While this approach had yielded important insights into disease outbreaks, until now it had not been used to calculate the speed of spread. The duo adapted current message-passing analysis to account for the mean delay in infection between individuals, at different degrees of connection from the central source.
Their updated equations allowed them to determine the times at which a simulated infection is most likely to arrive at certain individuals. Their showed excellent agreement with numerical simulations of real-world networks; even for densely populated communities, where webs of interaction become more complicated.
Moore and Rogers demonstrated the versatility of their approach by successfully modelling the particularly complex case in which individuals only become infected after interacting with multiple people with the disease. In addition, they showed that the time taken for an infection to spread throughout the bulk of a population shows no dependence on network size.
Rather, the jump from just a few, to many infected individuals can happen almost instantaneously. The duo hopes that their results will pave the way to more detailed multi-layered and time-varying models. If achieved, they predict that routes towards the development of monitoring and prevention protocols for real-world diseases could soon emerge.
The research is described in Physical Review Letters. The spread of infectious diseases can be unpredictable; fortunately, modeling techniques can help compensate for imperfect information gathered from large populations under difficult prevailing circumstances [ 30 ].
The infectious disease dynamics IDD approach, a mathematical technique, has developed into a rich interdisciplinary field. It is driven both by the desire for fundamental understanding and the need to use that understanding to aid public health decision making. Compared with traditional statistical models, the IDD model can not only describe the process of disease development and transmission and predict the state of disease occurrence but also evaluate the effects of various prevention and control measures and provide decision-making support regarding the measures need to prevent and control diseases.
The main idea underlying the CM model is to divide the population into several compartments, which, respectively represent agents in different disease states, and then dynamic equations of related variables are established by mathematical techniques.
Finally, the dynamic process of disease transmission can be modeled. For example, Kermack and McKendrick [ 31 , 32 ] predicted the number and distribution of cases of an infectious disease as it was transmitted through a population over time and proposed the classic SR model.
Unfortunately, the generalizability of this model is difficult to analyze, and a number of open questions remain regarding its dynamics. In a subsequent study, the metapopulation model MM was proposed based on the classic IR model. The modeling concept underlying the metapopulation model is to simulate the migration behavior of individuals between populations, and an SIR model or SIS model can be used to simulate the infectious disease transmission process within the population.
For example, Watts D. They found that when epidemics occur, the basic reproduction number R 0 may bear little relation to their final size. Next, Colizza V. The results provided a general theoretical understanding of the behavior of more realistic MMs. The individual-based model IM is a microsimulation model that mainly includes a cellular automata model and an agent-based model. The modeling concept underlying the IM model is that individuals are treated as cells or agents with a limited set of state and behavior rules.
For instance, Milne G. The results indicated that multiple social distancing measures applied early and continuously could be effective at interrupting the transmission of the pandemic virus for R0 values up to 2.
Moreover, for the network model NM , the main underlying modeling concept is to treat individuals in the population as nodes in a network, and the contact relationships between individuals are described by the edges between nodes in the network. For instance, Ajelli M. The results indicated that both models yielded epidemic patterns that were in very good agreement at the level of granularity accessible by both approaches. Riley [ 37 ] reviewed the application of four methods patch model, distance-transmission model, multigroup model, and NM to four diseases measles, foot-and-mouth disease, pandemic influenza, and smallpox.
The results showed that household demographics have an important impact on the spatial transmission of human diseases, such as smallpox, influenza, and other infectious diseases.
Given the characteristics of the spread of infectious diseases based on passenger flow through a rail transit station, passengers in a rail transit station and the SEIR model were chosen as the research subjects and the modeling method, respectively, for this study. Through the simulation of the spread of an infectious disease through the Tianyi station at Ningbo Rail Transit Line 1, the factors affecting the spread of the infectious disease, that is, the contact rate, the transmission ability, and the duration of the infectious disease, were quantitatively analyzed.
In the context of rail transit, the speed of the spread of the disease and the infection rate can be affected by multiple factors. The most immediate factors include the number of infectious persons and their distribution among the passengers, the transmission route and the transmissibility of the infectious disease, and the level of immunity. Based on the five routes of transmission, namely, contact transmission, aerosol transmission, water and food transmission, insect transmission and others, this manuscript dissects the external and internal factors and the passenger-related factors that affect the spread of infectious diseases through rail transit systems.
The external factors include the above transmission routes, while the internal factors include the temperature, humidity, facility layout, subway stations, air, hygienic conditions, density, and intensity of passengers in rail transit stations. The passenger-related factors include their basic characteristics, physical quality, behavioral habits, immunity, and medical history.
Based on the SI model, the SIS model can be used to simulate infectious diseases such as common influenza by adding the measurable characteristics of the disease, and considering whether individuals are susceptible or infected. To simulate the production of antibodies after recover and thus the acquisition of immunity to the disease, the SIR model was developed by introducing a third status, recovered.
Patients do not exhibit symptoms for a certain period as they move from the susceptible status to the infected status, during which time they do not spread the disease. To simulate the status of exposed but asymptomatic, the status exposed was introduced, and the SEIR model was developed. The SEIR differs from the SIR models in that it adds the duration of the disease and is more suitable for infectious diseases with no infectious potential during the incubation period.
To reflect the actual route selection behavior of rail transit passengers as much as possible and improve the reliability and effectiveness of the simulation results, the following assumptions were made:. The distributions of age, gender, health status and daily behavior of passengers in the rail transit station were investigated through a questionnaire survey. These data were combined with previous findings regarding relevant pedestrian behavior characteristics. The incubation period exposure period of the infectious disease was assumed to be 1—14 days, the recovered period was 30—60 days, and the transmissibility was 0.
The disease spreads through the contact message "infection". Thus, each passenger has four potential statuses: susceptible, exposed, infected, and recovered.
Passengers are initially susceptible. If they are exposed to the pathogen, they will enter the exposed status, which means that they display symptoms and produce antibodies against the pathogen. After a certain period, they will become susceptible again. Considering the given incubation period, the SEIR model was used to model passengers with four statuses: susceptible S , exposed E , infected I , and recovered R.
Each variable represents the number of passengers in the corresponding group. Its physical meaning is the protective efficacy of an individual after vaccination, and a larger value indicates that government interventions are more effective, and vice versa.
There is an important relationship between the validation of model parameters and the authenticity and effectiveness of the simulation results. The parameters include the time scale, agent static parameters, agent walking speed and characteristic parameters of disease spread. The specific values are as follows:. The static parameters of the agent: the space occupied by the agent on the ground is called the passenger space.
In the process of movement, the dynamics of the passenger space change in complex ways, and their specific sizes are difficult to measure. Therefore, only the static passenger space was considered in this study, which included the horizontal space and vertical space.
The former mainly consists of the shoulder width and the safety buffer, while the latter consists of the stride and the safety space. Many studies have reported human body measurements stratified by gender and age, as shown in Table 1. The other data used in this study are from relevant studies and experience.
The agent walking speed. In the simulation, the walking speed has a significant impact on the time to leave the station because age and gender affect walking speed [ 13 , 14 ]. Therefore, the following data were specifically set with reference to the statistical data published by Chinese and non-Chinese researchers on the walking speed of pedestrians, combined with the average walking speed of passengers at the Tianyi station measured with the preliminary survey, as shown in Table 2.
The structural characteristics of the rail transit station were mainly obtained from the indicator map and the field investigation to provide data to support the simulation. Spread parameters of the infectious disease. The spread parameters of the infectious disease reflect the process by which the virus is transmitted throughout the passenger flow and include the total population, infectivity, contact rate and average illness duration.
Based on historical data on the spread of infectious diseases, a model of an infectious disease at Tianyi station was constructed. The total number of passengers was set at Since There are 4 entrances and exits at Tingyi Station, and 4 entrances and exits are open under normal circumstances.
During the survey time, we took 15 minutes as a time interval to count the flow of pedestrians entering the station during working days and non-working days respectively, and finally summarized them in units of 1 hour. Hence, for the convenience of simulation statistics, the number of simulated people per unit hour was set to The SEIR model uses differential equations to reflect the relationship versus time of the number of individuals in each of the four different statuses, namely, S, E, I and R, which has certain practical value for the study of the spread of infectious diseases.
The static parameters and the walking speed of agents do not change with research objectives and environments; therefore, they can be regarded as constants and do not need to be calibrated.
The contact rate, infectivity coefficient, duration of exposure, and duration of infectiousness introduced are more abstract, which increases the difficulty of the practical application of the model. The accuracy of these parameters is the key to model construction and the correctness of the conclusion, but they differ across infectious diseases and transmission environments.
Therefore, optimizing these parameters was an important part of this work. Calibration experiments [ 38 — 40 ] were established for the SEIR model, and a table of the historical data was also established based on the collected epidemic data. The total number of passengers in the model was set at , and it was assumed that there was an infectious source in the target station.
He came up with an accurate global mobility network based on the flow of food distribution trucks throughout the country. But human nature also makes for kinks in the system. If Brockmann and his colleagues were better able to predict human responses to outbreaks, they might be able to build that into the model.
It might turn out, for example, that large groups of people running away from a disease could contribute to a faster spread. Hopefully, considering the spread of disease through a reorganized sense of time and space for the modern era could help cut off critical links in transmission.
And now that researchers understand how the underlying process works, the model could be applied to just about anything. AWS Deloitte Genpact. Events Innovation Festival.
NOTE: This is a simulation and it has been simplified to a small number of variables. These variables are applied uniformly across each person in the population. Real life is much more variable and complicated. Our goal is to help learners of all ages better understand how our actions impact the collective result and how mathematics can be used to better understand our world. Each time you click Calculate Results or Recalculate Results , the simulation recalculates.
For some cases the spread ends quickly. This is by chance given the variables. Click again for a new simulation. You may need to run multiple times to get a sense of the variability and impact of the settings. How Are Viruses Spread? As the world of science trains a careful eye on the development of the virus, the world of mathematics, too, can play a role. For example, the applet below can help predict the spread of a virus over a population.
Each person may pass on a germ or virus to others with whom they come in contact, but the disease will not be transmitted if the recipient has a resistance to the disease, has had a vaccination, or has already been infected. The spread of a virus is influenced by various factors.
Here are four factors:. A pandemic is an epidemic an outbreak of an infectious disease that spreads across international borders, or at least across a large region. Previous pandemics include the Swine Flu, cholera, and the Spanish Flu.
According to WHO, there were , confirmed cases and deaths globally at this same time. An epidemic is an abnormally high occurrence of a disease in a particular population or geographic area. On the other hand, a pandemic is a global epidemic that crosses international boundaries. The Antonine Plague This pandemic is believed to have been smallpox or measles that was brought to Europe by soldiers returning from the Near East. It may have killed as many as 5 million, and during a second outbreak in the middle of the third century, it was rumored that people a day were dying in Rome.
Cholera The first outbreak of cholera occurred in India in , and it became a pandemic by spreading from Bengal, across India, and to China and the Caspian Sea. The second pandemic of cholera affected Europe and North America in the late 's, and since then, there have been five other cholera pandemics.
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