Background
Respiratory infectious diseases, especially strains of influenza A and ADV such as H1N1, H7N9, ADV 7, and ADV 55, often lead to worldwide outbreaks and seriously endanger human health. For example, from late April to the end of 2009, the local H1N1 flu epidemic peaked in most countries, and approximately 70,000 laboratory-confirmed hospitalized patients and 2500 fatal cases were observed across 19 countries or administrative regions [
1,
2]. Epidemics of ADV infection often occur in healthy children or adults in closed or crowded settings (particularly in communities, military recruit training centres, hospitals, and chronic care facilities) worldwide [
3‐
6]. Fatality rates for untreated severe adenovirus-associated pneumonia or disseminated disease may exceed 50% [
7].
ADV 7 outbreaks are very common among military trainees in many countries [
8‐
13], most likely due to the close living quarters of trainees, the persistence of adenoviruses in the environment when infectious agents from epidemic areas enter the camp, the susceptibility of the general population to some variants [
14], and low vaccine coverage [
15]. These diseases can spread to create a large-scale outbreak in a very short period of time. Some viral strains can cause serious intrapulmonary infection and even lead to death. Therefore, determining the precise timing for disease control and adopting comprehensive scientific measures to control the spread of an epidemic are demanding challenges facing the public health systems of every country. To achieve the objectives discussed above, theoretical research on the dynamics of epidemic transmission is needed, and the time of control and the impact of measures on the attack rate must be quantitatively analysed.
Mathematical models of infectious diseases can help deepen our understanding of the epidemiological distribution of infectious diseases. Currently, the most commonly used model is the Susceptible-Exposed-Infectious-Recovered (SEIR) model, from which many models have been derived and widely adopted to analyse infectious outbreaks of Ebola, tuberculosis, and influenza, among other diseases [
16‐
18]. Indeed, the SEIR model has proven to be critical for revealing the epidemiological characteristics of infectious diseases. However, this model has some limitations in the analysis of outbreaks of respiratory infectious diseases. For example, SEIR-based models frequently assume that the effective contact rate (the number of people infected by one infector within the time unit when all exposed persons are susceptible) is a constant or a continuous function [
16‐
18], i.e., that infectors transmit the virus continuously. In reality, these contacts occur randomly, and time intervals exist between infection events. Furthermore, according to the SEIR model, as long as someone within the population is infected and the effective contact rate is greater than 0, an outbreak will be triggered, and the disease will spread continuously. However, again, in reality, even if someone in the population becomes infected, an epidemic outbreak may not occur, and even without human intervention, outbreaks typically end before all susceptible people become infected. For example, Justin L reported that 35% of students had an influenza-like illness during an H1N1 influenza outbreak in a middle school [
19]. Additionally, the SEIR model assumes that all infectors display the same epidemiological characteristics in their effective contact rates, incubation periods, symptom duration, and treatment duration, but these factors vary across patients. For example, Justin L reported that the 95% confidence interval was between 1.0 and 1.8 for the median incubation period for confirmed H1N1 influenza and between 1.7 and 2.6 for the development of symptoms [
19]. Another factor to consider is that the activities of the exposure population are not constant. For example, soldiers in military camps train together during the day, and at night, they rest in the dormitory with their squad unit. Thus, the close contacts of the infectors change over time, but the SEIR model fails to reflect this element.
To overcome the limitations of the SEIR model, we sought to establish an individual-level stochastic research model to simulate the spread dynamics of epidemic outbreaks in the real environment. Such models have been used in teaching and research related to the epidemiology of infectious diseases. For example, Eichner M used stochastic computer simulations to examine whether case isolation, contact tracing, and surveillance can extinguish smallpox outbreaks in highly susceptible populations within less than half a year without causing more than 550 secondary cases per 100 index cases [
20]. Salathe M modelled the spread of an infection in a “small-world” network based on computer simulations to assess how a personal opinion about vaccination affects the probability of disease outbreak. The study found that the inclusion of opinion formation led to frequent outbreaks in a homogeneously vaccinated population with vaccination coverage of less than 70% [
21]. Williams A employed a discrete time simulation environment to model a virtual town that experienced a bioterrorist attack of pneumonic plague and assessed the attack rate under the influence of a mass treatment centre and home isolation. They found that an attack rate of 93% was approximately equal to the expected theoretical attack rate if
R0 = 2.85 [
22]. In addition, Cremin I presented a teaching exercise in which an infectious disease outbreak was simulated over a five-day period and found substantial variation in the cumulative attack rate, with between 26 and 83% of the students uninfected at the end of each outbreak [
23]. Although these studies employed the concept of individual-level and random contact among people, they did not fully account for certain factors, such as differences in patient contact behaviour during day and night, the time of isolation, and the duration from onset to isolation, which influences morbidity. Therefore, a stochastic model for the prevention and control of outbreaks of respiratory infectious diseases in a military camp is still lacking.
We chose to use ADV 7, which has high incidence and poses serious health threats in the army, to establish a random collision model that simulates the complete occurrence and development of an ADV 7 outbreak with effective intervention measures. This model not only provided greater flexibility in setting the scope of the population’s activities and enabled the depiction of the transmission network of the outbreak but also permitted quantitative analysis of the impact of intervention measures, thereby providing a scientific basis for targeted prevention and control of the outbreak.
Discussion
In our study, we used an idea completely different from the SEIR model. The random collision model has the following specific advantages: 1. The model can more precisely describe the process of epidemic transmission. We used the individual subject as the study unit, and in the programme, each patient’s file contains a record of the times of infection, attack, isolation, and rehabilitation; susceptible persons he might have infected; and whether his range of activities was restricted during the day and night. Not only is this approach conducive to inquiring about the disease development process in each patient, but the transmission chain of the infection can be drawn, enabling in-depth analysis of the transmission path of an infectious disease in a crowd. 2. Randomization is in greater agreement with the transmission characteristics of an infectious disease. Contact between patients and susceptible persons is random rather than continuous. The effective contact rate, incubation period, treatment duration, and immunity of the patients are also in accordance with a random distribution. We randomly sampled from the probability distributions, shown in Table
1, that were obtained from the actual epidemic situation and distributed to each patient. This sampling can ensure the authenticity and scientific integrity of the research. 3. This method can be extended to other infectious diseases and their occurrence scenarios, e.g., tuberculosis outbreaks in schools or the spread of HIV among gay men; we can also set patient activities in programmes, such as in a school where students attend classes during the day and return home at night or board at the school. Furthermore, more complex factors affecting epidemic transmission can be integrated into the programme, providing flexibility and diversity that the SEIR and other traditional models cannot achieve.
Additionally, in the following four paragraphs, we discuss the impact of the four indicators, R0, TOI, IOI, and IR, on the attack rate in the outbreak.
Overall,
R0 was positively correlated with the attack rate at a PRCC of 0.61, which was the highest absolute value among the four parameters included in the comparison, indicating that
R0 has the strongest influence on the attack rate. Notably, the median attack rate did not continue to increase with an increase in
R0, which differs from the SEIR theory that an epidemic will be triggered once
R0 > 1 [
26]; rather, it began to increase dramatically at a certain critical point. The reason for this dramatic increase is that when the value of
R0 is small, even when a source of infection is present within the crowd, a patient’s ability to spread the disease is weak, and he will recover before the disease is transmitted to other susceptible people. As
R0 increases, the speed of disease transmission increases, and the cumulative effect is amplified in a manner that corresponds to the increase in the number of disease generations. This phenomenon reminds us that as long as appropriate preventive measures (such as health education and active circulation of indoor air) are taken to keep
R0 at a low level, serious disease outbreaks can be prevented.
Timely isolation of patients after an outbreak can very effectively control further outbreaks of an epidemic. Our results showed that when the R0 stays constant, with a delay in the TOI, the probability of a total patient number exceeding 80 people initially remains very low, then rises sharply, and finally reaches a high level and remains there. By contrast, SEIR theory posits that the attack rate will continue to rise as the TOI is delayed. This pattern reveals that when isolation treatment is carried out at the early/beginning stage of an outbreak, the attack rate can be controlled at a lower level; however, with postponement of isolation measures, the total number of patients will increase very quickly, and if isolation is initiated too late, outbreaks of the epidemic become extremely difficult to control. When the TOI occurred before the growth rate peaked, the growth rate displayed a phased trend of an initial increase, followed by a rapid decrease and a final slow decrease. This trend was observed because many people became infected before being isolated; although those who became sick after the TOI were isolated within 1 day of the onset of illness, those patients may have had contact with others and thus could have transmitted the virus. A small number of the individuals infected during this period will become new patients after the incubation period, which averages approximately 5 days.
Timely diagnosis and treatment of patients following early onset of the disease can reduce the number of susceptible people who are infected. The PRCC for the IOI was 0.45, indicating a strong positive correlation. Under a constant R0, the probability of an outbreak gradually increased as the IOI was extended. The median attack rate remained very low at first, but when the IOI reached a threshold, the attack rate increased rapidly and then slowed. However, according to SEIR theory, no such threshold exists. This trend suggests that we can effectively reduce the risk of an outbreak by taking isolation measures within a certain time frame. The earlier that detection, diagnosis, and isolation are performed, the greater the possibility that the disease attack rate will remain low.
Immunization is an effective approach for preventing infectious diseases. The PRCC between the IR and attack rate was − 0.27, indicating that the higher the IR among the population, the lower the attack rate will be. The results section shows that as the IR increased gradually from 0, the probability of outbreaks decreased steadily. In addition, the median attack rate continued to decrease rapidly until it reached a critical point, after which it remained at an extremely low level; this outcome diverges from the SEIR model, which posits that an epidemic will not occur once the IR increases to R0 < 1. This outcome occurs because when the IR increases to a certain extent, patients will not be able to continue to infect more susceptible people. This trend indicates that the outbreak of an epidemic can be efficiently restricted if the IR reaches a critical point. However, if it does not reach that critical point, disease prevention will be limited. In general, the relationships between the attack rate and the above four parameters were similar: all displayed a sharp rise in the attack rate after the parameter reached a certain critical value, indicating that the risk of an epidemic outbreak is manageable. Nonetheless, if the measures taken are not effective, the difficulty of controlling the outbreak will increase rapidly.
Although we present some original findings, our study has some limitations. For example, the suitability of the random collision model for diseases that have a more chronic prevalence among the population (such as tuberculosis and AIDS, among others) still requires further discussion, although the value of the SEIR model for these diseases has been confirmed. Because cluster outbreaks have fewer influencing factors and shorter durations, it is relatively easy to establish a random collision model. However, for certain other chronic diseases, modelling requires the consideration of various additional factors, including population migration, age structures, and government interventions. In addition, we calculated only the PRCCs between the attack rate and each parameter in the sensitivity analyses; we did not investigate the compounding effects of multiple parameters acting together on the attack rate, a topic that needs to be addressed in future studies.
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