the initial shortage of medical supplies and the lack of understanding of the epidemic propagation mode. The lockdown in Hubei and the outbreak of the initial disease had led to a serious shortage of viral nucleic acid test kits, which also affected the time required for diagnosis (1/σ) to a certain extent. Here, we put τ = (k, q, σ) into a parameter set:(2)τ=τa,10−22January2020,Wuhanwasnotlockeddown,τb,23−27January2020,thecitiesinHubeihavebeenlockeddown,τc,28January‐30April2020,afterHubeiwaslockeddown(6)
From the beginning of the lockdown in Wuhan to the lockdown in Xiangyang, some people in Hubei were out of the province, and some people were not susceptible compared with others in this epidemic. We remove these people from the susceptible (S) compartment:(3)φ=φa,somepeoplewholeftWuhancityon23January2020,φb,somepeoplewholeftHubeiprovinceon27January2020Based on the above analysis, a schematic flow diagram is created based on the dynamic process of the spread of COVID-19 infection in Figure1. In addition, considering the transmission characteristics of COVID-19, relevant population quarantine measures, suspected cases (part of them are influenza patients or common pneumonia patients, etc.), three classifications of confirmed patients, the difference between clinically diagnosed cases and confirmed cases in Wuhan, and the lockdown of Wuhan and Hubei, we constructed the following differential equations to describe the COVID-19 model:
(4)dSdt=−β1SpE+L+qI+kHN−β2SpE+L+qIN+ωηL,dEdt=β1SpE+L+qI+kHN+β2SpE+L+qIN−αE,dLdt=αρE+θ4σI−ω+ξL,dIdt=α1−ρE+ω1−ηL−σI−dI,dH1dt=θ1σI−δ1H1−d1H1,dH2dt=θ2σI+δ1λ1H1−δ2H2−d2H2,dH3dt=θ3σI+δ2λ2H2−δ3H3−d3H3,dRdt=1−∑i=14θiσI+δ11−λ1H1+δ21−λ2H2+δ3H3.Figure 1
Flow chart of compartments of the COVID-19 model.


### 2.2. The Basic Reproduction Number

The basic reproduction number (R0) is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual [51]. For threshold systems, if R0<1, it means that the infection cannot grow and the disease will die out. Conversely, if R0>1, then each infected individual produces more than one new infection on average, and the disease will become epidemic [52]. Of course, for nonthreshold systems, R0 is no longer the only measure of whether a disease is extinct. However, a smaller value of R0 can be advantageous to the control of the disease.We define the following:A=ω+ξσ+d−θ4σ1−η, B=ρω1−η+ω+ξ1−ρ, C=δ1+d1A, G=δ2+d2C, J=δ3+d3G, and K=θ1δ1λ1+θ2δ1+d1.By referring to the paper of Driessche and Watmough [33], R0i,i=1,2 is calculated through the ordinary differential equations as follows:
(5)R01=R11+R12+R13+R14+R15+R16,where
(6)R11=β1p/α,R12=β1ρσ+dθ4σ1−ρ/A,R13=β1qB/A,R14=β1kθ1σB/C,R15=β1kωρ1−ησK+σω+η1−ρK/G,R16=β1kBθ1δ1λ1+θ2δ1+d1σδ2λ2+θ3σδ1+d1δ2+d2/J,(7)R02=R21+R22+R23,where
(8)R21=β2p/α,R22=β2ρσ+d+θ4σ1−ρ/A,R23=β2qB/A.R01 represents the basic reproduction number of COVID-19, and the R02 represents the basic reproduction number of influenza or common pneumonia, etc.


### 2.3. Data Fitting and Parameter Estimation

#### 2.3.1. Data

The surveillance data of COVID-19 in Hubei province was reported by the Health Commission of Hubei Province (HCHP) [53] and WHO [54]. HCHP conducted daily briefings about COVID-19 data in all cities of Hubei province. Data information includes the cumulative number of reported cases (i.e., confirmed cases or confirmed cases plus clinical diagnosis cases), recovered cases, death cases, severe, and critical illness. The cumulative reported cases, severe cases, and critical cases data are from 10 January to 30 April 2020, a total of 112 days, and the death cases are from 10 January to 15 April 2020, a total of 97 days. The COVID-19 outbreak began in late January and was stabilized in April, which took nearly three months in Hubei.


#### 2.3.2. Parameter Estimation

We have determined the range of the transmission rate (βi,i=1,2) based on Abbott et al. [36] and Li et al. [23]. Referring to the results by Tang et al. [25], we set the corresponding range for mortality rate (d, d1, d2, and d3) and the proportion of infectious (ρ). The incubation period of the disease (1/α) is 5.1 days [55], so the transfer rate (from exposed individuals (E) to suspected patients (L) and clinically diagnosed cases (I)) is α=0.1961. We referred to the values of p, q, and k in Chowell et al.’s result [56] and obtained the appropriate ranges of p, q, and k. We considered the classification of different types of patients: mild, severe, and critical. And we also collected daily cumulative reported cases, severe cases, and critical cases from HCHP, calculated the proportions of the three types of patients in the confirmed cases, and further determined the value range of θ1, θ2, θ3, and θ4 (for details, see Table 2). According to the Statistical Communique on Hubei province’s National Economic and Social Development [57], we made assumptions about the range of the initial population of susceptible persons S0. According to data related to COVID-19 reported by HCHP [34], H20=7, R0=4, and D0=1, we set the range of H10 and H30. Table 2 sets the upper and lower bounds of other parameters required by the model.Table 2
Parameter estimates for COVID-19 model in Hubei.
ParameterDefinitionsValueSourceβ1a∈0.1,10Transmission rateday‐1individual‐11.8208Estimatedβ2a∈0.01,1Transmission rate0.5487Estimatedβ1b∈0.1,10Transmission rate1.1720Estimatedβ2b∈0.01,1Transmission rate0.3648Estimatedβ1c∈0.1,10Transmission rate0.8665Estimatedβ2c∈0.01,1Transmission rate0.5643Estimatedd∈0,0.0001Mortality rate of clinically diagnosed cases (day-1)3.8858×10−5Estimatedd1∈0,0.001Mortality rate of mild patients (day-1)2.0436×10−4Estimatedd2∈0,0.2Mortality rate of severely ill patients (day-1)4.5000×10−4Estimatedd3∈0,0.2Mortality rate of critically ill patients (day-1)0.0641Estimatedδ1∈0.1429,0.5Recovery rate of mild patients (day-1)0.2157Estimatedδ2∈0.05,0.5Recovery rate of severely ill patients (day-1)0.0606Estimatedδ3∈0.03,0.2Recovery rate of critically ill patients (day-1)0.0477Estimatedη∈0.01,0.25Self-healing ratio0.2117Estimatedka∈0.01,0.4Infectivity reduction factor0.0446Estimatedkb∈0.01,0.4Infectivity reduction factor0.0259Estimatedkc∈0.01,0.4Infectivity reduction factor0.0705Estimatedqa∈0.1,0.5Infectivity reduction factor0.1561Estimatedqb∈0.1,0.5Infectivity reduction factor0.2544Estimatedqc∈0.1,0.5Infectivity reduction factor0.2282Estimatedξ∈0.01,1System population reduction factor0.9700Estimatedλ1∈0.1,0.3Scale factor0.3000Estimatedλ2∈0.1,0.6Scale factor0.2499Estimatedω∈0.016,0.1Infectivity reduction factor0.0177Estimatedp∈0.1,0.5Infectivity reduction factor0.1364Estimatedρ∈0.1,1Proportion of the infectious0.9984Estimatedσa∈0.1,0.5Nucleic acid detection time (day-1)0.1472Estimatedσb∈0.1,0.5Nucleic acid detection time0.1309Estimatedσc∈0.1,0.5Nucleic acid detection time0.2576Estimatedθ1∈0.2381,0.3969Scale factor0.2411Estimatedθ2∈0.0471,0.0786Scale factor0.0786Estimatedθ3∈0.0148,0.0246Scale factor0.0246Estimatedθ4∈0.3,0.55Scale factor0.4870Estimatedφa∈0,2×107System discharges8.8731×106Estimatedφb∈1×107,5.907×107System discharges1.5910×107EstimatedS0∈1×104,5.927×107Initial susceptible population3.1372×107EstimatedE0∈2000,1×107Initial exposed population2.2380×103EstimatedL0∈0,1×106Initial suspected population55.5580EstimatedI0∈0,1×104Initial clinically diagnosed cases497.9463EstimatedH10∈0,41Initial mild patients5.7737EstimatedH30∈0,41Initial critical patients11.0952EstimatedHere, we will introduce the simulation process in detail:(1) Before Wuhan Was Locked Down (10 January-22 January 2020). During this period of time, many people have no sense of protection against COVID-19, and the phenomenon of human-to-human transmission has not been confirmed. The transmission rate (β1) and the infection rate of medical staff (k) are both high.(2) After Wuhan Was Locked Down and before Hubei Was Locked Down (23 January-27 January 2020). Wuhan was locked down at 10:00 a.m. on 23 January 2020, and the last city in Hubei (Xiangyang) was locked down at 0:00 a.m. on 27 January 2020. With the gradual adoption of lockdown measures across the province, the transmission rate (β1) will be reduced to a certain extent, but there will be a certain degree of shortage of medical supplies.(3) After Hubei Was Locked Down and before the Large-Scale Case Screening (28 January-11 February 2020). On 5 February 2020, the National Health Commission (NHC) issued the “Diagnosis and Treatment Protocol for COVID-19 (Trial Version 5)” [7]. At the time of lockdown in Hubei, due to the shortage of medical materials such as nucleic acid test kits, Hubei had adopted a different diagnosis method from other provinces, adding “clinically diagnosed cases” [58]. Other provinces still divided cases into “suspected cases” and “confirmed cases.” With the complete lockdown in Hubei, the transmission rate (β1) further decreased. With the further shortage of nucleic acid testing kits and medical supplies, the time required for nucleic acid testing (1/σ) increased, and the risk of infection between medical personnel and confirmed patients (k) increased.(4) After Adding Clinically Diagnosed Cases to Cumulative Reported Cases and before Removing Clinically Diagnosed Cases from Cumulative Reported Cases (12 February-19 February 2020). On 12 February 2020, NHC added clinically diagnosed cases to the cumulative cases. The cumulative number of reported cases in Hubei province increased to 13,332 on that day. And on 19 February 2020, clinically diagnosed cases had been excluded from the cumulative cases. Here, we take the same β, k, and q as the third paragraph (After Hubei Was Locked Down and before the Large-Scale Case Screening (28 January-11 February 2020)).Hence, the following equations could describe the dynamics of the cumulative reported cases (Y) in Hubei province:
(9)dYdt=θ1+θ2+θ3σI+δ1λ1H1+δ2λ2H2+α1−ρE+ω1−ηL,12February‐19February2020,θ1+θ2+θ3σI+δ1λ1H1+δ2λ2H2,othertime.(5) After the Large-Scale Case Screening (after 19 February 2020). With the improvement of nucleic acid detection kits and medical conditions, the screening speed of clinically diagnosed cases had been further improved, and the previous cumulative clinically diagnosed cases were screened on 19 February 2020, so the NHC decided to remove clinically diagnosed cases from the cumulative cases. Here, we take the same β, k, and q as the third paragraph (After Hubei Was Locked Down and before the Large-Scale Case Screening (28 January-11 February 2020)).To better fit the surveillance data, we divided the data into three stages from 10 January 2020 to 30 April 2020. These three stages set different transmission rates (βi,i=1,2), nucleic acid detection times (σ), the infection reduction factors between susceptible and confirmed cases (k), and the infection reduction factors between susceptible and clinically diagnosed cases (q). Different numbers of people leaving Hubei province are set on 23 and 27 January 2020 (i.e., the beginning and end of the second stage), and the rest parameters remain unchanged for the three stages.The number of the cumulative reported cases in the early stage of the outbreak of COVID-19 is small, but the number in the mid-term is relatively large. In the later stage, it gradually stabilizes; hence, the numerical curve fluctuates greatly. We fit four sets of data at the same time, and each set of data has fluctuation. The Chi-square value can make the error of small data as small as possible, and the error of relatively large data is allowed to be larger, which can improve the fitting effect of the model. So, the Chi-square value is selected to evaluate the reliability of the model [22], and 35 parameters and 6 initial values are estimated by the fminsearch optimization method [59, 60].
(10)J=∑i=1112Yti−Y∧ti2Y^ti+∑i=197Dti−D∧ti2D^ti+∑i=1112H2ti−H∧2ti2H^2ti+∑i=1112H3ti−H∧3ti2H^3ti.HereYti,i=1,2,⋯,112 represents the actual cumulative reported cases per day, and Y^ti,i=1,2,⋯,112 represents the corresponding fitted values. Dti,i=1,2,⋯,97 represents the actual number of deaths per day, and D^ti,i=1,2,⋯,97 represents the corresponding fitted values. H2ti,i=1,2,⋯,112 represents the actual number of severely ill patients per day, and H^2ti,i=1,2,⋯,112 represents the corresponding fitted values. H3ti,i=1,2,⋯,112 represents the actual number of critically ill patients per day, H^3ti,i=1,2,⋯,112 represents the corresponding fitted values, and J represents the total Chi-square value. Table 2 shows the parameter values and initial values of the model at different stages.


## 2.1. The COVID-19 Model

With the outbreak of COVID-19 in China, the decisive measures the Chinese government has taken effectively restrict the movement of people across provinces, especially in Hubei province [23]. On this account, we propose a compartmental dynamics model, and we divide COVID-19-related populations into nine epidemiological subgroups: susceptible (S); exposed (E); suspected and undetected (L); removed, such as influenza patients or common pneumonia patients (C); clinically diagnosed cases (I); mild patients (H1); severe patients (H2); critical patients (H3); and recovered (R). The total population size is denoted by N=S+E+L+I+H+R (H here is the total number of confirmed cases, H=H1+H2+H3). In order to make the model more suitable for the local situation in Hubei province, we put forward the following conditions:
(1)
Since we have only studied for a relatively short period of time, we ignore natural births and deaths, but we do not ignore deaths from COVID-19(2)
Asymptomatic transmission is a form of COVID-19 disease transmission [7]. For example, a person who contracts COVID-19 from someone with few or without symptoms may still be infected [50]. If we put the above examples into the model: exposed (E), suspected and undetected (L), clinically diagnosed cases (I), and confirmed cases (H), they might lead susceptible patients (S) to infect COVID-19 with different infection probabilities(3)
Novel coronavirus nucleic acid test-negative patients and a small number of COVID-19 self-healing patients from suspected and undetected (L) will still have a certain possibility of contacting COVID-19, so some people in this group will become susceptible (S)(4)
The city of Wuhan in Hubei province was locked down on 23 January 2020, and other areas in Hubei province were also locked down one after another. Xiangyang city, the last city in Hubei province, was locked down on 27 January 2020. Considering that the symptoms of COVID-19 are very similar to those of influenza patients or common pneumonia patients, etc. In addition, some suspected and undetected (L) are actually just influenza patients or common pneumonia patients, etc. So, we introduce (βi,i=1,2) the following. The β1 represents the transmission rate of COVID-19, and β2 represents the transmission rate of influenza or common pneumonia, etc.(1)βi=βia,10−22January2020,Wuhanwasnotlockeddown,βib,23−27January2020,thecitiesinHubeihavebeenlockeddown,βic,28January‐30April2020,afterHubeiwaslockeddown(5)
The probability of medical personnel contracting COVID-19 (k, q) before and after the lockdown in Hubei is different. The risk before the lockdown should be higher than that after it was taken, which is due to the initial shortage of medical supplies and the lack of understanding of the epidemic propagation mode. The lockdown in Hubei and the outbreak of the initial disease had led to a serious shortage of viral nucleic acid test kits, which also affected the time required for diagnosis (1/σ) to a certain extent. Here, we put τ = (k, q, σ) into a parameter set:(2)τ=τa,10−22January2020,Wuhanwasnotlockeddown,τb,23−27January2020,thecitiesinHubeihavebeenlockeddown,τc,28January‐30April2020,afterHubeiwaslockeddown(6)
From the beginning of the lockdown in Wuhan to the lockdown in Xiangyang, some people in Hubei were out of the province, and some people were not susceptible compared with others in this epidemic. We remove these people from the susceptible (S) compartment:(3)φ=φa,somepeoplewholeftWuhancityon23January2020,φb,somepeoplewholeftHubeiprovinceon27January2020Based on the above analysis, a schematic flow diagram is created based on the dynamic process of the spread of COVID-19 infection in Figure1. In addition, considering the transmission characteristics of COVID-19, relevant population quarantine measures, suspected cases (part of them are influenza patients or common pneumonia patients, etc.), three classifications of confirmed patients, the difference between clinically diagnosed cases and confirmed cases in Wuhan, and the lockdown of Wuhan and Hubei, we constructed the following differential equations to describe the COVID-19 model:
(4)dSdt=−β1SpE+L+qI+kHN−β2SpE+L+qIN+ωηL,dEdt=β1SpE+L+qI+kHN+β2SpE+L+qIN−αE,dLdt=αρE+θ4σI−ω+ξL,dIdt=α1−ρE+ω1−ηL−σI−dI,dH1dt=θ1σI−δ1H1−d1H1,dH2dt=θ2σI+δ1λ1H1−δ2H2−d2H2,dH3dt=θ3σI+δ2λ2H2−δ3H3−d3H3,dRdt=1−∑i=14θiσI+δ11−λ1H1+δ21−λ2H2+δ3H3.Figure 1
Flow chart of compartments of the COVID-19 model.


## 2.2. The Basic Reproduction Number

The basic reproduction number (R0) is the expected number of secondary cases produced, in a completely susceptible population, by a typical infective individual [51]. For threshold systems, if R0<1, it means that the infection cannot grow and the disease will die out. Conversely, if R0>1, then each infected individual produces more than one new infection on average, and the disease will become epidemic [52]. Of course, for nonthreshold systems, R0 is no longer the only measure of whether a disease is extinct. However, a smaller value of R0 can be advantageous to the control of the disease.We define the following:A=ω+ξσ+d−θ4σ1−η, B=ρω1−η+ω+ξ1−ρ, C=δ1+d1A, G=δ2+d2C, J=δ3+d3G, and K=θ1δ1λ1+θ2δ1+d1.By referring to the paper of Driessche and Watmough [33], R0i,i=1,2 is calculated through the ordinary differential equations as follows:
(5)R01=R11+R12+R13+R14+R15+R16,where
(6)R11=β1p/α,R12=β1ρσ+dθ4σ1−ρ/A,R13=β1qB/A,R14=β1kθ1σB/C,R15=β1kωρ1−ησK+σω+η1−ρK/G,R16=β1kBθ1δ1λ1+θ2δ1+d1σδ2λ2+θ3σδ1+d1δ2+d2/J,(7)R02=R21+R22+R23,where
(8)R21=β2p/α,R22=β2ρσ+d+θ4σ1−ρ/A,R23=β2qB/A.R01 represents the basic reproduction number of COVID-19, and the R02 represents the basic reproduction number of influenza or common pneumonia, etc.


## 2.3. Data Fitting and Parameter Estimation

### 2.3.1. Data

The surveillance data of COVID-19 in Hubei province was reported by the Health Commission of Hubei Province (HCHP) [53] and WHO [54]. HCHP conducted daily briefings about COVID-19 data in all cities of Hubei province. Data information includes the cumulative number of reported cases (i.e., confirmed cases or confirmed cases plus clinical diagnosis cases), recovered cases, death cases, severe, and critical illness. The cumulative reported cases, severe cases, and critical cases data are from 10 January to 30 April 2020, a total of 112 days, and the death cases are from 10 January to 15 April 2020, a total of 97 days. The COVID-19 outbreak began in late January and was stabilized in April, which took nearly three months in Hubei.


### 2.3.2. Parameter Estimation

We have determined the range of the transmission rate (βi,i=1,2) based on Abbott et al. [36] and Li et al. [23]. Referring to the results by Tang et al. [25], we set the corresponding range for mortality rate (d, d1, d2, and d3) and the proportion of infectious (ρ). The incubation period of the disease (1/α) is 5.1 days [55], so the transfer rate (from exposed individuals (E) to suspected patients (L) and clinically diagnosed cases (I)) is α=0.1961. We referred to the values of p, q, and k in Chowell et al.’s result [56] and obtained the appropriate ranges of p, q, and k. We considered the classification of different types of patients: mild, severe, and critical. And we also collected daily cumulative reported cases, severe cases, and critical cases from HCHP, calculated the proportions of the three types of patients in the confirmed cases, and further determined the value range of θ1, θ2, θ3, and θ4 (for details, see Table 2). According to the Statistical Communique on Hubei province’s National Economic and Social Development [57], we made assumptions about the range of the initial population of susceptible persons S0. According to data related to COVID-19 reported by HCHP [34], H20=7, R0=4, and D0=1, we set the range of H10 and H30. Table 2 sets the upper and lower bounds of other parameters required by the model.Table 2
Parameter estimates for COVID-19 model in Hubei.
ParameterDefinitionsValueSourceβ1a∈0.1,10Transmission rateday‐1individual‐11.8208Estimatedβ2a∈0.01,1Transmission rate0.5487Estimatedβ1b∈0.1,10Transmission rate1.1720Estimatedβ2b∈0.01,1Transmission rate0.3648Estimatedβ1c∈0.1,10Transmission rate0.8665Estimatedβ2c∈0.01,1Transmission rate0.5643Estimatedd∈0,0.0001Mortality rate of clinically diagnosed cases (day-1)3.8858×10−5Estimatedd1∈0,0.001Mortality rate of mild patients (day-1)2.0436×10−4Estimatedd2∈0,0.2Mortality rate of severely ill patients (day-1)4.5000×10−4Estimatedd3∈0,0.2Mortality rate of critically ill patients (day-1)0.0641Estimatedδ1∈0.1429,0.5Recovery rate of mild patients (day-1)0.2157Estimatedδ2∈0.05,0.5Recovery rate of severely ill patients (day-1)0.0606Estimatedδ3∈0.03,0.2Recovery rate of critically ill patients (day-1)0.0477Estimatedη∈0.01,0.25Self-healing ratio0.2117Estimatedka∈0.01,0.4Infectivity reduction factor0.0446Estimatedkb∈0.01,0.4Infectivity reduction factor0.0259Estimatedkc∈0.01,0.4Infectivity reduction factor0.0705Estimatedqa∈0.1,0.5Infectivity reduction factor0.1561Estimatedqb∈0.1,0.5Infectivity reduction factor0.2544Estimatedqc∈0.1,0.5Infectivity reduction factor0.2282Estimatedξ∈0.01,1System population reduction factor0.9700Estimatedλ1∈0.1,0.3Scale factor0.3000Estimatedλ2∈0.1,0.6Scale factor0.2499Estimatedω∈0.016,0.1Infectivity reduction factor0.0177Estimatedp∈0.1,0.5Infectivity reduction factor0.1364Estimatedρ∈0.1,1Proportion of the infectious0.9984Estimatedσa∈0.1,0.5Nucleic acid detection time (day-1)0.1472Estimatedσb∈0.1,0.5Nucleic acid detection time0.1309Estimatedσc∈0.1,0.5Nucleic acid detection time0.2576Estimatedθ1∈0.2381,0.3969Scale factor0.2411Estimatedθ2∈0.0471,0.0786Scale factor0.0786Estimatedθ3∈0.0148,0.0246Scale factor0.0246Estimatedθ4∈0.3,0.55Scale factor0.4870Estimatedφa∈0,2×107System discharges8.8731×106Estimatedφb∈1×107,5.907×107System discharges1.5910×107EstimatedS0∈1×104,5.927×107Initial susceptible population3.1372×107EstimatedE0∈2000,1×107Initial exposed population2.2380×103EstimatedL0∈0,1×106Initial suspected population55.5580EstimatedI0∈0,1×104Initial clinically diagnosed cases497.9463EstimatedH10∈0,41Initial mild patients5.7737EstimatedH30∈0,41Initial critical patients11.0952EstimatedHere, we will introduce the simulation process in detail:(1) Before Wuhan Was Locked Down (10 January-22 January 2020). During this period of time, many people have no sense of protection against COVID-19, and the phenomenon of human-to-human transmission has not been confirmed. The transmission rate (β1) and the infection rate of medical staff (k) are both high.(2) After Wuhan Was Locked Down and before Hubei Was Locked Down (23 January-27 January 2020). Wuhan was locked down at 10:00 a.m. on 23 January 2020, and the last city in Hubei (Xiangyang) was locked down at 0:00 a.m. on 27 January 2020. With the gradual adoption of lockdown measures across the province, the transmission rate (β1) will be reduced to a certain extent, but there will be a certain degree of shortage of medical supplies.(3) After Hubei Was Locked Down and before the Large-Scale Case Screening (28 January-11 February 2020). On 5 February 2020, the National Health Commission (NHC) issued the “Diagnosis and Treatment Protocol for COVID-19 (Trial Version 5)” [7]. At the time of lockdown in Hubei, due to the shortage of medical materials such as nucleic acid test kits, Hubei had adopted a different diagnosis method from other provinces, adding “clinically diagnosed cases” [58]. Other provinces still divided cases into “suspected cases” and “confirmed cases.” With the complete lockdown in Hubei, the transmission rate (β1) further decreased. With the further shortage of nucleic acid testing kits and medical supplies, the time required for nucleic acid testing (1/σ) increased, and the risk of infection between medical personnel and confirmed patients (k) increased.(4) After Adding Clinically Diagnosed Cases to Cumulative Reported Cases and before Removing Clinically Diagnosed Cases from Cumulative Reported Cases (12 February-19 February 2020). On 12 February 2020, NHC added clinically diagnosed cases to the cumulative cases. The cumulative number of reported cases in Hubei province increased to 13,332 on that day. And on 19 February 2020, clinically diagnosed cases had been excluded from the cumulative cases. Here, we take the same β, k, and q as the third paragraph (After Hubei Was Locked Down and before the Large-Scale Case Screening (28 January-11 February 2020)).Hence, the following equations could describe the dynamics of the cumulative reported cases (Y) in Hubei province:
(9)dYdt=θ1+θ2+θ3σI+δ1λ1H1+δ2λ2H2+α1−ρE+ω1−ηL,12February‐19February2020,θ1+θ2+θ3σI+δ1λ1H1+δ2λ2H2,othertime.(5) After the Large-Scale Case Screening (after 19 February 2020). With the improvement of nucleic acid detection kits and medical conditions, the screening speed of clinically diagnosed cases had been further improved, and the previous cumulative clinically diagnosed cases were screened on 19 February 2020, so the NHC decided to remove clinically diagnosed cases from the cumulative cases. Here, we take the same β, k, and q as the third paragraph (After Hubei Was Locked Down and before the Large-Scale Case Screening (28 January-11 February 2020)).To better fit the surveillance data, we divided the data into three stages from 10 January 2020 to 30 April 2020. These three stages set different transmission rates (βi,i=1,2), nucleic acid detection times (σ), the infection reduction factors between susceptible and confirmed cases (k), and the infection reduction factors between susceptible and clinically diagnosed cases (q). Different numbers of people leaving Hubei province are set on 23 and 27 January 2020 (i.e., the beginning and end of the second stage), and the rest parameters remain unchanged for the three stages.The number of the cumulative reported cases in the early stage of the outbreak of COVID-19 is small, but the number in the mid-term is relatively large. In the later stage, it gradually stabilizes; hence, the numerical curve fluctuates greatly. We fit four sets of data at the same time, and each set of data has fluctuation. The Chi-square value can make the error of small data as small as possible, and the error of relatively large data is allowed to be larger, which can improve the fitting effect of the model. So, the Chi-square value is selected to evaluate the reliability of the model [22], and 35 parameters and 6 initial values are estimated by the fminsearch optimization method [59, 60].
(10)J=∑i=1112Yti−Y∧ti2Y^ti+∑i=197Dti−D∧ti2D^ti+∑i=1112H2ti−H∧2ti2H^2ti+∑i=1112H3ti−H∧3ti2H^3ti.HereYti,i=1,2,⋯,112 represents the actual cumulative reported cases per day, and Y^ti,i=1,2,⋯,112 represents the corresponding fitted values. Dti,i=1,2,⋯,97 represents the actual number of deaths per day, and D^ti,i=1,2,⋯,97 represents the corresponding fitted values. H2ti,i=1,2,⋯,112 represents the actual number of severely ill patients per day, and H^2ti,i=1,2,⋯,112 represents the corresponding fitted values. H3ti,i=1,2,⋯,112 represents the actual number of critically ill patients per day, H^3ti,i=1,2,⋯,112 represents the corresponding fitted values, and J represents the total Chi-square value. Table 2 shows the parameter values and initial values of the model at different stages.


## 2.3.1. Data

The surveillance data of COVID-19 in Hubei province was reported by the Health Commission of Hubei Province (HCHP) [53] and WHO [54]. HCHP conducted daily briefings about COVID-19 data in all cities of Hubei province. Data information includes the cumulative number of reported cases (i.e., confirmed cases or confirmed cases plus clinical diagnosis cases), recovered cases, death cases, severe, and critical illness. The cumulative reported cases, severe cases, and critical cases data are from 10 January to 30 April 2020, a total of 112 days, and the death cases are from 10 January to 15 April 2020, a total of 97 days. The COVID-19 outbreak began in late January and was stabilized in April, which took nearly three months in Hubei.


## 2.3.2. Parameter Estimation

We have determined the range of the transmission rate (βi,i=1,2) based on Abbott et al. [36] and Li et al. [23]. Referring to the results by Tang et al. [25], we set the corresponding range for mortality rate (d, d1, d2, and d3) and the proportion of infectious (ρ). The incubation period of the disease (1/α) is 5.1 days [55], so the transfer rate (from exposed individuals (E) to suspected patients (L) and clinically diagnosed cases (I)) is α=0.1961. We referred to the values of p, q, and k in Chowell et al.’s result [56] and obtained the appropriate ranges of p, q, and k. We considered the classification of different types of patients: mild, severe, and critical. And we also collected daily cumulative reported cases, severe cases, and critical cases from HCHP, calculated the proportions of the three types of patients in the confirmed cases, and further determined the value range of θ1, θ2, θ3, and θ4 (for details, see Table 2). According to the Statistical Communique on Hubei province’s National Economic and Social Development [57], we made assumptions about the range of the initial population of susceptible persons S0. According to data related to COVID-19 reported by HCHP [34], H20=7, R0=4, and D0=1, we set the range of H10 and H30. Table 2 sets the upper and lower bounds of other parameters required by the model.Table 2
Parameter estimates for COVID-19 model in Hubei.
ParameterDefinitionsValueSourceβ1a∈0.1,10Transmission rateday‐1individual‐11.8208Estimatedβ2a∈0.01,1Transmission rate0.5487Estimatedβ1b∈0.1,10Transmission rate1.1720Estimatedβ2b∈0.01,1Transmission rate0.3648Estimatedβ1c∈0.1,10Transmission rate0.8665Estimatedβ2c∈0.01,1Transmission rate0.5643Estimatedd∈0,0.0001Mortality rate of clinically diagnosed cases (day-1)3.8858×10−5Estimatedd1∈0,0.001Mortality rate of mild patients (day-1)2.0436×10−4Estimatedd2∈0,0.2Mortality rate of severely ill patients (day-1)4.5000×10−4Estimatedd3∈0,0.2Mortality rate of critically ill patients (day-1)0.0641Estimatedδ1∈0.1429,0.5Recovery rate of mild patients (day-1)0.2157Estimatedδ2∈0.05,0.5Recovery rate of severely ill patients (day-1)0.0606Estimatedδ3∈0.03,0.2Recovery rate of critically ill patients (day-1)0.0477Estimatedη∈0.01,0.25Self-healing ratio0.2117Estimatedka∈0.01,0.4Infectivity reduction factor0.0446Estimatedkb∈0.01,0.4Infectivity reduction factor0.0259Estimatedkc∈0.01,0.4Infectivity reduction factor0.0705Estimatedqa∈0.1,0.5Infectivity reduction factor0.1561Estimatedqb∈0.1,0.5Infectivity reduction factor0.2544Estimatedqc∈0.1,0.5Infectivity reduction factor0.2282Estimatedξ∈0.01,1System population reduction factor0.9700Estimatedλ1∈0.1,0.3Scale factor0.3000Estimatedλ2∈0.1,0.6Scale factor0.2499Estimatedω∈0.016,0.1Infectivity reduction factor0.0177Estimatedp∈0.1,0.5Infectivity reduction factor0.1364Estimatedρ∈0.1,1Proportion of the infectious0.9984Estimatedσa∈0.1,0.5Nucleic acid detection time (day-1)0.1472Estimatedσb∈0.1,0.5Nucleic acid detection time0.1309Estimatedσc∈0.1,0.5Nucleic acid detection time0.2576Estimatedθ1∈0.2381,0.3969Scale factor0.2411Estimatedθ2∈0.0471,0.0786Scale factor0.0786Estimatedθ3∈0.0148,0.0246Scale factor0.0246Estimatedθ4∈0.3,0.55Scale factor0.4870Estimatedφa∈0,2×107System discharges8.8731×106Estimatedφb∈1×107,5.907×107System discharges1.5910×107EstimatedS0∈1×104,5.927×107Initial susceptible population3.1372×107EstimatedE0∈2000,1×107Initial exposed population2.2380×103EstimatedL0∈0,1×106Initial suspected population55.5580EstimatedI0∈0,1×104Initial clinically diagnosed cases497.9463EstimatedH10∈0,41Initial mild patients5.7737EstimatedH30∈0,41Initial critical patients11.0952EstimatedHere, we will introduce the simulation process in detail:(1) Before Wuhan Was Locked Down (10 January-22 January 2020). During this period of time, many people have no sense of protection against COVID-19, and the phenomenon of human-to-human transmission has not been confirmed. The transmission rate (β1) and the infection rate of medical staff (k) are both high.(2) After Wuhan Was Locked Down and before Hubei Was Locked Down (23 January-27 January 2020). Wuhan was locked down at 10:00 a.m. on 23 January 2020, and the last city in Hubei (Xiangyang) was locked down at 0:00 a.m. on 27 January 2020. With the gradual adoption of lockdown measures across the province, the transmission rate (β1) will be reduced to a certain extent, but there will be a certain degree of shortage of medical supplies.(3) After Hubei Was Locked Down and before the Large-Scale Case Screening (28 January-11 February 2020). On 5 February 2020, the National Health Commission (NHC) issued the “Diagnosis and Treatment Protocol for COVID-19 (Trial Version 5)” [7]. At the time of lockdown in Hubei, due to the shortage of medical materials such as nucleic acid test kits, Hubei had adopted a different diagnosis method from other provinces, adding “clinically diagnosed cases” [58]. Other provinces still divided cases into “suspected cases” and “confirmed cases.” With the complete lockdown in Hubei, the transmission rate (β1) further decreased. With the further shortage of nucleic acid testing kits and medical supplies, the time required for nucleic acid testing (1/σ) increased, and the risk of infection between medical personnel and confirmed patients (k) increased.(4) After Adding Clinically Diagnosed Cases to Cumulative Reported Cases and before Removing Clinically Diagnosed Cases from Cumulative Reported Cases (12 February-19 February 2020). On 12 February 2020, NHC added clinically diagnosed cases to the cumulative cases. The cumulative number of reported cases in Hubei province increased to 13,332 on that day. And on 19 February 2020, clinically diagnosed cases had been excluded from the cumulative cases. Here, we take the same β, k, and q as the third paragraph (After Hubei Was Locked Down and before the Large-Scale Case Screening (28 January-11 February 2020)).Hence, the following equations could describe the dynamics of the cumulative reported cases (Y) in Hubei province:
(9)dYdt=θ1+θ2+θ3σI+δ1λ1H1+δ2λ2H2+α1−ρE+ω1−ηL,12February‐19February2020,θ1+θ2+θ3σI+δ1λ1H1+δ2λ2H2,othertime.(5) After the Large-Scale Case Screening (after 19 February 2020). With the improvement of nucleic acid detection kits and medical conditions, the screening speed of clinically diagnosed cases had been further improved, and the previous cumulative clinically diagnosed cases were screened on 19 February 2020, so the NHC decided to remove clinically diagnosed cases from the cumulative cases. Here, we take the same β, k, and q as the third paragraph (After Hubei Was Locked Down and before the Large-Scale Case Screening (28 January-11 February 2020)).To better fit the surveillance data, we divided the data into three stages from 10 January 2020 to 30 April 2020. These three stages set different transmission rates (βi,i=1,2), nucleic acid detection times (σ), the infection reduction factors between susceptible and confirmed cases (k), and the infection reduction factors between susceptible and clinically diagnosed cases (q). Different numbers of people leaving Hubei province are set on 23 and 27 January 2020 (i.e., the beginning and end of the second stage), and the rest parameters remain unchanged for the three stages.The number of the cumulative reported cases in the early stage of the outbreak of COVID-19 is small, but the number in the mid-term is relatively large. In the later stage, it gradually stabilizes; hence, the numerical curve fluctuates greatly. We fit four sets of data at the same time, and each set of data has fluctuation. The Chi-square value can make the error of small data as small as possible, and the error of relatively large data is allowed to be larger, which can improve the fitting effect of the model. So, the Chi-square value is selected to evaluate the reliability of the model [22], and 35 parameters and 6 initial values are estimated by the fminsearch optimization method [59, 60].
(10)J=∑i=1112Yti−Y∧ti2Y^ti+∑i=197Dti−D∧ti2D^ti+∑i=1112H2ti−H∧2ti2H^2ti+∑i=1112H3ti−H∧3ti2H^3ti.HereYti,i=1,2,⋯,112 represents the actual cumulative reported cases per day, and Y^ti,i=1,2,⋯,112 represents the corresponding fitted values. Dti,i=1,2,⋯,97 represents the actual number of deaths per day, and D^ti,i=1,2,⋯,97 represents the corresponding fitted values. H2ti,i=1,2,⋯,112 represents the actual number of severely ill patients per day, and H^2ti,i=1,2,⋯,112 represents the corresponding fitted values. H3ti,i=1,2,⋯,112 represents the actual number of critically ill patients per day, H^3ti,i=1,2,⋯,112 represents the corresponding fitted values, and J represents the total Chi-square value. Table 2 shows the parameter values and initial values of the model at different stages.


## 3. Results

In this section, we first give the simulation comparison results, then calculateR0 for three different stages according to the parameter values fitted by the model. Secondly, since R0 is a key threshold for measuring the level of epidemic transmission, the sensitivity of each parameter in R0 is also worthy of our attention. Therefore, we will conduct a sensitivity analysis on R0 to find parameters that have a greater impact on R0.


### 3.1. The Result of Model-Based Estimates

We used the surveillance data found from the HCHP, the initial values, and the range of each parameter to perform the fminsearch optimization method. On this basis, the values of each parameter were continuously adjusted until the computer fit a suitable result. The fitting results of cumulative reported, severe cases, critical cases, and death cases are shown in Figures2–5.Figure 2
The cumulative reported and simulative cases in Hubei provinceYt.Figure 3
The severe cases and simulative cases in Hubei provinceH2t.Figure 4
The critical cases and simulative cases in Hubei provinceH3t.We fit four sets of data at the same time; from Figures2–5, we can see that the cumulative reported cases fit the best, and the other three sets of data can still fit the basic trend, because our model takes multiple actual factors into consideration, including suspected cases (part of them are influenza patients or common pneumonia patients, etc.), quarantine, patient classification (three types), clinically diagnosed cases, and lockdown of Wuhan and Hubei. The above considerations lead to high dimensions and many parameters of the model, so we need to fit multiple sets of data at the same time to verify the rationality of the model parameters. Therefore, the fitting result is obtained by cross-validation of the four sets of data.Figure 5
The death cases and simulative cases in Hubei provinceDt.


### 3.2. Results of Basic Reproduction Numbers

The results of basic reproduction numbers at the three different stages will be described in this part. From the COVID-19 model, we got that when Wuhan was not locked down (10 January-22 January),R0=3.1571 in Hubei province; after Wuhan was locked down but Hubei was not locked down (23 January-27 January), R0=2.0471; after Hubei was locked down (28 January-30 April), R0=1.5014.With the implementation of various control measures, the value of the basic reproduction number is constantly decreasing, which shows that timely lockdown, patient classification, and large-scale case screening are a series of effective control measures.


### 3.3. Sensitivity Analysis of Basic Reproduction Numbers

Sensitivity analysis of the basic reproduction number is performed in this section to determine several parameters that have the greatest influence on the prevalence and transmission of COVID-19. Through the sensitivity analysis of the basic reproduction numberR0, we can find the most sensitive parameter to R0. Then, we can reduce the basic reproduction number by making relevant measures to control the parameters and further achieve the purpose of disease control.According to the research method of Samsuzzoha et al. [61] on the sensitivity analysis of the basic reproduction number, we used the simulated parameter values (the third stage, i.e., after 28 January 2020) to perform a sensitivity analysis of the basic reproduction number. If a small change in a parameter can cause a large change in the value of the basic reproduction number, then this parameter is called a sensitivity factor, otherwise called an insensitive factor. The sensitivity indices of each parameter about the basic reproduction number R0 are shown in Table 3 (by increasing the value of the basic reproduction number R0 by 1%, observe the corresponding changes in the sensitivity index of the parameter).Table 3
The sensitivity indices of each parameter to the basic reproduction number.
ParameterSensitivity index toR0Percentage of corresponding change (%)β2c0.9996−1.0004p0.4020−2.4877qc0.0081−1.2392×102θ40.0046−2.1799×102ω0.0017−5.9956×102kc0.0011−9.3002×102θ18.2297×10−4−1.2151×103θ24.6762×10−4−2.1385×103λ22.1489×10−4−4.6535×104λ11.0618×10−4−9.4181×103θ31.4129×10−5−7.0778×103ξ−0.59041.6937ρ−0.47412.1093α−0.40202.4877σc−0.00801.2463×102η−0.00303.2885×102δ1−6.7225×10−41.4875×103δ2−3.1119×10−43.2134×103δ3−1.5198×10−46.5799×103From Table3, we can observe that β2c, p, qc, θ4, ω, kc, θ1, θ2, λ2, λ1, and θ3 have a positive impact on R0. ξ, ρ, α, σc, η, δ1, δ2, and δ3 have a negative impact on R0. Sensitivity analysis shows that the basic reproduction number is highly sensitive to β2c, p, ξ, ρ and α. Therefore, the lower transmission rate (β), the lower infectiousness reduction factor (p), and the lower proportion of the infectious 1−ρ can effectively reduce the basic reproduction number.From the sensitivity analysis of parameters, we can see thatk, q, and σ are not very sensitive which shows that their valuations have certain randomness and complexity. The parameters do not present in monotonous changes like increasing or decreasing. This situation appeared to show the complexity of the epidemic through transmission and the comprehensive impact of the number of patients in each stage, the limited degree of medical resources, the development of nucleic acid testing kits, and so on. So, when we use this model to describe the phenomenon, it can also reflect the complexity of the epidemic.


## 3.1. The Result of Model-Based Estimates

We used the surveillance data found from the HCHP, the initial values, and the range of each parameter to perform the fminsearch optimization method. On this basis, the values of each parameter were continuously adjusted until the computer fit a suitable result. The fitting results of cumulative reported, severe cases, critical cases, and death cases are shown in Figures2–5.Figure 2
The cumulative reported and simulative cases in Hubei provinceYt.Figure 3
The severe cases and simulative cases in Hubei provinceH2t.Figure 4
The critical cases and simulative cases in Hubei provinceH3t.We fit four sets of data at the same time; from Figures2–5, we can see that the cumulative reported cases fit the best, and the other three sets of data can still fit the basic trend, because our model takes multiple actual factors into consideration, including suspected cases (part of them are influenza patients or common pneumonia patients, etc.), quarantine, patient classification (three types), clinically diagnosed cases, and lockdown of Wuhan and Hubei. The above considerations lead to high dimensions and many parameters of the model, so we need to fit multiple sets of data at the same time to verify the rationality of the model parameters. Therefore, the fitting result is obtained by cross-validation of the four sets of data.Figure 5
The death cases and simulative cases in Hubei provinceDt.


## 3.2. Results of Basic Reproduction Numbers

The results of basic reproduction numbers at the three different stages will be described in this part. From the COVID-19 model, we got that when Wuhan was not locked down (10 January-22 January),R0=3.1571 in Hubei province; after Wuhan was locked down but Hubei was not locked down (23 January-27 January), R0=2.0471; after Hubei was locked down (28 January-30 April), R0=1.5014.With the implementation of various control measures, the value of the basic reproduction number is constantly decreasing, which shows that timely lockdown, patient classification, and large-scale case screening are a series of effective control measures.


## 3.3. Sensitivity Analysis of Basic Reproduction Numbers

Sensitivity analysis of the basic reproduction number is performed in this section to determine several parameters that have the greatest influence on the prevalence and transmission of COVID-19. Through the sensitivity analysis of the basic reproduction numberR0, we can find the most sensitive parameter to R0. Then, we can reduce the basic reproduction number by making relevant measures to control the parameters and further achieve the purpose of disease control.According to the research method of Samsuzzoha et al. [61] on the sensitivity analysis of the basic reproduction number, we used the simulated parameter values (the third stage, i.e., after 28 January 2020) to perform a sensitivity analysis of the basic reproduction number. If a small change in a parameter can cause a large change in the value of the basic reproduction number, then this parameter is called a sensitivity factor, otherwise called an insensitive factor. The sensitivity indices of each parameter about the basic reproduction number R0 are shown in Table 3 (by increasing the value of the basic reproduction number R0 by 1%, observe the corresponding changes in the sensitivity index of the parameter).Table 3
The sensitivity indices of each parameter to the basic reproduction number.
ParameterSensitivity index toR0Percentage of corresponding change (%)β2c0.9996−1.0004p0.4020−2.4877qc0.0081−1.2392×102θ40.0046−2.1799×102ω0.0017−5.9956×102kc0.0011−9.3002×102θ18.2297×10−4−1.2151×103θ24.6762×10−4−2.1385×103λ22.1489×10−4−4.6535×104λ11.0618×10−4−9.4181×103θ31.4129×10−5−7.0778×103ξ−0.59041.6937ρ−0.47412.1093α−0.40202.4877σc−0.00801.2463×102η−0.00303.2885×102δ1−6.7225×10−41.4875×103δ2−3.1119×10−43.2134×103δ3−1.5198×10−46.5799×103From Table3, we can observe that β2c, p, qc, θ4, ω, kc, θ1, θ2, λ2, λ1, and θ3 have a positive impact on R0. ξ, ρ, α, σc, η, δ1, δ2, and δ3 have a negative impact on R0. Sensitivity analysis shows that the basic reproduction number is highly sensitive to β2c, p, ξ, ρ and α. Therefore, the lower transmission rate (β), the lower infectiousness reduction factor (p), and the lower proportion of the infectious 1−ρ can effectively reduce the basic reproduction number.From the sensitivity analysis of parameters, we can see thatk, q, and σ are not very sensitive which shows that their valuations have certain randomness and complexity. The parameters do not present in monotonous changes like increasing or decreasing. This situation appeared to show the complexity of the epidemic through transmission and the comprehensive impact of the number of patients in each stage, the limited degree of medical resources, the development of nucleic acid testing kits, and so on. So, when we use this model to describe the phenomenon, it can also reflect the complexity of the epidemic.


## 4. Discussion

With the implementation of various control measures, the basic reproduction number was decreasing at different stages. But we do not know how much positive effect is brought by patient classification and lockdown. Next, we will discuss the following three situations in detail to introduce the impact on Hubei from 10 January to 18 April.


### 4.1. The Impact of Patient Classification on COVID-19

Assuming that patients are not classified, we modify the model. In this new model, we merge the previousH1, H2, and H3 into H, change the sum of θ1, θ2, and θ3 in the previous model to θ1=0.3443 in the current model, and change θ4 in the previous model to θ2=0.4870 in the current model, then modify δ2 in the previous model to δ=0.0593 in the current model. The model is described by the following ordinary differential equations:
(11)dSdt=−β1SpE+L+qI+kHN−β2SpE+L+qIN+ωηL,dEdt=β1SpE+L+qI+kHN+β2SpE+L+qIN−αE,dLdt=αρE+θ2σI−ω+ξL,dIdt=α1−ρE+ω1−ηL−σI−dI,dHdt=θ1σI−δH−d1H,dRdt=1−θ1−θ2σI+δH.After the patient is hospitalized, the hospital will adopt different treatment plans according to the different symptoms of the patients, which is equivalent to classifying the patients [7], so we discuss the situation of not classifying the case of the patients. In this case, we assume that the hospital does not classify and isolate clinically diagnosed cases according to the severity of pneumonia imaging features; in other words, the hospital treats all clinically diagnosed cases equally, and everyone will receive the subsequent nucleic acid test with equal probability. It will increase the probability of contact between COVID-19 carriers and non-COVID-19 carriers in clinically diagnosed cases and further increase the probability of infection between non-COVID-19 carriers.This series of reactions reflected in the model will leadθ1 to a further increase, and the corresponding θ2 will decrease to a certain extent. Therefore, we propose Δθii=1,2 to describe the change of scale factor (θ). If Δθ>0, it means that the value of θ will increase compared to the fitted value. Conversely, if Δθ<0, the value of θ will be reduced. The final cumulative reported cases are shown in Figure 6.Figure 6
The comparison chart of the cumulative reported cases if the patient is not classified.In Figure6, we can see that if the patients are not classified, the final disease scale will increase. If θ1=0.5443 and θ2=0.2870 (Δθ1=0.2, Δθ2=−0.2), the cumulative number of reported cases will reach 7.8255×104. If θ1=0.7443 and θ2=0.0870 (Δθ1=0.4, Δθ2=−0.4), the final scale of the disease will reach 9.9087×104, which is 1.45 times that of the actual cumulative number of reported cases.


### 4.2. The Impact of Lockdown on COVID-19

In this part, we will consider the impact of lockdown on the prevention and control of the epidemic in Hubei. In Figure1, the suspected patients who were introduced into compartment L will be taken to three different compartments after case screening. Among them, people from compartment L to compartment I represent a part of the suspected patients who have been identified as confirmed patients and should be hospitalized for isolation; in addition, people from compartment L to compartment C are the suspected patients who are not COVID-19 carriers, so they will be in strict home quarantine, and the probability that they will be infected in the short term will be ignored. In that case, they should be removed from the system. People from compartment L to compartment S denote that these suspected patients do not carry COVID-19, but they cannot be quarantined at home for some reasons, such as being a volunteer or medical staff. These people will continue to stay in the system. Therefore, when we do not take measures to lock down the city, we consider two situations from compartment L to compartment C. In the first case, we still remove the people who are in strict home quarantine from the system; in another case, these people are not removed from the system and are allowed to continue to stay in the system.


#### 4.2.1. Supposed Wuhan Was Locked Down, Hubei Was Not Locked Down

Here, we postulate that Wuhan was locked down on 23 January 2020, but Hubei province did not do a lockdown to control population movements. So, we use the parameters of the first stage (before Wuhan was locked down, i.e., 10 January-22 January 2020) and the second stage (after Wuhan was locked down and before Hubei was locked down, i.e., 23 January-27 January 2020) to simulate the cumulative reported cases and use the values of relevant parametersφa in the second stage to simulate the population movement before the lockdown of Wuhan. The fitting results are shown in Figure 7.Figure 7
The comparison chart of cumulative reported cases when discussing whether Wuhan and Hubei are locked down or not.


#### 4.2.2. Supposed Wuhan and Hubei Were Not Locked Down

From 10 January to 30 April, we assume that the entire Wuhan and Hubei provinces did not take a lockdown measure to control population movements. The parameter values of the first stage (before Wuhan was locked down, i.e., 10 January-22 January 2020) are used to simulate the cumulative reported cases, and the fitting results are shown in Figure7.From Figure7, when we do not remove the compartment C, we can see that the scale of the epidemic will reach 2.4677×105 on 18 February 2020, if Wuhan was locked down but Hubei was not locked down. But, if the suspected patients who do not carry COVID-19 after case screening are not quarantined at home, they will be the susceptible people again and still be in the system (the case of removing compartment C). At this time, if Wuhan and Hubei do not impose a lockdown measure, the confirmed cases will rise by 8.5097×106 on 18 February 2020, which is 138 times that of the actual number of reported cases.


### 4.3. The Impact of the Lockdown Time on COVID-19

According to the previous fitting results, after the lockdown of Wuhan, we removed8.8731×106 people (φa) from the system’s susceptible compartment; after the lockdown of Hubei, we removed 1.5910×107 people (φb) from the system’s susceptible compartment. Here, we introduce Δφ to characterize the change in the number of susceptible caused by the change of the lockdown time.IfΔφ>0, it means that the number of susceptible individuals has increased compared to the previous fitted value. If Δφ<0, it means that the number of susceptible individuals is reduced compared with the previous fitted value.
(12)Δφ=Δφa,withthechangeoflockdowntimeofWuhan,thechangevalue,Δφb,withthechangeoflockdowntimeofHubei,thechangevalue.


#### 4.3.1. Wuhan Lockdown Time Remains Unchanged

(1) The Lockdown Time of Hubei Was Delayed. The lockdown time of Xiangyang city is set as the lockdown time of Hubei province because Xiangyang city is the last city in Hubei province to be locked down. Here, we assume that the lockdown time of Hubei province is 30 January. Compared with the previous lockdown of Hubei on 27 January, during the three days from 27 January to 30 January, some people will not be able to stay in quarantine. Therefore, these people are added to the susceptible (number of susceptible on the 21st day, S21). Then, we discuss the impact of the different number of susceptible patients on the final disease scale. The fitting results are shown in Figure 8.Figure 8
The comparison chart of cumulative reported cases when Wuhan lockdown time remains unchanged, but Hubei lockdown time changed.(2) The Lockdown Time of Hubei Was Advanced. Based on the lockdown of Wuhan on 23 January, we assume that the lockdown time of Hubei province is two days earlier than the actual lockdown time. That is to say, Hubei province completed the lockdown on 25 January. Due to the advance of the lockdown, some people should stay in quarantine. We get rid of these people from the susceptible (S16) and discuss the impact of the different number of susceptible patients on the final disease scale. The fitting results are shown in Figure 8.From Figure8, if Hubei province completes the lockdown on 30 January, the final scale of the disease will increase, and when the number of susceptible people increases by 8 million, the final scale of the disease will be controlled at 1.7934×105. We can also see that if Hubei province completes the lockdown on 25 January, the final scale of the disease will become small and when the number of susceptible people is reduced by 4 million, the final disease scale will be controlled at 2.3002×104. Therefore, when the lockdown time in Wuhan is fixed, the advancement of the lockdown time in Hubei has a positive effect on disease control.


#### 4.3.2. Wuhan and Hubei Are All Locked Down in Advance or Delayed

(1) Wuhan Lockdown Time Was Advanced, and Hubei Lockdown Time Was also Advanced. In this part, we assume that Wuhan city was locked down on 20 January 2020, and Hubei was locked down on 23 January 2020. Compared with the actual situation, with the advancement of lockdown time, the population flow during the Spring Festival will decrease, which will further reduce the number of susceptible people. We get rid of these people from the susceptible (S11 and S14). For these people, we discuss the impact of different numbers on the final disease scale. The fitting results are shown in Figure 9.Figure 9
The comparison chart of cumulative reported cases when Wuhan and Hubei are all locked down in advance or delayed.(2) The Lockdown Time of Wuhan and Hubei Were Both Delayed. Supposing Wuhan was locked down on 25 January 2020, and Hubei was locked down on 30 January 2020. With the postponement of the lockdown time, the scale of population flow will expand, leading to more susceptible people than the actual situation. Thus, they have been added to the susceptible (S16 and S21). The fitting results are shown in Figure 9.If both Wuhan and Hubei are locked down in advance, the final scale of the disease will be well controlled, and when the number of susceptible people is reduced by 4 million, the final disease scale will be controlled at2.6501×104. On the contrary, if the lockdown time of Wuhan and Hubei are delayed, the final scale of the disease will increase, and when the number of susceptible people increases by 8 million, the final scale of the disease will be up to 1.6393×105, which is 2.41 times that of the actual number of reported cases.


#### 4.3.3. The Lockdown Time of Wuhan and Hubei in which One Is Advanced and Another Is Delayed

(1) Wuhan Lockdown Time Was Advanced, and Hubei Lockdown Time Was Delayed. We assumed that Wuhan was locked down on 20 January 2020, and Hubei was locked down on 30 January 2020 in this situation. With the early lockdown of Wuhan, we get rid of these people (Δφa=−2×106) from the susceptible S11, but the delay for the lockdown of Hubei will make more and more people become susceptible, so we added these people to the susceptible S21. The fitting results are shown in Figure 10.Figure 10
When the lockdown time of Wuhan and Hubei in which one is advanced and the other is delayed, the comparison chart of cumulative reported cases.(2) Wuhan Lockdown Time Was Delayed, and Hubei Lockdown Time Was Advanced. Presuming that Wuhan was locked down on 24 January 2020, and Hubei was locked down on 26 January 2020. With the postponement of Wuhan’s lockdown, some people will go out and become susceptible, and we add these people (Δφa=2×106) to the susceptible S15, but the advancement of the time for the lockdown of Hubei will cause a number of people to quarantine at home, which will decrease the number of susceptible people, so we remove these people from susceptible S17. The fitting results are shown in Figure 10.From Figure10, we can see that the scale of the disease is closely related to the change in the number of susceptible people. When Wuhan was locked down in advance (Δφa=−2×106), we made two different assumptions about the number of susceptible people caused by the delay in Hubei’s lockdown. If Δφb=3×106, the cumulative cases will be up to 8.9039×104; if Δφb=1×106, the final scale of the disease will be controlled at 5.9814×104. When Wuhan postponed the lockdown time of the city (Δφa=2×106), we also made two different assumptions. If Δφb=−1×106, the cumulative number of confirmed cases of the disease will reach 8.0203×104; if Δφb=−3×106, the final scale of the disease will be reduced to 5.5057×104. Through our multiple tests, if Δφa+Δφb<0, the cumulative cases will be less than the actual cases; if Δφa+Δφb>0, the number of the cumulative cases will be more than that of the actual cases.


### 4.4. Data Revision and Comparative Analysis

By referring to the results of Lipsitch et al. [62], we use an exponential function to revise the sudden increase of the number of confirmed cases on 12 February 2020 and allocate more than 10,000 people to the days from 10 January to 12 February. We assume that the number of daily new cases after 12 February is the real data, so we take t=34 from the following formula.
(13)Yt=eλt,λ=lnY34/34.Therefore, the revised cumulative number of confirmed cases from 10 January to 12 February can be obtained, and then, we can use the fminsearch optimization method to get Figure11.Figure 11
The revised cumulative reported and simulative cases in Hubei provinceYt.Comparing with Figure2, by calculating the minimum sum of the Chi-square, we get that the minimum sum of Chi-square in Figure 2 is 2.44×103, and the minimum sum of Chi-square in Figure 11 is 9.09×104. Therefore, the model of this paper is more suitable for describing the real reported data and less suitable for fitting smoothed data.


## 4.1. The Impact of Patient Classification on COVID-19

Assuming that patients are not classified, we modify the model. In this new model, we merge the previousH1, H2, and H3 into H, change the sum of θ1, θ2, and θ3 in the previous model to θ1=0.3443 in the current model, and change θ4 in the previous model to θ2=0.4870 in the current model, then modify δ2 in the previous model to δ=0.0593 in the current model. The model is described by the following ordinary differential equations:
(11)dSdt=−β1SpE+L+qI+kHN−β2SpE+L+qIN+ωηL,dEdt=β1SpE+L+qI+kHN+β2SpE+L+qIN−αE,dLdt=αρE+θ2σI−ω+ξL,dIdt=α1−ρE+ω1−ηL−σI−dI,dHdt=θ1σI−δH−d1H,dRdt=1−θ1−θ2σI+δH.After the patient is hospitalized, the hospital will adopt different treatment plans according to the different symptoms of the patients, which is equivalent to classifying the patients [7], so we discuss the situation of not classifying the case of the patients. In this case, we assume that the hospital does not classify and isolate clinically diagnosed cases according to the severity of pneumonia imaging features; in other words, the hospital treats all clinically diagnosed cases equally, and everyone will receive the subsequent nucleic acid test with equal probability. It will increase the probability of contact between COVID-19 carriers and non-COVID-19 carriers in clinically diagnosed cases and further increase the probability of infection between non-COVID-19 carriers.This series of reactions reflected in the model will leadθ1 to a further increase, and the corresponding θ2 will decrease to a certain extent. Therefore, we propose Δθii=1,2 to describe the change of scale factor (θ). If Δθ>0, it means that the value of θ will increase compared to the fitted value. Conversely, if Δθ<0, the value of θ will be reduced. The final cumulative reported cases are shown in Figure 6.Figure 6
The comparison chart of the cumulative reported cases if the patient is not classified.In Figure6, we can see that if the patients are not classified, the final disease scale will increase. If θ1=0.5443 and θ2=0.2870 (Δθ1=0.2, Δθ2=−0.2), the cumulative number of reported cases will reach 7.8255×104. If θ1=0.7443 and θ2=0.0870 (Δθ1=0.4, Δθ2=−0.4), the final scale of the disease will reach 9.9087×104, which is 1.45 times that of the actual cumulative number of reported cases.


## 4.2. The Impact of Lockdown on COVID-19

In this part, we will consider the impact of lockdown on the prevention and control of the epidemic in Hubei. In Figure1, the suspected patients who were introduced into compartment L will be taken to three different compartments after case screening. Among them, people from compartment L to compartment I represent a part of the suspected patients who have been identified as confirmed patients and should be hospitalized for isolation; in addition, people from compartment L to compartment C are the suspected patients who are not COVID-19 carriers, so they will be in strict home quarantine, and the probability that they will be infected in the short term will be ignored. In that case, they should be removed from the system. People from compartment L to compartment S denote that these suspected patients do not carry COVID-19, but they cannot be quarantined at home for some reasons, such as being a volunteer or medical staff. These people will continue to stay in the system. Therefore, when we do not take measures to lock down the city, we consider two situations from compartment L to compartment C. In the first case, we still remove the people who are in strict home quarantine from the system; in another case, these people are not removed from the system and are allowed to continue to stay in the system.


### 4.2.1. Supposed Wuhan Was Locked Down, Hubei Was Not Locked Down

Here, we postulate that Wuhan was locked down on 23 January 2020, but Hubei province did not do a lockdown to control population movements. So, we use the parameters of the first stage (before Wuhan was locked down, i.e., 10 January-22 January 2020) and the second stage (after Wuhan was locked down and before Hubei was locked down, i.e., 23 January-27 January 2020) to simulate the cumulative reported cases and use the values of relevant parametersφa in the second stage to simulate the population movement before the lockdown of Wuhan. The fitting results are shown in Figure 7.Figure 7
The comparison chart of cumulative reported cases when discussing whether Wuhan and Hubei are locked down or not.


### 4.2.2. Supposed Wuhan and Hubei Were Not Locked Down

From 10 January to 30 April, we assume that the entire Wuhan and Hubei provinces did not take a lockdown measure to control population movements. The parameter values of the first stage (before Wuhan was locked down, i.e., 10 January-22 January 2020) are used to simulate the cumulative reported cases, and the fitting results are shown in Figure7.From Figure7, when we do not remove the compartment C, we can see that the scale of the epidemic will reach 2.4677×105 on 18 February 2020, if Wuhan was locked down but Hubei was not locked down. But, if the suspected patients who do not carry COVID-19 after case screening are not quarantined at home, they will be the susceptible people again and still be in the system (the case of removing compartment C). At this time, if Wuhan and Hubei do not impose a lockdown measure, the confirmed cases will rise by 8.5097×106 on 18 February 2020, which is 138 times that of the actual number of reported cases.


## 4.2.1. Supposed Wuhan Was Locked Down, Hubei Was Not Locked Down

Here, we postulate that Wuhan was locked down on 23 January 2020, but Hubei province did not do a lockdown to control population movements. So, we use the parameters of the first stage (before Wuhan was locked down, i.e., 10 January-22 January 2020) and the second stage (after Wuhan was locked down and before Hubei was locked down, i.e., 23 January-27 January 2020) to simulate the cumulative reported cases and use the values of relevant parametersφa in the second stage to simulate the population movement before the lockdown of Wuhan. The fitting results are shown in Figure 7.Figure 7
The comparison chart of cumulative reported cases when discussing whether Wuhan and Hubei are locked down or not.


## 4.2.2. Supposed Wuhan and Hubei Were Not Locked Down

From 10 January to 30 April, we assume that the entire Wuhan and Hubei provinces did not take a lockdown measure to control population movements. The parameter values of the first stage (before Wuhan was locked down, i.e., 10 January-22 January 2020) are used to simulate the cumulative reported cases, and the fitting results are shown in Figure7.From Figure7, when we do not remove the compartment C, we can see that the scale of the epidemic will reach 2.4677×105 on 18 February 2020, if Wuhan was locked down but Hubei was not locked down. But, if the suspected patients who do not carry COVID-19 after case screening are not quarantined at home, they will be the susceptible people again and still be in the system (the case of removing compartment C). At this time, if Wuhan and Hubei do not impose a lockdown measure, the confirmed cases will rise by 8.5097×106 on 18 February 2020, which is 138 times that of the actual number of reported cases.


## 4.3. The Impact of the Lockdown Time on COVID-19

According to the previous fitting results, after the lockdown of Wuhan, we removed8.8731×106 people (φa) from the system’s susceptible compartment; after the lockdown of Hubei, we removed 1.5910×107 people (φb) from the system’s susceptible compartment. Here, we introduce Δφ to characterize the change in the number of susceptible caused by the change of the lockdown time.IfΔφ>0, it means that the number of susceptible individuals has increased compared to the previous fitted value. If Δφ<0, it means that the number of susceptible individuals is reduced compared with the previous fitted value.
(12)Δφ=Δφa,withthechangeoflockdowntimeofWuhan,thechangevalue,Δφb,withthechangeoflockdowntimeofHubei,thechangevalue.


### 4.3.1. Wuhan Lockdown Time Remains Unchanged

(1) The Lockdown Time of Hubei Was Delayed. The lockdown time of Xiangyang city is set as the lockdown time of Hubei province because Xiangyang city is the last city in Hubei province to be locked down. Here, we assume that the lockdown time of Hubei province is 30 January. Compared with the previous lockdown of Hubei on 27 January, during the three days from 27 January to 30 January, some people will not be able to stay in quarantine. Therefore, these people are added to the susceptible (number of susceptible on the 21st day, S21). Then, we discuss the impact of the different number of susceptible patients on the final disease scale. The fitting results are shown in Figure 8.Figure 8
The comparison chart of cumulative reported cases when Wuhan lockdown time remains unchanged, but Hubei lockdown time changed.(2) The Lockdown Time of Hubei Was Advanced. Based on the lockdown of Wuhan on 23 January, we assume that the lockdown time of Hubei province is two days earlier than the actual lockdown time. That is to say, Hubei province completed the lockdown on 25 January. Due to the advance of the lockdown, some people should stay in quarantine. We get rid of these people from the susceptible (S16) and discuss the impact of the different number of susceptible patients on the final disease scale. The fitting results are shown in Figure 8.From Figure8, if Hubei province completes the lockdown on 30 January, the final scale of the disease will increase, and when the number of susceptible people increases by 8 million, the final scale of the disease will be controlled at 1.7934×105. We can also see that if Hubei province completes the lockdown on 25 January, the final scale of the disease will become small and when the number of susceptible people is reduced by 4 million, the final disease scale will be controlled at 2.3002×104. Therefore, when the lockdown time in Wuhan is fixed, the advancement of the lockdown time in Hubei has a positive effect on disease control.


### 4.3.2. Wuhan and Hubei Are All Locked Down in Advance or Delayed

(1) Wuhan Lockdown Time Was Advanced, and Hubei Lockdown Time Was also Advanced. In this part, we assume that Wuhan city was locked down on 20 January 2020, and Hubei was locked down on 23 January 2020. Compared with the actual situation, with the advancement of lockdown time, the population flow during the Spring Festival will decrease, which will further reduce the number of susceptible people. We get rid of these people from the susceptible (S11 and S14). For these people, we discuss the impact of different numbers on the final disease scale. The fitting results are shown in Figure 9.Figure 9
The comparison chart of cumulative reported cases when Wuhan and Hubei are all locked down in advance or delayed.(2) The Lockdown Time of Wuhan and Hubei Were Both Delayed. Supposing Wuhan was locked down on 25 January 2020, and Hubei was locked down on 30 January 2020. With the postponement of the lockdown time, the scale of population flow will expand, leading to more susceptible people than the actual situation. Thus, they have been added to the susceptible (S16 and S21). The fitting results are shown in Figure 9.If both Wuhan and Hubei are locked down in advance, the final scale of the disease will be well controlled, and when the number of susceptible people is reduced by 4 million, the final disease scale will be controlled at2.6501×104. On the contrary, if the lockdown time of Wuhan and Hubei are delayed, the final scale of the disease will increase, and when the number of susceptible people increases by 8 million, the final scale of the disease will be up to 1.6393×105, which is 2.41 times that of the actual number of reported cases.


### 4.3.3. The Lockdown Time of Wuhan and Hubei in which One Is Advanced and Another Is Delayed

(1) Wuhan Lockdown Time Was Advanced, and Hubei Lockdown Time Was Delayed. We assumed that Wuhan was locked down on 20 January 2020, and Hubei was locked down on 30 January 2020 in this situation. With the early lockdown of Wuhan, we get rid of these people (Δφa=−2×106) from the susceptible S11, but the delay for the lockdown of Hubei will make more and more people become susceptible, so we added these people to the susceptible S21. The fitting results are shown in Figure 10.Figure 10
When the lockdown time of Wuhan and Hubei in which one is advanced and the other is delayed, the comparison chart of cumulative reported cases.(2) Wuhan Lockdown Time Was Delayed, and Hubei Lockdown Time Was Advanced. Presuming that Wuhan was locked down on 24 January 2020, and Hubei was locked down on 26 January 2020. With the postponement of Wuhan’s lockdown, some people will go out and become susceptible, and we add these people (Δφa=2×106) to the susceptible S15, but the advancement of the time for the lockdown of Hubei will cause a number of people to quarantine at home, which will decrease the number of susceptible people, so we remove these people from susceptible S17. The fitting results are shown in Figure 10.From Figure10, we can see that the scale of the disease is closely related to the change in the number of susceptible people. When Wuhan was locked down in advance (Δφa=−2×106), we made two different assumptions about the number of susceptible people caused by the delay in Hubei’s lockdown. If Δφb=3×106, the cumulative cases will be up to 8.9039×104; if Δφb=1×106, the final scale of the disease will be controlled at 5.9814×104. When Wuhan postponed the lockdown time of the city (Δφa=2×106), we also made two different assumptions. If Δφb=−1×106, the cumulative number of confirmed cases of the disease will reach 8.0203×104; if Δφb=−3×106, the final scale of the disease will be reduced to 5.5057×104. Through our multiple tests, if Δφa+Δφb<0, the cumulative cases will be less than the actual cases; if Δφa+Δφb>0, the number of the cumulative cases will be more than that of the actual cases.


## 4.3.1. Wuhan Lockdown Time Remains Unchanged

(1) The Lockdown Time of Hubei Was Delayed. The lockdown time of Xiangyang city is set as the lockdown time of Hubei province because Xiangyang city is the last city in Hubei province to be locked down. Here, we assume that the lockdown time of Hubei province is 30 January. Compared with the previous lockdown of Hubei on 27 January, during the three days from 27 January to 30 January, some people will not be able to stay in quarantine. Therefore, these people are added to the susceptible (number of susceptible on the 21st day, S21). Then, we discuss the impact of the different number of susceptible patients on the final disease scale. The fitting results are shown in Figure 8.Figure 8
The comparison chart of cumulative reported cases when Wuhan lockdown time remains unchanged, but Hubei lockdown time changed.(2) The Lockdown Time of Hubei Was Advanced. Based on the lockdown of Wuhan on 23 January, we assume that the lockdown time of Hubei province is two days earlier than the actual lockdown time. That is to say, Hubei province completed the lockdown on 25 January. Due to the advance of the lockdown, some people should stay in quarantine. We get rid of these people from the susceptible (S16) and discuss the impact of the different number of susceptible patients on the final disease scale. The fitting results are shown in Figure 8.From Figure8, if Hubei province completes the lockdown on 30 January, the final scale of the disease will increase, and when the number of susceptible people increases by 8 million, the final scale of the disease will be controlled at 1.7934×105. We can also see that if Hubei province completes the lockdown on 25 January, the final scale of the disease will become small and when the number of susceptible people is reduced by 4 million, the final disease scale will be controlled at 2.3002×104. Therefore, when the lockdown time in Wuhan is fixed, the advancement of the lockdown time in Hubei has a positive effect on disease control.


## 4.3.2. Wuhan and Hubei Are All Locked Down in Advance or Delayed

(1) Wuhan Lockdown Time Was Advanced, and Hubei Lockdown Time Was also Advanced. In this part, we assume that Wuhan city was locked down on 20 January 2020, and Hubei was locked down on 23 January 2020. Compared with the actual situation, with the advancement of lockdown time, the population flow during the Spring Festival will decrease, which will further reduce the number of susceptible people. We get rid of these people from the susceptible (S11 and S14). For these people, we discuss the impact of different numbers on the final disease scale. The fitting results are shown in Figure 9.Figure 9
The comparison chart of cumulative reported cases when Wuhan and Hubei are all locked down in advance or delayed.(2) The Lockdown Time of Wuhan and Hubei Were Both Delayed. Supposing Wuhan was locked down on 25 January 2020, and Hubei was locked down on 30 January 2020. With the postponement of the lockdown time, the scale of population flow will expand, leading to more susceptible people than the actual situation. Thus, they have been added to the susceptible (S16 and S21). The fitting results are shown in Figure 9.If both Wuhan and Hubei are locked down in advance, the final scale of the disease will be well controlled, and when the number of susceptible people is reduced by 4 million, the final disease scale will be controlled at2.6501×104. On the contrary, if the lockdown time of Wuhan and Hubei are delayed, the final scale of the disease will increase, and when the number of susceptible people increases by 8 million, the final scale of the disease will be up to 1.6393×105, which is 2.41 times that of the actual number of reported cases.


## 4.3.3. The Lockdown Time of Wuhan and Hubei in which One Is Advanced and Another Is Delayed

(1) Wuhan Lockdown Time Was Advanced, and Hubei Lockdown Time Was Delayed. We assumed that Wuhan was locked down on 20 January 2020, and Hubei was locked down on 30 January 2020 in this situation. With the early lockdown of Wuhan, we get rid of these people (Δφa=−2×106) from the susceptible S11, but the delay for the lockdown of Hubei will make more and more people become susceptible, so we added these people to the susceptible S21. The fitting results are shown in Figure 10.Figure 10
When the lockdown time of Wuhan and Hubei in which one is advanced and the other is delayed, the comparison chart of cumulative reported cases.(2) Wuhan Lockdown Time Was Delayed, and Hubei Lockdown Time Was Advanced. Presuming that Wuhan was locked down on 24 January 2020, and Hubei was locked down on 26 January 2020. With the postponement of Wuhan’s lockdown, some people will go out and become susceptible, and we add these people (Δφa=2×106) to the susceptible S15, but the advancement of the time for the lockdown of Hubei will cause a number of people to quarantine at home, which will decrease the number of susceptible people, so we remove these people from susceptible S17. The fitting results are shown in Figure 10.From Figure10, we can see that the scale of the disease is closely related to the change in the number of susceptible people. When Wuhan was locked down in advance (Δφa=−2×106), we made two different assumptions about the number of susceptible people caused by the delay in Hubei’s lockdown. If Δφb=3×106, the cumulative cases will be up to 8.9039×104; if Δφb=1×106, the final scale of the disease will be controlled at 5.9814×104. When Wuhan postponed the lockdown time of the city (Δφa=2×106), we also made two different assumptions. If Δφb=−1×106, the cumulative number of confirmed cases of the disease will reach 8.0203×104; if Δφb=−3×106, the final scale of the disease will be reduced to 5.5057×104. Through our multiple tests, if Δφa+Δφb<0, the cumulative cases will be less than the actual cases; if Δφa+Δφb>0, the number of the cumulative cases will be more than that of the actual cases.


## 4.4. Data Revision and Comparative Analysis

By referring to the results of Lipsitch et al. [62], we use an exponential function to revise the sudden increase of the number of confirmed cases on 12 February 2020 and allocate more than 10,000 people to the days from 10 January to 12 February. We assume that the number of daily new cases after 12 February is the real data, so we take t=34 from the following formula.
(13)Yt=eλt,λ=lnY34/34.Therefore, the revised cumulative number of confirmed cases from 10 January to 12 February can be obtained, and then, we can use the fminsearch optimization method to get Figure11.Figure 11
The revised cumulative reported and simulative cases in Hubei provinceYt.Comparing with Figure2, by calculating the minimum sum of the Chi-square, we get that the minimum sum of Chi-square in Figure 2 is 2.44×103, and the minimum sum of Chi-square in Figure 11 is 9.09×104. Therefore, the model of this paper is more suitable for describing the real reported data and less suitable for fitting smoothed data.


## 5. Conclusion

COVID-19 has now become a global epidemic, which has had a huge impact on all aspects of human beings. The outbreak broke out in Wuhan at the end of 2019 and the epidemic in Hubei had been fully controlled as early as late April 2020. The next time, although some small-scale epidemics still occurred in Beijing [63], Xinjiang [64], Liaoning [65], Hebei [66], Heilongjiang [67], Jilin [68], Jiangsu [69], Henan [70], Fujian [71], Inner Mongolia [72], and other places one after another, the cases can be cleared very quickly by taking measures like timely lockdown, patient classification, and large-scale case screening, which indicates that these actions are effective to control the epidemic. Obviously, the burden of treatment after the outbreak is far greater than the burden of disease prevention and control. Globally, as of 6:11 p.m. CET, 9 March 2022, the global cumulative confirmed cases number has exceeded 440 million, and the death cases number has exceeded 6 million [54]. In the meantime, the epidemic in foreign countries and other regions is still quite serious.In this study, taking into account the five factors of suspected cases, quarantine, lockdown of Wuhan and Hubei, patient classification, and clinically diagnosed cases, we built a COVID-19 model to simulate the spread of COVID-19 in Hubei.Firstly, we collected surveillance data published by the Hubei Provincial Health Commission and used the fminsearch optimization method to find the optimal parameter values. Next, the basic reproduction number from 10 January to 30 April (three stages) was calculated using the above parameter values. Before Wuhan was locked down (10 January-22 January),R0=3.1571; after Wuhan was locked down but Hubei was not locked down (23 January-27 January), R0=2.0471; after Hubei was locked down (28 January-30 April), R0=1.5014. From the results of the basic reproduction number, we can find that the control measures taken by Hubei government can curb the spread of COVID-19 to a certain extent. Our results of R0 are in agreement with other groups [12, 17, 19, 21, 28], and details can be seen in Table 1.After that, we performed a sensitivity analysis onR0, in which β, p, and 1−ρ were more sensitive than other parameters. Therefore, it can be concluded that strengthening personal protection [73], such as wearing masks, quarantining at home, and not going out unless necessary can effectively reduce the transmission rate (β). Keeping social distance and closing large entertainment venues [74] can reduce the infectivity reduction factor (p). Expanding the proportion and scope of nucleic acid testing, strengthening the test of close contacts and subclose contacts, and ensuring that all those in need are tested [2] can reduce the proportion of the infectious 1−ρ.Then, we discussed whether the patients were classified, whether Wuhan and Hubei were locked down, and the impact of the lockdown time adjustment on the final scale of the disease. Finally, we concluded that classifying and treating confirmed cases according to symptoms and locking down the city in Hubei as soon as possible can effectively reduce the number of cumulative confirmed cases, thereby further effectively curbing the spread of the epidemic. At present, the global epidemic is still very serious, and it is hoped that our research can provide some help for the epidemic prevention and give some references for other countries or regions.Because the time period we studied was in the early stage of the epidemic in Wuhan, Hubei, some follow-up factors were not involved in our paper, such as asymptomatic infections [75], vaccines [76], imported cases [77], and coronavirus variants [78]. Considering asymptomatic infections, Serhani and Labbardi [79] build an SIAQRD model and take asymptomatic people, the isolation of an infected person, and the quarantine of contacting people into consideration. The result illustrates that the level of containment is of great importance. Thinking about the impact of imported cases, Boldog et al. [80] establish a time-dependent compartmental model to estimate the cumulative number of cases in China and use the Galton-Watson branching process to model the initial spread of COVID-19. Among that, they think about the factor of imported cases and incorporate it into the formula of risk estimation. They finally find that the actual connectivity and the local reproduction number have an effect on assessing the risks of major outbreaks from imported cases outside China.In the future, we can further carry out follow-up research in these directions: sporadic cases brought by asymptomatic infected persons will lead to the local spread of the epidemic and the impact of vaccines and coronavirus variants on COVID-19.



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*Source: 8920117-2022-05-06.xml*