An Agent-based Approach for Modeling Dynamics of Contagious Disease Spread

Liliana Perez; Suzana Dragicevic

Disclosures

Int J Health Geogr 

In This Article

Methods

Communicable diseases are illnesses caused by an infectious agent that are highly contagious and may be transmitted from one individual to another one through direct contact. Individuals that make part of a human population are involved in a sequence of activities on a daily basis. Some of the activities are stationary and some are mobile. Stationary activities occur at fixed locations, such as a home, school, workplaces, commercial and shopping areas. At these geographical locations, individuals may interact among themselves in a group activity. Mobile activities are related to the daily commuting activities of individuals through the public transportation system. When a group disperses, an individual travels through space and time to a different location, often interacting and joining another group. The simulation of this population dynamic is essential to depict individuals' life path of movement through space and time. The disease propagation modeled in this study represents this movement path as a trajectory in space (movement from one place to another) through a transportation network and in time, expressed on hourly basis. In this fashion the daily activities of commuting, studying, working and leisure time are simulated.

The methodology for this study involved the development of a complex algorithm composed by two parts. The first was designed to describe communicable disease stages, which is the generic infection model. The second represents the rules that govern the life path behaviour of the agents and the infection behaviours that allow the transmission of the disease within a group of people in a city. Two scales are considered for the individual interactions in respect to transmission and propagation of the disease. One is at the individual scale considering the smallest space around a person when the disease can be transmitted. Second is the limit of boundaries of the city in which individuals move, live and interact with each other on a daily routine.

Generic Infection Model

Highly communicable diseases may spread by airborne droplets or through direct contact with nasal or throat secretions emitted through sneezing or coughing of infected persons.[36] Symptoms may vary between diseases, but they generally are divided in two stages. In the first stage, after the exposure to the infectious agent the symptoms are nothing special and used to be associated with a typical cold. The second stage initiates after a certain number of days and it is at that time when the specificity of the disease is very evident. After examining different compartmental models used in epidemiology, this simulation adopted the Susceptible-Exposed-Infectious-Removed (SEIR). The SEIR model successfully addresses a significant period of time during which an individual has been exposed to the infection but is not yet contagious: Exposed phase or latency period. The latency period is used to describe the period of time between exposure to the virus and the time the disease becomes apparent through symptoms and signs. The morbid period, also known as Infectious phase, is the period of time between the moment an individual starts being infectious until it is recovered (Figure 1).

Figure 1.

Different states of the SEIR infection model, to simulate the progress of and epidemic in a human population. LPi: latency period, IPi: infectious period, ti: first day that an individual is exposed to the virus for the first time, xLP: number of days for an exposed individual to become infective, and xIP: number of days for an individual to recover from the disease.

In this study, for simplicity reasons, the model is developed in such way that all recovered individuals cannot become infected again and they will remain immune. The initial conditions for the simulation are set up to represent a susceptible population, an immune population (due to previous communicable disease infection or vaccination), and infected population at the beginning of the outbreak. In order to design and implement the generic model, mathematical expressions are developed, so that the model can be used for different types of communicable diseases. As depicted in figure 1, latency period (LPi) and infectious period (IPi) are presented with following equations:

where (t i ) represents the one single day that an individual is exposed to the virus for the first time; (xLP) represents the number of days that have to elapse before the exposed individual to become infected; (xIP) corresponds to the number of days it takes for the individual to recover from the disease; (xLP) and (xIP) take different values for different diseases.

The rules that govern the unconscious biological process of disease spread are depicted in the flow diagram presented in Figure 2, where each individual is represented as an agent in the proposed model. Once an infectious individual arrives at a fixed location to perform any stationary activity (e.g., study, work, shopping, etc) a calculation is performed to determine the number of susceptible individuals within its perimeter to be contagious (Pc). This area is calculated based on a radius (r) of 1 m distance from the infected individual, considered as the smallest distance at which an individual can contract the disease and get infected.

Figure 2.

Flow diagram representing different infection phases.

After determining the susceptible individuals within the infected persons' surroundings (Pc), the disease is transmitted to some of them.

Agent Based Model

The model proposed in this study attempts to realistically represent the behaviour of individuals' daily path in an urban setting, as well as characterize the natural biological process of the disease spread among individuals (Figure 3). So that, it is required to maximize the simplicity regarding the interactions that one can account between agents, which then allow maximizing the understanding of their dynamics. The agent based model operates on discrete time steps during which a population of individuals, represented as agents, moves through a geographical space, where daily activities are performed. In order to represent a normal daily routine, it is assumed that the length time for out-of-house daily activities for an individual is 10 hours. Two hours are spent commuting via public transportation, and the other eight hours are spent either in work places, study places (high schools, universities, community colleges, etc) or doing some leisure activities at places like shopping malls.

Figure 3.

Process of the epidemic spread AB model for a single time step representing daily activity of individuals' activities and their interactions in an urban environment.

ABMs are best described in order of their major components: environment, agents and interactions. This model is implemented using georeferenced GIS data layers of an urban area in order to geographically represent the usual urban landscape where typical individuals' contact takes place. In order to account for some of the factors that may influence an epidemic in urban areas, the model was designed to include georeferenced information of population densities, different land uses and transportation networks. The flow diagram designed to represent the process of the disease propagation is used to explain the method for searching and querying the different geographic layers to determine the behaviour of the agents. As shown in figure 4(c), the agent-based model has four different georeferenced inputs; the first one (A) is the population of agents that represent the behaviour of individuals living in an urban area. The second (B) is the information regarding population density; this dataset allows the evaluation of the number of people per square meter and thereby greater probabilities of getting the infection will be assigned to individuals that perform any stationary activity in these areas. The third (C) is the transportation network, which is used by the agents to commute from one place to another within the municipality. Finally the fourth input (D) represents three different land use types: residential areas (houses, townhouses and apartments), work and study areas (institutional and commercial buildings), and entertainment areas (shopping malls).

Figure 4.

(a) Geographic area: City of Burnaby, Canada, with the relevant land use classes; (b) Geographic space of individual's interactions; (c) Geospatial data inputs.

Only one type of agent is used in this model to represent individuals and it has no attributes of age or gender. When the model is initialized and the agents are randomly added to residential areas, the georeferenced location of each individual is established and stored in the individuals' memory to recall its point of origin. After this initial stage, the agents simulate individuals' daily commuting to work places or study buildings. The entire susceptible population is divided into 70% of workers and 30% of students. After the first commuting hour, the workers go to work places and the students go to study areas. Figure 4(b) depicts the closed geographic space where individuals interact amongst themselves and transmit the disease. The movement of the individuals through the transportation network simulates the commuting behaviour, and was accomplished using a geographic information system (GIS) that holds the spatial locations of all the roads and stores the topological relationships between them. Therefore, if an agent needs to get to a destination, it needs to search the transportation network dataset in order to know which roads it has to travel along and then actually plot the shortest route to get to the desired point. The routing algorithm works by firstly building a list of coordinates which the agent must pass through to get to its destination and then moving along the planned route. The flow diagram shown in figure 5 illustrates the behaviour that determines different activities performed by diverse individuals.

Figure 5.

Flow diagram that characterizes daily activities of individuals within the city.

The agents are mainly distinguished by their health status (e.g. susceptible, exposed, infected, immune), movement rules, and mode of infection transmission. For the purpose of this model simulation, only public places (work places, schools and shopping malls) were considered for the propagation of the disease, this implies that the infection rules are not effective during the night time period in the residential areas or elsewhere.

An infection can only be transmitted from an infected person in a contagious stage to individuals that are healthy (susceptible) and within the established perimeter (Pc) for the disease to be communicable – explained in equation 3. The total number of individuals that are exposed within this perimeter (Pc) depends also on the population density where they are located at the moment of the interaction with and infected agent. If a group of susceptible individuals interact within the perimeter (Pc) of an infected individual, not all of them will contract the disease and therefore the rate of infection is determined using low, medium and high population densities, in order to calculate the number of individuals that are exposed to the disease. The fact that disease spread risk is considered to be higher for individuals living in highly dense populated urban areas is therefore taken in consideration.[37] Assuming a direct relationship between population and rate of infection, higher rates are given to those individuals located in areas with high density population. The flow diagram which describes the infection model and therefore the disease propagation among individuals is presented in Figure 6.

Figure 6.

Flow diagram for the infection rules that describe the disease propagation among individuals at physically fixed location.

The population of agents is held constant during a simulation run. Even if they have recovered, immune agents are not removed from the population. Agents are thus characterized by their location in the environment and by their internal state (status), which can be: susceptible, exposed, infected or recovered (immune). Once an agent is exposed, it remains infected for certain amount of days until it loses the infection status and recovers. This individual, therefore, remains immune for the rest of the simulation.

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