Long-term prediction of the sporadic COVID-19 epidemics induced by δ -virus in China based on a novel non-autonomous delayed SIR model.

With the outbreaks of the COVID-19 epidemics in several provinces of China, government takes prevention and control measures to contain the epidemics. It is more difficult to make the long-term prediction of the sporadic COVID-19 epidemics than widespread ones in that the former cannot obey the laws of the infectious disease well like the latter. In this paper, we make long-term predictions including end time and final size, peak and peak time of current confirmed cases and the number of accumulative removed cases of the sporadic COVID-19 epidemics in different regions of China by a novel non-autonomous delayed SIR compartment model (S-susceptible, I-infected, R-removed). The key contribution of this paper is that under the rigorous containments, we find transmission rate β ( t ) is approximately an exponential decreasing function with respect to time t, rather than a fixed constant. In addition, the removed rate γ ( t ) is approximately a piecewise linear increasing function instead of a linear increasing function which is (at + b)heaviside (t-14). First, according to the few data in the early stage, i.e., roughly the first 7 days, issued by the National Health Commission of China and local Health Commissions, we can accurately estimate these parameters, i.e., transmission and removed rates of the model. Then, by them, we accurately predict the evolution of the COVID-19 there. On the basis of them to predict Category A of the sporadic COVID-19 epidemics since July 20th, 2021 in this summer. The results agree very well to the actual ones. It is also adopted to predict Category B - - - the tour group epidemics since October 17th, 2021 and Category C - - - other sporadic epidemics since October 27th, 2021. The results show that although our method is simple and the needed data are very few, the long-term prediction of the sporadic COVID-19 epidemics in China is quite effective. We can use this novel non-autonomous delayed SIR model to accurately predict its end time and final size, peak and peak time of current confirmed cases and the number of accumulative removed cases in China. This work can help governments and policy-makers make optimal prevention and control policies for all cities and provinces to contain the COVID-19 epidemics, and prepare well for the resumption of work, production and classes in advance to reduce the economic and social losses.

Introduction

On July 20th, 2021, the COVID-19 suddenly broke out in Jiangsu Province, China. Health Commission of Jiangsu Province reported 7 new confirmed cases from 0: 00 to 24: 00 on July 20th, 2021, all of which occurred in Nanjing City. On July 28th, 2021, Health Commission of Yangzhou City reported 2 new confirmed cases. Then it spread to Hunan Province on July 29th, 2021, Henan and Hubei Provinces on July 31st, 2021, Fujian Province on September 10th, 2021 and Harbin City on September 21st, 2021. Experts of the World Health Organization said that the virus in these epidemics is -virus, which was formed by the mutation of the previous -virus. The main features of the -virus are faster transmission, stronger infectivity, shorter incubation period, atypical symptoms, faster intergenerational transmission and high virus load, which result in the new definition of close contacts. We call this round of epidemics as Category A of the sporadic COVID-19 epidemics in China. Due to the overseas input cases from the Ports in Ceke, Ejina Banner in Inner Mongolia Autonomous Region, which is close to the outside world, 1 new confirmed case was reported on October 13th, 2021. On October 19th, 2021, 4 new confirmed cases were reported in Gansu Province. It also spread to Shaanxi Province on October 17th, 2021, Ningxia Hui Autonomous Region on October 18th, 2021, Hebei Province on October 31st, 2021, Sichuan Province on November 1st, 2021, Henan Province on November 3rd, 2021 and Liaoning Province on November 4th, 2021 and other provinces. It spread to 11 provinces in just 7 days. This is the second round of epidemics which is called as Tour Group Epidemics and marked as Category B of the sporadic COVID-19 epidemics in China. The ensuing domestic epidemics still belongs to the overseas input outbreaks, but it had nothing to do with Category B. On October 27th, 2021, Heilongjiang Province reported 1 new confirmed case, followed by outbreaks in Jiangxi Province with report of 1 new confirmed case on October 30th, 2021, Hulunbuir City, Inner Mongolia Autonomous Region with report of 20 new confirmed cases on November 28th, 2021 and Xian City, Shaanxi Province with report of 3 new confirmed cases on December 9th, 2021, which are called as Category C of the sporadic COVID-19 epidemics in China. Once COVID-19 epidemics occurs, no matter it is widespread or sporadic, it will surely impose negativity upon all walk of life, particularly the economic life in the corresponding provinces and cities. Therefore it is urgent to propose a reasonable mathematical model to help to resume work, production and classes as soon as possible. Only in this way, the governments and policy-makers can make optimal emergency policies and measures for prevention and control, avoid the shortage of resources and get the economic and social life back to its right track.

The clinical symptoms of COVID-19 are different from those of common pneumonia in [1, 2]. In addition, Kumar et al. proposed a light-weight Convolutional Neural Network (CNN) with Modified-Mel-frequency Cepstral Coefficient (M-MFCC) using different depths and kernel sizes to classify COVID-19 and other respiratory sound disease symptoms such as Asthma, Pertussis, and Bronchitis in [3]. Many mathematicians have studied the characteristics and evolution of COVID-19, as well as its various influences [4, 5]. In addition, many mathematical models were used to study COVID-19 epidemics. Zahiri and Rafieenasab used the SIR model for the dynamics of an epidemic to provide an assessment of the COVID-19 in Iran [6]. Koziol, Stanislawski and Bialic [7] employed the nabla fractional-order difference defined by Grünwald–Letnikov formula to study the fractional-order generalization of the SIR epidemic model for predicting the spread of the COVID-19. Luo and Zhang [8] established the prediction models for the time series data of America by applying both the long short-term memory (LSTM) and extreme gradient boosting (XGBoost) algorithms. A non-autonomous SIR Model was used for studying the COVID-19 in some countries [9, 10]. SAIU model (S—susceptible or uninfected, A—asymptomatic, I—reported symptomatic infectious, U—unreported symptomatic infectious) was used for describing COVID-19 transmission by Piu and Jayanta [11]. In [12, 13], accurate prediction of NACPs and NADPs of COVID-19 in different countries were made. Alos [14], Lounis [15], Ferrari [16], and Sedaghat [17] used SIRD model to predict the spread trend of COVID-19 in some countries. Long-term forecast of COVID-19 was made in some regions at home and abroad in [18, 19]. The PSO algorithm was used for parameter estimation of SEIR model by He [20]. Annas [21] adopted SEIR model based on taking vaccination and isolation factors as model parameters, obtained the basic reproduction number and global stability of COVID-19 distribution mode by generating matrix method, and numerically simulated the secondary data of COVID-19 cases in Indonesia. Recently, Borah et al. [22] revisited the Bombay Plague epidemic of India and presents six fractional-order models (FOMs) of the epidemic based on observational data, used the plague results to verify that the method could be used to predict the evolution of the second wave of COVID-19 in India. Reis and Savi [23] proposed a dynamical map to describe COVID-19 epidemics based on the classical susceptible-exposed-infected-recovered (SEIR) differential model, incorporating vaccinated population. On this basis, the novel map represents COVID-19 discrete-time dynamics by adopting three populations: infected, cumulative infected and vaccinated. Gopal et al. [24] used the SEIR model to estimate and analyze the number of infected individuals during the second wave of COVID-19 in India. Wang et al. [25] reduced the SEIR model to a simple two-dimensional model and analyzed its multi-stability and chaos to help scientists further understand the complexity of COVID-19.

Although we cannot deny that scholars’ researches on COVID-19 are of great significance, most of them do not consider the time-delay or time-varying coefficients, which makes the application of the model somewhat lack accuracy and effectiveness. In this paper, we adopt a novel non-autonomous delayed SIR model to further improve the above methods, in which the finite difference method is used to estimate its parameters. Our research shows that the transmission rate is approximately an exponential decreasing function with respect to time t, rather than a fixed constant. The removed rate is approximately a piecewise linear increasing function instead of a linear increasing function which is ( +b)heaviside(t-14). They are very different from those of the previous SIR model [18, 19] and bring about the more accurate predictions of the COVID-19 epidemics. Due to the degree of illness and personal recovery, the first recovery may appear at different times, which it is roughly on the 14th day. It fits Category A of the sporadic COVID-19 epidemics well. After further estimation of the transmission and removal rates with the data of roughly the first 7 days, we find that the predicted result is consistent with the actual one of the total COVID-19 during the epidemics durations. Therefore, we use the data of roughly the first 7 days to make long-term predictions of Category A of the sporadic COVID-19 epidemics and can predict accurately their end time, final size, peak and peak time of current confirmed cases and the number of accumulative removed cases. What is more, the prediction results agree well to the actual evolution of these COVID-19 epidemics. Subsequently, fittings and parameter estimation are also carried out for Categories B and C of the sporadic COVID-19 epidemics and their long-term predictions are also made.

The structure of this paper is as follows. In Sect. 2, the SIR model is introduced. In Sect. 3, the data used in this paper are described. The fitting and parameter estimation of Category A are presented in Sect. 4. Long-term predictions and verifications of Category A are presented in Sect. 5. In Sect. 6, the long-term forecast of Categories B and C are described. The conclusion and discussion are presented in Sect. 7.

A novel non-autonomous delayed SIR model

In this section, we introduce a novel non-autonomous delayed SIR model. The SIR model is a compartmental model describing how a disease spreads among the populations. It is a set of general equations that explain the dynamics of an infectious disease. The subjects of SIR model are the susceptible, infected and removed cases. In addition, the removed group includes the dead and recovered cases. In the model, natural birth and death rates are not considered. The total population under study is presumed to be invariant and supposed to be N. Obviously, S(t)+I(t)+ . The number of people in each category is marked with the following symbols:

S(t): Susceptible, representing the number of people who do not have infectious diseases at time t, but are likely to have infectious diseases in the future.

I(t): Infected, representing the number of people who get infectious diseases at time t.

R(t): Removed, representing the accumulative or total number of the recovered and dead groups at time t.

In this paper, the highlights of the model lie in the following two aspects. First, time-delay is introduced to describe the incubation period of the -virus. Infected cases go through an incubation period of days before showing significant symptoms. Once symptoms appear, the infected person will seek treatment and be transformed into the confirmed case. Many works did not consider the effect of the incubation delay. But actually this delay is long, even up to more than 20 days, and its effect on the dynamic is crucial. Therefore, we have to introduce it into the model and consider its effect on the dynamics and evolution. In this paper, we take the mean value, roughly 3 days, as the incubation delay of the -virus. Second, through the analysis of the data reported by the National Health Commission of China, we find that both the transmission rate and the removed rate are functions that evolve gradually with time t. Through the estimations, we find that (d t) is approximately an exponential decreasing function of time t and ( +b)heaviside(t-14) is approximately a piecewise linear increasing function of time t.

According to analysis, we can get the novel non-autonomous time-delayed dynamic model of the sporadic COVID-19 epidemics in China as the following:In the above model, represents the rate of transmission for the susceptible to the infected and describes the removed rate of the infected cases. In DDEs (1), we take roughly 3 days as the incubation delay . According to the model, nearly equals the number of newly-increased cumulative confirmed cases of the present day divided by the product of the number of the infected and the number of the susceptible on the previous day, i.e., (t-) day. is approximately equal to the number of newly-increased removed cases on the present day divided by the number of infected cases on the present day. From DDEs. (1), we obtain the following expressions:

Data description

The sources of data are the National Health Commission and local Health Commissions in this paper. In addition, the actual data we use are from the beginning to the end of the sporadic COVID-19 epidemics for the local provinces and cities. These sources of websites are as follows: the website of the National Health Commission’s daily epidemic notification is (http://www.nhc.gov.cn/xcs/yqtb/list_gzbd.shtml). The website of Health Commission’s daily epidemic notification of Jiangsu Province is (http://wjw.jiangsu.gov.cn/col/col7290/index.html). The website of Health Commission’s daily announcements of Nanjing City is (http://wjw.nanjing.gov.cn/njswshjhsywyh/?id=xxgk_228). The website of Health Commission’s daily important news of Yangzhou City is (http://wjw.yangzhou.gov.cn/yzwshjh/ywkd/wjw_list_20.shtml). The website of Health Commission’s daily important issue of Hunan Province is (https://wjw.hunan.gov.cn/wjw/qwfb/yqfkgz_list.html). The website of Health Commission’s daily epidemic notification of Henan Province is (http://wsjkw.henan.gov.cn/ztzl/xxgzbdfyyqfk/yqtb/). The website of Health Commission’s daily important news of Hubei Province is (http://wjw.hubei.gov.cn/bmdt/dtyw/). The website of Health Commission’s daily epidemic situation dynamics of Fujian Province is (http://wjw.fujian.gov.cn/ztzl/gzbufk/yqtb/). The data of Harbin City are from the website of Health Commission’s daily epidemic situation dynamics of Heilongjiang Province (http://wsjkw.hlj.gov.cn/pages/5df84bfaf6e9fa23e8848a48?page=15). The website of Health Commission’s daily news issue of Gansu Province is (http://wsjk.gansu.gov.cn/wsjk/c112713/list.shtml). The website of Health Commission’s daily epidemic situation dynamics of Shaanxi Province is (http://sxwjw.shaanxi.gov.cn/sy/wjyw/). The website of Health Commission’s daily epidemic prevention and control dynamics of Ningxia Hui Autonomous Region is (https://www.nx.gov.cn/ztsj/zt/yqfkzccs/yqfkdt/). The website of Health Commission’s daily epidemic notification of Hebei Province is (http://wsjkw.hebei.gov.cn/html/yqtb/index.jhtml). The website of Health Commission’s daily epidemic notification of Sichuan Province is (http://wsjkw.sc.gov.cn/scwsjkw/gzbd01/ztwzlmgl.shtml). The website of Health Commission’s daily Health news of Liaoning Province is (http://wsjk.ln.gov.cn/wst_wsjskx/). The website of Health Commission’s daily epidemic notification of Heilongjiang Province is (http://wsjkw.hlj.gov.cn/pages/5df84bfaf6e9fa23e8848a48). The website of Health Commission’s daily announcements of Jiangxi Province is (http://hc.jiangxi.gov.cn/col/col38018/index.html). In addition, the website of Health Commission’s daily news issue of Inner Mongolia Autonomous Region is (http://wjw.nmg.gov.cn/xwzx/xwfb/).

Fitting and parameter estimation of the novel non-autonomous delayed SIR model for Category A

According to the analysis of the total data of the epidemics released by the National Health Commission of China and local Health Commissions, we estimate the transmission rate and removal rate of Jiangsu Province, Nanjing City, Yangzhou City, Fujian Province, Xiamen City, Putian City, Hubei Province, Hunan Province and Harbin City, which belong to Category A of the sporadic COVID-19 epidemics induced by -virus. The data are from the corresponding website of each province and city in Sect. 3. Based on the least square method, we use Isqnonlin function built in Matlab to carry out parameters inversion and get the optimal parameters. These results are perfect. We choose the data on the first day as the initial functions in [-,0] of S(t), I(t), R(t) in the DDEs (1). According to the time from the beginning to the end of the epidemics in each province or city, we take the corresponding overall data of the epidemics in these provinces and cities for fitting and parameter estimation of S(t), I(t) and R(t). Here, we present the function expressions of parameter estimation of and in Tables 1, 2, 3 and fitting curves of S(t), I(t) and R(t) in Figs. 1, 2, 3 about Category A of the sporadic COVID-19 epidemics.

Table 1

Estimated transmission and removal rates of Category A for Fujian Province, Xiamen and Putian Cities, respectively

	β(t)	γ(t)
Fujian Province	0.000001 exp(-0.524621t )	(0.005050 t−0.031555)*heaviside(t−14)
Xiamen City	0.000003 exp(-0.491934t )	(0.003339 t−0.003025)*heaviside(t−13)
Putian City	0.000005 exp(-0.590980t )	(0.010568 t−0.116416)*heaviside(t−13)

Table 2

Estimated transmission and removal rates of Category A for Jiangsu Province, Nanjing and Yangzhou Cities, respectively

	β(t)	γ(t)
Jiangsu Province	0.0000002 exp(-0.166473t )	(0.003505 t−0.064832)*heaviside(t−14)
Nanjing City	0.0000003 exp(-0.298808t )	(0.000459 t+0.010923)*heaviside(t−14)
Yangzhou City	0.000001 exp(-0.293799t )	(0.007479 t−0.127351)*heaviside(t−17)

Table 3

Estimated transmission and removal rates of Category A for Hubei Province, Hunan Province and Harbin City, respectively

	β(t)	γ(t)
Hubei Province	0.0000002 exp(-0.324836t )	(0.000001 t+0.010007)*heaviside(t−15)
Hunan Province	0.0000001 exp(-0.307025t )	(0.001825 t-0.022006)*heaviside(t−14)
Harbin City	0.0000003 exp(-0.385078t )	(0.019788 t−0.244697)*heaviside(t−14)

Fig. 1

Fitting curves of the susceptible, infected and removed in Fujian Province, Xiamen and Putian Cities, respectively

Fig. 2

Fitting curves of the susceptible, infected and removed cases in Jiangsu Province, Nanjing and Yangzhou Cities, respectively

Fig. 3

Fitting curves of the susceptible, infected and removed cases in Hubei Province, Hunan Province and Harbin City, respectively

The fitting and estimation results show that our method is quite simple and effective. From the results, we can see that the transmission rate is approximately exponential decrease with respect to time t rather than a low constant, and the removal rate is approximately a piecewise linear increasing function of time t. It indicates that under the rigorous containments, the prevalence level of COVID-19 is very low and tend to 0 exponentially in China. According to the results of parameter estimation, its numerical simulations of the susceptible, infected and removed people are quite consistent with the actual data. In spite of some errors in fitting in some epidemic regions due to the uncertainty of the outbreak of COVID-19, the detection of the emergency of the outbreaks and the mobility of the population, but this model can be widely applied to theory and practice.

Long-term predictions and verifications of Category A

According to the above fitting diagrams and parameter estimation results of each province and city during the epidemics, we can verify whether the model is accurate and predictable. In this section, we will take the data of the roughly the first 7 days to estimate the parameters and fit the evolution of the susceptible S(t), infected I(t) and removed R(t). The estimated results of the parameters are displayed in Table 4, and the fitting diagrams of S(t), I(t) and R(t) are described in Fig. 4. Furthermore, we make long-term predictions including their end time and final size, peak and peak time of current confirmed cases and the number of accumulative removed cases in Fujian Province, Xiamen City, Harbin City and Putian City, respectively. In addition, we verify their end time and final size, peak and peak time of current confirmed cases and the number of accumulative removed cases from the beginning to the end of the epidemics in each province or city. In other words, our long-term forecast is about the prediction of the above several important indexes from the beginning to the end of the epidemics for the corresponding province and city and the evolution of their overall COVID-19 epidemics. For example, for Fujian Province, the actual duration of the COVID-19 epidemic is from the beginning of the epidemic on September 10th, 2021 to the end of the epidemic on October 2nd, 2021. For Xiamen City, the actual duration of the COVID-19 epidemic is from the beginning of the epidemic on September 12th, 2021 to the end of the epidemic on October 2nd, 2021. For Putian City, the actual duration of the COVID-19 epidemic is from the beginning of the epidemic on September 10th, 2021 to the end of the epidemic on September 24th, 2021. For Harbin City, the actual duration of the COVID-19 epidemic is from the beginning of the epidemic on September 21st, 2021 to the end of the epidemic on October 5th, 2021. The comparisons of these predicted important indexes with those of actual data are executed. The results of the predictions are presented in Fig. 5. The comparisons between predicted and actual indexes are described in Table 5.

Table 4

Estimated transmission and removal rates of Category A for Fujian Province, Xiamen, Harbin and Putian Cities and with data of roughly the first 7 days, respectively

	β(t)	γ(t)
Fujian Province	0.00000156108 exp(-0.54225795539t)	(0.002 t+0.02)*heaviside(t−14)
Xiamen City	0.0000031634 exp(-0.49562623723t)	(0.002 t+0.02)*heaviside(t−13)
Harbin City	0.00000027654 exp(-0.38088054250t)	(0.005 t+0.02)*heaviside(t−14)
Putian City	0.00000422079 exp(-0.56218903537t )	(0.003 t+0.01)*heaviside(t−13)

Fig. 4

Fitting curves of the susceptible, infected and removed cases of Category A for Fujian Province, Xiamen, Harbin and Putian Cities with data of roughly the first 7 days, respectively

Fig. 5

Long-term predictions for Fujian Province, Xiamen, Harbin and Putian Cities for Category A of COVID-19, respectively

Table 5

Comparisons of the predicted and actual important indexes for Category A: Fujian Province, Xiamen, Harbin and Putian Cities, respectively

Indexes	Fujian Province	Xiamen City	Harbin City	Putian City
Predicted End time	2021.09.30	2021.10.01	2021.10.09	2021.09.26
Actual End time	2021.10.02	2021.10.02	2021.10.05	2021.09.24
Predicted Final size	459	233	85	217
Actual Final size	468	236	88	204
Predicted PeakofI(t)	445	223	82	214
Actual PeakofI(t)	445	223	83	199
Predicted Peak time of I(t)	2021.09.23	2021.09.24	2021.10.04	2021.09.22
Actual Peak time of I(t)	2021.09.23	2021.09.24	2021.09.30	2021.09.22

The actual meaning presented in Fig. 5 is as follows: here’s the main idea. Taking Fujian Province as an example, we select the data from September 10th to September 15th, and use this model to estimate parameters and fit them. According to the results in Table 4 and Fig. 4, it is very potentially accurate to use this model to fit the data by adjusting the initial values and bounds of the parameters with roughly the first 7 days. Then, the evolution of the COVID-19 epidemics after September 16th is predicted after substituting the estimated parameters into the original model and simulating it. As shown in Fig. 5a–c, since we make the prediction from the first day of the outbreak, we choose the data of the first day as the initial functions in of S(t), I(t) and R(t), i.e., actual meaning of the first green asterisk represented. The remaining green asterisks represent the actual data of the first 6 days, the red asterisks represent the actual data after the 6th day, and the solid blue curves describe the predicted results.

According to the comparison results in Fig. 5 and Table 5, we find that the difference between the predicted end time and the actual end time is 4 days in Harbin City, only 2 days in Fujian Province and Putian City, and the better result is that the difference is only 1 day in Xiamen City. There are only 6.37% and relative errors between the predicted final size and the actual final size in Putian and Harbin Cities, respectively and a better result is that the relative errors are only and in Fujian Province and Xiamen City, respectively. There is only 7.54% relative errors between the predicted peaks of I(t) and the actual one in Putian City. The better result is that the difference in Harbin is only 1 person. The perfect result is that the predicted peaks of I(t) of Fujian province and Putian City is completely consistent with the actual ones. The difference between the predicted peak time of I(t) and the actual peak time of I(t) is 4 days in Harbin City. Satisfactorily, the predicted peak time of I(t) of Fujian Province, Xiamen and Putian Cities are completely consistent with the actual peak time of I(t). In summary, we can see that the predicted end time, final size, peak and peak time of current confirmed cases and the number of accumulative removed cases have low relative errors compared with those of the actual data. It presents the effectiveness, predictability and simplicity of this novel non-autonomous delayed SIR model for the sporadic COVID-19 epidemics induced by -virus in China.

Long-term predictions of Categories B and C

Through the verification of the long-term predictions based on the above-mentioned parameter estimation of Category A of the sporadic COVID-19 epidemics, the consistency and effectiveness of long-term predictions of this novel SIR model are presented. In this section, we will use this model to estimate the parameters of Categories B and C of the sporadic COVID-19 epidemics, plot the fitting diagrams of S(t), I(t) and R(t) in each province and city, and finally make the long-term forecasts of their COVID-19 epidemics’ evolutions. Due to different natural conditions, economic status, mobility of people and living habits in different COVID-19 epidemic regions, the evolution of the COVID-19 epidemics is also different. Therefore, we will take the data of the first 7 days or roughly the first 7 days for fitting and prediction. For Category B, we mainly analyze the sporadic COVID-19 epidemics in Gansu Province since October 19th, 2021, Hebei Province since October 31st, 2021, Sichuan Province since November 1st, 2021, Henan Province since November 3rd, 2021 and Liaoning Province since November 4th, 2021. For Category C, we make long-term predictions on the sporadic COVID-19 epidemics in Heilongjiang Province since October 27th, 2021, Jiangxi Province since October 31st, 2021, Inner Mongolia Autonomous Region since November 28th, 2021 and Shaanxi Province since December 9th, 2021. Particularly, due to the small number of newly confirmed cases in Shaanxi province from December 9th, 2021 to December 17th, 2021, it disobeys the evolutive law of the COVID-19 epidemics in Shaanxi Province. Therefore, we choose the data on December 18th, 2021 to start its prediction and select the data on December 15th, 2021 as the initial function. In addition, their parameter estimation results are presented in Table 6, the fitting curves of S(t), I(t) and R(t) are presented in Fig. 6. The methods used in Table 6 and Fig. 6 are similar to Table 4 and Fig. 4, respectively. The predicted diagrams are presented in Fig. 7. The comparisons of actual and predicted important indexes are shown in Table 7. Similarly, the data come from the corresponding Websites of Health Commission for each province and city. In addition, the long-term forecast in Fig. 7 is also an important indexes for each province from the beginning to the end of the COVID-19 epidemics.

Table 6

Estimated transmission and removal rates of Categories B and C for Heilongjiang, Hebei, Sichuan, Jiangxi, Henan, Gansu and Liaoning Provinces, Inner Mongolia Autonomous Region and Shaanxi Province with data of roughly the first 7 days, respectively

	β(t)	γ(t)
Heilongjiang Province	0.00000137006 exp(-0.46314793551t)	(0.0015 t+0.02)*heaviside(t−14)
Hebei Province	0.00000006465 exp(-0.2566155128t)	(0.01100809472 t-0.08038774832)*heaviside(t−8)
Sichuan Province	0.00000013395 exp(-0.28522643703t )	(0.001614 t+0.091598)*heaviside(t−9)
Jiangxi Province	0.00000010433 exp(-0.16912955145t )	(0.004038 t+0.114095)*heaviside(t−6)
Henan Province	0.00000009141 exp(-0.23847161652t )	(0.03 t+ 0.02)*heaviside(t-13)
Gansu Province	0.0000002 exp(-0.25t)	(0.0008t+0.02)*heaviside(t-14)
Liaoning Province	0.00000030627 exp(-0.25446657329t )	(0.01 t+0.002)*heaviside(t-14)
Inner Mongolia Autonomous Region	0.00000121783 exp(-0.31292064703t )	(0.003 t+0.002)*heaviside(t-14)
Shaanxi Province	0.00000036516 exp(-0.24254950696t )	(0.0027 t+0.0005)*heaviside(t-13)

Fig. 6

Fitting curves of the susceptible, infected and removed cases of Categories B and C for Heilongjiang, Hebei, Jiangxi, Sichuan, Henan, Gansu and Liaoning Provinces, Inner Mongolia Autonomous Region and Shaanxi Province with data of roughly the first 7 days, respectively

Fig. 7

Long-term predictions for Heilongjiang, Hebei, Gansu and Liaoning Provinces, Inner Mongolia Autonomous Region and Shaanxi Province in Categories B and C of the sporadic COVID-19 epidemics, respectively

Table 7

Comparisons of the predicted and actual important indexes for Categories B and C: Heilongjiang, Hebei, Gansu and Liaoning Provinces, Inner Mongolia Autonomous Region and Shaanxi Province, respectively

	Heilongjiang	Hebei	Gansu	Liaoning	Inner Mongolia	Shaanxi
Indexes	Province	Province	Province	Province	Autonomous Region	Province
Predicted end time	2021.11.15	2021.11.21	2021.11.14	2021.11.28	2021.12.28	2022.01.24
Actual end time	2021.11.15	2021.11.14	2021.11.08	2021.11.27	2021.12.17	2022.01.20
Predicted final size	283	166	141	336	640	2304
Actual final size	277	132	144	308	560	2080
Predicted peak of I(t)	270	126	120	280	581	1613
Actual peak of I(t)	258	121	132	287	534	1824
Predicted peak time of I(t)	2021.11.09	2021.11.11	2021.11.03	2021.11.17	2021.12.11	2022.01.04
Actual peak time of I(t)	2021.11.11	2021.11.13	2021.11.05	2021.11.18	2021.12.11	2022.01.06

According to the comparison results in Fig. 7 and Table 7, we find that the differences between the predicted end time and the actual end time are 11 days, 7 days, 6 days and 4 days in Inner Mongolia Autonomous Region, Hebei, Gansu and Shaanxi provinces, respectively, the better result is that the difference is only 1 day in Liaoning Province. The perfect result is that the predicted end time of Heilongjiang Province is completely consistent with the actual end time. The evolutions of the COVID-19 epidemics are also different due to different natural status, economic conditions, mobility of people and living habits in corresponding provinces and cities, so there are some relative errors for predictions of the COVID-19 final size in Hebei Province. There are 14.286%, 10.769% and 9.09% relative errors between the predicted final size and the actual final size in Inner Mongolia Autonomous Region, Shaanxi and Liaoning Provinces, respectively. The better result is that the relative errors are only 2.166% and in Heilongjiang and Gansu Provinces, respectively. There are and 8.8% relative errors between the predicted peaks of I(t) and the actual ones in Shaanxi, Gansu and Inner Mongolia Autonomous Region, respectively, and a better result is that the relative errors between Heilongjiang and Hebei Provinces are only 4.65% and 4.13%, respectively. The ideal result is only relative errors in Liaoning Province. Moreover, the differences between the predicted peak time of I(t) and the actual peak time of I(t) are only 2 days in Heilongjiang, Hebei, Gansu and Shaanxi Provinces. Satisfactorily, that of Liaoning Province is only 1 day. Moreover, the perfect result is that the predicted peak time of I(t)of Inner Mongolia Autonomous Region is completely consistent with the actual peak time of I(t). In conclusion, the above results further indicate the importance and effectiveness of our method.

Conclusion and discussion

In this paper, we use a novel non-autonomous delayed SIR model to make the long-term forecasts of the sporadic COVID-19 epidemics induced by -virus in China. It displays the effectiveness and simplicity of this novel SIR epidemic model in the predication of the COVID-19 and other infectious diseases. According to the analysis of the evolutions of distinct COVID-19 epidemics, the transmission rate is approximately an exponential decreasing function of time t, and the removed rate is approximately a piecewise linear increasing function of time t. In addition, we use a small amount of data to estimate these two parameters and fit the number of the susceptible, the infected and the removed respectively, and finally use these estimated results to make the long-term predictions of sporadic COVID-19 epidemics in China. Moreover, according to the comparisons of predicted and actual important indexes, our method is quite accurate for these important indexes including the end time and final size, peak and peak time of current confirmed cases and the number of accumulative removed cases. It indicates that our method is very simple and effective in long-term predictions of the sporadic COVID-19 epidemics in China. More importantly, the reliability of our forecasts can also help the governments and policy-makers to prepare the necessary resources for the epidemics in real time, and strengthen the implementation of prevention and control measures and strategies in time. In addition, it can help the governments, policy-makers and the people to make the optimal plans for the resumption of work, production and classes. We believe that under the timely and rigorous prevention and control of the government, the COVID-19 epidemics in China can be well contained in a very short period.