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Modeling and Forecast of COVID-19 Vaccinations
Population Size in China: A Non-Clinical Study
Wenbin Du ( duwb@swufe.edu.cn )
Southwestern University of Finance and Economics https://orcid.org/0000-0001-7604-3902
Fengrui Hua
Southwestern University of Finance and Economics
Shengyuan Xu
Tohoku University: Tohoku Daigaku
You Wu
Southwestern University of Finance and Economics
Research
Keywords: COVID-19 vaccine, time series analysis, Python language, autoregressive integrated moving
average (ARIMA) model, forecast
DOI: https://doi.org/10.21203/rs.3.rs-729132/v1
License: This work is licensed under a Creative Commons Attribution 4.0 International License.
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1
Modeling and Forecast of COVID-19 Vaccinations population size in China: a
non-clinical study
Wenbin Du1*, Fengrui Hua2, Shengyuan Xu3, You Wu4
1 Research Institute of Social Development, Southwestern University of Finance and
Economics, No. 555, Liutai Road, Wenjiang District, Chengdu 611130, China. Tel:
+86-13669357615; Fax: +86-28-87092257; Email: duwb@swufe.edu.cn
2 Research Institute of Social Development, Southwestern University of Finance and
Economics, No. 555, Liutai Road, Wenjiang District, Chengdu 611130, China. Tel:
+86-15034084982; Fax: +86-28-87092257; Email: huafengrui@smail.swufe.edu.cn
3 Graduate School of Information Sciences, Tohoku University, 6-3-09 Aoba,
Aramakiaza Aobaku, Sendai, 980-8579, Japan. Tel: +81-08033363888; Fax:
+81-22-795-5815; Email: kyos163@163.com
4 School of Accounting, Southwestern University of Finance and Economics, No. 555,
Liutai Road, Wenjiang District, Chengdu 611130, China. Tel: +86-13717647930; Fax:
+86-28-87092257; Email: waiola1999@sina.com
Corresponding author:
Wenbin Du, Research Institute of Social Development, Southwestern University of
Finance and Economics, No. 555, Liutai Road, Wenjiang District, Chengdu 611130,
China.
Tel: +86-13669357615;
Fax: +86-28-87092257;
Email: duwb@swufe.edu.cn
2
Abstract
BACKGROUND
Since its outbreak in December 2019, severe acute respiratory syndrome
coronavirus-2, the virus responsible for the COVID-19 pandemic, has considerably
affected the worldwide population. Health authorities and the medical community
identify vaccines as an effective tool for managing public health.
METHODS
In this study, the autoregressive integrated moving average (ARIMA) model built-in
Python was adopted to establish the COVID-19 vaccination forecast model. In this
study, the sample data were selected from the Our World in Data website. COVID-19
vaccinations administered daily in China from December 16, 2020 to March 21, 2021
were analyzed to establish an autoregressive integrated moving average (ARIMA)
model.
RESULTS
The built-in ARIMA module function of Python was used, and the optimum model
was ARIMA (3, 2, 3) according to the established time series analysis. The analysis
showed that the predicted COVID-19 vaccination uptake supplemented well with the
actual values with a small relative error.
CONCLUSIONS
This indicated that the ARIMA(3, 2, 3) model could be used to forecast the number of
COVID-19 vaccinations in China.
CONTRIBUTION
3
In this study, the ARIMA model has demonstrated high goodness of fit for forecasting
the COVID-19 vaccinations, which can be used in the short-term prediction of the
monitored COVID-19 vaccination data sequence, providing a reference for the
establishment of immune barrier and the supply of vaccines in China.
Keywords:
COVID-19 vaccine, time series analysis, Python language, autoregressive integrated
moving average (ARIMA) model, forecast
4
Background
The COVID-19 pandemic is causing an unprecedented effect on global health and
economics. When safe and efficient vaccines and treatments were unavailable,
nonpharmaceutical interventions were used to reduce transmission and the burden of
the pandemic. However, most of these measures involved huge economic costs.
Therefore, effective COVID-19 vaccines have been urgently needed to lower the
morbidity and mortality of COVID-19 (Li et al., 2020). The COVID-19 pneumonia
outbreak at the beginning of 2020 involved the first COVID-19 cases caused by the
novel β-coronavirus severe acute respiratory syndrome CoV-2 (SARS-CoV-2)
(Tregoning et al., 2020). The genetic information was disclosed on January 10, 2020,
54 days after the first case was reported. On March 13, 2020, 63 days after the
SARS-CoV-2 virus was sequenced, the news was released that the first dose of the
human vaccine was under testing. The Strategic Advisory Group of Experts on
Immunization of the World Health Organization has outlined the value framework for
COVID-19 vaccine allocation and priority, revealing the core principles of vaccine
distribution (World Health Organization, 2020). These guidelines require further
specifications and should be targeted to each country and their local conditions,
including the intensity of the pandemic, target of pandemic control, supply of
vaccines, and population eligible for vaccination. As the first epicenter country of
COVID-19 pneumonia, China has invested heavily in its vaccination, being the main
participator in the COVID-19 vaccine development for controlling the pandemic,
wherein its vaccines were provided by the government, vaccine makers, and
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nongovernmental organizations (World Health Organization, 2020). On January 24,
2020, the Chinese Center for Disease Control and Prevention successfully separated
the first COVID-19 virus strain. On June 19, 2020, the first mRNA vaccine was
approved for a clinical test. As of February 25, 2021, there had been four types of
COVID-19 vaccines, three inactivated vaccines, and one adenovirus vector vaccine.
At 24 o’clock of March 20, 2021, the reported number of administered COVID-19
vaccinations reached 74.956 million in China (CCTV News, 2021). In this study, the
autoregressive integrated moving average (ARIMA) model built-in Python was
adopted to establish the COVID-19 vaccination forecast model. Time series analysis
and short-term forecast were performed for daily data of COVID-19 vaccinations
administered in China from December 16, 2020, to March 21, 2021. By forecasting
the number of COVID-19 vaccinations and capturing the variation rule of the time
series of vaccinations using the ARIMA model, the evolution rule of the time series
could be extended, thereby accurately predicting the short-term COVID-19 vaccine
demand and providing a reference for vaccine supply in China.
Data source and modeling principle
Data source
In this study, the sample data were selected from the Our World in Data website
（https://ourworldindata.org/coronavirus-testing#china）, established by the Oxford
University, on the basis of the decades-long data involving the human living standards
of various countries. The website provides the daily updated number of COVID-19
vaccinations and the complete COVID-19 dataset on its COVID-19 Explorer. The
6
data were collected by browsing the official public information, and specifically, the
data for China were sourced from the information published by the National Health
Commission of the People’s Republic of China. In this study, the daily vaccination
uptake in China from December 16, 2020, to March 21, 2021, updated by the Our
World in Data website, was used for modeling and the short-term forecast.
Method
ARIMA model
The ARIMA model is mostly used to analyze the nonstationary, nonseasonal time
series (Box, Jenkins & Reinsel, 2010), which is the most commonly used model for
fitting nonstationary series. It can be further divided into the autoregressive (AR),
moving average (MA), and autoregressive moving average (ARMA) models
(Kantelhardt, et al, 2002). They are explained below in detail:
AR model
An AR model with the following structure is called the -order AR model and
denoted as AR(). The mathematical formula is as follows:
= ∅+ ∅+ + ∅ + ⋯ + ∅ + (1)
Here, the value of the random variable at moment t is the multiple linear
regression of the first terms , …, and . The error term is the
current random interference , which is a zero-mean white noise sequence. When
∅ = 0, the model is called the centralized AR(ρ) model.
MA model
7
A MA model with the following structure is called the qth-order AR model and
denoted as MA(q). The mathematical formula is as follows:
= + − − − ⋯ − (2)
Here, the value of the random variable at moment t is the multiple linear
function of the first q random interferences , …, . The error term is the
current random interference , which is a zero-mean white noise sequence.
Specifically, when μ = 0, the model is called as the centralized MA(q) model.
ARMA model
An ARMA model with the following structure is denoted as ARMA(p, q) and its
mathematical formula is as follows:
= ∅+ ∅ + ∅ + ⋯ + ∅ + − − − ⋯ ⋯ −
(3)
The ARMA model is used for stationary time series, which is a combination of the
AR and MA models. It is supposed that is the observed time series, is the
white noise sequence, and is the mixed process of autoregression and moving
average. When the orders of AR and MA are p and q, respectively, the model is
denoted as ARMA(p, q).
ARIMA model
If the time series is nonstationary, its difference should be considered to transform
it into a stationary one, which is then followed by the forecast using the ARMA(p, q)
model. These processes constitute the ARIMA(p, d, q) model wherein p represents the
number of autoregressive terms, d the times of difference of the time series, and q the
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number of moving average terms. The structure of the ARIMA(p, d, q) model is as
follows:
Φ(Β)∇= Θ(Β) (4)
Ε()= 0, ()=
, Ε()= 0, ≠ (5)
Ε= 0, ∀< (6)
∇= (1 − Β) (7)
Φ(Β)= 1 − ∅Β − ⋯ − ∅Β − ⋯ − ∅Β (8)
Θ(Β)= 1 − ∅Β − ⋯ − Β (9)
ARIMA modeling steps
For time series analysis, the ARIMA model forecasts through the following four
steps: time series stationarity test, model recognition and order determination, model
validation, and model forecast. These specific processes are explained below:
Series establishment and stationarity test
The time series ARIMA model must be established on the basis of a stationary time
series as it cannot capture nonstationary ones. Therefore, the stationarity test is needed
at first. The typical methods of stationarity tests include the sequence diagram,
autocorrelation, and unit root tests. In the sequence diagram test, the mean and
variance curves of the time sequence data are plotted to determine if the time series is
stationary by observing its trend. In the autocorrelation test, the autocorrelation
function (ACF) figure is used. For nonstationary data, the ACF figure will approach 0
slowly or not at all. For the unit root test, the solved augmented Dickey-Fuller (ADF)
value will be used to determine whether the sequence data are significant in the given
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confidence interval. If so, the sequence data are stationary. The ADF test assumes that
the time series has a unit root, i.e., the time series is nonstationary with periodic
fluctuations. If the test result indicates that the absolute value of T is lower than the
observation level of 1%, 5%, or 10% and the significance probability ρ value is
larger than 0.05, the sequence is deemed as nonstationary, and the original hypothesis
cannot be rejected. Similar to Eq. (10), the mean of the random series is expressed as
follows:
= =∫()
(10)
where represents the expectation of the random series at moment t.
If the time series is determined to be nonstationary, i.e., periodic or with a marked
trend, the difference of the original sequence data will be determined. The following
Eq. (11) gives the d difference equation:
∇=(1 − Β)=∑ (−1)
(11)
Here, indicates the stationary sequence after difference; Β is the delay operator,
indicating that the times of time index difference is d for the series.
Model recognition and order determination
After d times of difference, the ACF and partial autocorrelation function (PACF)
tests on the stationary time series are commonly performed to obtain the number of
layer p and order q for ARIMA. By observing the truncation and trailing of the ACF
and PACF tests, the model is fitted. The fitting results are compared and adjusted
accordingly for the preliminary construction of one or more suitable ARIMA models.
If both the ACF and PACF tests of the time series are trailing, it can be determined
10
that the forecast model is ARMA(p, q). The values of p and q are investigated
gradually from the lower orders, and the smallest p and q are selected that can
minimize the values of the Akaike information criterion (AIC) and Bayesian
information criterion (BIC). Eq. (12) shows the mathematical expression of the ACF:
ρk = ∑ (− )/
∑ (− )
(12)
where represents the value of the random series at moment t, and represents
the mean of the random series.
Eq. (13) displays the mathematical expression of the PACF:
= −∑
,
− 1 ∑ ,
(13)
where = , , whereas indicates the ACF of the random
series.
Model validation
Before establishing the ARIMA model to forecast the number of administered
COVID-19 vaccinations, the reasonability of the model should be evaluated. The
purpose of model evaluation is to avoid p and q deviations determined by figure
reading and ensure the precision of the forthcoming vaccination forecast. The white
noise and normality tests are typically used to examine the residual sequence. If p
values for the fitting test statistics are all considerably larger than the significance
level of 0.05, the residual sequence can be observed as a white noise sequence. If the
fitted model cannot pass the test, the model will be reselected for fitting. In the
normality test, the obtained data are, however, assessed to see whether they satisfy the
normal distribution. The standardized residual Q-Q plot of the ARIMA model is
11
drawn to determine the reasonability of the established forecast model.
Model forecast
The main function of the time series analysis is to explain the autocorrelation of a
time series with mathematical models, thereby predicting the future variation rule of
the series. In this study, the daily data of the COVID-19 vaccinations administered in
China from December 16, 2020 to March 21, 2021 will be selected to find the
optimum forecast model and make five-day projections. By comparing the predicted
value with the actual data, the forecast performance will be assessed.
Result
Time series establishment and stationarity test
The data of the daily COVID-19 vaccinations administered in China from
December 16, 2020 to March 21, 2021 were first used to create the time series plot.
Figure 1(a) shows the date on the horizontal axis and the number of new COVID-19
vaccinations each day on the vertical axis. The vaccination data clearly showed
sustainable growth, which is typical of a nonstationary series.
[Figure 1 about here] Time series plots of the COVID-19 vaccination data in
China. (a) Time series plot of the original COVID-19 vaccination data, (b) Time
series plot of the COVID-19 vaccination data after first-order difference, (c) Time
series plot of the COVID-19 vaccination data after second-order difference
12
The original time series was nonstationary, which required the operation of the
difference to be stationary. It was found that after trial, the first- and second-order
differences could remove the trend of the original series, making it stationary. The
time series plots after the difference are shown in Figures 2(b) and 2(c). In addition,
after k=3, the autocorrelation and partial autocorrelation coefficients gradually
decreased and fell into the confidence interval, indicating that the current time series
turned stationary (Figure 2(c)).
[Figure 2 about here] Plots of autocorrelation and partial autocorrelation before
and after difference. (a) Plots of autocorrelation and partial autocorrelation for the
original series, (b) Plots of autocorrelation and partial autocorrelation after the
first-order difference, (c) Plots of autocorrelation and partial autocorrelation after the
second-order difference.
The unit root (ADF) method was then adopted for the stationarity test. As shown in
Table 1, the t statistic of the ADF test targeting the time series data of administered
COVID-19 vaccinations in China was 0.519; p was 0.985; and 1%, 5%, and 10%
thresholds were −3.503, −2.893, and −2.584, respectively. Moreover, p = 0.985 > 0.1,
so the original hypothesis could not be rejected, and the series was not stationery. The
first-order difference of the series was then performed, followed by the ADF test,
resulting in p = 0.845 > 0.1. Therefore, the original hypothesis still could not be
rejected, indicating a nonstationary series. The second-order difference was then
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performed, followed by ADF test, resulting in p = 0.000 < 0.01, indicating 99% or
higher chance of rejecting the original hypothesis. At this moment, the series was
stationary. Therefore, the parameter d of the ARIMA model was 2.
[Table 1 about here] Table 1. ADF test results
Model recognition and order determination
According to the ACF and PACF plots of given in Figure 2(c), the
autocorrelation plot after the second-order difference showed a significantly nonzero
autocorrelation number of 2 or 3 with the trailing property, so q = 2 or 3. In addition,
the partial autocorrelation plot showed a PACF of 2 or 3; therefore, p = 2 or 3. From
the ADF test results, when the times of difference was 2, the series was stationary, so
d value was 2. The preliminary models were thus ARIMA(2, 2, 2), ARIMA(2, 2, 3),
ARIMA(3, 2, 2), and ARIMA(3, 2, 3). Parameter estimation and model validation
were then performed to further determine the optimum model. The AIC, BIC, and
Hannan-Quinn information criterion (HQIC) were used to select the best p and q
values. The equation of AIC and BIC is as follows:
(, )= ln
+ 2( + + 1)/ (14)
If (, )= min
, (, ), the order of the ARMA model is decided as (p,
q), and
is the maximum likelihood estimate of the corresponding sequence.
Therefore, when selecting the parameters for ARMA(p, q), AIC and BIC were
combined, and the relatively optimum model was searched according to the
minimization principle of AIC, BIC, and HQIC (Burnham & Anderson, 2004). From
the lowest order, the AIC, BIC, and HQIC values of the preliminary models were
14
calculated, and the AIC thermodynamic diagram was created so as to determine the
orders.
[Table 2 about here] Table 2. AIC, BIC, and HQIC values of the ARMA model
As indicated in Table 2, the parameters of p and q were selected, which
corresponded to the minimum AIC, BIC, and HQIC values. The research results
showed that ARMA(3, 3) model had the lowest AIC, BIC, and HQIC values, which
were 2316.015, 2336.361, and 2324.233, respectively. Similarly, from the
order-determining AIC thermodynamic diagram (Figure 3), we noted that the
optimum model occurred when (p, q) values were (3, 3). Therefore, when p = 3 and q
= 3, the AIC, BIC, and HQIC values reached the minima. Considering that the times
of difference was d = 2, ARIMA(3, 2, 3) was determined to be a better model for
forecasting the number of COVID-19 vaccinations, which can be expressed as
follows:
= 2829.663 + 0.374 ∗ − 0.217 ∗ + 0.388 ∗ − 0.504 ∗ +
0.585 ∗ − 0.944 ∗ (15)
[Figure 3 about here] AIC thermodynamic diagram.
Model validation
After model establishment, the residual sequence needs to be checked to determine
if it only contains white noise. If not, it indicates that there still exists useful
information in the residual, demanding further revision of the model until the residual
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sequence is a white noise sequence. The validation process was as follows. First,
when the size of the fitting model was determined, the significance of the model was
assessed (α = 0.05). It was p = 0.14, much larger than 0.05. Therefore, the original
hypothesis was acknowledged, assuming that the time series was a white noise
sequence, i.e., a randomly generated series with no temporal correlation. Second, the
autocorrelation plot of the residual was created, and the pure randomness of the
residual was tested. The results of the residual autocorrelation test of ARIMA(3, 2, 3)
are shown in Figure 4(b), where the ACF is basically in the 95% confidence interval,
and the residual is white noise. Finally, the Q-Q plot of the ARIMA model was drawn
(Figure 4(c)). The points in the middle are almost overlapping with the straight line,
proving that the residual of the forecast model ARIMA(3, 2, 3) followed the normal
distribution, and thus, the established ARIMA(3, 2, 3) was reasonable.
[Figure 4 about here] Results of the ARIMA model validation. (a) Plot of residual
sequence, (b) Autocorrelation and partial autocorrelation plots of the residual
sequence, (c) Q-Q plot of the standardized residual for the ARIMA model.
Model forecast
The forecast ( ) function from the StatsModels library of Python was used for
constructing the model combining the values of p, d, and q. Specifically, the
ARIMA(3, 2, 3) model for forecasting COVID-19 vaccinations was established and
five-day projections were made. The modeling code is as follows:
16
data=data.astype('float64')
model=ARIMA(data,order=(p,d,q)).fit()
predict_data=model.predict()
forecast=pd.Series([int(each) for each in model.forecast(5)[0]])
dates=pd.date_range(final_data['date'].values[-1],periods=6)[1:]
forecast.index=dates
The predicted values, actual values, absolute errors, relative errors, and variation
coefficients are given in Table 3. The predicted trend was observed to match with the
actual trend with a good fitting. The average absolute error was −148175.2, the
average relative error was 10.06%, and the average variation coefficient was 5.34%.
The forecast of the model was within an acceptable range. The formula for calculating
the average relative error is as follows:
=
∑|
|
(16)
The formula of the variation coefficient is as follows:
=
(17)
In Eq. (17), represents the standard deviation and represents the mean.
[Table 3 about here] Table 3. Predicted results by the ARIMA(3, 2, 3) model
Figure 5 shows that the five-day projections (March 22, 2021 to March 26, 2021)
match well with the actual data of COVID-19 vaccination administration, showing a
high accuracy of the forecast. In addition, the number of administered vaccinations
shows a steady upward trend.
[Figure 5 about here] Performance of COVID-19 vaccination forecast.
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Discussion
The data of COVID-19 vaccinations daily administered in China from December
16, 2020 to March 21, 2021 were selected for a time series analysis and fitting of the
optimal model of ARIMA(3, 2, 3). The model was then used to predict the number of
five-day vaccinations from March 22 to March 26, 2021, the results of which were
compared with the actual data. The average relative error was shown to be 10.06%,
whereas the average variation coefficient was 5.34%. The analysis showed that the
predicted COVID-19 vaccination uptake supplemented well with the actual values
with a small relative error. This indicated that the ARIMA(3, 2, 3) model could be
used to forecast the number of COVID-19 vaccinations in China. The ARIMA model
performs statistical analysis based on historic data. However, the vaccination speed
and number were influenced by other factors such as the relevant policies and vaccine
production scales. Historically, vaccine effectiveness often drops when the vaccines
are ultimately rolled out into the general population because clinical trials occur under
controlled and ideal conditions and may not capture all heterogeneity within the
population (Fedson, 1998). This would raise the question of whether to vaccinate with
currently available vaccines or wait for one with a higher effectiveness against that
particular variant Therefore, the ARIMA model was only applicable to short-term
vaccination forecasts (Baden et al., 2021; Polack et al., 2021). For more accurate
prediction, the inclusion of new data and adjustment of the model parameters are
needed to adapt to the actual vaccination scenario.
18
Conclusion
In summary, the ARIMA(3, 2, 3) model has demonstrated high goodness of fit for
forecasting the COVID-19 vaccinations in China, which can be used in the short-term
prediction of the monitored COVID-19 vaccination data sequence, providing a
reference for the establishment of immune barrier and the supply of vaccines in
China.
List of abbreviations
autocorrelation function (ACF)
augmented Dickey-Fuller (ADF)
Akaike information criterion (AIC)
autoregressive (AR)
autoregressive integrated moving average (ARIMA)
Bayesian information criterion (BIC)
Hannan-Quinn information criterion (HQIC)
moving average (MA)
partial autocorrelation function (PACF)
19
Declarations
Ethics approval and consent to participate
Not applicable
Consent for publication
Not applicable
Availability of data and materials
The data that support the findings of this study are available from Our World in Data
but restrictions apply to the availability of these data, which were used under license
for the current study, and so are not publicly available. Data are however available
from the authors upon reasonable request and with permission of Our World in Data.
Competing interests
The authors declare that they have no competing interest.
Funding
This study was supported by the Introducing Talents Initiated Project of Southwestern
University of Finance and Economics (Grant No. 230600001002020015).
Authors' contributions
WD, FH, SX conceived and designed the experiments. WD analyzed the data. WD
wrote the paper. FH contributed to the revised manuscript. SX and YW collected the
data. All authors read and approved the final manuscript.
Acknowledgements
Not applicable
20
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22
Tables:
Table 1. ADF test results
Difference
order
t p
Threshold
1% 5% 10%
0 0.519 0.985 -3.503 -2.893 -2.584
1 -0.706 0.845 -3.503 -2.893 -2.584
2 -5.54 0 -3.503 -2.893 -2.584
Table 2. AIC, BIC, and HQIC values of the ARMA model
Model AIC BIC HQIC
ARMA(3,3) 2316.015 2336.361 2324.233
ARMA(2,2) 2319.739 2334.999 2325.903
ARMA(2,3) 2320.167 2337.97 2327.358
ARMA(3,2) 2325.573 2343.377 2332.765
Table 3. Predicted results by the ARIMA(3, 2, 3) model
Date Predict
value
Actual
value
Absolute
error
Relative
error%
Variation
coefficient%
20210322 2010823 1973905 36918 1.87 0.93
23
20210323 2248285 2077238 171047 8.23 3.95
20210324 2418990 2270243 148747 6.55 3.17
20210325 2527634 2816476 -288842 10.25 5.40
20210326 2645064 3453810 -808746 23.41 13.26
Note: Absolute error = Actual value－Predict value, whereas Relative error=Absolute
error/Actual value
Figure 1
Time series plots of the COVID-19 vaccination data in China. (a) Time series plot
of the original COVID-19 vaccination data, (b) Time series plot of the COVID-19
vaccination data after first-order difference, (c) Time series plot of the COVID-19
vaccination data after second-order difference
(a)
24
(b)
(c)
Figure 2
Plots of autocorrelation and partial autocorrelation before and after difference.
(a) Plots of autocorrelation and partial autocorrelation for the original series, (b) Plots
of autocorrelation and partial autocorrelation after the first-order difference, (c) Plots
of autocorrelation and partial autocorrelation after the second-order difference.
25
(a) (b)
(c)
Figure 3
AIC thermodynamic diagram.
26
Figure 4
Results of the ARIMA model validation. (a) Plot of residual sequence, (b)
Autocorrelation and partial autocorrelation plots of the residual sequence, (c) Q-Q
plot of the standardized residual for the ARIMA model.
(a)
(b)
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(c)
Figure 5
Performance of COVID-19 vaccination forecast.
