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Evaluation and Prediction of COVID-19 Prevention and Control Strategy Based on the SEIR-AQ Infectious Disease Model
Based on the SEIR model, which takes into account prevention and control measures, prevention and control awareness, and economic level and medical level indicators, this paper proposes an infectious disease model of “susceptible-exposed-infected-removed-asymptomatic-isolated” (short for SEIR-AQ) to assess and predict the development of the COVID-19 pandemic with different prevention and control strategies. The kinetic parameters of the SEIR-AQ model were obtained by fitting, and the parameters of the SEIR-AQ model were solved through the Euler method. Furthermore, the effects of different countries’ prevention and control strategies on the number of infections, the proportion of isolation, the number of deaths, and the number of recoveries were also simulated. The theoretical analysis showed that measures such as isolation for prevention and control and medical tracking isolation had a significant inhibitory effect on the development of the COVID-19 pandemic, among which stratified treatment and enhanced awareness played a key role in the rapid regression of the peak of COVID-19-infected patients. Conclusion of the Simulation. The SEIR-AQ model can be used to evaluate the development status of the COVID-19 epidemic and has some theoretical value for the prediction of COVID-19.
Coronavirus disease 2019 is a lung disease caused by SARS-CoV-2, which is highly lethal and infectious . Some scholars have clinically analyzed the causes of death in patients with confirmed COVID-19  and identified factors associated with the death of patients with COVID-19 pneumonia caused by the novel coronavirus SARS-CoV-2 . With 130566186 cumulative diagnoses and 2842363 cumulative deaths worldwide as of April 4, 2021, COVID-19 not only impacted the global economy but also deeply affected the governance of all countries in the world.
For COVID-19 epidemic prediction, most scholars used the classical SIR and SEIR models proposed by Beretta and Takeuchi  and Cooke and van den Driessche  to infer the COVID-19 peak time and maximum number of confirmed cases based on the existing data. Yu et al.  evaluated and predicted COVID-19 based on a SIR model with time-varying parameters to obtain expected inflection points and maximum number of confirmed cases. Wei et al.  and Geng et al.  studied the effect of prevention and control isolation measures on the development trend of the COVID-19 epidemic based on the SEIR model and concluded that strict prevention and control isolation measures can slow down the development trend of the COVID-19 epidemic. Wang et al.  established the SEIADR model by introducing asymptomatic infected individuals on the traditional SEIR model. Also, they predicted the development of the COVID-19 epidemic in Hubei Province, which had better fitting effect compared with the SEIR model. Shao et al.  used the classical SEIR model to conduct a predictive analysis of COVID-19 in Shandong Province and Korea, comparing the impact of control measures on the spread of COVID-19. Li et al.  fitted the COVID-19 regeneration coefficient () curve based on the SEIR model to predict and analyze the development trend of the COVID-19 epidemic in Hubei Province, China, America, India, Italy, and Iran and also predicted that the spread of the epidemic in Hubei Province would be better controlled compared with that of foreign countries. Lin , Chen et al. , and Ansumali et al.  introduced asymptomatic infected individuals based on the SEIR model, which led to a significant improvement in the fitting and prediction performance of the SEIR model. Ivorra et al.  developed the -SEIHRD model, where asymptomatic infected patients and medical conditions were taken into account, which could predict hospital bed demand more accurately. Pai et al.  improved the SEIR model by integrating government control policies, public health, and other factors to analyze the impact on the development trend of COVID-19 in India.
Although the above-mentioned studies have achieved certain effects, they only consider the latent infectious capacity of COVID-19-infected people, the infectious capacity of asymptomatic patients, prevention and quarantine measures, etc. In fact, many other factors including different medical levels, economic levels, and prevention and control awareness also have a great impact on the transmission of COVID-19. Therefore, based on the SEIR model, we divided susceptible people, contacts, and infected people into isolated and exposed states and then introduced hospitalized patients and asymptomatic patients as well as indicators such as prevention and control measures, prevention and control awareness, economic level, and medical level to construct a more interpretive SEIR-AQ paradigm. The development trend of COVID-19 at different levels was simulated by adjusting the parameters of prevention and control measures, prevention and control awareness, economic level, and medical level.
2. Establishment of the SEIR-AQ Model
The traditional SEIR model divides the population into susceptible people (), contacts (), infected people (), and recovered people (). However, the SEIR-AQ paradigm adds isolation of susceptible people (), isolation of contacts (), isolation of infected people (), asymptomatic patients (), and hospitalized patients (). Now, according to the proportion of , , and , we converted the isolated infected people, unisolated infected people, and asymptomatic patients into hospitalized patients . The SEIR-AQ paradigm assumes that the infected and quarantined people are not infectious during the isolation period, and the infected people are immune to cure. The parameter is the ratio of the transmission capacity of the unisolated contacts to the unisolated infected people , and the parameter is the ratio of the transmission capacity of the asymptomatic patients to the unisolated infected people . Therefore, the SEIR-AQ paradigm in this paper is more interpretive and adaptable. The warehouse conversion relationship of the SEIR-AQ paradigm is shown in Figure 1.
, , , and are the isolation ratio, the infection probability, the contact rate, and the effective contact coefficient, respectively, and is the effective contact rate. The conversion rate from unisolated susceptible people () to isolation of , , and unisolated contacts () is , , and , respectively. At the same time, considering the impact of unisolated infected people (), , and on susceptible populations, there is also isolation of that is retransformed into at a rate of . The natural death rate of is . Therefore, the governing equation for the number of susceptible people is
is the quarantine release rate, taking (the quarantine duration is 14 days).
The SEIR-AQ model considers different prevention and control measures, prevention and control awareness, economic level, and medical level:
, , and are the rates of conversion of to , , and , respectively, taking (the incubation period is 7 days); , , , and are the death rates of , , , and ; , , and are the rates of isolation of to , , and , respectively; and are the rates at which turns into and ; , , , and are the recovery rate of , , , and ; , , , and are the coefficient of the recovery rate of , , , and ; and is the probability that the infected people have symptoms.
In fact, will be immediately sent to a designated hospital for isolation and treatment during the COVID-19 epidemic; thus, will all be converted into in this model based on the SEIR-AQ model paradigm. Considering that the asymptomatic patients would not be taken to the hospital until any symptom was shown, we removed the asymptomatic patients into the relationship between the hospitalized patients and the infected people. , if confirmed, will be directly sent to the hospital and converted into . Therefore, the relationship that is converted to is removed. Based on the modified SEIR model established by Cao et al.  comprehensively considering the transmission characteristics of COVID-19, the population transformation relationship of SEIR-AQ, which is suitable for analyzing the COVID-19 epidemic situation, is shown in Figure 2. In Figure 2, the gray lines represent the transformation relationship deleted from the original and the red line represents the increased part converted from to .
At this time, the SEIR-AQ model is
The parameter represents the isolation ratio, represents the probability of infection, represents the contact rate, represents the rate of death due to illness, represents the rate where is converted to , represents the rate of recovery, represents the probability that the infected people have symptoms, represents the rate of converted to , represents the ratio of the transmission capacity of compared with , and represents the natural mortality rate.
3. Global Stability of the Equilibrium Point
In the proof of this paper, the following equation is considered apart from :
According to model (4), we can express as Then, the derivation with respect to is computed as
It shows that is an increasing function as the density of tends to go infinity, which implies that
According to equation (5), we can obtain
So the equilibrium point exists.
is the basic reproduction number of model (4). Then, there comes Proposition 1.
Proposition 1. Model (4) admits a disease-free equilibrium . If , then model (4) admits a unique endemic equilibrium : Next, the global stabilities of the equilibria of model (4) around the disease-free equilibrium and the endemic equilibrium are investigated in Theorem 2, respectively.
Theorem 2. If , then the disease-free equilibrium of model (4) is globally asymptotically stable. If , then the unique endemic equilibrium of model (4) is globally asymptotically stable.
4. Model Parameter Assignment and Verification
To further illustrate the applicability of the SEIR-AQ model, we select COVID-19 data from China, America, Brazil, and India for analysis. The numbers of BRIC, China, Brazil, and India share the similar economic level while America boosts higher level. At the same time, China and India have relatively higher population density. America and Brazil have similar populations and prevention and control awareness. Based on the SEIR-AQ model and the values of different parameters, the impact of different prevention and control measures, prevention and control awareness, economic level, and medical level on the development trend of the COVID-19 epidemic is simulated.
4.1. Parameter Setting and Fitting
The initial values of the SEIR-AQ model were referenced to the single-day confirmed data of COVID-19 in four countries from January 23, 2020, to November 10, 2020. The parameters were estimated in conjunction with relevant literature. The initial values are shown in Table 1, and the parameter values are shown in Table 2. The evaluation index of the SEIR-AQ model fit was the coefficient of determination , which is shown in Table 3.
From Figure 3, it can be concluded that the number of single-day confirmed cases of the COVID-19 outbreak in China increased rapidly at the beginning of the outbreak, peaked in February 2020, and then decreased rapidly to a stable and manageable state. The COVID-19 outbreak in America started in April and soon reached to top on April 5 and July 20, respectively. The overall trend of single-day confirmed cases continued to increase. Meanwhile, when it comes to Brazil and India, the number continued to decrease after the peak at August 1 and September 26, respectively, and continued to decrease thereafter. The predictions based on the SEIR-AQ model fit basically matched the actual epidemic development trend, and the SEIR-AQ model fitting evaluation index determination coefficient is greater than 85%, indicating that the SEIR-AQ model has a significant fitting effect and can predict the development trend of the COVID-19 epidemic.
5. Further Discussion of the SEIR-AQ Model
In this paper, we analyze and study COVID-19 based on the SEIR-AQ model, theoretically analyze the law of COVID-19 evolution, and analyze the influence of different countries’ prevention and control measures, prevention and control awareness, economic level, and medical level on their COVID-19 evolution, in which the peak number of new confirmed cases in each country when the parameters change is shown in Table 4.
5.1. Assessment of the Impact of Prevention and Control Measures and Prevention and Control Awareness on the COVID-19
Since the emergence of the COVID-19 epidemic, the Chinese government has decisively adopted strict prevention and control measures to reduce and increase . Under the positive instructions of the Chinese government, the citizens gradually raise their awareness of epidemic prevention and control, which reduced and minimized the development of clustered epidemics. As a result, the COVID-19 epidemic tends to be in a more controllable and stable state. In the initial period of the COVID-19 outbreak, only few effective prevention and control measures were taken by the United States government. This then lead to large and small which finally resulted in a significant increase in the number of the COVID-19 confirmed cases in a single day. Relatively weak awareness of prevention and control among U.S. nationals led to large . Although America strengthened its prevention and control measures in the later period, it had missed the prevention and control window period, which led to the development of the COVID-19 epidemic. In April 2020, the COVID-19 epidemic developed on a large scale in Brazil. The lack of timely adoption of prevention and control measures proposed by WHO indirectly led to lower awareness of prevention and control among the Brazilian population, which increased and and lowered . The number of single-day confirmed cases in Brazil continued to hit a record high. But in June, the government introduced compulsory epidemic prevention measures, such as the adoption of entry restrictions. The number of single-day confirmed cases has gradually decreased since the beginning of August. With only a few hundred cumulative cases, India has adopted prevention and control measures relatively early. Since a city closure policy was implemented on March 25, 2020, the country went into emergency closure. Considering that India boosts a huge population and most of Indians lack enough literacy level and awareness of prevention and control, high-density aggregation of people have caused the virus to spread across the country, resulting in large , small , and small , which promotes the outbreak of India’s COVID-19. In the theoretical analysis, we simulate the development trend of the COVID-19 epidemic under different prevention and control measures by changing , , and , which are used to evaluate the impact of prevention and control measures on the development trend of the COVID-19 epidemic. This is shown in Figures 4–6.
As can be seen from Figure 4,when is 1.5 times the actual situation in each country, the number of single-day confirmed cases in China will reach a peak of about in mid-February 2020. In comparison, China reached the earliest and the number of confirmed cases per day tends to be zero at the end of March and remains stable. In America, the number of single-day confirmed cases has continued to increase after reaching . The peak in Brazil has reached about and then gradually decreased. In India, the number of single-day confirmed cases reached in August. It will continue to increase to and then gradually decreased. When comparing the original curve () with the different values of the quarantine ratios for each country, it can be observed in Figure 5 that there is a significant delay in the peak number of single-day confirmations at each stage in America, and the peak number of single-day confirmations in the other three countries is correspondingly earlier. The time is correspondingly advanced. As increases, the peak number of confirmed cases in a single day decreases. When the quarantine ratio of each country is 0.5 times the actual situation, the number of single-day confirmed cases in China will reach a peak of about in February 2020 and stabilize in March. Around June 2020, the number of confirmed cases in America will reach and continue to increase to reach the second-stage peak of about . The number of COVID-19 confirmed cases in Brazil and India reached in a single day in August 2020, respectively, but both will continue to increase to their respective peaks. According to Figure 6, at of 0.25 times, it takes only one month for the number of confirmed cases in China to reach the highest peak of in a single day, while it takes only one month for Brazil and India to reach in mid-to-late July. They need to continue to increase for about a month before reaching their respective peaks. The second-phase peak of the epidemic in America was about . It is expected to continue to increase on a single day.
In China, the government has adopted greater and more rapid prevention and control measures. Chinese people have a strong awareness of prevention and control. So the development of COVID-19 reaches its peak fastest and is rapidly stable. In Brazil and India, the prevention and control measures and people’s awareness of prevention and control were relatively weak so that the COVID-19 epidemic reached its peak later. In the United States, the prevention and control measures and people’s awareness of prevention and control are weak and continue to increase after reaching the peak of the stage. It indicates that strict and timely prevention and control measures are taken. The higher the awareness of prevention and control, the more effectively the development of the COVID-19 epidemic can be suppressed.
5.2. Assessment of the Impact of Medical and Economic Levels on the COVID-19
As of May 31, 2020, China have allocated a total of ¥162.4 billion in COVID-19 prevention and control at all levels of finance to ensure that financial support for COVID-19 prevention and control measures is in place. After the COVID-19 outbreak, the Chinese government has effectively increased the recovery rate through a large-scale free nucleic acid test, thereby ensuring receiving timely treatment of patients. America invested $8.3 billion in the prevention and control of COVID-19 in the early stage of the COVID-19 outbreak. However, the number of single-day confirmed cases in America reached a peak after April 5, 2020, due to the limited testing capacity. To prevent the collapse of the medical system, the U.S. government adopted restrictions on testing, resulting in low . Then, the number of single-day confirmed cases fluctuated up and down at the peak. Brazil’s uneven distribution of medical resources, insufficient reserves, and a fragile medical system have led to low , which directly led to the acceleration of the increase in the number of single-day confirmed cases of the COVID-19 epidemic. India’s weak basic medical facilities and social medical security system, low virus detection capacity, small size of land area, high population density, and lack of medical equipment have led to the inability to treat infected patients in a timely manner, resulting in low . This has led to a rapid increase in the number of single-day confirmed cases in June 2020. In the theoretical analysis, we simulate the development trend of the COVID-19 epidemic under different medical and economic levels by changing which is used to evaluate the impact of the medical level and economic level on the development trend of the COVID-19 epidemic, as shown in Figure 7.
It can be seen from Figure 7 that under the same scenario (2.0 times the actual ), the COVID-19 epidemic in China will reach a peak of in early February 2020, with the number of single-day confirmed cases stabilizing around March. The number of single-day confirmed cases in Brazil will reach a peak of about in July 2020 and stabilize after December. The number of single-day confirmed cases of COVID-19 in India will reach in mid-April and then continue to increase to a peak of . As in the three countries increases, the time to reach the peak of the number of single-day confirmed cases will be correspondingly advanced and the peak will be correspondingly smaller, with the most significant effect in India. Around July 2020, the number of single-day confirmed cases in America reached , which dropped slightly after reaching the peak of the second phase. However, the number of single-day confirmed cases increased significantly in October.
The Chinese government has invested heavily in the prevention and control of the COVID-19 epidemic and quickly adopted prevention and control measures. The COVID-19 epidemic has stabilized quickly. In Brazil and India, the number of single-day confirmed cases have peaked and stabilized due to fragile medical systems and other reasons. In the United States, although a large amount of money was invested in prevention and control, the uneven distribution of follow-up medical resources has led to a continuous increase in the number of single-day confirmed cases. It shows that increasing capital investment in epidemic prevention and control and sufficient medical resources are conducive to controlling the development trend of the COVID-19 epidemic.
In this paper, the transmission dynamic characteristics of COVID-19 are analyzed in four countries based on the SEIR-AQ model: China, America, India, and Brazil. As the economic level, medical level, prevention and control awareness, and prevention and control policies adopted by the four countries China, America, India, and Brazil are significantly different, we fully compare the differences in dynamic parameters such as , , , and among the above four countries and focus on the effects of prevention and control measures, awareness of prevention and control, economic level, and medical level on these parameters. A series of evidence shows that the control of the development of the COVID-19 epidemic requires the local government to quickly take prevention and control measures to minimize the number of exposed people in order to reduce the exposure rate and increase the isolation rate. On this basis, it should also increase financial investment in improving the economic level and medical level to increase the recovery rate, to realize the rapid conversion of confirmed patients to the recovered population. In addition, it is also crucial to raise awareness of prevention and control.
The labeled dataset used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare no conflicts of interest.
This work is supported by college students’ Innovative Entrepreneurial Training Plan Program Grant No. 202010295015Z and the National Natural Science Foundation of China under Grant 61702225.
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