Spatial Diffusion: Conceptualizations and Formalizations

Kathleen Hornsby

National Center for Geographic Information and Analysis

and

Department of Spatial Information Science and Engineering

University of Maine

Orono, ME 04469-5711

kathleen@spatial.maine.edu

Introduction

The concept of a phenomenon spreading through geographic space is considered in many diverse subject areas such as the spread of infectious disease, growth of an urban center, the spread of wildfires, diffusion of innovation, and ripple effects in the natural world. And yet, a thorough understanding of the mechanisms of spread remains elusive. "The dynamics by which a phenomenon originally located at one point becomes transferred to another is a question which is as difficult to answer as it is easy to pose" (Cliff et al, 1981 p. 191). Our current information systems and tool kits are not able to represent easily a dynamic process like spatial diffusion. The linking of environmental models, often the workhorses that describe spread, to geographic information systems (GISs) still seems to be some way from being a smooth operation and contributes to the hurdles that users face as they attempt to use a GIS in their research (Nyerges 1993; Wesseling et al, 1996). The move away from stand-alone GISs towards open or interoperating systems also suggests that the capability to represent basic spatial processes such as diffusion will become increasingly important.

A consideration of the underlying concepts of spatial diffusion and how humans reason about diffusion in space, is important as a first step in building a conceptual model for spreading phenomenon. Conceptual models form the basis for the development of computational models upon which information systems are built. The conceptual model needs to reflect the spatial patterns of spread and yet, must also reflect how different user groups perceive and understand the process. For instance, an individual can takes steps to avoid the spread of an infectious disease by keeping away from an infected individual, whereas an epidemiologist might not be concerned with individual cases, but rather is interested in the overall distribution of illnesses in a geographic area in order to better plan control measures such as vaccination programs. These naive views can be captured in conceptual schemata and provide an important foundation for future implementations in a GIS. This research abstract describes a study which will develop the conceptual models for how humans reasons about the process of spatial diffusion in space, and extend this work with a formal specification of the model based on an algebraic approach.

Classes of Spatial Diffusion

Cliff et al. (1981) classified spatial diffusion into four basic categories which represent the characteristics of the spread. This categorization into classes comes from a familiarity with spatial diffusion as garnered from the results of research involving spreading phenomena. Firstly, expansion diffusion is the term given to that class of spread where the spreading phenomena has a source and diffuses outwards into new areas. The spread of a wildfire or diffusion of an innovation are examples of expansion diffusion. Secondly, relocation diffusion describes the spread that occurs when the spreading phenomena moves into new areas, but leaves behind its origin or source. A common example of relocation diffusion is that of migration, for instance the movement of persons from rural to urban areas. The third category is that of contagious diffusion. The spread of an infectious disease, such as measles, that requires direct contact between individuals for infection to occur, is commonly given as an example of this type of spread. Finally, spatial diffusion may also occur through an ordered sequence of classes or places, and may be described as hierarchical diffusion. For instance, Gould et al.(1991) describes the spread of AIDS from large urban centers to smaller towns in the U.S. as an example of hierarchical diffusion. Figure 1 illustrates these four categories of spatial diffusion.

Figure 1. Types of spatial diffusion: (a) expansion diffusion, (b) relocation diffusion, (c) contagious diffusion, and (d) hierarchical diffusion (Cliff et al. 1981).

This classification of spatial diffusion into four basic types, is a starting point to describing the form which this process takes. It provides an overall framework, but is devoid of any consideration of how humans reason about diffusion. We can extend this analysis by looking at the objects and operations that work together to create the process of spread from a human perspective, and consider what is the integrating framework between geographic space, the process, the entities that are affected by the process. That is, whether certain characteristics are shared among the classes depending on the user perspective or whether certain types of spread are a subset or superset of the others. We can also consider how geographic space is treated in each case, for instance, how is diffusion affected by constraints to space or barriers? From this work, a conceptual schemata for spatial diffusion will be developed.

Conceptual Model for Spatial Diffusion

The methods used by domain scientists to represent the process over geographic space gives important insight into the different perspectives held with regard to the process of diffusion. A survey of the literature detailing approaches to the study of the spread of various phenomenon has revealed that most studies fall into one of the four following classes: (1) studies that utilize statistical or mathematical methods to predict or describe the spread; (2) mapping spatial and temporal distributions of an entity; (3) cell-based simulations of spread; and (4) network approaches to representing spread. The method of choice typically reflects the scientist's perspective of how the process is taking place. For instance, in studies where contagious diffusion is involved, many researchers choose cellular simulations to simulate the process of spread and capture the dynamic nature of the interactions between neighboring cells. On the other hand, studies attempting to shed light on the spatial distribution of an entity, may focus on mapping the boundary or extent of the entity.

Further efforts to establish how humans reason about diffusion in geographic space can be highlighted from a consideration of contagious diffusion and expansion diffusion. For instance, the transmission of measles is commonly given as an example of contagious diffusion (Cliff et al, 1981; Sattenspiel 1990). Direct contact between a carrier infected with measles and a susceptible individual, is all that is necessary to be at risk from infection. Based on a simple SIR (susceptible, infective, removal) model we can express the process of the spread of measles as shown in Figure 2.

Figure 2. Simplified representation of spread of measles in a population.

Abstracting from the specific domain of the spread of an infectious disease, we note that contagious diffusion requires a population of "susceptible" entities, a source of infection, and direct contact between entities to conduct the spread. The focus of this class of diffusion, is on the individual entity, and the interaction is conceptualized as taking place between entities. At this point, the characteristics of geographic space have not been defined and this will be the next step in extending the conceptual model.

From the entity view embodied by contagious diffusion, the set of underlying concepts for expansion diffusion may be quite different. Recall that expansion diffusion is defined as that group of spreading phenomenon that has a source and diffuses outwards from the source. The spread of a fire, or pollution being emitted from a point source are examples of this type of spatial diffusion. For this class of spread, one conceptual schema may no longer focus on individual objects coming into contact with each other, but instead now focus on the extent of the spread, the continuous zone or area that the spreading phenomenon covers. In this case, we do not concentrate our efforts on specifying say, the spread of fire between blades of grass, but rather what area has been burned, or where is the wildfire spreading. The user may be only interested in areas or regions, rather than individual entities.

This work can be extended to include hierarchical and expansion diffusion in order to develop a comprehensive model that will incorporate the conceptual primitives and give a framework for the general process of spread.

Formal Specification of Conceptual Models

Formalizations of the conceptual models based on algebraic specifications will test the expressions used to portray the models and verify if they are behaving as expected, and are thus, a correct representation of reality. Typically, a phenomenon or procedure is translated into a programmed prototype that uses a formal language with mathematically defined semantics that express the real world entity. Frank and Kuhn (1995) give several important reasons why such specifications are useful:

Work at the Technical University of Vienna on formalization of various conceptual models for GIS has used with success, a functional programming language to test their hypotheses (Car and Frank 1995; Frank and Kuhn 1995). A functional programming environment is also being tested in this study, in an attempt to use formal specifications to express the components of the conceptual model.

Concluding Remarks

This research abstract outlines the needs for developing a better understanding of how users reason about processes in geographic space, and focuses in particular on the process of spatial diffusion. The task of defining a model which embodies the properties of spread, the space in which it takes place, and the objects upon which it acts, and combining this knowledge with how humans understand and perceive the process will lead to the ability to formalize this representation in an information system such as a GIS. Ultimately, it is hoped that this effort will allow researchers to derive computational models which can form the tools that will aid scientists investigating spreading phenomena.

References

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Cliff, A. D., P. Haggett, J. K. Ord and G. Versey (1981). Spatial Diffusion: An Historical Geography of Epidemics in an Island Community, Cambridge University Press.

Frank, A. U. and W. Kuhn (1995). Specifying Open GIS with Functional Languages. Advances in Spatial Databases, 4th International Symposium, SSD `95, Portland, ME, Springer.

Gould, P., J. Kabel, W. Gorr and A. Golub (1991). AIDS: Predicting the next map. Interfaces 21(3): 80-92.

Guttag, J. V., J. J. Horning and J. M. Wing (1985). Larch in Five Easy Pieces, Digital Systems Research Center.

Nyerges, T. L. (1993). Understanding the Scope of GIS: Its Relationship to Environmental Modeling. Environmental Modeling with GIS. M. F. Goodchild, B. O. Parks and L. T. Steyaert, Oxford University Press: 75-93.

Sattenspiel, L. (1990). Modeling the Spread of Infectious Disease in Human Populations. Yearbook of Physical Anthropology 33: 245-276.

Wesseling, C. G., D.-J. Karssenberg, P. A. Burrough and W. P. A. V. Deursen (1996). Integrating Dynamic Environment Models in GIS: The Development of a Dynamic Modelling Language. Transactions in GIS 1(1): 40-48.