Monitoring scheme for early detection of coronavirus and other respiratory virus outbreaks.

In December 2019, an outbreak of pneumonia caused by a novel coronavirus (severe acute respiratory syndrome coronavirus 2 [SARS-CoV-2]) began in Wuhan, China. SARS-CoV-2 exhibited efficient person-to-person transmission of what became labeled as COVID-19. It has spread worldwide with over 83,000,000 infected cases and more than 1,800,000 deaths to date (December 31, 2020). This research proposes a statistical monitoring scheme in which an optimized np control chart is utilized by sentinel metropolitan airports worldwide for early detection of coronavirus and other respiratory virus outbreaks. The sample size of this chart is optimized to ensure the best overall performance for detecting a wide range of shifts in the infection rate, based on the available resources, such as the inspection rate and the allowable false alarm rate. The effectiveness of the proposed optimized np chart is compared with that of the traditional np chart with a predetermined sample size under both sampling inspection and 100% inspection. For a variety of scenarios including a real case, the optimized np control chart is found to substantially outperform its traditional counterpart in terms of the average number of infections. Therefore, this control chart has potential to be an effective tool for early detection of respiratory virus outbreaks, promoting early outbreak investigation and mitigation.

Introduction

Background

In December 2019, an outbreak of mysterious pneumonia from an unidentified origin occurred in Wuhan, China. Chinese health authorities identified a novel coronavirus (severe acute respiratory syndrome coronavirus 2 [SARS-CoV-2]) that was responsible for the outbreak (World Health Organization, 2020b). Coronaviruses are a large family of viruses that cause illnesses ranging from the common cold to more severe diseases, such as middle east respiratory syndrome (MERS-CoV) and severe acute respiratory syndrome (SARS-CoV). SARS-CoV-2 exhibited efficient person-to-person transmission of what became labeled as coronavirus disease 2019 (COVID-19), which quickly led to a worldwide outbreak of potentially fatal viral pneumonia. COVID-19 has spread around the world with over 83,000,000 infected cases and more than 1,800,000 deaths to date (December 31, 2020), and further dissemination through air travel is likely (Goscé, Phillips, Spinola, Gupta, & Abubakar, 2020). As a result, the World Health Organization (WHO) declared COVID-19 a pandemic on March 11, 2020 (World Health Organization, 2020a). A timeline of crucial early events related to SARS-CoV-2 is shown in Fig. 1 (Health, 2020, National Health Commission of China, 2020).

Fig. 1

A timeline of early stages of the COVID-19 outbreak.

The attack rate (i.e., how rapidly the disease is spreading) of a virus is indicated by its reproductive number (R0). A recent study estimated the R0 for COVID-19 to be between 2.24 and 3.58 (Zhao et al., 2020). Per this estimate, on average, every case of COVID-19 would create two to three new cases, exhibiting 2 to 3 times greater transmissibility than seasonal influenza viruses (Zhang et al., 2017). Furthermore, the mortality rate of COVID-19 is currently estimated at around 3% (Wang, Horby, Hayden, & Gao, 2020). For comparison, the mortality rate of seasonal flu is less than 0.1% (Centers of Disease Control and Prevention, 2019), but the mortality rate is approximately 10% for SARS-CoV and 34% for MERS (Jiang, Rayner, & Luo, 2020). Early detection and response to epidemics and pandemics, including quarantine of patients with confirmed infections and observation of those who have had close contact with infected patients, can help to mitigate outbreaks, lowering the attack rate and the total number of deaths (Bauer, 2015). In this analysis, we present how early detection of important respiratory virus outbreaks could be achieved through use of an optimized np control chart at a worldwide network of sentinel airports to improve the quality of surveillance.

Statistical process control

Using effective surveillance tools is essential for the early detection of outbreaks of coronaviruses and other respiratory viruses. When used for outbreak detection, statistical process control (SPC) charts have been proven to be effective, easy to implement, and inexpensive (Wiemken et al., 2017, Woodall, 2006). SPC charts were originally developed by Walter Shewhart in the 1920s for monitoring production processes (Montgomery, 2019). Since the early 1990s, there has been a growing interest in applying these charts to healthcare (Ahamed et al., 2018, Lawson et al., 2012), including those related to the detection and monitoring of outbreaks (e.g., Baker et al., 2018, Shu et al., 2014, Sogandi et al., 2019, Xie et al., 2010). Sogandi et al. (2019) proposed a Bernoulli state-space model for monitoring multi-stage medical processes. The proposed model performed well under different shifts and was able to identify the out-of-control stage efficiently. Grigg (2019) discussed the problem of maintaining patient ordering according to the treatment timeline for different charts. They recommended that compromising on the fullness of presentation of the historical data is the best way to preserve patient ordering on any chart. Gould and Wang (2017) proposed an effective method for routine monitoring of safety information for programs that include blinded trials. A comprehensive literature review of the various applications of SPC charts in healthcare can be found in Suman and Prajapati, 2018, Tennant et al., 2007, and May, Simpson, Hart, Rowett, and Perrier (2009).

A control chart is a visual tool that can provide early identification of statistically significant changes in data. For effective process monitoring, several studies have proposed to optimize the parameters of different types of charts in various applications. For instance, Rahim and Sultan Khalaf (1997) presented an optimal design of exponentially weighted moving average (EWMA) chart parameters using genetic algorithms. The results showed that the optimal design reduces the false alarm probability (i.e., the probability that the control chart gives an out-of-control signal, while the process is actually in control) and is powerful in detecting serious shifts. Haridy, Wu, Khoo, and Yu (2012) proposed an algorithm for the optimal design of a Syn-np chart which combines the synthetic chart and the np chart. The proposed chart was more effective than the np chart by 73% and the synthetic by 31%. Shamsuzzaman, Khoo, Haridy, and Alsyouf (2016) proposed an optimization design of the combined Shewhart chart and EWMA chart. The charting parameters and the allocation of detection power between the elements of both charts were optimized based on the loss function. Muhammad, Yeong, Chong, Lim, and Khoo (2018) developed an algorithm for the optimization of coefficient of variation (CV) control chart. The results revealed that the proposed optimized CV chart outperforms five existing CV charts in literature in almost all scenarios.

For the early detection of outbreaks of coronaviruses and other respiratory viruses, this study proposes a monitoring scheme that utilizes an attribute chart — namely, the np chart — with optimized parameters. Furthermore, we introduce the average number of infections (ANI) as an effective measure of the overall detection speed of the control chart for monitoring the infection rate. Finally, the use of the proposed monitoring scheme is illustrated by different scenarios.

The proposed monitoring scheme, which is shown in Fig. 2 , can be used as a monitoring tool at selected metropolitan airports where checkpoints are established. Sampling is actually a common practice in airports when 100% is impossible due to the limited resources (Bauer, 2015, Civil Aviation Authority, 2017). By the proposed scheme, airports would screen passengers to detect fevers potentially related to respiratory viruses. A typical screening procedure would include thermal screening by measuring the skin temperature using various tools, such as thermal cameras, thermal imaging, or forehead thermometer guns (Air Technology, 2020). Alternatively, if available, airports could use automated temperature screenings using artificial intelligence (GovInsider, 2020). Surveillance for other symptoms, such as cough or shortness of breath, might also be incorporated into the model to determine the likelihood of respiratory virus infection. If outbreak onset is already known to have occurred, it would be important to increase the inspection rate (i.e., increase the sample size n and decrease the sampling interval h) if 100% inspection is impossible so that a greater percentage of infected passengers could be evaluated. However, the proposed monitoring scheme can also be used continuously where either rational subgrouping or 100% inspection is adopted (Air Technology, 2020).

Fig. 2

The proposed monitoring scheme.

Implementation and design of the np chart

The np control chart is an attribute chart that can be used to monitor the number of infections (d) found in a sample of arriving passengers, which is assumed to follow a binomial distribution. The process being monitored is considered to be in control if d satisfies LCL ≤ d ≤ UCL, where LCL and UCL are the lower control limits and upper control limits of the np chart. In other words, if d ≤ LCL, then a downward p shift will be signaled and if d ≥ UCL, then an upward p shift will be signaled. However, this analysis focuses on designing a one-sided upper control chart that only detects an increase in the infection rate since decreasing the number of infections where d ≤  is the desirable target. The np control chart process monitoring is implemented as follows:

A sample of n passengers is taken at the end of each sampling interval h and the number of infections, d, is counted for this sample. The policymakers could then make real-time decisions on managing the infected cases, such as quarantine.

The resulting d is plotted for each sample on the np chart.

If d ≥ UCL, then a potential outbreak is declared and in this case, a 100% inspection is recommended. Otherwise, the process is in control, and step 1 is repeated for the next sample.

The charting parameters (i.e., n, h and UCL) need to be decided in an optimal and effective way. With the aim of carrying out the optimal design for the np chart, several specifications need to be set. The design specifications of this study are summarized below:

is the in-control infection rate;

τ is the allowable minimum value of the in-control average time to signal ();

r is the inspection rate; and

is the maximum out-of-control infection rate

The abovementioned specifications are commonly used to design attribute control charts (Bourke, 1991, Gan, 1993, Haridy, Rahim, Selim, Wu, & Benneyan, 2017, Reynolds & Stoumbos, 1998, Wu et al., 2004). The in-control infection rate is assumed to be known, as it is considered to be the baseline or expected infection rate that does not require investigation. In addition, the value of τ is set based on the requirements of the false alarm rate that is deemed to be acceptable and can be managed by airports. The sampling rate r is determined according to the availability of resources such as manpower and inspection tools (e.g., thermometers and testing kits). The maximum out-of-control infection rate, , is decided based on the shift size the authority is interested to detect.

The measure of performance

The performance of a control chart is often evaluated using different measures of performance. A measure called the average time to signal (ATS), which is the expected time from when a shift with a particular size occurred until the control chart indicates an out-of-control signal (i.e., outbreak), is usually recommended (Li, Zou, Gong, & Wang, 2014). Nevertheless, it is not easy to predict the size of an outbreak. Therefore, in this study, we introduce a performance measure that is the average number of infections (ANI) to evaluate the overall performance of the proposed np chart over a wide range of shifts in the infection rate. The ANI is actually a weighted average of the out-of-control ATS values over different shifts in an infection rate, hence it is a better measure for the overall performance of a control chart.

When an increasing shift in the infection rate occurs, the infection rate will change from p 0 to p. The ANI is the average number of infections that occurred over a shift range of prior to control chart detection. The infection rate is considered to be in control when p = p 0 and out of control when with a maximum infection rate at p = p.

If N is the number of arrivals per unit time and ATS(p) is the out-of-control ATS value that corresponds to a particular infection rate p, then the ANI produced by a control chart across the p range () can be calculated as follows:where f(p) is the probability density function of p which is assumed to follow uniform distribution in this research and can be estimated as follows:

The out-of-control ATS(p) at a particular infection rate p can be calculated as follows:

N in Eq. (1) is assumed to be constant. As a result, it may be removed while not affecting the optimization design and comparative study.

The optimization model

This section presents the optimization model and algorithm for the np chart. The optimal design is carried out based on the four design specifications listed in Section 2. The optimization procedure to compute the optimal parameters of the np chart in minimizing ANI as the objective function is given as follows: where n is the independent variable, while the h and UCL are the dependent variables on the n, r and specified value of τ, respectively. The above-mentioned model will provide the optimal values of n, h and UCL that will minimize ANI over a shift range of (), and meanwhile, ensure that the in-control ATS 0 is greater than or equal to a predefined value of τ. The ATS 0 represents the expected time the control chart takes to give a false alarm signal. The ATS 0 of the np chart can be calculated as follows:where α is the probability that the np chart gives an out-of-control signal when the infection rate is actually in-control. α can be determined as follows:

The optimization design of np chart is implemented as follows:

Specify the design specifications p 0, τ , r and p .

Initiate ANI min variable to store the minimum value of ANI and set the initial value of ANI min to very large number.

Search the optimal value of n, starting with n = 1 and increase its value in an increment of 1.

For each n, find h (=n / r) that satisfies constraint (5).

For each pair of (n, h), find α using Eq. (6) where ATS 0 = τ (i.e., α = h / τ) and then the value of UCL using Eq. (7) so that constraint (4) can be fulfilled.

For the identified n, h and UCL, find the corresponding value of ANI using Eq. (1).

If the calculated ANI is less than the current ANI, replace the latter by the former and the current values of n, h and UCL are stored as temporary optimal solution.

For each trail n value, step 4 will be repeated until ANI cannot be further minimized. The optimization algorithm is terminated if the ANI keeps increasing for 60 consecutive iterations. The optimal np charting parameters n, h and UCL will be the values that produce the minimum ANI, while satisfying constraints (4), (5).

The optimization algorithm of the np chart is summarized as shown in Fig. 3 .

Fig. 3

The optimization algorithm of the np chart.

The above search mechanism is reliable and straightforward as the only independent design variables, n, is integral, and therefore all its possible values can be examined. It can complete the optimization design of the np chart in a few seconds of CPU time on a personal computer. In addition, the results can be used to study the effect of the sample size on the performance of the np chart. C programming language was used to code the design algorithm of the np chart. It can be obtained from authors upon request.

Comparative studies

This section shows the results of optimizing the charting parameters of the np chart, including n, h and UCL. It also conducts a comparative study between the optimized np chart and the traditional np chart for one real case and five simulated scenarios. The optimized np chart proposed in this study is named as npoptimal chart whereas, the traditional np chart is referred to as the nptraditional chart. Both ANI and ATS are used as measures of performance to compare the npoptimal and nptraditional charts and to attain a clear conclusion on how the sample size affects the performance of the monitoring chart.

Comparison under real case

This study is conducted from late December 2019 through January 2020 at an international airport with limited resources that do not allow 100% inspection. The name of the airport is not disclosed due to confidentiality reasons. The screening procedure is performed by checking the temperature using thermometer guns. A symptomatic passenger will be detected if he has a significant fever (i.e., his temperature exceeds 100.4F). Based on the available resources, the airport can only inspect 100 arrivals every hour (i.e., r = 100/hour) and it can handle one false alarm signal every 27 days on average (i.e., τ = 648 h). The in-control infection rate that does not reflect a potential outbreak (, whereas the maximum out-of-control value of the infection rate the airport is interested to detect is 10 times the in-control infection rate (i.e.,). It is worthy to mention that the shift is assumed to follow a uniform distribution. Fixed values of n and h are used as design specification for the nptraditional chart without any optimization. Thus, we can consider that nptraditional inspects a sample of 100 arrivals per hour. Contradictory, both n and h are optimized under the given inspection rate r in this study. As highlighted previously, the optimal design of the np chart came up with the optimal combination of n, h and UCL, which produces the minimum ANI while satisfying the constraints (4) and (5). Applying the optimization algorithm, we found that the optimal charting parameters of the npoptimal chart are to inspect a sample of size 185 (i.e., n = 185) at every time interval of 1.85 h (i.e., h = 1.85hr) using UCL = 6. The values of the charting parameters and the corresponding ANI values for both charts are shown below:

nptraditional chart: n = 100, h = 1, UCL = 5 and ANI = 0.2469.

npoptimal chart: n = 185, h = 1.85, UCL = 6 and ANI = 0.1248.

Fig. 4 shows the effect of n on the ANI values. It can be seen that the ANI values have a decay and rise pattern till they reach one point (i.e., the optimal sample size n = 185) where the ANI value does not go lower any further. Fig. 4 also shows that the proposed npoptimal chart outperforms nptraditional chart in terms of ANI under the same design specifications.

Fig. 4

The values of ANI against n.

The values of ANI are compared based on a defined relative performance index (RPI) which can be calculated as:

RPI shows the percentage of improvement achieved by the npoptimal chart compared to nptraditional chart. Under the studied case, the RPI shows an improvement of 98% (in the ANI of the npoptimal chart compared to that of the nptraditional chart. One interpretation of this result is that inspecting a sample of 185 every 111 minutes can detect an outbreak almost two times faster than inspecting a sample of 100 every hour while satisfying the same constraints on the false alarm rate and inspection rate.

As shown in Fig. 4, there are several valley points (VPi). These valley points are always the local minima on the curve of ANI against n, and ANI is actually a concave-upward function of n at these valley points. This is due to the fact that the UCL at a valley point is always the tightest, and the corresponding ATS 0 is just slightly larger than the specified τ. It also results in the smallest ANI in the neighborhood of a valley point. If the sample size n is increased by one from the sample size at a valley point, the UCL has to be increased by one in order to meet the constraint (4). Consequently, the in-control ATS 0, as well as the ANI, will increase sharply. Therefore, the optimal sample size (n) is identified as one of the valley points. For instance, the optimal sample size (n = 185) is associated with the 4th valley point (VP4).

Moreover, the nptraditional and npoptimal charts are compared in terms of the out-of-control average time to signal ATS. Fig. 5 shows the performance of both np charts in terms of the normalized ATS (ATS nptraditional / ATS npoptimal). As can be seen from Fig. 5, the npoptimal chart is more effective than nptraditional chart for detecting p shifts over almost the whole given range. Also, it can be noted that as the shift increases, the nptraditional chart performs roughly similar to the npoptimal chart. In other words, the superiority of the npoptimal chart over the nptraditional chart decreases with increasing the shift size in the infection rate.

Fig. 5

The normalized ATS of the nptraditional and npoptimal charts.

Sensitivity analysis

In most processes, the process shift usually follows a specific probability distribution. However, as Siddall (1983) pointed out, if there is uncertainty about a random variable except for its bounds, then uniform distribution might be an excellent option to represent that variable. Many researchers designed control charts assuming that the process shift follows uniform distribution (Castagliola, Celano, & Psarakis, 2011, Sparks, 2000, Domangue and Patch, 1991), while others used beta distribution (Ou, Wu, & Goh, 2011) and Rayleigh distribution (Haridy, Maged, Kaytbay, & Araby, 2017, Wu et al., 2004) to describe the process shift.

In this section, a sensitivity analysis is conducted for the case in Section 5.1 (i.e., τ = 648 h, p 0 = 0.01, r = 100/hour and p = 10p 0) to study how the charts will perform if the estimated distribution of the p shift is not uniform. The nptraditional and npoptimal charts are designed for three other cases in which p shift follows a beta distribution as shown in cases 1, 2 and 3 of Table 1 . The probability density function of the beta distribution can be determined as follows:

Table 1

Control Charts under Different Distributions of p Shift.

Case	Distribution	Distributionparameters		Chart	n	h	UCL	ANI	RPI
		a	b
1	Betaskewed to right	2	4	nptraditional	100	1	5	0.572	120%
				npoptimal	128	1.28	5	0.260
2	Betasymmetrical	3	3	nptraditional	100	1	5	0.167	80%
				npoptimal	185	1.85	6	0.093
3	Betaskewed to left	4	2	nptraditional	100	1	5	0.082	24%
				npoptimal	128	1.28	5	0.066

The skewness of a beta distribution depends primarily on the parameters a and b. If (a < b), the probability distribution of the p shift will be skewed to right (Fig. 6 (a)) and most of the shifts cluster to the lower end. If (a > b), the probability distribution of the p shift will be skewed to the left (Fig. 6(c)), and most of the shifts cluster to the upper end. Finally, if (a = b), the distribution of the p shift will be symmetric (Fig. 6(b)). Cases 1, 2, and 3 in Table 1 serve as representatives of different types of non-uniform probability distributions of p shift.

Fig. 6

Three Beta Probability Density Functions of p Shift.

The RPI values in Table 1 show that, under any probability distributions of p shift, the npoptimal chart always outperforms the nptraditional chart. The superiority of the npoptimal chart over the np chart is more significant when f(p) is skewed to the right (case 1). This finding is justifiable as the npoptimal chart uses a relatively large sample size (n = 100), making it less sensitive for detecting large p shifts. When the beta distribution is symmetrical (case 2), it can also be observed that the RPI value for case 2 when the beta distribution is symmetric is close to that of the uniform distribution in Section 5.1. It can be concluded that the npoptimal chart always considerably outperforms the nptraditional chart regardless of the probability distribution of the p shift. The distribution of p shift may only influence the degree of the superiority of the poptimal chart over the nptraditional chart.

Comparison under different scenarios

The performance of the nptraditional and npoptimal charts is further compared under five more scenarios with different design specifications to demonstrate the improvement that can be achieved by optimizing n. The overall improvement is represented in terms of RPI, which is calculated using Eq. (8). The results are shown in Table 2 .

Table 2

A comparison of the nptraditional and npoptimal charts for sampling inspection under five different scenarios.

Scenario	Chart	pmax	τ	po	r	n	h	UCL	ANI	RPI
I	nptraditional	5po	300	0.03	40	40	1	5	0.739	151%
	npoptimal					119	2.975	8	0.294
II	nptraditional	15po	300	0.03	40	40	1	5	0.331	37%
	npoptimal					32	0.8	4	0.241
III	nptraditional	15po	900	0.005	120	120	1	4	0.186	92%
	npoptimal					164	1.367	4	0.097
IV	nptraditional	5po	900	0.03	20	20	1	4	3.968	505%
	npoptimal					134	6.7	9	0.656
V	nptraditional	5po	900	0.03	40	40	1	6	2.202	437%
	npoptimal					119	2.975	9	0.410

As it can be observed from Table 2, the RPI values shows the superiority of the npoptimal chart over the nptraditional chart throughout the five scenarios. For instance, in case 4 where r = 20, τ = 900, p 0 = 0.03 and p = 5p 0, the npoptimal chart is able to make a reduction by 505% in the average number of infections (ANI) compared to the nptraditional chart. This indicates that the npoptimal chart is substantially more powerful for detecting the entire range of the shifts under such design specifications.

Comparison of detection speed

This section shows a comparison of the detection speed of the nptraditional and npoptimal charts under the same scenarios shown in Table 2. For each scenario, 30 samples with a random number of infections (d) are generated by simulation using inverse transform method. It only requires the sample size and infection rate in order to simulate d. The first 15 samples are generated to satisfy the in-control condition, and the rest are generated to be out of control. Both samples follow a binomial distribution with the same n but different infection rates (i.e., p 0 for the in-control and i × p 0 for the out-of-control where i represents the increase in the shift). The values of i are indicated in Table 3 . For each chart, the same sample size in Table 2 is used. For example, the first 15 samples in scenario I when using the nptraditional chart are generated using a binomial distribution B(40, 0.03), while the other 15 samples are generated with B(40, 0.06) (i.e., using a sample size of n = 40 and an infection rate of p = 2 × 0.03 = 0.06).

Table 3

A comparison of the detection speed of the nptraditional and npoptimal charts.

Scenario	Chart	Shift (i)	p (=i × p0) after shift	Detection sample
I	nptraditional	2	0.06	23
	npoptimal		0.06	18
II	nptraditional	5	0.15	22
	npoptimal		0.15	19
III	nptraditional	8	0.04	23
	npoptimal		0.04	17
IV	nptraditional	3	0.09	27
	npoptimal		0.09	16
V	nptraditional	4	0.12	24
	npoptimal		0.12	17

Table 3 shows the sample at which both charts will detect the shift (i.e., the detection sample). The detection sample in Table 3 indicates that the npoptimal chart always gives an out-of-control signal faster than the nptraditional chart. This demonstrates that the former has a better detection speed than the latter and consequently it is adopted for the early detection of an outbreak. Fig. 7 illustrates the detection speed of both the nptraditional and npoptimal charts under the settings given in scenarios I-V in Table 3. It is obvious that the npoptimal chart always gives an out-of-control signal before the nptraditional chart. This demonstrates that the former has a better detection speed than the latter and consequently it is adopted for the early detection of an outbreak.

Fig. 7

A comparison of the detection speed of the two np charts under five simulated scenarios.

Comparison under 100% inspection

In this section, the performance of the nptraditional and npoptimal charts is compared under 100% inspection using the same design specification in Section 5.1 (i.e., τ = 648 h, p 0 = 0.01 and p = 10p 0) and assuming that the airport has sufficient resources to carry out such inspection. A predetermined n of 100 arrivals is used for the nptraditional chart, while it is optimized in the design of the npoptimal chart. In 100% inspection, optimizing the sample size n means adjusting the grouping of the inspected units (Montgomery, 2019, Reynolds & Stoumbos, 1999). The optimization model of the np chart under 100% inspection can be formulated as follows:

Objective: Minimize ANI

Constraint: ATS 0 ≥ τ

Design variables: n and UCL

For 100% inspection, n and UCL are the only parameters to be optimized as there is no sampling interval h. In addition, there is no constraint on the inspection rate r as the sampling inspection is no longer implemented.

The values of the charting parameters and corresponding ANI values for both nptraditional and npoptimal charts are indicated below:

nptraditional chart: n = 100, UCL = 2 and ANI = 6.4344.

npoptimal chart: n = 40, UCL = 1 and ANI = 3.4073.

The RPI =  indicates that the npoptimal chart has a better overall performance than the nptraditional chart by 89%. Fig. 8 shows the performance of both np charts in terms of the normalized ATS (ATS nptraditional/ATS npoptimal). It is clear that the npoptimal chart outperforms the nptraditional chart for detecting the whole range of p shifts. In the meantime, Fig. 8 indicates that the superiority of the npoptimal chart over the nptraditional chart increases with increasing the shift size in the infection rate. This result is justifiable because the npoptimal chart uses a sample size (n = 40) smaller than that of the nptraditional chart (n = 100).

Fig. 8

The normalized ATS of the nptraditional and npoptimal charts under 100% inspection.

The performance of the nptraditional and npoptimal charts is further compared under the same five scenarios in Section 5.2 under 100% inspection. The same design specifications (τ, p 0 and p) for each scenario are used. The design specifications, charting parameter, ANI and RPI of both charts are all shown in Table 4 for each scenario.

Table 4

A comparison of the nptraditional and npoptimal charts for 100% inspection under five different scenarios.

Scenario	Chart	pmax	τ	po	n	UCL	ANI	RPI
I	nptraditional	5po	300	0.03	40	2	5.202	26%
	npoptimal				9	1	4.127
II	nptraditional	15po	300	0.03	40	2	9.020	172%
	npoptimal				9	1	3.310
III	nptraditional	15po	900	0.005	120	1	4.269	20%
	npoptimal				74	1	3.537
IV	nptraditional	5po	900	0.03	20	2	6.976	0%
	npoptimal				20	2	6.976
V	nptraditional	5po	900	0.03	40	3	7.804	11%
	npoptimal				20	2	6.976

The overall performance of the npoptimal chart, in terms of ANI, is always better than, or at least equal to, that of the nptraditional chart across the five scenarios. RPI values illustrate the improvement in the overall detection effectiveness that can be achieved when the npoptimal chart is used instead of the nptraditional chart for each scenario.

Conclusion

This paper proposes a monitoring scheme for early detection of outbreaks caused by coronaviruses and other important respiratory viruses. For respiratory viruses with high transmissibility and mortality rates, such as SARS-CoV-2, early detection of important clusters of infections provides critical information to public health representatives and policymakers. In particular, early detection of respiratory virus activity at sentinel airports would allow for near real-time decisions about the potential need for specific virologic testing, patient quarantine, travel restriction, and other important outbreak investigation and mitigation measures.

The proposed monitoring scheme suggests using an optimized np control chart for monitoring the infection rate of respiratory viruses. In this scheme, clinical symptoms that are simple to monitor are used as surrogates for infection. The suggested optimized np chart is compared with the traditional np chart under both sampling inspection and 100% inspection using different settings. The results reveal that the former substantially outperforms the latter for detecting a wide range of shifts in the infection rate. Furthermore, the optimized np chart is as simple as the traditional np chart to implement. An out-of-control signal on the np chart could be a potential outbreak. The design and implementation of the developed np chart are simple, especially for healthcare practitioners without a background in control charts. Adopting the proposed monitoring scheme in sentinel airports could help identify the origin of the virus, compare infection rates at different locations, and initiate early mitigation measures.

In this research, the number of infections d is assumed to follow a binomial distribution. It would be worthwhile to conduct a sensitivity analysis in future research by assuming different distributions of d. Furthermore, the number of arrivals per unit time (N) is assumed to be constant; however, it might vary over time in airports. In this case, further study can be conducted employing N as a measure of risk by determining the exact average number of infections (ANI). This adjustment will allow decision-makers to manage the infected cases effectively by providing the required resources and responses, if an outbreak exists.

CRediT authorship contribution statement

Salah Haridy: Conceptualization, Methodology, Coding, Analysis, Results, Writing. Ahmed Maged: Methodology, Results, Writing. Arthur W. Baker: Methodology, Writing. Mohammad Shamsuzzaman: Methodology, Writing. Hamdi Bashir: Methodology, Writing. Min Xie: Writing, Editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.