



ORIGINAL ARTICLE 

Year : 2021  Volume
: 14
 Issue : 8  Page : 364374 

EWMA control chart based on its first hitting time and coronavirus alert levels for monitoring symmetric COVID19 cases
Areepong Yupaporn^{1}, Sunthornwat Rapin^{2}
^{1} Department of Applied Statistics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand ^{2} Industrial Technology Program, Faculty of Science and Technology, Pathumwan Institute of Technology, Bangkok 10330, Thailand
Date of Submission  20Apr2021 
Date of Decision  02Jul2021 
Date of Acceptance  06Jul2021 
Date of Web Publication  23Jul2021 
Correspondence Address: Sunthornwat Rapin Industrial Technology Program, Faculty of Science and Technology, Pathumwan Institute of Technology, Bangkok 10330 Thailand
Source of Support: The research was funding by King Mongkut’s University of Technology North Bangkok Contract no. KMUTNB61KNOW014, Conflict of Interest: None  3 
DOI: 10.4103/19957645.321613
Objective: To define the alert levels for the total number of COVID19 cases derived by using quantile functions to monitor COVID19 outbreaks via an exponentially weighted movingaverage (EWMA) control chart based on the first hitting time of the total number of COVID19 cases following a symmetric logistic growth curve. Methods: The cumulative distribution function of the time for the total number of COVID19 cases was used to construct a quantile function for classifying COVID19 alert levels. The EWMA control chart control limits for monitoring a COVID19 outbreak were formulated by applying the delta method and the sample mean and variance method. Samples were selected from countries and region including Thailand, Singapore, Vietnam, and Hong Kong to generate the total number of COVID19 cases from February 15, 2020 to December 16, 2020, all of which followed symmetric patterns. A comparison of the two methods was made by applying them to a EWMA control chart based on the first hitting time for monitoring the COVID19 outbreak in the sampled countries and region. Results: The optimal first hitting times for the EWMA control chart for monitoring COVID19 outbreaks in Thailand, Singapore, Vietnam, and Hong Kong were approximately 280, 208, 286, and 298 days, respectively. Conclusions: The findings show that the sample mean and variance method can detect the first hitting time better than the delta method. Moreover, the COVID19 alert levels can be defined into four stages for monitoring COVID19 situation, which help the authorities to enact policies that monitor, control, and protect the population from a COVID19 outbreak. Keywords: COVID19 alert levels; Symmetric pattern of the total number of COVID19 cases; Monitoring COVID19 situation; EWMA control chart
How to cite this article: Yupaporn A, Rapin S. EWMA control chart based on its first hitting time and coronavirus alert levels for monitoring symmetric COVID19 cases. Asian Pac J Trop Med 2021;14:36474 
How to cite this URL: Yupaporn A, Rapin S. EWMA control chart based on its first hitting time and coronavirus alert levels for monitoring symmetric COVID19 cases. Asian Pac J Trop Med [serial online] 2021 [cited 2023 Jan 27];14:36474. Available from: https://www.apjtm.org/text.asp?2021/14/8/364/321613 
1. Introduction   
Coronavirus disease 2019 (COVID19) is caused by a singlestranded RNA virus, SARSCOV2, which was firstly reported in Wuhan, Hubei, China, in the end of 2019. Common symptoms of COVID19 infection are fever, fatigue, body ache, dry cough, appetite loss, and loss of the sense of smell^{[1],[2]}. The COVID19 outbreak has affected the interaction and activities of people abiding by the strict measures to control COVID19 outbreaks, such as maskwearing, washing hands often, physical and social distancing, quarantining after traveling both between and within countries. COVID19 has caused a new norm in the human lifestyle and has affected the economic and social systems of most countries in the world.
Emerging technologies are being deployed to combat the COVID19 pandemic. For example, smartphone applications are being used for connecting infected patients and the management information systems of hospitals. The rapid deployment of applications for selfscreening, selfhealth checkups, and COVID19 hospital information systems, along with surveillance, testing, tracing, quarantining, and hospital management to combat the COVID19 outbreak have been forthcoming^{[3]}. Artificial intelligence is being used in systems to detect SARSCOV2, fever, and other suspected symptoms of COVID19. Telemedicine and mobile applications in response to the COVID19 situation have been deployed to enable consultations on medical conditions via video calls with the physicians. In addition, drones and robotics are being used in hazardous areas infected with SARSCOV2^{[4]}. The Internet of Things is being used for COVID19 patients to connect to hospital networks, while blockchain is being used for effective supply chain management. Moreover, cloud computing and big data have been deployed to store and provide data storage capacity for information about the COVID19 situation^{[5]}.
Plans and policies have been laid out to handle, control, and monitor the COVID19 outbreak. Recognizing the pattern of the total number of COVID19 cases is important for acting on the policies to control COVID19 outbreaks. The levels of COVID19 severities which are called the COVID19 alert levels have been defined by the authorities based on the COVID19 situation. Moreover, the statistical control chart has been applied to monitor and control the COVID19 outbreak. Thus, many researches have been conducted on the symmetric pattern of the total number of COVID19 cases, COVID19 alert levels, and statistical control charts for monitoring the COVID19 outbreak. For example, the total number of COVID19 cases in many countries, such as Thailand, Singapore, Sweden and Norway, follows a logistic growth curve, which is a symmetric pattern^{[6],[7]}. The number of cases relative to the maximum cumulative infected, the total confirmed, and the total recovered cases have been fitted to a logistic growth curve model for the COVID19 suppression dynamics for each region in China^{[8]}. Moreover, several countries and states have classified the COVID19 alert levels for monitoring the COVID19 situation and applied measures that people must abide by to protect the population from COVID19 outbreak spikes. For example, New Zealand classified COVID19 severity with a COVID19 alert system^{[9]}. The state of Alaska classified three statewide alert levels for evaluating the COVID19 transmission risk^{[10]}. The UK defined COVID19 alert levels to communicate the current risk of COVID19 transmission to the population^{[11]}.
Statistical control charts such as the exponentially weighted movingaverage (EWMA) control chart and the proportional control chart have been adopted to monitor COVID19 outbreaks. The EWMA control chart is being used for monitoring and producing warnings of COVID19 outbreaks in some major states in India^{[12]}, as well as to monitor and identify the control limits of the death variation of the COVID19 situation in Pakistan^{[13]}. The proportional control chart is being used to monitor the COVID19 positivity rate, COVID19 healing rate, and COVID19 mortality rate over time^{[14]}. Using quantitative methods to set the COVID19 alert levels is a useful approach for supporting the authorities when making decisions about launching measures to protect the population during a COVID19 outbreak. Therefore, the objective of the present research is to apply a quantitative and probability method (a quantile function) that is an inversion of the cumulative distribution function of the time over the total number of COVID19 cases to set the COVID19 alert levels when monitoring COVID19 outbreaks by using a EWMA control chart. The two methods for estimating the expected value and variance of the COVID19 cases as the population for the EWMA statistic and the results from the EWMA control chart based on the optimal first hitting time were evaluated and compared for COVID19 outbreaks in Thailand, Singapore, Vietnam, and Hong Kong where the pattern of the total number of COVID19 cases follows a logistic growth curve, which is a symmetric pattern.
2. Subjects and methods   
This section contains the data collection protocol, mathematical and statistical methods, probabilistic analysis, COVID19 alert levels, and the EWMA control chart for describing and monitoring a COVID19 outbreak.
2.1. Data collection
The data for this research is the total number of COVID19 cases over time. The total number of COVID19 cases, which are secondary data, were collected and downloaded from the website Worldometer^{[15]}. The total number of COVID19 cases was specifically selected as the sample because the theories and analyses reported herein that comprise the main purpose for this research are based on the symmetric growth curve. The collected data of the total number of COVID19 cases was complete with no missing values. The sampled countries or region for this research (Thailand, Singapore, Vietnam, and Hong Kong) were specifically selected because their total numbers of COVID19 cases follow a symmetric growth curve (the logistic growth curve), and are suitable for estimation based on the symmetric pattern of the total number of COVID19 cases^{[6]} for the duration of February 15, 2020, to December 16, 2020.
2.2. The model for describing the behavior of a COVID19 outbreak
The behavior of a COVID19 outbreak can be represented by a logistic differential equation of the total number of COVID19 cases^{[6],[7]}. The total number of COVID19 cases can be described by using a logistic growth curve that is the solution for the logistic differential equation. Let I (t) be the total cases at any time t≥0 and C be the carrying capacity of the environment. Thus, the logistic differential equation can be modeled as
where r is an intrinsic growth rate.
One of the variables in Equation (1), I (t), can be evaluated as a logistic growth curve (a symmetric growth curve) as follows:
where I_{0} is the initial total number of COVID19 cases.
The logistic growth curve in Equation (2) has an inflection point or median at C/2. Moreover, the parameters for the logistic growth curve consist of C, k and r. The leastsquareserror method can be adopted to estimate these parameters as follows:
where e (t) is the error between actual and estimated values at time t=0, 1, …, n and Î(t) is the estimated logistic growth curve for time t=0, 1, …, n. Hence, the partial derivative with respect to parameters C, k and r can be applied to respectively estimate these parameters as:
The parameters C, k and r can be solved from the system of Equation (3).
The accuracy of the estimation parameters and estimated logistic growth curve which represents the total number of COVID19 cases is measured as the Root Mean Square Percentage Error (RMSPE).
The explanation of the independent variable (time) in relation to the dependent variable (estimated logistic growth) can be measured by using the coefficient of determination (R^{2}).
2.3. The probability density function, expected value, and variance of the time for the total number of COVID19 cases
The cumulative distribution function (CDF: F) of the time for the total number of COVID19 cases is defined as:
The probability density function (PDF: f) of the time for the total number of COVID19 cases is a derivative of the CDF expressed as:
The quantile function (QF: Q_{p}) with probability p is defined as
Q_{p}= min{tϵ Real:F(t)≥p};pϵ(0,1)
In other words, Q_{p} gives the minimum value of time t where the cumulative distribution function exceeds the probability p.
Especially,
The expected value E(T) and variance Var (T) of the time (T) for the total number of COVID19 cases can be respectively calculated by using the moment generating function (m), Beta function (B), and Gamma function (Γ) as:
Next, the estimate expected value and estimate variance of the total COVID19 cases are evaluated.
2.4. Estimating the expected value and variance of the total number of COVID19 cases
These can be calculated via two methods.
2.4.1. The delta method^{[16]}
Obviously, the total number of COVID19 cases I is a function of time t, i.e., I=I(t). The respective estimated expected value and variance of random variable I based on a Taylor series expansion about t=E(T) are:
Therefore, the second order approximation for the expected value of I is:
The first order approximation for the variance of I is:
2.4.2. The sample mean and variance method
The expected value of I can be estimated by using the sample mean as:
The variance of I can be estimated by using the sample variance as:
2.5. The exponentially weighted moving average control chart for monitoring a COVID19 outbreak
The EWMA control chart^{[17],[18],[19]} is based on EWMA statistic z (t) for smoothing parameterλϵ (0, 1], which can be written as:
In this research, the EWMA control chart was applied for monitoring the total number of COVID19 cases. Thus, the assumption is that the I(t)'s are independent identically distributed random variables with expected value E(I(t))=E(I) and variance
Therefore, expected value E (z) of the EWMA statistic is:
E(z)=λE(l)+(1λ)E(z(t1))
E(z)=E(l)
and variance of the EWMA statistic is:
The control limits of the EWMA control chart consist of:
Upper control limit (UCL)=E (z)+dσ_{z}
Center line (CL)=E (z)
Lower control limit (LCL)=E (z)dσ_{z}
where d is the width of the control limit.
Moreover, the total number of COVID19 cases is estimated via logistic growth, which is a gradual increase in the first period, exponential growth in the middle period, and slow growth in the end period. To monitor the total number of COVID19 cases, the first hitting time (FHT), or first passage time^{[20],[21],[22]}, is the run length of the number of samples at which point the EWMA control chart gives the first signal over the upper control limit. The first hitting time of the EWMA control chart is defined as:
FHT=inf{t>0  z(t)>UCL}
The first hitting time for monitoring a COVID19 outbreak through the EWMA control chart should be as large as possible because it reveals the efficiency of controlling the COVID19 outbreak.
2.6. The COVID19 alert levels
Here, we present the COVID19 alert levels for monitoring a COVID19 outbreak. The COVID19 alert levels are important for the authorities to make decisions on monitoring, controlling, and protecting the population during a COVID19 outbreak. In this research, the COVID19 alert levels were classified into four levels. However, the total number of COVID19 cases for a logistic growth curve can be described by a symmetric growth curve.
[Figure 1] shows a plot of the quantile functions corresponding to the COVID19 alert levels. The total number of COVID19 cases starts increasing over the range (Q_{0.01}, Q_{0.25}) in alert level 4 (L_{4}) with very high outbreak severity. Subsequently, the total number of COVID19 cases exponentially increases over the range (Q_{0.25}, Q_{0.50}) in alert level 3 (L_{3}) with high outbreak severity. After the inflection point (median), the total number of COVID19 cases starts decreasing over the range (Q_{0.50}, Q_{0.75}) in alert level 2 (L_{2}) with medium outbreak severity. Following this, the total number of COVID19 cases significantly decreases and then becomes stable over the range (Q_{0.75}, Q_{0.99}) in alert level 1 (L_{1}) with low outbreak severity.  Figure 1: The COVID19 alert levels and a symmetric growth curve (logistic growth curve) of the total number of COVID19 cases.
Click here to view 
3. Results   
3.1. The COVID19 outbreak in Thailand
As shown in [Figure 2]A, the estimated logistic growth curve for the total number of COVID19 cases in Thailand from February 15, 2020, to December 16, 2020, is:  Figure 2: (A) Estimated total number of COVID19 cases and quantile function of the time for the total number of COVID19 cases in Thailand and (B) the probability density function and cumulative density function of the time for the total number of COVID19 cases in Thailand. The EWMA control chart for monitoring a COVID19 outbreak in Thailand: The expected value and variance estimated via (C) the delta method and (D) the sample mean and variance method. FHT: First hitting time; λ: Smoothing parameter. UCL: Upper control limit; CL: Center line; LCL: Lower control limit.
Click here to view 
and the quantile functions of the total number of COVID19 cases are 0.96, 37.44, 48.91, 60.36 and 96.84.
As shown in [Figure 2]B, the PDF of the time for the total number of COVID19 cases in Thailand is positively skewed. The probability of the total number of COVID19 cases in Thailand from February 15, 2020, to December 16, 2020, reached the maximum value at around day 50 and approached zero at around day 120. The expected value and variance of the times for the COVID19 outbreak in Thailand are 319.08 and 1.52e+04, respectively. The sample value and variance of the COVID19 outbreak in Thailand are 2.89e+03 and 1.38+06, respectively.
As shown in [Figure 2], the EWMA control chart was applied for monitoring the total number of COVID19 cases in Thailand from February 15, 2020, to December 16, 2020. The results from the experiment, the first hitting time based on the delta method (method 1) was 0 for all values of the smoothing parameter [Figure 2]C whereas the sample mean and variance method (method 2) [Figure 2]D for smoothing parameter λ=0.3 and d=2 more effectively detected the first hitting time (280 days) than with the other smoothing parameter values. Hence, the optimal first hitting time on the EWMA control chart for monitoring a COVID19 outbreak in Thailand is approximately 280 days for λ=0.3 and d=2.
3.2. The COVID19 outbreak in Singapore
As shown in [Figure 3]A, the estimated logistic growth curve for the total number of COVID19 cases in Singapore from February 15, 2020, to December 16, 2020, is:  Figure 3: (A) Estimated total number of COVID19 cases and quantile function of the time for the total number of COVID19 cases in Singapore and (B) the probability density function and cumulative density function of the time for the total number of COVID19 cases in Singapore. The EWMA control chart for monitoring a COVID19 outbreak in Singapore: The expected value and variance estimated via (C) the delta method and (D) the sample mean and variance method. FHT: First hitting time; λ: Smoothing parameter. UCL: Upper control limit; CL: Center line; LCL: Lower control limit.
Click here to view 
and the quantile functions of the total number of COVID19 cases are: 8.71, 74.25, 100.32, 126.39 and 209.35.
As shown in [Figure 3]B, the PDF of the total number of COVID19 cases in Singapore is positively skewed. The probability of the total number of COVID19 cases in Singapore from February 15, 2020, to December 16, 2020, reached the maximum value at around day 100 and approached zero at around day 290. The expected value and variance of the times for the COVID19 outbreak in Singapore are 200.57, and 7.40e+03, respectively. The sample value and variance for the COVID19 outbreak in Singapore are 3.79e+04, and 5.13e+08, respectively.
As shown in [Figure 3], the EWMA control chart was applied for monitoring the total number of COVID19 cases in Singapore from February 15, 2020, to December 16, 2020. The results from the experiment, the first hitting time based on the delta method (method 1) was 0 for all values of the smoothing parameter [Figure 3]C whereas the sample mean and variance method (method 2) [Figure 3]D for smoothing parameter λ=0.3 and d=2 more effectively detected the first hitting time (208 days) than for the other smoothing parameter values. Thus, the optimal first hitting time on the EWMA control chart for monitoring the COVID19 outbreak in Singapore is approximately 208 days for λ=0.3 and d=2.
3.3. The COVID19 outbreak in Vietnam
As shown in [Figure 4]A, the estimated logistic growth curve for the total number of COVID19 cases in Vietnam from February 15, 2020 to December 16, 2020, is:  Figure 4: (A) Estimated total number of COVID19 cases and its quantile function in Vietnam and (B) The probability density function and cumulative density function of the time for the total number of COVID19 cases in Vietnam. The EWMA control chart for monitoring a COVID19 outbreak in Vietnam: The expected value and variance estimated via (C) the delta method and (D) the sample mean and variance method. FHT: First hitting time; λ: Smoothing parameter. UCL: Upper control limit; CL: Center line; LCL: Lower control limit.
Click here to view 
and the quantile functions of the total number of COVID19 cases are: 66.42, 119.69, 178.16, 236.63 and 422.74.
As shown in [Figure 4]B, the PDF of the total number of COVID19 cases in Vietnam is quite symmetric. The probability of the total number of COVID19 cases in Vietnam from February 15, 2020, to December 16, 2020, reached the maximum value at around day 170. The expected value and variance of the time for the total number of COVID19 cases in Vietnam are 240.17 and 1.69e+04, respectively. The sample value and variance of the COVID19 outbreak in Vietnam are 646.02, and 2.11e+05, respectively.
As shown in [Figure 4], the EWMA control chart was applied for monitoring the total number of COVID19 cases in Vietnam from February 15, 2020, to December 16, 2020. The results from the experiment, the first hitting time based on the delta method (method 1) was 0 for all values of the smoothing parameter [Figure 4]C whereas the sample mean and variance method (method 2) [Figure 4]D detected first hitting times of 198, 286, and 0 for λ=0.7 and d=2, respectively. For smoothing parameter λ=0.3, 0.7 and 0.9, the sample mean and variance method more effectively detected the first hitting time (286 days) than for the other smoothing parameter values. Hence, the optimal first hitting time on the EWMA control chart for monitoring the COVID19 outbreak in Vietnam is approximately 286 days for λ=0.7 and d=2.
3.4. The COVID19 outbreak in Hong Kong
As shown in [Figure 5]A, the estimated logistic growth curve for the total number of COVID19 cases in Hong Kong from February 15, 2020, to December 16, 2020, is:  Figure 5: (A) Estimated total number of COVID19 cases and its quantile function in Hong Kong and (B) the probability density function (PDF) and cumulative density function (CDF) of the time for the total number of COVID19 cases in Hong Kong. The EWMA control chart for monitoring a COVID19 outbreak in Hong Kong: The expected value and variance estimated via (C) the delta method and (D) the sample mean and variance method. FHT: First hitting time; λ: Smoothing parameter. UCL: Upper control limit; CL: Center line; LCL: Lower control limit.
Click here to view 
and the quantile functions of the total number of COVID19 cases are: 21.24, 127.04, 173.63, 220.22 and 368.50.
As shown in [Figure 5]B, the probability density function of the total number of COVID19 cases in Hong Kong is quite symmetric. The probability of the total number of COVID19 cases in Hong Kong from February 15, 2020, to December 16, 2020, reached the maximum value at around day 160. The expected value and variance of the time for the COVID19 outbreak in Hong Kong are 319.08, and 1.52e+04, respectively. The sample value and variance of the COVID19 outbreak in Hong Kong are 2.91e+03, and 5.12e+06, respectively.
As shown in [Figure 5], the EWMA control chart was applied for monitoring the total number of COVID19 cases in Hong Kong from February 15, 2020, to December 16, 2020. The results from the experiment, the first hitting time based on the delta method (method 1) was 0 for all values of the smoothing parameter [Figure 5]C whereas the sample mean and variance method (method 2) [Figure 5]D detected first hitting times of 202, 290, and 298 for λ=0.3, 0.7 and 0.9, respectively. For smoothing parameter λ=0.9 and d=2, the sample mean and variance method more effectively detected the first hitting time (298 days) than for the other smoothing parameter values. Thus, the optimal first hitting time on the EWMA control chart for monitoring the COVID19 outbreak in Hong Kong is approximately 298 days for λ=0.9 and d=2.
Here, we compare, discuss, and suggest applying the EWMA control chart and parameter estimation for monitoring symmetric patterns of the total number of COVID19 cases for the sampled countries and region: Thailand, Singapore, Vietnam, and Hong Kong.
[Table 1] reports the estimated parameters, EWMA control chart, and the COVID19 alert levels for monitoring COVID19 outbreaks to illustrate the levels of severity in each country. The delta model for estimating the expected value and the variance of the total number of COVID19 cases could not detect the first hitting time on the EWMA control chart. However, the mean and variance model with the optimal smoothing parameter for the EWMA control chart of the total number of COVID19 cases could detect the following first hitting times: 280 days with λ=0.3 for the COVID19 outbreak in Thailand, 208 with λ=0.3 for the COVID19 outbreak in Singapore, 286 days with λ=0.7 for the COVID19 outbreak in Vietnam, and 298 days with λ=0.9 for the COVID19 outbreak in Hong Kong. Furthermore, when using smoothing parameter λ=0.3 with the EWMA control chart for monitoring the COVID19 outbreaks, the maximum first hitting times were 280, 208, 202, and 198 days in Thailand, Singapore, Hong Kong, and Vietnam, respectively. These results imply that Thailand can more effectively hold the COVID19 outbreak time before breaking the upper control limit of the EWMA control chart and thus better protect its population from COVID19 outbreaks compared to the other countries or region. Moreover, the total number of COVID19 cases escalate rapidly in COVID19 alert level 4, thus signifying that policies for controlling a COVID19 outbreak, such as lockdown, social distancing, etc., should be strictly encouraged. COVID19 total cases are exponentially increasing for alert level 3, which suggests that the policies mentioned for level 4 should be strictly maintained to suppress the total number of COVID19 cases. For alert level 2, the COVID19 total cases are starting to gradually decrease after having passed the inflection point at Q_{0.05}. However, applying the policies mentioned previously should still be strictly applied to suppress the total number of COVID19 cases. For alert level 1, the total number of COVID19 cases is low or almost nonexistent, and so considerations in adjusting public health and social measures are recommended.  Table 1: Comparison of the COVID19 outbreaks in sampled countries and region.
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4. Discussion   
The scope of this research was the study of probabilistic methods for the symmetric pattern of the total number of COVID19 cases. The findings of this research are restricted to investigating the PDF of the symmetric pattern of the total number of COVID19 cases. The samples are from only four countries or region: Thailand, Singapore, Vietnam, and Hong Kong. The outbreak time range was restricted to the period of February 15, 2020, to December 16, 2020. The approaches outlined in this research could be applied to symmetric patterns of the total number of COVID19 cases in other countries. Moreover, we only used the EWMA control chart for monitoring the COVID19 outbreaks. The application of the approach in this research on COVID19 outbreaks in other countries has been left for the future. Asymmetric patterns for the total number of COVID19 cases could be forecasted and estimated by using the PDF. Furthermore, other control charts such as the modified EWMA and cumulative sum, etc., for monitoring COVID19 outbreaks could be applied to classify the COVID19 alert levels.
Probabilistic methods such as the PDF, quantile functions, and optimal first hitting times were used to set COVID19 alert levels for the total number of COVID19 cases with a symmetric pattern. The EWMA control chart was adopted to monitor and set the alert levels for COVID19 outbreaks. Symmetric patterns of the total number of COVID19 cases in Thailand, Singapore, Vietnam, and Hong Kong, were investigated. The sample mean and variance method and the delta method were used for estimating the mean and variance to detect the first hitting time on the EWMA control chart, of which the former method detected the first hitting time better than the latter. Four COVID19 alert levels were defined for monitoring COVID19 outbreaks based on quantile functions: (Q_{0.01}, Q_{0.25}) for level 4, (Q_{0.25}, Q_{0.50}) for level 3, (Q_{0.50}, Q_{0.75}) for level 2, and (Q_{0.75}, Q_{0.99}) for level 1. For example, the quantile functions for Thailand were Q_{0.01}=0.96, Q_{0.25}=37.44, Q_{0.50}=48.91, Q_{0.75}=60.36, and Q_{0.99}=96.84. However, for this research, the delta method which was used for estimating the mean and variance to compare the sample mean and variance method. The sample countries for this research: Thailand, Singapore, Vietnam, and Hong Kong, the delta method could not detect the first hitting time on the EWMA control chart but the delta method could detect the first hitting time on the EWMA control chart for the other countries.
The COVID19 alert levels identified by using the probabilistic methods mentioned earlier are to help the authorities to enact policies that monitor, control, and protect the population from a COVID19 outbreak. For example, a strict lockdown policy should be employed for a COVID19 alert level 4 situation to flatten the total number of COVID19 cases following a symmetric logistic growth curve while relaxation of the policy should be applied for a COVID19 alert level 1 situation. Moreover, a new COVID19 wave can be identified by the EWMA control chart when the monitoring shows that the infection rate is outofcontrol, i.e., exceeds the upper control limit.
Conflict of interest statement
We declare that we have no conflict of interest.
Funding
The research was funding by King Mongkut’s University of Technology North Bangkok Contract no. KMUTNB61KNOW014.
Authors’ contributions
All authors substantially contributed to drafting, revising the article as well as the final version of the manuscript. R.S. contributed to the conception and design of the study and the theoretical proof. Y.A. provided data and performed the data analysis. Both R.S. and Y.A. authors contributed to analyze, interpret, and summarize the results of the study.
Supplementary Materials [Additional file 1]
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[Figure 1], [Figure 2], [Figure 3], [Figure 4], [Figure 5]
[Table 1]
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