Abstract
There are many prevention and control measures for emerging infectious diseases. This paper divides the effects of these measures into two categories. One is to reduce the infection rate. The other is to use diagnosis rate to reflect the decreases of the infection source. The impacts of measures intensity, diagnosis rate, and the start time of taking measures on emerging infectious diseases with infectious capacity during the incubation period are considered comprehensively by using a differential equation model. Results show that for each diagnosis rate, the number of infections and deaths has a phase change structure with respect to the measures intensity. If the measures intensity is less than the value of the phase change point, the epidemic will break out whenever measures are taken. If the measures intensity is greater than the value of the phase change point, the epidemic can be controlled when the measures are taken timely. But if measures are not taken in time, epidemic will also break out. The location of the phase change point is related to the diagnosis rate. For the different measures intensity and the diagnosis rate, this paper gives a method to judge whether the spread of the corresponding epidemic can be controlled.
1. Introduction
The term “emerging infectious diseases” was first proposed by Lederberg et al. which refers to the new infectious diseases appearing in the last 20 years, including those caused by newly identified species of pathogens or pathogens affecting a new population, as well as reemerging infections [1]. Due to the lack of awareness of emerging infectious diseases, people are unable to formulate prevention and control strategies in time, resulting in a widespread of emerging infectious diseases and a large number of infected people. Moreover, human beings do not have natural immunity to resist emerging infectious diseases, which will seriously harm health. The variety of emerging infectious diseases has attracted worldwide attention, and researchers have strengthened their research on emerging infectious diseases.
The compartment model in epidemiology is a powerful mathematical tool for describing and analyzing the complex behavior of epidemic. In 1927, Kermack and McKendrick proposed the famous SIR model [2]. Due to the different transmission characteristics of infectious diseases, many new epidemic models, such as SIRS model [3–6], SEIR model [7–10], and SEIRS model [7, 10–13] have emerged on the basis of SIR model in terms of its transmission mechanism. It is of great theoretical value and practical significance to establish epidemic model and to analyze the spreading mechanism of emerging infectious diseases.
Among the many emerging infectious diseases, there is a category of epidemic with an incubation period, such as AIDS, influenza A (H1N1), SARS, Ebola virus disease, and COVID-19. Since patients in the incubation period do not show any symptoms and are infectious, it is difficult to control the epidemic. Therefore, in recent years, scholars have paid attention to the emerging infectious diseases with an incubation period [14–16]. Mukandavire et al. proposed an epidemic model of discrete delay differential equations to study the impact of public health education on the spread of AIDS [17]. Naresh et al. used the stability theory of differential equation and numerical simulation to carry out model analysis, and the results showed that contact tracing is of great help in reducing the spread of AIDS [18]. In 2006, Lekone et al. established the SEIR model of Ebola virus transmission and used the maximum likelihood estimation method and Monte Carlo simulation method for parameter estimation [19]. Then, Diaz et al. considered the impact of hospitalization and the interaction of a susceptible individual with a deceased but infectious individual on the spread of Ebola [20]. To understand the evolution of the 2009 influenza A (H1N1) pandemic in local areas of Japan, Saito et al. studied the importance of population migration between regions [21]. Due to the strong infectivity of COVID-19, it has spread all over the world and has been studied from different aspects. Various transmission models of COVID-19 have been proposed to predict and control the spread of the disease [22–26]. Luo et al. proposed that contact tracing isolation and household quarantine are other prevention and control measures for emerging infectious diseases and then play a crucial role in controlling COVID-19 [27]. Lin et al. used the SEIR model to describe the process of the COVID-19 outbreak and comprehensively considered the effects of extended vacation, travel restrictions, hospitalization, and isolation measures on the epidemic, providing an important reference for understanding the trend of the epidemic [28]. Luo et al. improved the topological structure of complex networks by isolating high-degree models. Meanwhile, they found the isolation time threshold of taking measures will infect the final size of infectious diseases [29]. Nanshan Zhong’s team predicted the epidemic trend of COVID-19 under public health intervention in China [30]. At present, the research on epidemic models involves various aspects, such as the study of the influence of population age structure on the epidemic [31, 32] and the study of perturbing specific parameters in the model [33–37].
Based on the experience of the prevention and control of several major emerging infectious diseases in the past, timely formulation of prevention and control strategies can effectively curb the spread of epidemics. However, due to the unpredictability of emerging infectious diseases, prevention and control measures are often unable to be implemented at the initial moment. Therefore, this paper establishes an SEIHR model to study the impact of phased prevention and control measures on emerging infectious diseases with infectious capacity during the incubation period.
This paper is organized as follows: Section 2 introduces a transmission dynamics model of emerging infectious diseases during the incubation period. Section 3 presents numerical results and analysis, and Section 4 concludes this paper.
2. Model
There is a lack of understanding of emerging infectious diseases, and the development of vaccines usually needs to take a long time, so the epidemic can be controlled only through prevention and control measures. People are often required to wear masks, close crowded public place, cancel mass gatherings, and make fewer trips outside. The effect of such measures is to reduce contact, which can reduce infection rate and thereby achieve the aim of epidemic prevention and control. Another type of measures is reflected in isolation, such as the isolation of confirmed infections and the isolation of close contacts of confirmed infections. The purpose is to reduce the number of infection sources. Therefore, in the prevention and control of the epidemic, the measures adopted can be divided into two types: one is reflected by reducing the infection rate and the other is reflected by reducing the number of infection sources.
Measures intensity is used to reflect the reduction of infection rate, and the infection rate becomes times of the original infection rate. Use the diagnostic rate to reflect the reduction in the number of infection sources. Infected people are quarantined after diagnosis to reduce the number of infections. Consider an emerging infectious disease during the incubation period; the individuals in a population can be classified into susceptible , incubation period , infected but not quarantined , hospitalized , recovered , and died . The transformation relationship of populations is shown in Figure 1.
Based on the above transformation diagram, the differential equation model can be established as follows:. Here, represents the contacts rate and represents the measures intensity, and the infection rate will be times of the original infection rate if measures are taken ( means that there is no measure). The susceptible individual will be infected with probability in contact with the incubation individuals or the infected but not quarantined individuals. Let be the decreasing proportion of infectivity in the incubation period . The incidence rate of infections with incubation period is . Symptomatic infections are diagnosed with probability and then hospitalized for isolation treatment. The self-healing rate of infected but not quarantined is , and the mortality rate is . The recovery rate of hospitalized is , and the mortality rate is .
3. Results
In what follows, we use model (1) to study the impacts of , and start time of taking measures on the epidemic. Take COVID-19 as an example; according to [38], when , the parameter values are estimated as follows: , and .
Because the understanding of the characteristics of emerging infectious diseases and the extent of their harm increases over time, measures to reduce the infection rate are generally not taken at the onset of an outbreak. Here, measures to reduce the infection rate are considered from time ; that is, before time , . First, the evolution processes of all kinds of individuals in model (1) are considered after the parameter value is given. Take the parameter . When and , the results are as shown in Figure 2.
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One can see from Figures 2(a) and 2(b) that the epidemic size and deaths will eventually stabilize. Figure 2(c) shows that the peak of the current number of infections appears on the 58th day. From Figure 2(d), the peak of the current number of people carrying viruses (the number of infection sources) appears on the 50th day and then gradually decreases to 0, so that there is no infection source. We stipulate that the epidemic will end when the number of infections that can infect others is 0, so the corresponding epidemic end time is 123 days.
When and , the epidemic final size, the number of deaths, and the duration of the epidemic can be obtained. Next, we consider the effect of . When , the corresponding epidemic final size, death number, and the duration of the epidemic are as shown in Figure 3.
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Figure 3(a) shows that when no measures are taken to reduce the infection rate , the proportion of epidemic final size is very high (reaching 82%). With the increases of the measures intensity, the epidemic final size decreases. It is worth noting that the epidemic final size has a phase change structure; that is, there is a measures intensity . When the measures intensity is less than , the epidemic final size is at a high level. Conversely, when the measures intensity is greater than , the epidemic final size drops rapidly. In Figure 3(a), is about 0.55. The epidemic final size at is reduced from 7.14 million to 5,178 at , a decrease of 7.13 million.
Figure 3(b) shows that with the increase of measures intensity, the number of deaths decreases, which is similar to Figure 2(a). The number of deaths also has a phase change structure, and the phase change point is the same as that of Figure 2(a). The number of deaths at is reduced from 217,380 to 157 at , a decrease of 217223.
In Figure 3(c), there is a peak in the duration of the epidemic and the corresponding value of is the same as that of the phase change point in Figures 3(a) and 3(b). When the measures intensity is less than the phase change point, the number of infections will be large. Due to the high infection rate, the susceptible will be infected quickly in a short time and the number of infections will increase rapidly. Therefore, the number of susceptible will be small, and the potential for the infections number to increase will be small. According to the law of transmission, those infected individuals eventually become immune or die, so the epidemic will be over. The duration of the epidemic is 302 days when the number of infections is high, that is, when the measures intensity is 0. At the phase change point, the duration of the epidemic is 10706 days. This is because although the number of infections is large, with the increases of the measures intensity, the infection rate decreases, so the number of susceptible is relatively large and the number of infections increases slowly. Therefore, the epidemic will last a long time. In Figure 3(b), this corresponds to a high mortality rate (more deaths). When the measures intensity is greater than the phase change point, the duration of the epidemic decreases with the increases of the measures intensity. This is due to the increased measures intensity reduces the infection rate to a level where the number of infections is very low. Thus, the duration of the epidemic is shortened. This corresponds to the low mortality in Figure 3(b).
According to Figure 3, both the epidemic final size and deaths have obvious phase change points and the measures intensity can be roughly divided into three regions: the conservative prevention and control area that is much larger than the phase change point; the development prevention and control area that is larger than the phase change point and close to the phase change point; and the out-of-control area that is smaller than the phase change point. The phase change point divides the duration of the epidemic into two different change directions.
For the purpose of epidemic prevention and control, weak measures intensity can be selected in the conservative prevention and control area. In this region, the epidemic can be contained due to strict measures. Further strengthening of prevention and control measures is not very meaningful, which is equivalent to excessive protection.
If the prevention and control capabilities are insufficient or economic development and other factors are considered, at least the prevention and control measures should be within the development control area. Although the prevention and control effect of the epidemic is not as good as the conservative prevention and control area, it will not cause a large-scale outbreak of the epidemic. But the duration of the epidemic will be longer than that of the conservative area.
The measures in the out-of-control area are not strong enough, and the epidemic will break out. In the out-of-control area, if the measures are weak, the end time of the epidemic will be advanced, but the consequence is that most individuals are infected and there are many deaths.
Figure 3 shows the results at . Next, we consider the impact of the start time of taking measures on the evolution of the epidemic. When , the epidemic final size, the death number, and the duration of the epidemic are as shown in Figure 4.
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It can be seen from Figure 4(a) that as long as the measures are taken in time, there will be a phase change point . If the measures intensity is less than , no matter when the measures are taken, the epidemic final size will be large (about 82% of the total number of individuals). Meanwhile, the epidemic final size has nothing to do with the start time of taking measures. That is to say, if the control measures are not strict enough, no matter what time to start taking measures, it will eventually lead to epidemic outbreak. If the measures intensity is greater than , it means that the measures are stricter and the epidemic can be controlled to a certain extent. From the perspective of the epidemic final size, the earlier the measures are taken, the better the control effect. However, further strengthened measures intensity in this region does not contribute much to the reduction of infected. It can also be found from Figure 4(a) that the phase change point of the measures intensity corresponding to the different start time of taking measures is the same; that is, the position of the phase change is constant with respect to the start time of taking measures. It can also be seen from Figure 4(a) that when the measures are taken later, such as when the measures are taken on the 110th day, the phase change structure will disappear. At this time, regardless of the measures intensity, the epidemic final size and number of deaths are at higher values. At this time, it is too late to take measures. No matter how strict the measures are, the epidemic cannot be controlled.
The property of Figure 4(b) is exactly the same as that of Figure 4(a). When it is not too late to take action, the number of deaths at different moments has a phase change point. And the position of the phase change point is constant with respect to the start time of taking measures.
As can be seen from Figure 4(c), if the measures intensity is greater than the phase change point , the earlier the measures are adopted, the shorter the duration of the epidemic will be. However, if the measures intensity is less than the phase change point, then the time to start taking measures has little effect on the duration of the epidemic.
In Figure 4, the phase change point of the measures intensity corresponding to the epidemic final size and number of deaths is independent of the start time of taking measures, and it will be studied whether the phase change point is related to the diagnosis rate.
Taking the parameters , the results of the epidemic final size and death number are as shown in Figure 5.
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According to Figure 4 and 5, in the case of taking measures timely, for each diagnosis rate, the measures intensity has a phase change structure relative to the number of infections and deaths. The phase change structure has nothing to do with the start time of taking measures but is related to the diagnosis rate. The higher the diagnosis rate, the lower the measures intensity corresponding to the phase change point.
In addition, for different diagnosis rates, there is a maximum start time that can make Figure 5 appear as the phase change structure. If the start time of taking measures is greater than this value, there will be no phase change structure, that is, no matter how strong the measures are, the epidemic cannot be controlled. And the greater the diagnosis rate, the greater the maximum value of the start time that can make Figure 5 appear as the phase change structure. By improving the diagnosis rate, you can strive for the longest possible period of time, so that the start time of taking measures is located in this period of time, which is conducive to epidemic control.
Figures 4 and 5 show that if measures are taken in time, different diagnosis rates correspond to different phase change points. Because for each diagnosis rate, the phase change will occur when measures are taken at 50 days, and it corresponds to the phase change point of this diagnosis rate. Therefore, when the start time of taking measures is the 50th day, it is general to study the position of the phase change point at different diagnosis rates. When , the corresponding phase change points are as shown in Figure 6.
The curve in Figure 6 is a dividing line. The two regions correspond to the combination of the diagnosis rate and measures intensity for whether the epidemic is controllable or not. Region I is the controlled area, and region II is the out-of-control area. The value of needs to be within region I and as far as possible away from the line to control the outbreak. The closer the value of is to the dividing line, it is easier to enter the out-of-control area.
Region I has a characteristic that under the condition of , the increase of the diagnosis rate can allow a relatively low measures intensity and the increase of the measures intensity can allow a fairly small diagnosis rate. In other words, when the diagnosis rate is not large enough, the measures intensity can be increased to achieve the purpose of controlling the epidemic, and with the increase of the diagnosis rate, the measures intensity can be appropriately weakened and the control of the epidemic can still be guaranteed.
It can be seen from Figures 4 and 5 that for the measures intensity in area I, the phase change does not occur at any start time of taking measures. When the start time is not timely, no phase change structure occurs and the epidemic final size and death number are both large. Next, the impact of the start time of taking measures on the results will be discussed.
First of all, take parameters . Note that are in area II. The effects of different start time of taking measures on the epidemic size and death number are as shown in Figure 7.
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In Figure 7, when is in area II, no matter when the measures are taken, there are many infected individuals and deaths. Although increasing the measures intensity can reduce the epidemic size and death number, the overall number of infections and deaths is too large. When is in area I, the epidemic size and death number also have a phase change structure for the start time of taking measures. That is, there is , so that when , the epidemic size and death number are very small, which can be understood as the epidemic is under control. When , delaying the start time of the measures by one day can increase the epidemic size and death number. When , the epidemic size and death number quickly reach the maximum and the epidemic outbreaks. In other words, when belongs to area I, as long as the start time of taking measures is timely, the epidemic can be controlled.
Figure 7 is the result of . Next, the effects of the start time of taking measures on the epidemic size and death number when will be considered. Results are shown in Figure 8.
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It is important to note that the values of are , and ; these values are in area I. As can be seen from Figure 8, for each value of the diagnosis rate, both the epidemic size and death number correspond to the start time of the measures . When the actual start time for taking measures is earlier than this value, the epidemic can be controlled. Moreover, the higher the diagnosis rate, the greater the , indicating that there is an opportunity to take measures later if the diagnosis rate is high.
4. Conclusion
Many measures are taken in the prevention and control of emerging infectious diseases with infectious capacity during the incubation period. This paper classifies the effects of these measures into two categories: One is reflected in the reduction of the infection rate, such as curb population flow, close crowded public place, cancel mass gatherings, make fewer trips outside, and wear masks. Another type of measures is reflected in the isolation, such as the isolation of confirmed infections and the isolation of close contacts of confirmed infections. Such measures are mainly to reduce the number of infection sources. Isolation is related to the diagnosis rate of emerging infectious diseases. The faster the diagnosis, the higher the diagnosis rate and the greater the reduction in the number of infection sources. Therefore, the effect of such measures can be reflected by the diagnosis rate. In this way, the preventive and control measures for emerging infectious diseases are reflected in two categories: one is to reduce the infection rate and the other is to increase the diagnosis rate. In addition, due to cognitive or objective ability, it is not guaranteed that effective measures can be introduced at the outset of the epidemic, so the start time for taking measures is also an important factor.
The differential equation is used to describe the propagation of emerging infectious diseases with infectious capacity during the incubation period. We introduce the measures intensity to reflect the effect of measures such as reducing the infection rate. On this basis, the impacts of the measures intensity, diagnosis rate, and the start time for taking measures on the epidemic transmission are studied.
The results show that no matter what the diagnosis rate is, there is a phase change structure in the epidemic size and death number in terms of the measures intensity. If the actual measures intensity is less than the intensity of the phase change point, the epidemic cannot be controlled no matter when the measures are taken. Conversely, if the measures intensity is greater than the intensity of the phase change point, as long as measures are taken in time, the epidemic can be controlled. But if the measures are taken too late, the epidemic cannot be controlled.
The location of the phase change point is related to the diagnosis rate. Based on whether the epidemic can be controlled, the space between the measure intensity and the diagnosis rate is divided into two parts: one is the controllable area and the other is the out-of-control area. To control the epidemic, the diagnosis rate and measures intensity need to be within the controllable area.
In addition, if measures are not taken in time, the epidemic cannot be controlled regardless of the measures intensity. Therefore, it is necessary to take measures as soon as possible, and it is effective to improve the diagnosis rate or reduce the infection rate.
In this study, the diagnosis rate and measures of intensity are fixed throughout the epidemic period without considering the change over time. So this is a problem that can be studied in future work.
Conflicts of Interest
The author declares no conflicts of interest.
Acknowledgments
This work was funded by the National Nature Science Foundation of China (Grant 11471151).