Spatial epidemiology issues are outstandingly important, particularly the viral spread through populated areas is believed to be one of the major concerns [1]. The incidence and prevalence of infectious diseases in a given population, with varied geographic and demographic settings, need to be analyzed over the spatial and temporal domain in order to build dynamic models that provide a global insight of outbreaks' behaviour.

Transmission of an infectious disease may occur through several pathways: by means of contact with infected individuals, by water, airborne inhalation, or through vector-borne spread. However, for the purpose of this study, the direct contact of susceptible individuals with an infected one will be considered as the main transmission medium of contagious diseases. Therefore, it is assumed that infectious diseases are diffused from individual to individual following a network of contact between them. Since this contact usually takes place in a geographical space, it is fairly natural to expect that the space plays an important role in the dynamics of infectious diseases [2]. Clear evidences that some infectious diseases in humans populations spread geographically are the three well-known recent examples of communicable disease spatial advance in the United Kingdom [3] and Canada [4, 5]. For this reason, it is required to understand the complex dynamics of contagious illnesses given certain spatial environments. Some of the most well known mathematical approaches are the differential equation models (DE) [6], and mean-field type models (MF) [7], which have not taken into account spatial and temporal factors such as variable population density and dynamics, and they also ignore space implications within the system. The neglect of the spatial component in the formulation of epidemic models can be solved by describing the spatial behaviour with the use of complex systems theory approaches.

One of the challenges that face geographers, epidemiologists and computer scientists working in the field of spatio-temporal modeling, is trying to understand the complexity of the spread of diseases. The search for an understanding of the non-linear behaviour of epidemics' spread and their causes in order to control them, has resulted in several attempts to model and predict the pattern of many different communicable diseases through a population. Models can be defined as an abstraction of the real world, regardless of type or complexity, they are basically simplifications of a real-life system, which can contain only some of the essential elements of it – as determined by the researcher -, models are not exact reproductions of reality and can be interpreted by different people in different ways [8]. In spatial epidemiology, models have been primarily used to facilitate an understanding of the complexity of the interaction between the spread of a disease among different individuals and its impact on society.

It is for these reasons that the objective of this study is to develop and implement an agent-based modeling approach for the spread of a communicable disease. The theoretical framework will be implemented in a case study of measles to allow the creation, representation and execution of a communicable disease propagation simulation over space and time and in an urban environment. One of the most important factors that this study considered is the complexity of mobile individuals in an urban setting with transportation network, their exchanges during the commuting time and some of the possible interactions among them in specific locations such as work places, schools, university and shopping malls where people flow and where their contacts and interactions are dynamic.

### Representation of Space and Time in Epidemic Modeling

Epidemics have been modeled making use of many different types of models, from those that are purely mathematical to the spatially explicit ones. The mathematical modeling of epidemics has been the subject of a number of studies over the past century [9]. The formulation of these classic epidemic models enable the simulation of events for which laboratory experiments could not be conducted easily. The main assumption of this kind of models is that the population, in which a pathogenic agent is active, comprises different subgroups of individuals and they examine only the temporal dynamics of the infection cycle.

### Classic Epidemic Spread Models

Traditional epidemiology models represent epidemics of communicable disease using a population-based, non-spatial approach. The conceptual framework for this approach is rooted in the general population model which divides a population into different population segments [10]. Nowadays, epidemiology has known numerous disease-spreading models; one of the most famous models is the stochastic model introduced by Kermack and McKendrick (1927) [11], followed by others more or less sophisticated. The simplest model of epidemic spread which employs deterministic ordinary differential equations, is based on the separation of the total population into two groups: "Susceptible" (those individuals who are potentially capable of contracting the disease), and "Infected" (those individuals who are capable of spreading the disease). Due to this division of the population the model is called "SI". There are other epidemic models also based on the classification of the total population (SIR: Susceptible-Infected-Recovered, SEI: Susceptible-Exposed-Recovered, and SEIR: Susceptible-Exposed-Infected-Recovered). These deterministic models assume that populations are completely mixed and ignore spatial effects of spread epidemics; also interaction between individuals is neglected since they model populations as continuous entities [12]. The SI, SIR, SEIR, SIS and SIRS models fail to effectively model spatial aspects of the spread of an epidemic, the individual contact process, and the effects of individual behaviours, among others [12]. For this reason, the development of new mathematical and computing methodologies are necessary.

### Complex Systems Approaches for Epidemic Spread Models

Cellular automata (CA) theory has been used for modeling location-specific characteristics of susceptible populations together with stochastic parameters that capture the probabilistic nature of disease transmission [13, 14]. However, the representation of individuals' movement and interactions over the space was no presented. This is an important factor to consider in highly contagious diseases and therefore this methodology gave way to a new approach. Agent-based modeling (ABM), is also a bottom-up approach, similar to CA models, but has the advanced capability of tracking the movement of a disease and the contacts between each individual in a social group located in a geographic area [15, 16]. The potentials that ABM possess to model epidemic spread, have been used in epidemiology to study and track the movement of infected individuals and their contacts in a social system [17, 18].

Agent-based models allow interaction among individuals and are capable to overcome the limitations of different approaches such as cellular automata and classical epidemic models, permitting to study specific spatial aspects of the spread of epidemics and addressing naturally stochastic nature of the epidemic process. Consisting of a population of individual actors or "agents", an environment, and a set of rules [19], actions in ABM take place through the agents, which are simple, self-contained programs that collect information from their surroundings and use it to determine how to act [20]. Modeling in epidemiology using an agent-based approach pursues the progression of a disease through each individual (thus populations become highly heterogeneous by health status during simulations), and tracks the contacts of each individual with others in the relevant social networks and geographical areas (e.g., co-workers, schoolmates). All the rules for individual agent movement (e.g., to and from workplace and/or school) and for contacts with and transmissions to other people are explicit [21].

ABMs and their ability to produce emergent macro-effects from micro-rules have served as a cornerstone for the development of different methodological frameworks in epidemiology [16]. Epidemiologic applications using ABM approach are mostly designed to allow epidemiological researchers to do a preliminary "what-if" analysis with the purpose of assessing systems' behaviour under various conditions and evaluating which alternative control policies to adopt in order to fight epidemics such as smallpox [22–24]. Although these models effectively track the progression of the disease through each individual, and track the contacts of each individual with others in the relevant system (social or natural), they need to add physical infrastructures such as road networks, and real geographic environments to account for more complex interactions among susceptible and infected individuals. Another important application of ABM in epidemiology is the modeling of vector-borne diseases and the changes in their incidence that are attributable to climatic changes. These models have been developed in order to allow the evaluation of impacts of climate change on vector borne diseases like malaria, as well as the a priori evaluation of environmental management-based interventions [25].

The spread of human epidemics strongly relies on the structure of the underlying social network, and it has become clear that differently structured networks lead to different types of epidemiology [26, 27]. By modeling the correlations between individuals, it is possible to understand the role of spatial heterogeneity in spreading dynamics. The previous statements have lead to the development of different models in order to depict the spatial behaviour of diverse infectious diseases through structured and realistic urban networks, for example, influenza [10, 18, 28–31], Mumps [32, 33], West Nile virus [34, 35], Tuberculosis (TB) [16], Lassa virus [18], among others. Some of the models mentioned represent the spatial distribution and mobility of individuals making it possible to model the spatial heterogeneity in the disease transmission. Nonetheless, one of their drawbacks is the lack of use of real landscape structures and integration with geospatial data and geographic information systems (GIS) to represent the continuous environment where the discrete individuals interact.