key: cord-0311266-t1dgsmg6
authors: Min, L.
title: Modelling, Simulations and Analysis of the First and Second COVID-19 Epidemics in Beijing
date: 2021-07-07
journal: nan
DOI: 10.1101/2021.07.04.21259205
sha: 2a46607dae09e921d604ff4da976589f504893aa
doc_id: 311266
cord_uid: t1dgsmg6

To date, over 182 million people on infected with COVID-19. It causes more 3.9 millions deaths. This paper introduces a symptomatic-asymptomatic-recoverer-dead differential equation model (SARDDE). It gives the conditions of the asymptotical stability on the disease-free equilibrium of SARDDE. It proposes the necessary conditions of disease spreading for the SARDDE. Based on the reported data of the first and the second COVID-19 epidemics in Beijing and simulations, it determines the parameters of SARDDE, respectively. Numerical simulations of SARDDE describe well the outcomes of current symptomatic and asymptomatic individuals, recovered symptomatic and asymptomatic individuals, and died individuals, respectively. The numerical simulations suggest that both symptomatic and asymptomatic individuals cause lesser asymptomatic spread than symptomatic spread; the blocking rates of about 80% and 97.5% to the symptomatic individuals cannot prevent the spread of the first and second COVID19 epidemics in Beijing, respectively. Virtual simulations suggest that the strict prevention and control strategies implemented by Beijing government are not only very effective but also completely necessary. The numerical simulations suggest also that using the data from the beginning to the day after about 14 -- 17 days at the turning point can estimate well the following outcomes of the two COVID-19 academics, respectively. It is expected that the research can provide better understanding, explaining, and dominating for epidemic spreads, prevention and control measures.

In December 2019, a novel coronavirus-induced pneumonia (COVID-19) broke out in Wuhan, Hubei. Now over 182 million people on infected with COVID-19 have been identified worldwide. It causes more 3.9 millions deaths. COVID-19 affects more than 220 countries and regions including Antarctica.

One of the reasons of such a tragedy is that people in some countries do not pay attentions to theoretical analysis and estimations for COVID-19 epidemic. In fact mathematical models for epidemic infectious diseases have played important roles in the formulation, evaluation, and prevention of control strategies. Modelling the dynamics of spread of disease can help people to understand the mechanism of epidemic diseases, formulate and evaluate prevention and control strategies, and predict tools for the spread or disappearance of an epidemic [1] .

Since the outbreak of COVID-19 in Wuhan, many scholars have published a large numbers of articles on the modeling and prediction of COVID-19 epidemic (for examples see [2] [3] [4] [5] [6] [7] [8] [9] ). It is difficult to describe well the dynamics of COVID-19 epidemics. In a Lloyd-Smith et al's paper, it described nine challenges in modelling the emergence of novel pathogens, emphasizing the interface between models and data [10] .

On Jan. 19, two Beijingers returning from Wuhan were diagnosed with COVID-19. That triggered the first wave of COVID-19 in Beijing. During the first wave of COVID-19, a total of 420 locally diagnosed cases were Figure 1 : Flowchart of disease transmission among susceptible population S, current symptomatic infected individuals I, current asymptomatic but infected individuals I a recovered symptomatic infected individuals I r , recovered asymptomatic but infected individuals I ra , and died individuals D.

First, the symptomatic infected individuals (I) and the asymptomatic but infected individuals (I a ) infect the susceptible population (S) with the probabilities of β 11 and β 21 , respectively, making S become symptomatic infected individuals, and with the probabilities of β 12 and β 22 , respectively, making S become asymptomatic individuals. Then, a symptomatic individual is cured at a rate κ, an asymptomatic individual returns to normal at a rate κ a . An infected individual dies at a rate α. Here all parameters are positive numbers. Assume that the dynamics of an epidemic can be described by m time intervals. At ith interval, the model has the form:

dI a dt = θ 2 (i)(β 12 I + β 22 I a )S − κ a (i)I a (1b)

where θ 1 (i) ′ s and θ 1 (i) ′ s (i = 1, . . . , m) represent blocking rates to symptomatic and asymptomatic infections, respectively. Then system (1) has a disease-free equilibrium: Then equation (1) has a disease-free equilibrium:

The stability of system (1) is determined by the first two equations (1a) and (1b). Denote in (1a) and (1b)

,

a 21 = θ 2 (i)β 12 S,

a 22 = θ 2 (i)β 22 S − κ a (i).

Then at the disease-free equilibrium of system(18), the Jacobian matrix of (18a) and (18b) is J = a 11 a 12 a 21 a 22

Solving the corresponding eigenequation obtains 2 eigenvalues:

x 1,2 = 1 2 a 11 + a 22 ± (a 11 + a 22 ) 2 − 4(a 11 a 22 − a 12 a 21 )

Therefore it obtains the following:

Theorem 1 Suppose that a 11 , a 12 , a 21 and a 22 are defined by (3)- (6) . Then the disease-free equilibrium E of system (1) is globally asymptotically stable if, and only if, the following inequalities hold:

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The copyright holder for this preprint this version posted July 7, 2021. ; https://doi.org/10.1101/2021.07.04.21259205 doi: medRxiv preprint (a) R 1 local a 11 + a 22 < 0,

(b) R 2 local a 12 a 21 − a 11 a 22 < 0.

If an epidemic can occur, thenİ

This implies that

Solving the above inequalities gives the following Theorem 2 If system (1) satisfies the following inequalities

then a disease transmission will occur.

Based on the reported clinical COVID-19 epidemic data from January 19 to June 8, 2020 in Beijing [11] , this Section will discuss the applications of the above theoretical results. Numerical simulations and drawings are performed by using MATLAB software programs. The first 50 days' reported clinical data on current confirmed infection cases, and the reported clinical data on recovered cases of the COVID-19 epidemic in Beijing [11] are shown in Figs. 2(a) and 2(b) 1 . The number of current symptomatic infected individuals is showed in Fig3(a) by circles. The numbers of cumulative recovered symptomatic infected individuals, and cumulative died infected individuals are showed in Fig3(b) by circles and stars respectively.

The number of current infected individuals was risen rapidly in the first 4 days (see Fig. 2(a) ). The number of current infected individuals reached the highest 295 on the day 24th, February 12 and then after the day 31th, February 19, declined rapidly (see Fig. 2 (a) and 3(a)).

Observe from the Figs. 3(a) and 3(b) that the overall changes in the number of current confirmed infections are not subject to the law of exponential changes, but the data can be approximated in good agreement with 8 straight lines in log scale (see Fig. 3 ). This phenomenon can be explained as: different medical measures prevention and control strategies have been adopted at the different 8 time intervals. On the day 86th, April 15, there are 3 Chaoyang district infected people coming back Beijing form foreign country which makes calculated blacking rates to rise. Therefore the i in SARDDE model (1) should be chosen as i = 1, 2, . . . , 8. 1 In the cases that some reported data crossed one day, we assign approximately numbers according to the ratios of time intervals.

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The copyright holder for this preprint this version posted July 7, 2021. 

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The copyright holder for this preprint this version posted July 7, 2021. ; Table 1 The data of the first COVID-19 epidemic in Beijing on different days and corresponding calculated parameters of SARDDE. Where NCSII and NCDI represent the numbers [11] of current symptomatic infected individuals and current died individuals, respectively; NRSII the number [11] of recovered symptomatic infected individuals. 

First it needs to determine the parameters κ(i), κ a (i) and α(i). There are different methods for calculating the recovery rate κ(i) in a specific time interval. It seems to be reasonable that we stands for κ(i) via using the number of the recovered patients to divide the days of patients stayed in the hospital during the ith time.

Denote s 1 (i) and s 2 (i) to be the days that the old patients and the new patients stayed in the hospital during ith time interval. Denote R(i) and d(i) to be the numbers of the recovered patients and died patients during ith time interval, respectively. Similar to the formulas given in Ref. [12] , R(i) and d(i) can be defined by

Since there is no information on recovered asymptomatic infected individuals, we take

That is, an asymptomatic infected individual will recover in average 7 days. The calculated κ(i) ′ s and α(i) ′ s are shown in the 7 ∼ 8 columns in Table 1 .

Second it needs to determine the parameters β ′ ij s in SARDDE. One can assume that S = 1 because the effects of S can be deleted by calculated β ′ ij s. This makes the calculated β ′ ij s have general sense. Using the practical data of the first COVID-19 epidemic in Beijing [11] (also see the second row in Table 1 ) selects following initial condition:

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The copyright holder for this preprint this version posted July 7, 2021. ; https://doi.org/10.1101/2021.07.04.21259205 doi: medRxiv preprint (I(0), I a (0), I r (0), I ra (0), D(0)) = (2, 0, 0, 0, 0).

Substitute parameters κ(1), α(1), θ 1 (1) and θ 2 (1) listed in Table 1 into system (1). Using a minimization error square criterion:

determines k ′ ij s. A group (β 11 , β 12 , β 21 , β 22 ) that makes δ be "smallest" (considering continued simulations) are Third it needs to determine:

Let I c (t i ) to be the number of the reported current symptomatic infected individuals at t i in the first Beijing CONVID-19. Let I cr (t i ) and D c (t i ) to be the numbers of the reported cumulative recovered infected and died individuals at t i , respectively. Using the minimization error square criterion:

determines the θ 1 (i) ′ s and θ 2 (i) ′ s. The calculated results are shown in Table 1 . The corresponding simulation results of system (1) are shown in Figs. 3(a) and 3(b). Observe that the simulation results of SARDDE model (1) describe well the dynamics of the first COVID-19 epidemic in Beijing.

(1) On the day 0, day 3, and day 140, the numbers of simulated current symptomatic individuals are approximately the same as those of the actual reported numbers. On the day 12, day 36, day 75 and day 97, there are only one or two differences. On the day 24, and day 52, there are 3 and 5 differences.

(2) On the day 0, day 3, day 12 and day 140, the numbers of simulated cumulative recovered symptomatic individuals are approximately the same those of the actual reported numbers.

On the day 52, and day 97, there are 2 and 3 differences, respectively. On the day 24, day 36 and day 73, there are 5 ∼ 7 differences, respectively.

(3) The all numbers of practical and simulated cumulative died individuals are approximately the same as the actual reported numbers on the day 0, day 3, day 12, day 24, day 36, day 52, day 73, day 97 and day 140.

(4) There is no information on the current symptomatic infected and recovered symptomatic infected individuals. But it has reported that after the 73 day, April 1, there is no symptomatic infected individuals until the day 143, June 11 [11] . Our simulation results shows that on the day 73, the number of the simulated current symptomatic infected individuals was less than one (≈ 0.7), which seems to explain the actual report data. [11] .

(5) Computed results (see (16)) of the transmission rates β ′ ij s show that the ratio of the transmission rates of asymptomatic and symptomatic individuals infecting susceptible population to become symptomatic individuals is about 0.159 (β 21 : β 11 ). This suggests that asymptomatic individuals cause lesser symptomatic spread than symptomatic individuals do.

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The copyright holder for this preprint this version posted July 7, 2021. ; https://doi.org/10.1101/2021.07.04.21259205 doi: medRxiv preprint Table 2 The criterions of the asymptotical stability and disease spreading of the disease-free equilibrium of SARDDE at eight time intervals. (16)) also show that the ratios of transmission rates of asymptomatic and symptomatic individuals infecting susceptible population to become asymptomatic and symptomatic individuals are about 0.646 (β 12 :β 11 ) and 0.667 (β 22 :β 21 ), respectively. This suggests that both symptomatic and asymptomatic individuals cause lesser asymptomatic spreads than symptomatic spreads.

(5) The criterions (7) and (8) of the asymptotical stability of the disease-free equilibrium of SARDDE at eight time intervals are listed in the 5th ∼ 8th columns in Table 2 . It shows that until the blocking rates (1 − θ 1 , 1 − θ 2 ) reach to (98.09%, 99%), the disease-free equilibrium becomes globally asymptotical stability. The conditions (9) and (10) of disease spreading are listed in the last two columns in Table 2 . It shows also that until the blocking rates (θ 1 , θ 2 ) reach to (98.09%, 99%), the spreading of COVID-19 epidemic can be blocked . Now assume that after the day 24th, February 12, it still keeps the blocking rates Furthermore assume that after the day 52, March 11, it still keeps the blocking rates (1 − θ 1 (5), 1 − θ 2 (5)) ≈ (98.3%, 99%), the cure rates (κ(5), κ a (5)), and the died rate α(5) until the day 140th, June 8. The simulation results of SARDDE are shown in Figs 5(a) and 5(b). Observe that the numbers of the current symptomatic and asymptomatic infected individuals are both less than one, respectively; The numbers of cumulative recovered symptomatic and died individuals are about 411 and 11, respectively. The results suggest that using the data before the day 52h (about 17 days after the turning point) can approximately estimate the following outcome of the the first COVID-19 academic in Beijing.

In summary, SARDDE model (1) can simulate the outcomes of the first COVID-19 epidemic in Beijing. The calculated equation parameters can help us to understand and explain the mechanism of epidemic diseases and control strategies for the event of the practical epidemic.

A total of 335 locally symptomatic cases and 50 locally asymptomatic cases were reported during the 2th wave COVID-19 epidemics. After 56 days, all symptomatic and asymptomatic patients were cured. The medical personnel has realized the zero infection. This event of Xinfadi COVID-19 epidemic provides a valuable example of accurate preventing and controlling strategies and excellent clinical treatments.

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The copyright holder for this preprint this version posted July 7, 2021. 19) 10(1.29) 20(2.08) 30(2.18) 40(2.28) 50(3.09) 60(3.19) 70(3.29) 80(4.08) 90(4.19) 100(4.29) 110(5.09) 120(5.19) 130(5.29) 

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Similar to Section 2.1 the transition among these states is governed by the following rules (Flowchart of the rules is shown in Fig.6 , where S represents susceptible population.) Figure 6 : Flowchart of disease transmission among susceptible population S, current symptomatic infected individuals I, current asymptomatic but infected individuals I a recovered symptomatic infected individuals I r , recovered asymptomatic but infected individuals I ra .

Similar to Section 2.1, assume that the dynamics of an epidemic can be described by m time intervals. At ith interval, the model has the form:

Then system (18) has a disease-free equilibrium:

The stability of system (18) is determined by the first two equations (18a) and (18b). Denote in (18a) and (18b):

,

Then at the disease-free equilibrium of system(18), the Jacobian matrix of (18a) and (18b) has the form J = a 11 a 12 a 21 a 22 .

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Solving the corresponding eigenequation obtains 2 eigenvalues:

x 1,2 = 1 2 a 11 + a 22 ± (a 11 + a 22 ) 2 + 4(a 12 a 21 − a 11 a 22 ) .

Therefore it obtains the following:

Theorem 3 Suppose that a 11 , a 12 , a 21 and a 22 are defined by (20)-(23). Then the disease-free equilibrium E of system (18) is globally asymptotically stable if, and only if, the following inequalities hold:

If an epidemic can occur, thenİ

This implies that

Solving the above inequalities gives the following then a disease transmission will occur.

Until June 10, 2020, the Beijing whole city continual 56 days has no new reports of the locally confirmed COVID-19 cases. There have been 11 districts in all 15 districts which continually have no reported locally COVID-19 case over 100 days. However in June 11, Xinfadi in Tongzou district appeared a COVID-19 confirmed case. Thus has caused the second wave COVID-19 epidemic in Beijing.

Based on the reported clinical COVID-19 epidemic data from June 11 to August 6, 2020 in Beijing [11] , this Section will discuss the applications of above theoretical results. Figure 7 (a) show that the reported data on current confirmed symptomatic infection cases [11] . Figure 7 (b) show that the reported data on cumulative recovered symptomatic infection cases [11] . Figure 8(a) show that the reported data on current confirmed asymptomatic infection cases [11] . Figure 8(b) show that the reported data on cumulative recovered asymptomatic infection cases [11] .

The evolution of the current symptomatic infected individuals, and the current asymptomatic infected individuals are shown in Fig. 9 (a) by circles and diamonds, respectively. The evolution of cumulative recovered symptomatic infected individuals, and cumulative recovered asymptomatic infected individuals are shown in Fig.  9 (b) by circles and diamonds, respectively.

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The copyright holder for this preprint this version posted July 7, 2021. Observe from Fig.9 that the overall changes in the number of current confirmed infections are not subject to the law of exponential changes, but the data can be approximated in good agreement with 7 straight lines in log scale (see Fig.9 ). This phenomenon can be explained as: different medical measures and prevention and control strategies have been adopted at different 8 time intervals. Therefore the i in SARDDE model (18) satisfies i = 1, 2, . . . , 7.

Based on the reported clinical COVID-19 epidemic data from June 11 to August 6, 2020 in Beijing [11] , this Section will discuss the applications of above theoretical results. The numbers of current symptomatic infected individuals, and current asymptomatic but infected individuals are showed in Fig. 9 (a) by circles and diamonds, respectively. The numbers of current recovered symptomatic infected individuals, and current recovered asymptomatic but infected individuals are showed in Fig. 9 (b) by circles and diamonds, respectively.

The number of current infected individuals was risen rapidly in the first 4 days (see Fig. 9(a) ). The number of current infected individuals reached the highest 326 on the day 19th, June 30 (see Fig. 9 (a)), and then after the day 27th, July 7, declined rapidly. The corresponding cumulative number of recovered symptomatic and asymptomatic individuals has risen rapidly after the day 27th, July 7 (see Fig. 9(b) ). Observe from the Figs. 9(a) and 9(b) that the overall changes in the number of current firmed infections are not subject to the law of exponential changes, but the data can be approximated in good agreement with 7 straight lines in log scale (see Fig. 9 ). This phenomenon can be explained as: different medical measures and prevention and control strategies have been adopted at the different 7 time intervals. Therefore the i in model (18) satisfies i = 1, 2, . . . , 7.

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The copyright holder for this preprint this version posted July 7, 2021. ; Table 3 The data of the second wave COVID-19 epidemics on 7 different days and corresponding calculated parameters of SARDDE model (18). Where NCSII and NCAII represent the numbers of the current symptomatic infected individuals and the current asymptomatic infected individuals, respectively; NRSII and NRAII represent the numbers of the cumulative recovered symptomatic infected individuals and the asymptomatic infected individuals over the ith interval. First it needs to determine the parameters κ(i), κ a (i). Denote s 1 (i) and s 2 (i) to be the days that the old patients and the new patients stayed in the hospital during ith time interval. Denote R(i) and R a (i) to be the numbers of the recovered symptomatic patients and asymptomatic patients during ith time interval, respectively. Then κ(i) and κ a (i) can be defined by

The calculated κ(i) ′ s and κ a (i) ′ s are shown in Table 3 .

Second it needs to determine the parameters β ′ ij s in system (18). One can assume that S = 1 because the effects of S can be deleted by calculated β ′ ij s. This makes the calculated β ′ ij s have general sense. Using the practical data of Xinfadi COVID-19 epidemic (see the second line in Table 3 ) selects following initial condition (I(0), I a (0), I r (0), I ra (0)) = (1, 0, 0, 0).

(30) Substitute parameters κ(1), κ a (1), θ 1 (1) and θ 2 (1) listed in Table 3 into system (18). Using a minimization error square criterion:

determines β ′ ij s. A group β ′ ij s that makes δ be "smallest" (considering continued simulations) are (β 11 , β 12 , β 21 , β 22 ) = (1.4492, 0.072824, 0.13108, 0.72824).

The simulations of system (18) with above equation parameters are shown in Figs. 9(a) and 9(b). Observe that the simulation results are in good agreement with the reported first 4 days' clinical data (see the solid and dash lines in Figs. 9(a) and 9(b) ).

Third it needs to determine: θ 1 (i), θ 2 (i), i = 2, 3, . . . , 7. Denote t 1 = 10, t 2 = 19, t 3 = 27, t 4 = 34, t 5 = 44, t 6 = 56.

Let I c (t i ) and I ca (t i ) to be the numbers of the current symptomatic and asymptomatic infected individuals at time t i , respectively; I cr (t i ) and I cra (t i ) to be the numbers of the cumulative recovered symptomatic and asymptomatic infected individuals at time t i , respectively. Using the minimization error square criterion:

determines the θ 1 (i) and θ 2 (i). The calculated results are shown in Table 3 .The corresponding simulation results of system (18) are shown in Fig. 9 (a) and 9(b). Observe that the simulation results of model (18) describe well the dynamics of the second COVID-19 epidemic in Beijing.

(1) On the days 0, 3, 10, 29, 44 and 56, the numbers of the practical and simulated current symptomatic individuals are approximate the same. On the days 27 and 56, they have only one difference. On the days 34, they have 9 differences.

(2) On the days 0, 3, 10, 19, 44 and 56, the numbers of the practical and simulated current asymptotic individuals are approximate the same. On the days 27 and 34, they have only one difference.

(3) On the days 0, 3, 10 and 19, the numbers of practical and simulated cumulative recovered symptomatic individuals are approximate the same, respectively. On the day 27, it has 4 differences. On the day 34, it has 12 differences. On the day 44, it has 3 differences. On the day 56, it has only one difference.

(4) On the days 0, 3, and 19, the numbers of practical and simulated cumulative recovered asymptomatic individuals are approximate the same. On the days 10, 27, 44 and 56, they have only one difference. On the day 34, it has one two differences.

(5) Computed results (see (32)) show that the ratio of the transmission rates of asymptomatic and symptomatic individuals infecting susceptible population to become symptomatic individuals is about 9% (β 21 : β 11 ). This suggests that asymptomatic individuals cause lesser symptomatic spread than symptomatic individuals do.

(6) Computed results (see (32)) also show that the ratios of the transmission rates of asymptomatic and symptomatic individuals infecting susceptible population to become asymptomatic and symptomatic individuals are about 5% (β 12 : β 11 ) This suggests that symptomatic individuals cause lesser asymptomatic spread than symptomatic spread. (7) The criterions of the stability of the disease-free equilibrium of system (18) at 7 time intervals are listed in Table 4 . It shows that until the blocking rates (1 − θ 1 , 1 − θ 2 ) reach to ( 99.869%, 98.69%), the disease-free equilibrium becomes globally asymptotical stable. The blocking rates (97.46%, 96.94%) cannot prevent the spread of the second COVID-19 epidemic in Beijing.

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The copyright holder for this preprint this version posted July 7, 2021. ; https://doi.org/10.1101/2021.07.04.21259205 doi: medRxiv preprint Table 4 The criterions of the asymptotical stability and disease spreading of the disease-free equilibrium of system (18) at 7 time intervals. Now assume that it keeps still the blocking rates (1 − θ 1 (3), 1 − θ 2 (3)) ≈ (97.46%, 96.94%) and the cure rates (κ(3), κ a (3)) until the day 56, August 6. The simulation results of system (18) are shown in Fig.10 . Observe that on the day 56, the numbers of the current symptomatic and asymptomatic infected individuals reach to 1210 and 89, respectively; The numbers of cumulative recovered symptomatic and asymptomatic infected individuals reach to 23 and 82, respectively.

Furthermore assume that after the day 34th, July 14, it still keeps the blocking rates (1 − θ 1 (5), 1 − θ 2 (5)), the cure rates (κ(5), κ a (5)) until the day 56th, August 6. The simulation results of system (18) are shown in Figs. 11(a) and 11(b) . Observe that on the day 56, the numbers of the current symptomatic and asymptomatic infected individuals are about 29 and 2, respectively. The numbers of cumulative recovered symptomatic and 16 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.

The copyright holder for this preprint this version posted July 7, 2021. asymptomatic individuals are about 309 and 47, respectively. The results suggest that using the data before the day 34 (about two weeks after the turning point) can approximately to estimate the following outcome of the the second COVID-19 academic in Beijing.

The main contributions of this paper are summarized as follows:

(1) Proposed the SARDDE models ( (1) and (18)) with 4 or 5 states: current symptomatic and asymptomatic infected individuals, cumulative recovered symptomatic and asymptomatic infected individuals, died individuals.

(2) Provided the criterion inequalities for the asymptotical stability of the disease free equilibrium point of SARDDE (see Theorem 1 and Theorem 3).

(3) Given the criterion inequalities for epidemic transmission (see Theorem 2 and Theorem 4) of the symptomatic and asymptomatic infections.

(4) Using the reported clinic data and the model simulations results are depicted for biologically significant model parameters.

(5) In systems (1) and (18), assume, respectively, that after the day 24th and the day 19th if still keeps the blocking rates (θ 1 (3), θ 2 (3)), the cure rates (κ(3), κ a (3)) and the died rate α(3) until the day 140th, June 8 and the day 56, August 6. Virtual simulations of systems (1) and (18) suggest that even the a blocking rate to symptomatic individuals reaches to about 90%/97%, the two COVID-19 epidemics can still spread and reach very height levels (see Figs. 4 and 10) . Therefore the strict prevention and control strategies implemented by Beijing government is not only very effective but also completely necessary. (6) Simulations showed that using the data form the beginning to the day after about two weeks from the turning points, we can estimate well or approximately the following outcomes of the first or second COVID-19 17 . CC-BY 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)

The copyright holder for this preprint this version posted July 7, 2021. ; https://doi.org/10.1101/2021.07.04.21259205 doi: medRxiv preprint academics in Beijing. (7) The selections of the transmission rates β ′ ij s are difficult because different combinations of β ′ ij s can produce very closed simulation errors. Agreements of followed simulations are used to judge the reasonableness of the selected β ′ ij s. (8) In the case of the first COVID-19 academic in Beijing, The proposed SARDDE is simpler that our previous one [7, 8] . The determinations of the SARDDE parameters only used one assumption (13) . However it can better describe and explain the practical reported data [11] although the lack of the data of the asymptomatic infected individuals. In the case of the second COVID-19 epidemic in Beijing, the numerical simulations show that the four variables of model (18) describe and explain well the real world data [11] . In both cases, the numerical simulations of the SARDDEs accurately predict the long-terms real world data [11] .

Because not all infected people can go to the hospital for treatment and be confirmed at the first time. In some cases: adequate resources, no shortage of beds and medical treatment advantages, patients may be left behind when they are discharged from the hospital. Therefore, it does not have very important practicality, that the simulation results of the model are required accurately describe every datum reported on the epidemic. Longterm accumulated data, such as the total number of patients and the number of deaths may eliminate short-term deviations. Therefore, the accuracy of predicting long-term epidemics should be the standard for evaluating the rationality of the selected model and unknown model parameters. It is expected that the research can provide better understanding, explanation, and dominating the spread and control measures of epidemics.

The author declares no potential conflict of interest.

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The copyright holder for this preprint this version posted July 7, 2021. ; https://doi.org/10.1101/2021.07.04.21259205 doi: medRxiv preprint

Epidemic Modeling: An Introduction

Modified SEIR and AI prediction of the epidemics trend of covid-19 in China under public health interventions

Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak

Real-time forecasts of the COVID-19 epidemic in China from

An updated estimation of the risk of transmission of the novel coronavirus (2019-nCov)

Study on transmission model of the novel coronavirus

A SIR model including asymptomatic virus carriers with application (in Chinese)

Simulations and estimations of COVID-19 epidemics in Beijing and Shanghai

COVID-19 outbreak reproduction number estimations and forecasting in Marche, Italy

Nine challenges in modelling the emergence of novel pathogens

Epidemic bulletin (in Chinese)

Acucurate simulations and analysiss of Xinfadi COVID-19 epidemics in Beijing

Modelling, simulations and analysis of the first COVID-19 epidemic in Beijing

The author declares no potential conflict of interest.

The previous versions of the manuscript have been published online [12, 13] .