Research on safety strategies for nucleic acid testing in sudden epidemic conditions based on a SEIARD dynamic model

Abstract

Infectious diseases have caused enormous disasters in human society, and asymptomatic carriers are an important challenge in our epidemic prevention and control process. Nucleic acid testing has played an important role in rapid testing for asymptomatic individuals. How to carry out nucleic acid testing in a scientific manner is a practical problem encountered in normal production and life. Based on the real COVID-19 epidemic data from Shanghai, we established a susceptible-exposed-infected-asymptomatic-recovered-death (SEIARD) dynamic model. The least squares method was used to fit the data and estimate the unknown parameters β and E(0) in the model, and MATLAB software was employed to simulate the development of the epidemic. The data fitting results indicated that the SEIARD model can better describe the early development patterns of the epidemic (R² = 0.98; MAPE = 2.54%). We calculated the basic reproduction number of the Shanghai epidemic as R₀ = 2.86. As the frequency of nucleic acid testing increased, the basic reproduction number R₀ continued to decrease. When there is one latent carrier and one asymptomatic carrier in the nucleic acid testing team, the number of queues is directly proportional to the number of infected individuals, the nucleic acid testing team increases by 50 people, and the number of new asymptomatic cases increases by approximately 4 people. If both susceptible individuals (S) and asymptomatic patients (A) are not wearing masks, the infection rate reaches approximately 7%; after wearing masks, the final infection rate is less than 1% at 1.5 m between two people. The queue spacing is inversely proportional to the number of infected individuals. With a distance of d = 1 m, a nucleic acid testing team of 100 people added 8% of the infected individuals; when d = 1.5 m, fewer than 2% of the newly infected individuals. The results confirmed that controlling the queue size for nucleic acid testing, strictly wearing masks, and maintaining a queue spacing of more than 1.5 m are safe and effective nucleic acid testing strategies. Our findings are also applicable to the prevention of other newly emerging infectious diseases.

Introduction

Infectious diseases have endangered humanity for thousands of years and are among the most important threats to human survival and health. Well-known infectious diseases, including Pestis (black death), smallpox, cholera, tuberculosis, malaria HIV, Ebola, and dengue fever, have caused significant harm to humans. In the twenty-first century, we face attacks from emerging infectious diseases. In 2003, the SARS virus ravaged China^1,2. In 2009, the H1N1 influenza virus caught the attention of researchers^3,4. From 2012 to 2015, Middle East respiratory syndrome (MERS) caused tens of thousands of infections^5,6. The greatest shock has been the worldwide outbreak of novel coronavirus pneumonia (COVID-19) since December 2019. As of December 30, 2022, there are 650 million confirmed cases of COVID-19 and 6.64 million deaths worldwide⁷. In addition, the COVID-19 pandemic has caused a major economic crisis worldwide and has severely affected people’s normal work and daily lives. The leading economic organisations in the world, such as the IMF, UNIDESA, and OPEC, projected that the world economy would experience the worst recession after the Great Depression in 2007⁸. According to the latest statistics from the UNIDO Industrial Production Index (IIP), data from 49 countries revealed an economic loss of 87% of world value-added production. Compared with the data from the IIP for December 2019, there was a reduction in industrial production of an average of 20% in 93% of countries worldwide⁹. The COVID-19 epidemic has caused severe disasters to people worldwide.

The outbreak of infectious diseases urgently requires mathematical models to study the development process of the disease, reveal its epidemic patterns, predict its changing trends, analyse the causes and key factors of disease epidemics, and seek optimal strategies for prevention and control. Among the numerous mathematical models, dynamic models have played crucial roles in disease prevention and control. During the COVID-19 epidemic, many dynamic models have been established, such as susceptible-infected-recovered (SIR) models^10–12, susceptible-exposed-infected-recovered (SEIR) models^13–15, and other improved models (considering the various characteristics of COVID-19 disease and the policies implemented) for epidemic transmission prediction and evaluating the effectiveness of prevention and control measures^16–18. With the emergence of many asymptomatic patients, susceptible-exposed-infected-asymptomatic-recovered (SEIAR) models have been established to simulate the early period of COVID-19 cases^19–22. Several studies have reported that asymptomatic infections in China, England, and Indonesia have spread, creating a high prevalence of infection^23–25. Markets and crowded places have a high risk of COVID-19 transmission by asymptomatic infection because their status is undetected.

To control the presence of asymptomatic infections, rapid testing for asymptomatic and symptomatic individuals can reduce the basic reproduction number²⁶. With the development of medicine, nucleic acid testing has played a crucial role in screening and isolating asymptomatic individuals and preventing further disease outbreaks^27,28. All organisms except prions contain nucleic acids, including deoxyribonucleic acid (DNA) and ribonucleic acid (RNA). The novel coronavirus is a virus that contains only RNA. The specific RNA sequence in a virus is a marker used to distinguish the virus from other pathogens. Nucleic acid testing determines whether there is nucleic acid from foreign invading viruses in respiratory tract samples, blood or the stool of patients to determine whether they are infected by COVID-19. Therefore, once a patient tests positive for nucleic acid, the presence of the virus in the patient's body can be confirmed²⁹. Through population nucleic acid testing, cases, especially asymptomatic infections, can be detected as early as possible, risk areas and key populations can be identified, and targeted control measures can be taken promptly to prevent the spread of the epidemic.

However, after the policy of universal nucleic acid testing was proposed, while medical staff worked hard to prevent the virus, the virus also found the best time to spread from point to local and then to the overall situation. According to an epidemiological investigation in China, after controlling personnel mobility, among the many routes of virus transmission, the proportion of infections during nucleic acid testing is the highest, reaching 26%³⁰. To avoid virus spread during nucleic acid testing, more comprehensive research on testing strategies is needed.

Therefore, we established a SEIARD dynamic model that includes asymptomatic carriers. However, there are several significant differences from previously published studies^19–21, as follows: (1) We took into account the impact of nucleic acid testing on asymptomatic carriers and the spread of the epidemic; additionally, we have studied the effectiveness of nucleic acid testing in controlling the epidemic by comparing the basic reproduction number R_0. (2) We consider the late stage of the incubation period infectious. (3) We use a variable coefficient function to estimate the recovery rate of infected individuals. (4) We simulated the transmission of epidemics with different numbers of latent individuals, team spacings, team sizes, and wearing of masks on the nucleic acid testing team to provide safe nucleic acid testing strategies for the prevention and control of sudden infectious diseases.

Materials and methods

Data sources

Urban population density and high population mobility are more likely to lead to the rapid spread of infectious diseases. Therefore, we take Shanghai, a metropolis in China, as an example to simulate a safe nucleic acid testing strategy. The COVID-19 epidemic data come from the daily epidemic report issued by the Shanghai Municipal Health Commission (https://wsjkw.sh.gov.cn/yqtb/index_2.html). We collated the number of newly confirmed cases, new asymptomatic infections, existing confirmed cases, cumulative confirmed cases, cumulative cured cases and cumulative deaths in Shanghai from March 19 to April 30, 2022, totalling 43 days.

SEIARD model

The population is divided into six parts according to the process of disease transmission: susceptible (S), exposed (E), infected (I), asymptomatic (A), recovered (R) and dead (D). The model flowchart is shown in Fig. 1. Before establishing the model, we make the following assumption:

1. The population types are evenly mixed, without considering the large-scale influx and outflow of the population. In the short term, the natural population birth and death rates are not considered.

2. Assume that the population is susceptible and considers only the transmission mode from person to person. At present, there is no evidence that novel coronaviruses are transmitted to animals³¹.

3. The incubation period has a certain degree of infectivity. Several studies have shown that patients with COVID-19 are highly infectious during the incubation period^32,33.

4. Asymptomatic carriers with a proportion of 1-P and asymptomatic carriers are infectious.

5. The recovered person will have antibodies in the short term and will no longer consider the possibility of secondary infection³⁴.

Fig. 1.

SEIAR model with asymptomatic infected individuals.

The above structure is represented by a dynamic model as follows:

$$\left\{\begin{array}{c}\frac{\mathit{dS}}{\mathit{dt}}=\frac{-c\beta S[\sigma E+\eta A+I]}{N},\\ \frac{\mathit{dE}}{\mathit{dt}}=\frac{c\beta S[\sigma E+\eta A+I]}{N}-\omega E,\\ \frac{\mathit{dA}}{\mathit{dt}}=(1-p)\omega E-{\gamma }_{1}A,\\ \frac{\mathit{dI}}{\mathit{dt}}=p\omega E-{\gamma }_{2}I,\\ \frac{\mathit{dR}}{\mathit{dt}}={\gamma }_{1}A+\kappa {\gamma }_{2}I,\\ \frac{\mathit{dD}}{\mathit{dt}}=(1-\kappa ){\gamma }_{2}I,\\ \end{array}\right)$$ 1

In the model, c is the daily average contact rate of infected individuals, β is the probability of infection for infected individuals, σ is the probability attenuation factor for latent infection, η is the attenuation factor of the transmission probability of asymptomatic carriers, $\mathrm{\omega }$ is the rate at which latent individuals transform into infected individuals, P is the proportion of overt infected individuals,${\gamma }_1$ is the recovery rate of asymptomatic infected individuals (equivalent to the nucleic acid testing rate; once asymptomatic carriers are found, they are isolated), ${\gamma }_2$ is the recovery rate of dominant infected individuals, and 1-k is the mortality rate of illness. Owing to the low mortality rate of asymptomatic infections, it can be ignored.

In this model, the basic reproduction number $R_0$ is calculated by the next generation regeneration matrix, and the order $x={(E,A,I,S)}^T,$ subsystem of Model (1) can be rewritten as Ref.³⁵:

$$\frac{\mathit{dx}}{\mathit{dt}}=F(x)-V(x)$$

wherein

$$F(x)=\left(\begin{array}{c}\frac{c\beta S(\sigma E+\eta A+I)}{N}\\ 0\\ 0\\ 0\end{array}\right),V(x)=\left(\begin{array}{c}\omega E\\ -(1-P)\omega E+{\gamma }_{1}A\\ -P\omega E+{\gamma }_{2}I\\ c\beta S(\sigma E+\eta A+I)/N\end{array}\right)$$

The full derivatives of F(x) and V(x) with respect to x are calculated separately, and the following equation is obtained:

$$F=\left(\begin{array}{cccc}\frac{c\beta \sigma S}{N}& \frac{c\beta \eta S}{N}& \frac{c\beta S}{N}& \frac{c\beta (\sigma E+\eta A+I)}{N}\\ & & 0& \\ & & 0& \\ & & 0& \end{array}\right),$$

$$V=\left(\begin{array}{cccc}\omega & 0& 0& 0\\ -(1-P)\omega & {\gamma }_1& 0& 0\\ -P\omega & 0& {\gamma }_2& 0\\ c\beta \sigma S/N& c\beta \eta S/N& c\beta S/N& c\beta (\sigma E+\eta A+I)/N\end{array}\right),$$

$R_0$ was calculated via the following formula:

$$R_0=\rho (F{V}^{-1})=c\beta \left\{\frac{\sigma }{\omega }+\frac{\eta (1-p)}{{\gamma }_1}+\frac{p}{{\gamma }_2}\right\}.$$ 2

If $R_0$ > 1, the epidemic disease continues; if $R_0$ < 1, the epidemic disease ends, and the greater the reduction in $R_0$ is, the greater the control effect.

Statistical analysis

This study used Berkeley Madonna and lstOpt software to estimate the model parameters in various scenarios. The fourth-order Runge–Kutta–Fehlberg method with a tolerance of 0.001 was used to solve the differential equations, and the optimisation algorithm used was the Universal Global Optimization Algorithm (UGO1)³⁵. MATLAB 2016 was used for data fitting and the production of related graphics. Moreover, the accuracy of model prediction is evaluated via the coefficient of determination (R²), mean error rate (MAPE) and root mean square error (RMSE). The smaller the MAPE and RMSE are and the larger the R² is, the better the model fitting effect; the formula is as follows³⁶:

$$R^2=1-\frac{\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{\sum _{i=1}^n{(y_i-{\overline{y}}_i)}^2},MAPE=\frac{1}{n}\sum _{i=1}^n\frac{|y_i-{\widehat{y}}_i|}{y_i},RMSE=\sqrt{\frac{\sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}{n}},$$ 3

In the model, ${\widehat{y}}_i$ represents the theoretical data predicted by the model on day $i$, $y_i$ represents the actual epidemic data on day $i$, ${\overline{y}}_i$ represents the average value of the actual data, and $n$ represents the total number of days for analysing the data.

Epidemic prediction via the SEIARD model

Parameter estimation

The recovery and mortality rate data were calculated based on real epidemic data from March 19th to April 30th, 2022, in Shanghai. Due to the reversal of the migration rate on April 16th, we adopted a segmented fitting method using an exponential function $\kappa (t)=\frac{K_m}{1+a{\text{e}}^{b(t-1)}}$. In the model, K_m is the maximum recovery rate of diagnosed patients under current medical conditions, a is the exponential growth coefficient, and b is the exponential growth rate of the recovery rate.

Based on real data, we obtain the K_m value. Then, using the least squares method to obtain the values of unknown parameters a and b in the model, the expression of the confirmed patient recovery rate is obtained as follows:

$$\kappa (t)=\left\{\begin{array}{c}1-\frac{0.93}{1+18e^{-0.14(t-1)}},\begin{array}{cc}& \text{Before April 16th}\end{array}\\ \frac{0.65}{1+30e^{-0.12(t-1)}},\begin{array}{ccc}& & \text{After April 16th}\end{array}\\ \end{array}\right)$$ 4

The data fitting results are shown in Fig. 2.

1. The mean incubation period (1/ω) was 5.2 days (95% CI 4.1‒7.0)²¹, ω = 1/5.2;

2. According to a previous report³⁷, the effective daily contact rate of an infected person is 7 people, c = 7;

3. Once symptomatic infected individuals are diagnosed, they are isolated, and the average time from onset to diagnosis was 5 days in Shanghai, so the removal rate of I was γ₂ = 1/5. As asymptomatic infected individuals are not easily found and isolated, the recovery duration of A was 10 days¹⁹, so the removal rate of A was γ₁ = 1/10.

4. The transmission rate of latent and asymptomatic carriers is lower than that of overt carriers³⁷; assume that $\mathrm{\sigma }$= $\mathrm{\eta }=0.7$;

Fig. 2.

Fitting of the confirmed case recovery rate function.

Next, the initial data of five warehouses are put into the SEIAR model and the total population of Shanghai. With the help of Berkeley Madonna software, the least squares method is used to fit the data and estimate the unknown parameter in the model: β and E(0), the optimal values of {β, E(0)} can be obtained by solving Equation (5):

$$\begin{array}{c}\underset{\beta ,E(0)}{min}||I(\beta ,c,\sigma ,\eta ,{\gamma }_1,{\gamma }_2,\kappa (t))-I_{\mathit{data}}{||}_2+||A(\beta ,c,\sigma ,\eta ,{\gamma }_1,{\gamma }_2,\kappa (t))-A_{\mathit{data}}{||}_2\\ \begin{array}{ccc}& & +||R(\beta ,c,\sigma ,\eta ,{\gamma }_1,{\gamma }_2,\kappa (t))-R_{\mathit{data}}{||}_2\end{array}\\ \end{array}$$ 5

The specific parameter estimation results are shown in Table 1.

Table 1.

Meaning and estimated values of parameters in the model.

Parameter	Meaning	Estimated value	Source
$\beta$	Initial infection rate	0.04	Data fitting
$\omega$	latency coefficient	1/5.2	21
$c$	Average daily contact rate of infected individuals	7	37
${\gamma }_1$	removal rate of A	1/10	19
${\gamma }_2$	removal rate of I	1/5	19
$\sigma$	Latent infectious attenuation factor	0.7	37
$\eta$	Asymptomatic carrier infectious attenuation factor	0.7	37
$p$	The proportion of infected individuals	0.2	Data calculation
${\kappa }_m$	The maximum proportion of confirmed patients recovering	0.93	Data calculation
N	The total population of Shanghai	2489.43万	Official
S(0)	Number of susceptible individuals at the initial moment	N-E-I-A-R	Real data
E(0)	Number of patients in the initial incubation period	2284	Data fitting
I(0)	Number of initial infected individuals	22	Real data
A(0)	Number of asymptomatic infected individuals at the initial time	494	Real data
R(0)	Number of removed individuals at the initial time	4435	Real data

Model fitting

The fitting results of the data from March 19th to April 30th are shown in Fig. 3. The simulation prediction result is relatively good. The error between the daily number of new confirmed cases and the cumulative number of confirmed cases is relatively small. We also calculated the R² values, which are 0.93 and 0.98, which effectively describe the development of the epidemic.

Fig. 3.

Fitting results of daily new confirmed cases, new asymptomatic patients, cumulative confirmed cases, and cumulative removed cases in Shanghai from March 19 to April 30, 2022.

The data fitting in the second graph of Fig. 3 is not very good, due to the rapid development of asymptomatic carriers and implementation of nucleic acid testing, leading to the discovery of many asymptomatic carriers daily.

We also provided a detailed comparison of the cumulative number of confirmed cases predicted by the model in the last 5 days (April 26th to April 30th). The results are shown in Table 2. Table 2 shows that the model predicts an average error rate of 2.54% in the last 5 days. Overall, the SEIARD model can better describe the early development patterns of the epidemic.

Table 2.

Errors in Model Prediction for Accumulated Confirmed Cases.

Date	April 26th	April 27th	April 28th	April 29th	April 30th
Real data	49,519	50,811	56,300	57,550	58,339
Predictive data	51,213	51,833	56,862	56,471	55,765
Absolute value of error	1694	1022	562	1079	2574
Error rate	3.42%	2.01%	1.00%	1.87%	4.41%
Evaluation criterion	MAPE = 2.54%, RMSE = 1550.46

The impact of nucleic acid testing on the spread of the epidemic

According to Formula (2) and the parameter estimates, we calculate the basic reproduction number of the Shanghai epidemic as R₀ = 2.86. After nucleic acid testing was performed, asymptomatic carriers were quickly detected. When γ₁ = 1/7, R₀ = 2.39; when γ₁ = 1/5, R₀ = 2.08; when γ₁ = 1/3, R₀ = 1.78. As the frequency of nucleic acid testing increases, the basic reproduction number continues to decrease, indicating the effectiveness of nucleic acid testing in epidemic prevention and control (Fig. 4).

Fig. 4.

Trend of basic reproduction number changes with nucleic acid testing frequency (1/day).

Simulation study on the nucleic acid detection strategy

The impact of latent individuals on the spread of the epidemic

The latent period patients include two parts: asymptomatic infected individuals and those who remain latent after infection. We take A = 1 person, E = 1, 2, and 3 people to simulate the spread of the epidemic in team. The number of newly infected individuals is shown in Fig. 5 and Table 3.

Fig. 5.

Changes in the number of new asymptomatic infections among patients with different incubation periods.

Table 3.

Impact of the number of latent individuals on a 100-person nucleic acid testing team on the spread of the epidemic.

Asymptomatic patient	Latent period patients	Newly infected individuals	Newly added asymptomatic carriers	Infection rate (%)
A = 1	E = 1	1	8	9
	E = 2	1	9	10
	E = 3	1	10	11

We observed that when there was one asymptomatic carrier and one latent patient on a 100-person nucleic acid testing team, the infection rate reached 9%. As the number of latent patients increases, the infection rate also increases.

The impact of queue spacing on the spread of the epidemic

During the period of national nucleic acid testing, personnel are highly concentrated, which is the best time for the virus to search for a large number of senses from point to local, then to the whole, and from point to surface. Therefore, we need to accurately detect the optimal distance between nucleic acid testing teams efficiently and orderly to complete nucleic acid testing without leaving room for virus transmission.

We simulate the spread of the epidemic for different spacings, where d represents the distance between two people on a team. When d = 1 m, d = 1.5 m and d = 2 m, the simulation results are shown in Fig. 6.

Fig. 6.

Changes in the number of new asymptomatic infected individuals under different queue spacings.

Figure 6 shows that the larger the queue spacing is, the fewer infected individuals there are, and the queue spacing is inversely proportional to the number of infected individuals. When A = 1 person and E = 1 person with a queue spacing of d = 1 m, a nucleic acid testing team of 100 people added 8% of the infected individuals (1 infected person, 7 asymptomatic infected persons) (Table 4). When d = 1.5 m, fewer than 2% of the individuals are newly infected; when d = 2 m, fewer than 1% are newly infected. Therefore, to ensure the safety of nucleic acid testing and improve testing efficiency, a team spacing of 1.5 m is recommended.

Table 4.

Impact of the queue spacing of 100 nucleic acid testing teams on the spread of the epidemic.

Asymptomatic patient	Queue spacing (m)	Newly infected individuals	Newly added asymptomatic carriers	Infection rate (%)
A = 1, E = 1	< 1	1	7	8
	1–1.5	0	< 2	< 2
	1.5–2	0	< 1	< 1

Wearing masks and their impact on the spread of the epidemic

Wearing masks is currently one of the simplest and most effective measures for epidemic prevention. We simulate the spread of the epidemic for different mask-wearing situations. Situation 1: Both susceptible person, S, and infected person, I, are not wearing masks; the chance of infection is 90%; Situation 2: Susceptible person, S, wears a mask and infected person, I, does not wear a mask; the chance of infection is 30%.Situation 3: Susceptible person, S, wears a mask and infected person, I, wears a mask; the chance of infection is 5%³⁸.

Table 5 shows that when both susceptible individuals S and infected individuals are not wearing masks, the infection rate reaches approximately 7%, which means that, out of a 100-person nucleic acid team, 7 people will be infected. After wearing masks, the final infection rate was less than 1% at 1.5 m. Thus, wearing a mask during nucleic acid testing can effectively protect oneself.

Table 5.

Impact of wearing masks in a queue of 100 nucleic acid testing teams with a spacing of 1.5 m on the spread of the epidemic.

Asymptomatic patient	Mask wearing condition	Newly infected individuals	Newly added asymptomatic carriers	Infection rate
A = 1, E = 1	Situation 1	0	7	7%
	Situation 2	0	< 1	< 1%
	Situation 3	0	< 1	< 1%

The impact of queue size on the spread of the epidemic

During the entire nucleic acid testing period, we often saw long queues at the community entrance or nucleic acid testing points. We simulate the spread of the epidemic for different queue sizes, N = 50, N = 100, and N = 150. The simulation results are shown in Fig. 7.

Fig. 7.

Changes in the number of new asymptomatic infected individuals under different queue sizes.

When there is one latent carrier and one asymptomatic carrier in the nucleic acid testing team, the more people queue up, the more infected they become. When N = 50, the infection rate reaches approximately 6% at a distance of less than 1 m. When N = 100, the infection rate reaches approximately 10%. The number of people queuing is directly proportional to the number of infected individuals. When the number of nucleic acid test teams increases by 50 people, the number of new asymptomatic cases increases by approximately 4 people. Therefore, it is necessary to conduct nucleic acid testing in batches and strictly queue them up according to the prescribed number of people.

Discussion

In recent years, the world has experienced many epidemic impacts caused by various viruses. Nucleic acid testing is an important means for the prevention and control of infectious diseases and can detect, diagnose, and treat infected individuals early, minimising the risk of infecting others.

In this study, we investigated the effects of nucleic acid testing on asymptomatic carriers and the epidemic spread based on Shanghai COVID-19 data. Compared with the models of Niu et al.¹⁹, Liu et al.²⁰ and Zha et al.²¹, the SEIARD model considers the impact of nucleic acid testing on the epidemic spread. We studied the effectiveness of nucleic acid testing in controlling the epidemic by comparing the basic reproduction number R_0. In addition, we considered that the late stage of the incubation period was infectious and used a variable coefficient function to estimate the recovery rate of infected individuals. The data fitting results indicate that the SEIARD model can better describe the early development patterns of the epidemic (R² = 0.98; MAPE = 2.54%).

We calculated the basic reproduction number of the Shanghai epidemic as R₀ = 2.86. The Shanghai epidemic strain is the Omicron BA.2.76 variant, which has a shorter incubation period, stronger transmission ability, and faster transmission speed³⁹. Our research also revealed this, with a rapid increase in asymptomatic infections. Another study revealed that 93% of asymptomatic infected individuals were affected during the Shanghai epidemic⁴⁰. We also simulated the impact of nucleic acid testing on the epidemic spread by calculating R₀; as the frequency of nucleic acid testing increased, the basic reproduction number R₀ continued to decrease. Nucleic acid testing aims to 'clean up' hidden infected individuals in the crowd to quickly control the infection source and cut off the transmission chain⁴¹.

Furthermore, we studied the safe implementation strategy of nucleic acid testing.

Based on our research findings, we propose the following safe nucleic acid testing strategies. First, nucleic acid testing should be performed in batches, preferably with a testing team of less than 50 people. We found that the number of people queuing up is directly proportional to the number of infected individuals. When there were two latent patients on the team, the number of nucleic acid test samples increased by 50 people, and the number of new asymptomatic cases increased by approximately 4 people. Sometimes there is far more than one latent patient on the team⁴². Second, strictly match with masks, preferably N95 medical masks. When there is one latent patient on the team, if both susceptible individuals (S) and asymptomatic patients (A) are not wearing masks, the infection rate reaches approximately 7%; after wearing masks, the infection rate is less than 1% at a distance of 1.5 m. Many studies have reported that wearing a mask is an effective way to prevent infection⁴³. In particular, the N95 mask is the most effective. A medical protective mask (N95) refers to a mask filter material that, under the testing conditions specified by the National Institute for Occupational Safety and Health (NIOSH) standard, has a filtration efficiency greater than or equal to 95% for nonoily particles (such as dust, acid mist, paint mist, and microorganisms) and can block the splashing and penetration of liquids such as blood⁴⁴. Third, maintain a safe distance of at least 1.5 m. Many infectious diseases, especially respiratory diseases, can be transmitted through viruses, droplets, and aerosols in the air. Therefore, avoiding close contact and maintaining a safe distance is very effective. We found that when there were two latent patients on the team, wearing a mask combined with a queue spacing of d = 1.5 m, resulted in fewer than 2% of newly infected individuals. Several studies have also reported the effectiveness of maintaining social distancing in controlling epidemics^45,46. Fourth, during the nucleic acid testing process, individuals do not encounter any items, leave quickly after completion, and disinfect and wash their hands.

The world should work together to address some basic phenomena to combat future pandemics like COVID-19. The government and policy-makers should emphasise health protection and ensure economic support for both the supply and demand sides. We need to develop better healthcare systems and sophisticated technologies to predict, detect and treat patients in outbreaks so that they do not become pandemics. In the bottom line, we should be prepared for this before the next time something such as this happens because prevention is always better than a cure.

Conclusion

In this study, we investigated the effects of nucleic acid testing on asymptomatic carriers and the epidemic spread based on Shanghai COVID-19 data. A SEIARD model was established, and the data fitting results indicate that the model can better describe the early development patterns of the epidemic. Then, we calculated the basic reproduction number of the Shanghai epidemic as R₀ = 2.86 and simulate the impact of nucleic acid testing on the epidemic by calculating R₀. As the frequency of nucleic acid testing increases, the basic reproduction number R₀ continues to decrease, indicating the effectiveness of nucleic acid testing. Finally, we studied the implementation strategy of nucleic acid testing safety. The results confirmed that controlling the queue size for nucleic acid testing, strictly wearing masks, and maintaining a queue spacing of more than 1.5 m are safe and effective nucleic acid testing strategies. Our findings are also applicable to preventing other newly emerging infectious diseases.

During the spread of the epidemic, the model parameters have changed over time with the implementation of various intervention measures, such as the contact rate c and the removal rate ${\gamma }_1$ of asymptomatic carriers. They are not constants but variable coefficient functions. In our model, we used only an exponential function to fit the recovery rate of infected individuals. Hence, we can use variable coefficient functions to estimate important parameters in the next research step. Furthermore, inspired by the literatures^47,48, the contact of the crowd is random in the network, and we can also study the dynamic models of network propagation and stochastic models. The spread of viruses in the body is also an interesting direction worth considering.

Appendix: Proof of mathematical properties of models

Theorem 1

With every non-negative initial condition, the solutions of the fractional model (6) are non-negative and bounded.

Proof:

The original model considering birth and mortality rates is as follows:

$$\left\{\begin{array}{c}\frac{\mathit{dS}}{\mathit{dt}}=\mathrm{\Lambda }-c\beta \text{S[}\sigma \text{E +}\eta \text{A + I] -}\mu \text{S,}\\ \frac{\mathit{dE}}{\mathit{dt}}=c\beta \text{S[}\sigma \text{E +}\eta \text{A + I] - (}\omega +\mu )\text{E,}\\ \frac{\mathit{dA}}{\mathit{dt}}=(1-p)\omega \text{E - (}{\gamma }_1+\mu )\text{A,}\\ \frac{\mathit{dI}}{\mathit{dt}}=p\omega \text{E - (}{\gamma }_2+\mu +\alpha )\text{I,}\\ \frac{\mathit{dR}}{\mathit{dt}}={\gamma }_1\text{A +}{\gamma }_2\text{I -}\mu \text{R,}\\ \end{array}\right)$$ 6

wherein $\mathrm{\Lambda }$ is the population input rate and $\mu$ is the natural mortality rate, $\alpha$ is the disease mortality rate, other parameters, please refer to the main text. Using method given in Zakaria Yaagoub et al.⁴⁹, we define

$N=S+E+I+A+R$,adding up several equations in model (6) yields:

$$N^{\prime }=\mathrm{\Lambda }-\mu N-\alpha I<\mathrm{\Lambda }-\mu N$$

Therefore,$\underset{t\to +\infty }{lim}supN\le \frac{\mathrm{\Lambda }}{\mu }.$

The solution of model (6) will ultimately enter or remain at $\mathrm{\Omega }\in R_{+}^5$, where

$$\mathrm{\Omega }=\left\{(S,E,I,A,R)\in \left(R_{+}^5:S+E+I+A+R\le \frac{\mathrm{\Lambda }}{\mu }\right)\right\}$$

it is the forward invariant set of model (6). As a result, system’s non-negative solutions are bounded. The basic regeneration number of the system was calculated by formula,

$$R_0=c\beta \frac{\mathrm{\Lambda }}{\mu }\left\{\frac{\sigma }{\omega }+\frac{\eta (1-p)}{{\gamma }_1}+\frac{p}{{\gamma }_2+\alpha }\right\}.$$

The existence of equilibrium point

Theorem 2

There are at most two equilibrium points in system (6).

1. when R₀ ≤ 1, the system (6) only had a disease-free equilibrium point $P_0=(\frac{\mathrm{\Lambda }}{\mu },0,0,0,0).$

2. when R₀ > 1, the system (6) not only has a disease-free equilibrium point P₀, but also a unique endemic equilibrium point $P_{}^{\ast }=(S_{}^{\ast },E_{}^{\ast },A_{}^{\ast },I_{}^{\ast },R_{}^{\ast }).$

Proof:

Directly calculate the disease-free equilibrium point of the model and obtain $P_0=(\frac{\mathrm{\Lambda }}{\mu },0,0,0,0).$

The equilibrium point of endemic diseases $P_{}^{\ast }=(S_{}^{\ast },E_{}^{\ast },A_{}^{\ast },I_{}^{\ast },R_{}^{\ast }).$ should satisfy the following equation:

$$\left\{\begin{array}{c}\mathrm{\Lambda }-c\beta \text{S}_{}^{\ast }(\sigma \text{E}_{}^{\ast }+\eta \text{A}_{}^{\ast }\text{+ I}_{}^{\ast })-\mu \text{S}_{}^{\ast }=0,\\ c\beta \text{S}_{}^{\ast }(\sigma \text{E}_{}^{\ast }+\eta \text{A}_{}^{\ast }\text{+ I}_{}^{\ast })-(\omega +\mu )\text{E}_{}^{\ast }=0,\\ (1-p)\omega \text{E}_{}^{\ast }-({\gamma }_1+\mu )\text{A}_{}^{\ast }=0,\\ p\omega \text{E}_{}^{\ast }-({\gamma }_2+\mu +\alpha )\text{I}_{}^{\ast }=0,\\ {\gamma }_1\text{A}_{}^{\ast }+{\gamma }_2\text{I}_{}^{\ast }-\mu \text{R}_{}^{\ast }=0,\\ \end{array}\right)$$ 7

From the fourth equation, we can obtain

$$E_{}^{\ast }=\frac{{\gamma }_2+\mu +\alpha }{p\omega }I_{}^{\ast },$$ 8

Substitute Eq. (8) into the third equation to obtain

$$A_{}^{\ast }=\frac{(1-p)({\gamma }_2+\mu +\alpha )}{({\gamma }_1+\mu )p}I_{}^{\ast },$$ 9

Then bring equation (9) into the fifth equation, we can obtain

$$R_{}^{\ast }=\left[\frac{{\gamma }_1(1-P)({\gamma }_2+\mu +\alpha )}{({\gamma }_1+\mu )p\mu }+\frac{{\gamma }_2}{\mu }\right]I_{}^{\ast },$$ 10

Add the first equation and the second equation together to obtain

$$\mathrm{\Lambda }-\mu S_{}^{\ast }-(\omega +\alpha )E_{}^{\ast }=0,$$ 11

Bring Eq. (8) into Eq. (11), we can get

$$S_{}^{\ast }=\frac{\mathrm{\Lambda }}{\mu }-\frac{(\omega +\mu )({\gamma }_2+\mu +\alpha )}{p\omega \mu }I_{}^{\ast },$$ 12

Finally, bring $E_{}^{\ast },A_{}^{\ast },S_{}^{\ast }$ into the first equation and combined with R_0, we can obtain

$$I_{}^{\ast }=\frac{\mathrm{\Lambda }(R_0-1)p\omega }{R_0(\omega +\mu )({\gamma }_2+\mu +\alpha )}.$$

when $I_{}^{\ast }>0$, equivalent to $R_0>1$, the system (6) only had a disease-free equilibrium point $P_{}^{\ast }=(S_{}^{\ast },E_{}^{\ast },A_{}^{\ast },I_{}^{\ast },R_{}^{\ast })$.

Theorem 3

When R₀ ≤ 1, local asymptotic stability of disease-free equilibrium point P₀; when R₀ > 1, disease-free equilibrium point P₀ is unstable.

Proof:

According to the method described in the literature⁵⁰, it can be concluded that the Jacobian matrix of system (6) at the disease-free equilibrium point $P_0=\left(\frac{\mathrm{\Lambda }}{d},0,0,0,0\right)$ is

$$J(p_0)=\left(\begin{array}{ccccc}-\mu & -c\sigma \beta \frac{\mathrm{\Lambda }}{\mu }& -c\eta \beta \frac{\mathrm{\Lambda }}{\mu }& -c\beta \frac{\mathrm{\Lambda }}{\mu }& 0\\ 0& c\sigma \beta \frac{\mathrm{\Lambda }}{\mu }-(\omega +\mu )& c\eta \beta \frac{\mathrm{\Lambda }}{\mu }& c\beta \frac{\mathrm{\Lambda }}{\mu }& 0\\ 0& (1-p)\omega & -({\gamma }_1+\mu )& 0& 0\\ 0& p\omega & 0& -({\gamma }_2+\mu +\alpha )& 0\\ 0& 0& {\gamma }_1& {\gamma }_2& -\mu \end{array}\right),$$

The equation has eigenvalues

$${\lambda }_1={\lambda }_2=-\mu <0,{\lambda }_3=-({\gamma }_1+\mu )<0.$$

The other two eigenvalues ${\lambda }_4,{\lambda }_5$ satisfy the equation

$${\lambda }^2+a_1\lambda +a_2=0,$$

wherein, $a_1=\omega +\alpha +{\gamma }_2+2\mu -c\sigma \beta \frac{\mathrm{\Lambda }}{\mu },$

$$a_2=({\gamma }_1+\mu )({\gamma }_2+\alpha +\mu )\left(\omega +\mu -c\sigma \beta \frac{\mathrm{\Lambda }}{\mu }\right)-\frac{c\eta \beta (1-p)\mathrm{\Lambda }\omega }{\mu }-\frac{c\beta p\mathrm{\Lambda }\omega }{\mu },$$

According to Viete theorem⁵¹, it can be concluded that

$${\lambda }_4+{\lambda }_5=(\omega +\alpha +{\gamma }_2+2\mu )-c\sigma \beta \frac{\mathrm{\Lambda }}{\mu },$$

$${\lambda }_4{\lambda }_5=({\gamma }_1+\mu )({\gamma }_2+\mu +\alpha )\left(\omega +\mu -c\sigma \beta \frac{\mathrm{\Lambda }}{\mu }\right)-\frac{c\eta \beta (1-p)\mathrm{\Lambda }\omega }{\mu }-\frac{c\beta p\mathrm{\Lambda }\omega }{\mu }$$

$$\begin{array}{cc}=({\gamma }_1+\mu )({\gamma }_2+\mu +\alpha )(\omega +\mu )(1-R_0)& \end{array}$$

when $R_0=\frac{c\beta \mathrm{\Lambda }}{\mu }\left\{\frac{\sigma }{\omega +\mu }+\frac{\eta (1-p)\omega }{(\omega +\mu )({\gamma }_1+\mu )}+\frac{p\omega }{(\omega +\mu )({\gamma }_2+\mu +\alpha )}\right\}<1,$

We have

$$\begin{array}{c}c\sigma \beta \mathrm{\Lambda }({\gamma }_1+\mu )({\gamma }_2+\mu +\alpha )+c\eta \beta \mathrm{\Lambda }(1-p)\omega ({\gamma }_2+\mu +\alpha )+c\beta \mathrm{\Lambda }({\gamma }_1+\mu )p\omega \\ \begin{array}{cccc}& & & \begin{array}{ccc}& & \begin{array}{ccc}& & \end{array}\end{array}\end{array}<\mu (\omega +\mu )({\gamma }_1+\mu )({\gamma }_2+\mu +\alpha )\\ \end{array}$$

Namely

$$c\sigma \beta \frac{\mathrm{\Lambda }}{\mu }+c\eta \beta \frac{\mathrm{\Lambda }(1-p)\omega }{\mu ({\gamma }_1+\mu )}+c\beta \frac{\mathrm{\Lambda }p\omega }{\mu ({\gamma }_2+\mu +\alpha )}<\omega +\mu$$

.

Hence,

$$c\sigma \beta \frac{\mathrm{\Lambda }}{\mu }<\alpha +{\gamma }_2+2\mu +\omega$$

.

So we can get ${\lambda }_4+{\lambda }_5=c\sigma \beta \frac{\mathrm{\Lambda }}{\mu }-(\omega +\alpha +{\gamma }_2+2\mu )<0.$

Since when $R_0<1,{\lambda }_4·{\lambda }_5>0,$ so ${\lambda }_4<0$ and ${\lambda }_5<0$, all eigenvalues of the system of equations are less than zero. According to the Lyapunov first method: when R₀ ≤ 1, local asymptotic stability of disease-free equilibrium point P_0; when R₀ > 1, disease-free equilibrium point P₀ is unstable_.

Theorem 4

When R₀ ≤ 1, global asymptotic stability of disease-free equilibrium point P₀. R₀ > 1, global asymptotic stability of endemic equilibrium point P^*.

Proof:

When R₀ ≤ 1, we construct the Lyapunov function of the system

$$V=E+\frac{\omega +\mu }{p\omega }I+\frac{\omega +\mu }{(1-p)\omega }A.$$

The total derivative along model (6) is

$\begin{array}{c}{V}^{\prime }=E^{\prime }+\frac{\omega +\mu }{p\omega }{I}^{\prime }+\frac{\omega +\mu }{(1-p)\omega }{A}^{\prime }\\ =c\beta S(\sigma E+\eta A+I)-(\omega +\mu )E+\frac{\omega +\mu }{p\omega }[p\omega E-({\gamma }_2+\mu +\alpha )I]+\\ \begin{array}{cc}& \end{array}\frac{\omega +\mu }{(1-p)\omega }[(1-p)\omega E-({\gamma }_1+\mu )A]\\ \end{array}$

$$=c\beta S(\sigma E+\eta A+I)-\frac{(\omega +\mu )({\gamma }_2+\mu +\alpha )}{p\omega }I+(\omega +\mu )E-\frac{(\omega +\mu )({\gamma }_1+\mu )}{(1-p)\omega }A$$

$$=c\beta S\sigma \frac{({\gamma }_2+\mu +\alpha )}{p\omega }I+c\beta S\eta \frac{(1-p)({\gamma }_2+\mu +\alpha )}{({\gamma }_1+\mu )p}I+c\beta SI-\frac{(\omega +\mu )({\gamma }_1+\mu )}{(1-p)\omega }A$$

$$\le \frac{(\omega +\mu )({\gamma }_2+\mu +\alpha )c\beta \mathrm{\Lambda }I}{\mu P\omega }\left[\frac{\sigma }{(\omega +\mu )}+\frac{\eta (1-P)\omega }{({\gamma }_1+\mu )(\omega +\mu )}+\frac{p\omega }{(\omega +\mu )({\gamma }_2+\mu +\alpha )}\right]$$

$$-\frac{(\omega +\mu )({\gamma }_2+\mu +\alpha )}{p\omega }I$$

$$=\frac{(\omega +\mu )({\gamma }_2+\mu +\alpha )}{p\omega }I(R_0-1)$$

when $R_0\le 1,V^{\prime }\le 0,$ when $V^{\prime }=0$ only if $I=0$, while $t\to +\infty ,$$E(t)\to 0,A(t)\to 0,I(t)\to 0,S(t)\to \frac{\mathrm{\Lambda }}{\mu }.$ Also $P_0\left(\frac{\mathrm{\Lambda }}{d},0,0,0,0\right)$ is the only equilibrium point, according to the LaSalle invariant set principle⁵²: The disease-free equilibrium point P₀ is globally asymptotically stable.

Using a similar method, we can prove that R₀ > 1, global asymptotic stability of endemic equilibrium point P^*. For a more detailed method, we can review references^16,17.

Author contributions

Yan-bin Du conceived the idea, performed the statistical analysis and wrote the main manuscript text. Hua Zhou is the guarantor of the overall content. All authors revised and approved the final manuscript.

Funding

This work was supported by Key Scientific and Technological Projects in Henan Province (grant number 242102320071).

Data availability

Data were obtained from the daily epidemic report issued by the Shanghai Municipal Health Commission, which is publicly available (https://wsjkw.sh.gov.cn/yqtb/index_2.html).You can also contact the corresponding author.

Competing interests

The authors declare no competing interests.