- Abstract -

A spatially explicit computer model is developed to examine the dynamic spread of fox rabies across the state of Illinois and to evaluate possible disease control strategies. The ultimate concern is that the disease will spread from foxes to humans through the pet population.

Modeling the population dynamics of rabies in foxes requires comprehensive ecological and biological knowledge of the fox and pathogenesis of the rabies virus. Variables considered including population densities, fox biology, home ranges, dispersal rates, contact rates, and incubation periods, can greatly effect the spread of disease. Accurate reporting of these variables is paramount for realistic construction of a spatial model. The spatial modeling technique utilized is a grid based approach that combines the relevant geographic condition of the Illinois landscape (typically described in a georeferenced database system) with a nonlinear dynamic model of the phenomena of interest in each cell, interactively connected to the other appropriate cells (usually adjacent ones).

The resulting spatial model graphically links data obtained from previous models, fox biology, rabies information and landscape parameters using various hierarchical scales and makes it possible to follow the emergent patterns and facilitates experimental stimulus/result data collection techniques. Results of the model indicate that the disease would likely spread among the native healthy fox population from East to West and would occur in epidemiological waves radiating from the point of introduction; becoming endemic across the State in about 15 years. Findings also include the realization that while current hunting pressures can potentially wipe out the fox in the State, some level of hunting pressure can be effectively utilized to help control the disease.

Spatially explicit modeling of complex environmental problems is essential for developing realistic descriptions of past behavior and the possible impacts of alternative management policies . Past ecosystem scaled model development has been limited by the conceptual complexity of formulating, building, and calibrating intricate models. This has lead to a general recognition of the need for collaborative modeling projects . A graphically based, spatial modeling environment (SME) has been developed at the University of Maryland to address the conceptual complexity and collaborative barriers to spatio-temporal ecosystem model development . The modeling environment links icon-based graphical modeling environments (e.g. STELLA) with parallel supercomputers and a generic object database . It allows users to create and share modular, reusable model components, and utilize advanced parallel computer architectures without having to invest unnecessary time in computer programming or learning new systems .

The application of this system of collaborative spatial modeling was first made to the Atchafalya Estuary in Louisiana in the early 1990’s. The initial modeling process began with the division of the Estuary waterscape into square kilometer cells. Dynamic models of the waterway were then constructed in each of the cells. The 2-dimensional models were then compiled and integrated by the Spatial Modeling Environment (SME). The required spatial data varied with each location and was supplied by digital maps from a standard geographical information system. The completed model was used to predict the water depths of the estuary by determining the number and density of existing plant species distribution. Aerial photographs were used to calibrate and verify the accuracy of the model (similar models are being used at the South Florida Water District).

More recent applications of the technique have been developed at the Geographical Modeling Systems Laboratory at the University of Illinois. The large and relatively undisturbed landscape of Army training grounds were the focus of efforts to determine the impact of both civilian and military activities on indigenous endangered species (Military training landscapes are managed differently than contiguous and surrounding lands. While neighboring lands are often highly managed for agriculture or human habitation, the training land is likely to have been maintained in a more natural state.). Three such studies are worth noting here (http://blizzard.gis.uiuc.edu).

The first of these addressed the plight of the endangered sage grouse in the semi-desert landscape of the Yakima Training Center near Yakima, Washington. The impact of vehicular training on the long term population levels of the grouse was modeled. Ten meter square cells were used, each cell containing an elaborate model of sage grouse behavioral patterns as developed by an interdisciplinary team of researchers. The model run covered a multiple year period and seasonal variations in the model were noteworthy. Conclusions that the vehicular training tended to compress the vegetative food stocks and frighten nesting birds were made along with the realization that slight timing adjustments in the training schedule could be shown to leave the grouse population virtually unaffected .

The second model was developed in response to noted population declines of the desert tortoise (gopherus agassizii), a herbivorous reptile inhabiting the Ft Irwin training grounds in the Mojave Desert of California. The model was prepared using the collaborative spatial modeling techniques. More than 3 thousand interacting one kilometer square cells, each with an interconnected dynamic model (using monthly computational time steps) and a series of 20 digital raster based maps were used to help describe desert tortoise behavior across the desert landscape. Results of the model show that the natural population levels of the tortoise was substantially higher than the current population levels found. Civilian access to the grounds and static structures such as powerlines and telephone poles also had surprising impacts. The poles provided unnatural perches for predatory birds .

The third and most extensive model was designed to assess the land management impacts on two endangered warbler species that migrate from Mexico to the training grounds at Ft. Hood, Texas. Vegetation type and age, the presence of fire and of parasitic birds played major roles in the population dynamics. This model was made web-accessible to landscape management officials at Fort Hood and their plans for vegetative conservation and possible fire control strategies to protect the warbler populations are aided by the model .

The collaborative spatial modeling approach is currently being used for the development of a spatially derived watershed model and in the development of an urban growth model. These models are being developed to enable land use planners to project and comprehend the complex interactions of ecological and socially scaled ecosystems. These two models broaden the scope of spatial modeling to include the human dynamic. The hope is that this type of model will provide a more realistic and visual basis for deciding when technological, ecological or sociological responses are most appropriate.

In this paper, the collaborative spatial modeling approach is applied to the problem of the spread of infectious disease. The dynamic spread of fox rabies across the state of Illinois is modeled to determine the spatial patterns of the spread of the disease and to evaluate possible disease control strategies. The practical concern is that the disease will spread from foxes to humans through the pet population. (Although the chance of this transmission is small , the level of concern following just one case can trigger an expensive and lasting control program.) Following a more detailed description of the collaborative spatial modeling technique, the fox rabies case-study model will be described. Previous models of fox rabies transmission will introduce the pathogenesis of the rabies disease and the biological characteristics of the fox modeled. A description of the dynamic behavioral model follows with assumptions, spatial characteristics and the georeferencing approach taken described in a section on model design. The spatially explicit results and implications of the fox rabies model conclude our discussion.

Structuring large-scale models into smaller distinct modules is a well-recognized method for reducing modeling complexity . Ecosystem models with a modular hierarchical structure should be closer to natural ecosystem structure than procedural models , since the component populations of ecosystems are themselves complex hierarchical systems with their own internal dynamics . The Spatial Modeling Environment (SME) is a hierarchically based modular modeling environment that facilitates collaborative model construction, since teams of specialists can work independently on different modules of large-scaled models with minimal risk of interference. The developed modules can also be archived into libraries and serve as a set of templates to speed future model development .

SME is designed to support a range of platforms, both in the front-end model development module and in the back-end parallel computing module, this helps to maximize accessibility and collaboration. The main components of the SME have been described as View, ModelBase and Driver architecture by it’s developers . Figure 1 describes the basic architectural modules of the SME.

Figure 1. View, ModelBase and Driver Architecture for the Spatial Modeling Environment .

View

The View component of the SME is used to graphically construct dynamic, front-end modules. Although the SME can utilize a number of graphic modeling tools such as STELLA, EXTEND, SimuLab, or VenSim we will focus our discussion on STELLA, the environment chosen for our Fox Rabies case study model.

STELLA is a graphically based dynamic simulation software based on Jay Forrester's systems dynamics language . It is one of an expanding number of dynamic computer modeling languages that uses icons and symbols to communicate a model’s structure. STELLA was the first and has a good mix of simplicity in manipulating model components and power of model expression. Icons include reservoirs representing stocks of resources and "pipes" and "valves" representing flows and controls between those reservoirs, each with an associated user defined equation . When the STELLA model definitions are complete, the ‘sub’ model can be run. Variables of interest can be scaled and plotted in various formats to help visualize model behavior. Using iconographic modeling techniques greatly increase the ease with which the model can be changed and calibrated. The effects of changes made can be viewed immediately; allowing the user to concentrate on modeling instead of computational details, greatly reducing model development time .

The importance of the simplicity of iconography cannot be overstressed. Large and complex ecosystem models, are simply too complicated for the single modeler approach. It is extremely difficult for a single modeler to fully understand each dynamic interaction required in complex ecosystem models. And if a set of contextual experts are forced to interact with a single modeler, they tend not believe that their applicable expertise has been adequately captured by the process and the model results are discounted. STELLA's iconographic modeling system enables a modular modeling approach, where a team of experts work simultaneously on a single study problem. This "knowledge capturing" relies on an easy to understand modeling iconography. With STELLA as the basis of the ‘view’ modeling component a team oriented effort is possible, in which each expert participant can see that their knowledge has been appropriately captured and embedded in the final spatially explicit model. In fact, experts involved in these processes have sometimes gained insights into their own area of interest, and consensus among experts has been easily and publicly achievable.

ModelBase

When the STELLA modules are complete, the Module Constructor translates the component modules into a text-based Modular Modeling Language (MML). The MML modules can then be archived in the ModelBase to be accessed by other researchers, or used immediately to construct a working spatial simulation .

Code Generators

Code Generators are the mechanisms that convert MML modules into a C++ object hierarchy that is incorporated into the simulation driver. Final spatial output configuration and modification takes place during this step. The configuration of the output defines the model variables to be displayed or mapped. Configuration also includes setting space and time functions for each cell. This associates each MML object with a spatially oriented cell in a coordinate based (raster) Geographic Information Systems (GIS) map. The maps are used to supply baseline or initial landscape or variable conditions (the existing state of the system) and also to display the core set of modeled variables ).

Driver

The driver is a distributed object-oriented environment that incorporates the set of code modules that perform the spatial simulation. The code generator produces a set of code modules which are transferred to the target platform, compiled, and linked with the local driver modules to produce a working spatial simulation. The code generators also produce a set of simulation resource files that are used for runtime configuration of model parameters, input, output, and other simulation parameters. The driver then handles input-output of parameter, database, and GIS files and execution of the simulation .

The collaborative modeling approach incorporating the Spatial Modeling Environment was applied to the development of a spatially explicit simulation of the spread of the fox rabies disease in the state of the Illinois. A team of interdisciplinary researchers was assembled to tackle the problem. Programmatic areas of concern represented include: University of Illinois Departments of Geography, Veterinary Bio-Sciences and Natural Resources, the Illinois Natural History Survey and US Army Construction Engineering Research Laboratory. The main modeling effort took place over a course of 6 months at the University of Illinois and the results of the modeling effort are described below.

Fox Rabies in Illinois

The epidemiology of fox rabies is intimately linked with fox behavior. Foxes produce their young in Spring and juveniles migrate each Fall and early Winter, adults will also migrate out of their home range if their population density is sufficiently high. This migratory behavior becomes the vehicle for wide-spread transmission of disease because the behavior of the infectious animal becomes erratic and combative and the disease is then spread during contact with healthy foxes through biting. The incubation period for fox rabies varies from 14 to 90 days, ending in clinical illness. An animal may be infectious for up to a week before the onset of symptoms and remains infectious until death . Model parameters such as the effective biting rate and the actual length of the infected and infectious periods are difficult to determine in the field. We have used the best available data and determined an effective biting coefficient by trial and error comparisons of fox densities gained from the literature.

A complete model of the fox and fox behavior might include a set of sex differentiated age cohorts. We found however, that the history of the disease and of fox behavior could be adequately represented by a simple four stock model of both healthy and sick juveniles and adults. The model includes both deterministic and stochastic components and can be adapted to any disease that possesses spatial dynamics by simply adjusting the input data. The results of our epidemic model indicate that the incidence of fox rabies can be decreased with an intervention strategy such as hunting. However, the results also indicate that the current fox hunting pressures, coupled with the introduction of the rabies disease would lead to long term elimination of the fox in Illinois. Our results suggest that a reduced hunting pressure can leave a sustainable fox population in spite of the occasional introduction of the disease from surrounding areas. The disease can also be controlled by aerial deposition of baited vaccines over a large area. The model indicates the spatial dynamics of diseased foxes and thus allows the most judicious and least expensive aerial deployment of the vaccine.

Previous Fox Rabies Models

Dynamic models of rabies in wildlife populations have been proposed by others . These models have focused on the spatial spread of disease and potential impact of various control measures. But the addition of a spatial component to the disease dynamic is, in our opinion, a critical component. Spatial components can more easily explain variation in the rate of disease spread through a population , as well as provide a more holistic view of the dynamic interaction of animal, disease and landscape. Since wildlife populations are not indolent and are typically in a perpetual state of flux, contact rates between diseased and healthy animals depend to some extent on spatially derived information. David, et al. (1982) proposed a simple model of vulpine rabies which included much of the same biological components we utilize in our model: reproduction, dispersal, and spatial distribution . The spatial component of the David model is not linked to habitat resources however, and the display mechanisms of the SME offer a much more explicit depiction of possible scenarios.

Vulpine rabies poses a serious problem in Europe due to increasingly large fox populations and its zoonotic potential, increasing the probability of human contact in heavily populated areas . Fox densities in Bristol, England for example, range from 1.82 to 3.64 foxes per square kilometer over a home range size of 0.45 square kilometers . Compared with much lower densities in the United States, 0.15 foxes per square kilometer over a larger home range size of 9.6 square kilometers . In the U.S., rabies in the red fox, Vulpes vulpes, has reached epidemic levels in western Alaska and northern New York.

Previous linear models using data collected from European fox populations show a dramatic decrease in the number of foxes when rabies is introduced into a healthy population. These decreases reduce the population below an apparent disease threshold and the disease is shown to die out . These models typically demonstrate an inversely proportional relationship between infected and healthy foxes when rabies is first introduced into the population. As the disease becomes established, the number of infectious foxes increases as the susceptible fox population decreases . Murray (1987) describes these density decreases as "breaks", where the population becomes too low for the disease to persist in the environment. Gardner et al. (1990) concluded that the disease could be eradicated from a fox population when fox numbers are reduced to a critical level below the carrying capacity . Most rabid epizootics do not drive fox populations to extinction however.

Several models demonstrated that both healthy and infected fox populations stabilized over a period of 20 to 30 years . At this time healthy populations reached levels that were half of the total carrying capacity and infected foxes were reduced below ten percent of the total population . Other models have concluded that rabies, a cyclical virus, can reemerge between 3.9 to 5 years after a period of quiescence .

We concluded that vulpine rabies can be viewed as a cyclical, non-linear disease. When a susceptible population becomes infected, it decreases the healthy population but does not eliminate it. When the population rebuilds to a critical mass the disease is then able to reestablish itself and the cycle begins again. In this way vulpine rabies is an epidemiological disease. This concept is important for the development of a spatial model that describes the spread of the disease over a landscape and for the evaluation of possible control measures.

The Rabies Virus

Canine rabies transmitted to humans has been reduced in the United States, although it is still a factor in over 75,000 human cases worldwide and is still considered a human health issue . The disease, like many communicable diseases, appears to occur in cyclical waves. Its spread can best be understood through comprehensive study of the behavior of its mammalian hosts and the pathogenesis of the virus . Typical host populations are heterogeneous in nature and field studies are difficult. There appears to be a hierarchy of susceptibility to rabies with foxes, wolves and coyotes being the most susceptible . The fox adds to this complexity with shy and elusive behavior . Although foxes do not typically interact with humans as frequently as other medium sized mammals, they do come in contact with feral felines and canines. This contact increases the risk of stray cats and dogs contracting the rabies virus and that risk places humans and domestic pets at risk.

The primary mode of rabies transmission is through the bite of an infected animal. To a lesser extent, scratching and licking can also transmit the disease. The virus replicates at the site of entry and once it reaches a sufficient titer the virus travels via the neural pathways to the brain . The virus titer is defined as the smallest amount of virus per unit volume capable of producing infection . The virus then travels from the central nervous system via peripheral nerves to the salivary glands where it continues to multiply. Shedding of the virus in the saliva may occur before the appearance of clinical signs . The incubation period, the time from inoculation to the appearance of clinical signs, can vary depending on the site of entry and its proximity to the central nervous system as well as the amount of virus entering the wound site . Early clinical signs may be subtle and depend on where the virus is most concentrated in the central nervous system. There are two clinical forms of rabies. The furious form affects the limbic system and, thus, the animal’s behavior . The dumb or paralytic form causes depression and lack of coordination (MacDonald 1980). Once clinical signs of rabies develop, death usually occurs in 7-10 days .

After the onset of clinical disease, foxes exhibit overt behavioral changes such as restlessness, pacing, and loss of appetite followed by either aggression or confusion, depending on the clinical form of the virus. The furious form of rabies will result in aggressive behavior, which encourages transmission of the disease. A fox with the paralytic form of rabies will become lethargic and confused and may only bite if provoked or approached by others. The final stages of either form of the disease are seizures and coma, followed by death. Normal foxes may "shy away" from rabid foxes thereby reducing their risk of infection .

Fox Biology

The red fox, Vulpes vulpes, is distributed throughout much of North America. Within the United States, the red fox has extended its range into forested areas where wolves and coyotes have been reduced or eliminated and where forests have been cleared .

Variations in fox population densities are closely linked to their social organization, which is in turn linked to the type of food supply being exploited, and to the threat of predation [MacDonald, 1980 #39. According to research conducted at the Illinois Natural History Survey and a study of foxes in the North Central United States that includes data for parts of Illinois , fox numbers have dramatically decreased in Illinois since the 1970’s . This population decline may be connected to an increase in predator populations (e.g. coyotes) and a reduction in available habitat coupled with interspecies competition and hunting pressure . Archery deer hunter surveys indicated a steady decline in fox populations between 1991 and 1996. Hunters in 1991 observed 10 to 12 foxes per 1000 observation hours, while an average of 5 foxes per 1000 hours was observed in 1996 .

The basic social unit of the red fox is typically a group of 3 or 4 breeding adults, and their juvenile offspring . In cases where a territory includes several adults, there is normally one male and a variable number of closely related vixens . Average litter size is 6 pups and 4 pups generally survive until the time of dispersal .

Foxes are solitary, nocturnal foragers, which exploit available food supply, within a fairly well defined home range. Members of a group tend to follow each other from one resource patch to another and will eventually end up very close to the original point of departure towards the end of each night . The vixens have individual ranges that overlap with each other and are encompassed by the home range of the male fox, which essentially defines the territory of the group. Juveniles will exploit a limited number of resource patches close to their dens and within the home range of their parents, gradually expanding their ranges and separating themselves through the late summer until the time of dispersal . Group ranges inevitably overlap to some degree, since an individual fox will normally cover less than half of its range in one night . Encounters between foxes of different social groups in these overlapping areas will undoubtedly result in territorial conflicts when they occur at resource nodes.

Foxes that disperse from their home territories will normally travel in a relatively straight line until they find another territory that is available for occupancy . If they are unable to find another territory within the dispersal season they become transients, forced to move continually. Hunting is the primary source of fox mortality, accounting for approximately 80% of deaths . Hunting pressure is the primary mechanism for producing available territory during dispersal, and permitting the majority of young foxes to establish a home range. Hunting season in the state of Illinois lasts November 10 through the end of January.

Foxes appear to be very susceptible to the rabies virus . A dose of less than 10 mouse intra-cerebral lethal dose-50 (MICLD50) killed 40% of foxes and 80-100 MICLD50 killed all study animals injected . Foxes bitten by other animals, such as skunks, frequently fail to become infectious as they develop neurological symptoms and die before the virus ever reaches the salivary glands. Foxes also produce a low enough level of virus in their saliva that they are limited in the species in which they are capable of causing infection. Sikes found only 2 of 24 study animals had saliva virus levels greater than 1000 MICLD50 . The virus multiplies in the salivary glands and higher levels of virus are usually present in the salivary gland than in the brain .

Rabid foxes typically remain in their territories, but they do spend time resting at the peripheries, where they are more likely to come in contact with foxes from neighboring groups. Fox contact behavior however remains the most important unknown parameter in the spread of fox rabies. . Foxes exhibit different

social behaviors, which may be density dependent. At lower densities animals may be solitary or live in pairs. At higher densities, loose family groups occur and are generally comprised of one male, several females and their offspring. Data from Sheldon (1950) indicates that females determine the home range of a family group and that males are residents for only part of the year. Evidence of communal denning was also found and it appears that foxes are gregarious during denning season .

Model Design

The View, ModelBase, Driver approach within the SME (see Figure 1) was utilized to develop our spatially explicit model of the spread of fox rabies in Illinois. The model includes variables relating to: population densities, home ranges, contact rates, and incubation periods, reproduction, dispersal, and spatial distribution, along with natural and un-natural mortality rates each of which can greatly effect the spread of disease.

As noted previously, the View component of the SME utilized was STELLA, developed by High Performance Systems Inc. The ModelBase was derived in the SME as described above and the Driver used was the Geographic Resources Analysis Support System (GRASS) a GIS environment developed at the US Army Construction Engineering research Laboratory . A simplified flow chart of the spatial modeling procedure used in the development of the fox rabies model can be seen in Figure 2.



Figure 2. The collaborative spatial modeling process used for the Illinois fox rabies.

In this grid-based approach, each 6 mile square cell defined by the GIS maps used is considered a typical fox home range. Each grid cell contains a highly nonlinear STELLA model that simulates the dynamic interaction and movement of foxes in one month increments. The cellular model includes variables describing the propensity for fox immigration and emigration (based on fox population density) between adjacent cells. Each cellular model is automatically parameterized using the GIS maps for the area of concern. The GIS maps in our case referenced land-use conditions in Illinois that were used to develop fox carrying capacities in of each 1610 cells that describe the State. Georeferenced maps were also used for the initial introduction of three diseased foxes along the State’s Eastern boundary (the disease appears to be spreading from East to West in the U.S.). The model collection is then run on a workstation computer for a model run-time of 25 years.

Cellular Model

Figure 3 describes the iconographic STELLA model of fox population dynamics. The four main variables measured – adult foxes, juvenile foxes, adult sick foxes and juvenile sick foxes are represented as stock variables. The flow variables regulate the additions and subtractions to the stocks that take place at each time step (in this model, one month) and the rate variables help determine the amount of flow and changes in the flow variables. For the population dynamics model, flows and rates include: births of juvenile foxes, a death rate for each stock, emigration (out of) and immigration (into) each stock from adjoining cells, and a maturation of juvenile foxes into adulthood. A more detailed explanation of the model follows.



Figure 3. The individual cell STELLA model diagram for fox population dynamics. Stocks or State Variables represent the current status of the system and are measurable and (in this case) conserved variables. Flows or Control Variables represent the action or change in a State Variable. Rates or Transforming Variables help to control the flow of information to the state variable by chnging or regulating flow.

Birth Rate

Separate stocks of adult males and females and juvenile were established. This enabled the inclusion of juvenile dispersal behaviors in the fall. Young are born in the spring(March - month 3). Litter sizes are randomly generated from a normal distribution of 6 kits per vixen as the mean and +/- .5 as the standard deviation. Not all females reproduce, for reasons of simple infertility and the hierarchical dominance of other vixens. For Illinois fox populations, we used a 95% vixen reproduction rate (Fig. 3) based on Storm’s model analysis .

Maturation Rate

Juveniles mature to adulthood in January (month 1) (Fig. 3). All juveniles who survive the first year become adults regardless of true chronological age. Juvenile animals disperse in the fall (months 9,10 and 11). As these animals are seeking uninhabited territories they are more likely to experience mortality from automobiles and to encounter other foxes in territorial disputes. The model begins at t = 1 (month, January), therefore the initial population of juveniles is 0.

The model allows for the stocks to contain fractional foxes. Such a feature lets us represent the random circulation of the occasional healthy and sick foxes and the random entry of a healthy fox into the state from Indiana. The State’s Eastern boundary is a river which inhibits fox crossings. These random circulation of both healthy and sick animals affects the course of the epidemic slightly, allowing the population to recover from decimation due to the disease.

Infection Rate

Infected adult and juvenile populations are developed from contact with other sick animals (Fig. 3).

Bacon and MacDonald reported that contact rates greatly influence the spread of rabies. They concluded that a contact rate of 2.9 rendered the fox population extinct. A contact rate greater than 1 is necessary for the disease to persist in the environment . In our model, 3 infected foxes migrating into Illinois from the eastern border at three different locations are used to initially infect the state currently healthy population. These rabid foxes are introduced into the model at the sick adult stock (ADULT SICK) and their entry location is determined from a map (MAP INFECTED). In the absence of specific data for rabies contact rates, inferences were made based on the history of fox biology and the behavior of the rabies virus. Repeated runs of the model revealed an appropriate range for the biting coefficient, the number to be assumed constant in converting healthy to sick animals. Foxes infected with the rabies virus are contagious for only a short period of time, typically 10-15 days. The rate of infection is dependent upon the rate of encounters with other rabid foxes. The equation that determines the rate at which adult foxes become infected is based on the law of mass action and can be characterized as:

ADULT GET SICK = (ADULT * ADULT_SICK * ADULT_BITE_C) + (JUV_SICK * ADULT * ADULT_BITE_C)

The law of mass action is used describes the average behavior of a system that consists of many interacting parts. This concept was first derived for use with complex chemical systems but has been found to be a reliable way of capturing the spread of disease in epidemiological and ecological models . The bite coefficient (ADULT or JUV BITE C) used in this model is very low (0.015). This reflects the fact that not all encounters between infected and non-infected foxes will result in biting. The proportion of biting encounters used, is the same for adults and juveniles.

Mortality Rate

Fox mortality (Fig. 4) is comprised of natural death rates, death rates due to hunting pressure and death rates that are caused by population pressures (density dependent death rates (DDD)). Death rates are derived from Storm, (1976). Storm used tagging and electronic tracking devices to monitor the movement of foxes in Illinois on a monthly basis.

The compilation of this data and the description of the causes of mortality in the tagged foxes is used to develop the natural and hunting pressure death rates. Storm’s report of an overall 81% annual mortality rate ("Full Hunt") is built into the model. Natural deaths were reported to account for 51% mortality rate over the course of the year and are noted be caused by roadkill, disease and starvation . Our model allows the natural death rate to reduce the population and then the hunting death rate is applied to the remaining population. Use of a ‘switch’ allows hunting mortality to be turned on or off, making it possible to examine the effects of hunting on the population level and the transmission rate of rabies.



Figure 4. The individual cell model diagram for density dependent mortality rates (DDD).

A density-dependent death rate was also constructed to prevent unrealistically high fox populations from occurring in any cell. A multiplier (230%) was applied to the migratory carrying capacity to determine the maximum density or ultimate cell capacity (MAX DENSITY CC). As the population of a cell moves toward the maximum density, density dependent deaths (DDD) are applied to the migratory population (the population of the cell that exceeds the migratory carrying capacity). This function simulates increased mortality rates that result from overcrowding and reflects pressure from other infectious diseases and increasingly scarce food resources.

The DDD is only employed in the model when the gap population (cell population minus migratory carrying capacity) is a positive number. Equations for the density dependent death rate in adults (DR DD ADULT) includes natural death rates and deaths due to hunting:

DR DD Adult = IF CELL POP > 0 THEN DDD * (ADULT / CELLPOP) + (DRNAT + DR HUNT) * (CELLPOP * DDD * DT) * (ADULT / CELLPOP) ELSE 0

This will produce a death rate that is dependent upon the density of the cell and approaches 1 if the cell reaches the density limit.

Migration Rate

The initial stock of adult foxes in each cell is set at 80% of the migratory carrying capacity (MIGRATORY CC). The migratory carrying capacity is derived from land use characteristics in each cell and read from the prepared GIS maps. It describes the healthy carrying capacity of foxes and determines migration tendency. When the population of foxes exceeds MIGRATORY CC, there is pressure for the extra foxes to move. The average carrying capacity for the individual cells in Illinois is approximately 32 animals. (The basis for these calculations are explained in subsequent sections).

Model Assumptions

1. There is no significant difference in the behavior of males and females regarding interaction outside the family unit and dispersion. The behavior of foxes is not significantly different when foxes outside the family unit are encountered.
2. Large rivers are not a barrier to migration; it is assumed that foxes will use bridges, swim, or walk across the frozen surface during the winter.
3. Rabies is always fatal; every fox that contracts rabies will die.
4. All rabid foxes are equal sources of infection, all foxes during the rabid phase will behave similarly.
5. Rabies incubation and infectious periods are the same length of time; there is no significant difference in the time periods of incubation and being infectious.
6. All surviving juvenile foxes mature to adulthood at 11 months.
7. There is no significant difference in the amount of resources each fox uses, regardless of age.
8. The fox population has a definable maximum. We used the maximum densities of foxes found in Great Britain .
9. The model starts with 3 rabid foxes along the Illinois/Indiana border.

1. Georeferencing

Red foxes, Vulpes vulpes, live in a variety of habitats . According to the Illinois Natural History Survey, foxes avoid forested areas and interior urban areas . Foxes use forested areas for migration but they are commonly avoided due to coyote competition. In general, red foxes are found in open croplands, grasslands or pasture, using sloped areas for den sites. Urban edge areas and farmsteads are important habitat for the red fox due to the abundance of prey and forage in these areas. Gosselink (1998) estimated the average east central Illinois red fox density at 3 adult foxes per 10 square miles; a breeding pair, plus 1 non-breeding (juvenile) fox . An average litter size is estimated at 6 pups, so approximately 9 foxes are estimated to occupy every 10 square miles in Illinois.

This estimate of the average healthy carrying capacity in the state helped to create a habitat-weighted fox carrying capacity map for the state of Illinois. This was done using data collected by the Illinois Natural History Survey in the Land Cover of Illinois GIS based mapping projects . Urban edge cells were assigned a fox habitat value of 5 (urban edge is defined as any areas within 0.5 kilometers of an urban-non-urban interface) and all interior urban cells were assigned a value of 0. Forest cells were assessed a value of 1 and remaining cells, including wetlands, croplands, and grass/pasture lands were assigned a range of values from 2-3 depending on slope.

The Land Cover of Illinois map data was aggregated into 6 by 6 mile cells, statewide. The habitat suitability values of the 6x6 cells were determined by the mean value the aggregated smaller units. The mean habitat suitability value for a central Illinois cell is 1.977. The minimum habitat value in Illinois is 1.312 (downtown areas of Chicago). The maximum habitat value is 4.33 (suburban Chicago area). A healthy carrying capacity map was the produced by incorporating the habitat suitability map data values with the average carrying capacity calculation:

HCC_map = 9_{(foxes/10sq.mi}) * 3.6_(sq.mi.)*[Habitat Suitability]/1.977_{(mean habitat suitability)}

In this new carrying capacity map, the minimum healthy carrying capacity cell value is 21.50 foxes per cell and the maximum healthy carrying capacity cell value is 70.84 foxes per cell. This map was exported into a GRASS raster based format and imported into the Spatial Modeling Environment (SME) for use with the STELLA fox model.

2. Spatial Characteristics

Adult foxes do not always remain in the territory they chose during juvenile dispersal. Adults are known to disperse only in the fall and winter, as juveniles do, so it’s possible for incoming juveniles to displace some adult foxes. To simulate these characteristics and to simplify intercellular movement, the spatial model randomly assigns the direction of travel and limits travel to the four main compass directions. A fox that moves into a habitat cell that is at the carrying capacity (as derived from the map) forces the movement of a fox out of the same cell.

The dispersal of foxes is essentially a function of two parameters: foxes emigrating (out of a cell), and foxes immigrating (into a cell). These parameters control the geographic movement of foxes throughout the spatial structure of the model. Figure 5 describes the cellular emigration functions of each stock in each of the cellular models. These functions become spatial in the SME environment when cell carrying capacities are applied to the spatial components.

1. Emigration

Emigration is a function of the spatial resource limits of each cell. In this model, the carrying capacity of a cell is a fixed amount and has been determined by land use cover characteristics. The relative population of foxes in each cell fluctuates with births, deaths and the immigration/emigration. Foxes emigrate when the calculated population of foxes in the cell exceeds the carrying capacity of that cell. This simulates the relationship between fox populations and resource availability. When the relative fox population of an area exceeds the resources available, there is pressure for a part of the population to move. The characteristics of the emigrating population in the model are determined by the characteristics of the total population that occupies the cell at time t1. Total population of each cell is a summation of the four stock variables in the cellular model: ADULTS, JUVENILES, ADULTS SICK and JUVENILES SICK. The proportion of each stock to total cell population (CELL POP) determines the number of emigrants from each stock variable. For example if a cell consists of 10 adult foxes, 15 juvenile foxes, 2 sick juvenile and 3 sick adults, the total cell population is summed to 30 foxes. This is compared to the migratory carrying capacity of the cell, in this case 20. This means that there are 10 foxes too many in the cell and 10 will be forced to move the next time step. Of the 10, 10/30 will be adults, 15/30 will be juveniles, 2/30 will be rabid juveniles and 3/30 will be rabid adults. These are calculated as EM ADULT, EM JUV, EM ADULT SICK, and EM JUV SICK.

Once the total number and type of emigrants for each stock is determined, directional preferences must be calculated. Foxes sometimes travel great distances to find suitable and available habitat. It appears however, that in most instances foxes choose home ranges based on availability and not attractiveness. For this reason, a directional preference was assigned randomly for each group of emigrants. These random assignments only occur when certain landscape and time considerations are met. There has to be an available pool of emigrants (from the EM calculations) to move, and it must be during September through December since foxes typically travel during the fourth quarter of each year. If each of these conditions is met, directional distributions are randomly assigned; i.e. each emigrating fox (in each of the four stock variables) is assigned to one of the four geographic coordinates North, East, South, and West (Figure 5).



Figure 5. Model computation of random directional assignments for each emigrating stock (Juveniles-JM, Sick Juveniles-JSM, Adults-AM, and Adult Sick-ASM). For example, JSRS - Juvenile Sick Random South, calculates what percentage of the sick juvenile foxes that emigrate will move to the cell south of the reference cell)

The directional assignment ratios are then applied to the stocks to determine the number and direction of foxes moving out of each cell at each time step (Figure 6).



Figure 6. The individual cell model diagram of fox emigration calculations for each stock (JUV OUT< JUV SICK OUT< ADULT SICK OUT and ADULT OUT). It includes a monthly counter (MONTH) and directional assignments calculated in Figure 5. For example, JS GO E – tells SME the number of sick juvenile foxes that are moving to the cell east of the reference cell. Note that the Adult Sick emigrants are not constrained by a monthly timer. This is to simulate the erratic behavior that rabid foxes exhibit.

The total number assigned to each direction then moves and becomes part of the immigrating population for the adjacent cell. For example, the total number of foxes moving in a northerly direction from the base cell, will then become FOXES IN, or additional foxes that have accessed the northern cell from the south. As a result the emigration function becomes the dynamic movement of the foxes across the landscape and the main driver for the immigration function of each cell.

2. Immigration

Immigration, driven primarily by the emigration function, is the number of foxes in each of the four primary stocks that have been added to the cell at each computational time step. If the incoming foxes plus foxes in the cell, exceed the carrying capacity of the cell then the emigration function is re-activated and the movement process begins anew. If the carrying capacity has not been exceeded then there is no pressure for any foxes to leave the base cell and none will emigrate.

3. Model Constraints

The spatial dynamics of the spread of the fox rabies disease in Illinois provides an interesting picture. Figure 7 displays four maps produced form from the full 25 year run of the model showing the spread of the disease among the originally healthy, un-hunted fox population.



Figure 7. Selected panels from the various periods throughout the 25 year model run of the spread of the disease through the healthy, un-hunted population of Illinois foxes (t=xy years).

Moving from left to right, the initial map depicts the state of the system at present – that is the initial number of healthy foxes across the state of Illinois – black areas are indicative of cells with low fox populations. The initial map also gives the reader an indication of the scale of the typical fox home range (each pixel in the picture represents the 6 square mile cell size in the model) and the complexity of the computational problems involved.

The second map shows the incipient waves of disease that are created by the introduction of just three rabid foxes in the first month of the model run (these three rabid foxes are introduced into the model only once). Easily visible at the eastern border of the State, the dark areas in this map are areas of minimal fox survival and the map displays the dramatic impact that rabies can have on a healthy population.

After 2.6 years the advance of the first wave of disease is clearly recognized and in the third map, the disease has spread halfway across the state, and is calculated to be advancing at about 24 miles per year. Also visible at this point is the reintroduction of the disease at its origin. This wave phenomena has also been noted in the construction of other dynamic epidemic disease models .

The fourth panel (7 years) describes a mature disease and the wave of disease is in its epizootic stage. The traveling wave is continuously repeated, with each successive wave peak of the disease more spatially diffused in the East-West direction of the advance. This traveling epizootic wave can be seen as the slow cycle on the annual cycle in Figure 8 below. The successive waves develop because the disease does not eradicate the entire population and some of the disease remains viable in the surviving foxes. The disease is unable to move at this point, however, because the population of available foxes is not large enough to encourage much migration. Once the population builds to critical mass however, migratory behavior resumes and the disease begins to spread again, perhaps augmented by the occasional healthy immigrant from Indiana, they re-grow to a substantial healthy population but again are infected by the strays heading East from the disease front. The epidemic builds more slowly in each ensuing repetition since it begins with a smaller healthy population each time. Finally, although difficult to display in a static format, the disease becomes endemic...without epidemic pulse. In this stage the number of diseased foxes in the state is nearly constant (5,586).

Mapped images are extremely powerful for displaying the spatial interactions and dynamic movement of the rabies disease. Although difficult to represent in static format the animations of these images provide a strong case for the use of spatial simulation modeling for numerous applications. The mapped images will also become important in future work regarding the most efficient disease control strategies. Below is a more quantified approach for displaying the results of our calibrated model runs is discussed below.

1. Rabies Pressure

Although hunting pressure now exists on an apparently declining Illinois fox population, we thought the presentation of our results would be clearer if we started the model with the fox population at or near the 1970 estimated mean level of about 88,000. We did lower the initial population in the model but the long term results of subsequent runs were unaffected by this kind of change in the initial conditions. For the first part of our model runs we choose to suspend the effects of hunting to allow us to gauge the impact of the disease alone.

Healthy fox populations were set initially at 80% of the carrying capacity of the cell. The model results show that the healthy, un-hunted population settles to a reasonable fluctuation between 40,000 and 118,000 with a mean level of 79,000 (Figure 8). The total population cycle shown in this figure is caused by the birth of a large number of juveniles every March and the automatic redefinition of the surviving juveniles to adults in the ensuing January.

1. The effect of disease alone

Our next step was to introduce the disease to a healthy cyclically stable, un-hunted fox population. Since rabies is noted to be spreading westward from Pennsylvania, we choose to introduce three rabid adult foxes in the first month of the 25 year run. We choose to separate the effects of hunting from those of the disease. The results are shown in Figure 8. Note that the fox population declines to an average steady cycle with a mean of about 22,000 foxes, this is a 72 percent reduction form initial population levels. From this model run, our estimate of the effect of rabies is that the population will suffer a severe setback (a reduction to one quarter of the healthy mean) the fox does not disappear from the landscape. To ecologists, this is the good news: the fox is not eliminated by the disease. To public health officials, the news is not so good: diseased foxes remain in the state albeit at much reduced levels, and therefore some threat of spread of the disease into the pet population remains.



Figure 8. The model results for a healthy population of Illinois fox and for the same population responding to the introduction of the three rabid foxes in the first month of a 25 year period. These results are for the condition where the hunting pressure was entirely suspended.

2. Hunting Pressure

We then introduced the full effect of the estimated current level of hunting. The results indicate that the healthy fox population disappears over a 25 year period at current rates. Our hunting rates are constant from year to year however, an assumption which cannot be totally accurate. Some of the fox hunting is for pelts and that portion (unknown) of the hunting pressure would fluctuate with the price of pelts and the availability of foxes.

The dynamics of the fox under the pressure of full scale hunting is illustrated in Figure 9. The Illinois fox population, subjected to current hunting pressures, declines to essentially zero over the ensuing 25 years.

We made several more runs with the hunting pressure reduced to one-half and then one-quarter of the full scale current hunting pressure. As one reduces the hunting pressure, the fox populations approaches the no-hunting, cyclic result mentioned above, with most of the recovery made when the hunting pressure was halved. These results are shown below.



Figure 9. The model results for a healthy population of Illinois foxes (no rabies disease) with and without hunting pressure.

3. Combined Effects

As noted, current hunting pressures are sufficient to essentially eradicate the Illinois fox population in about 25 years, with or without introduction of the disease. The current hunting pressure is so severe that the introduction of the disease would have virtually no effect. This is predictable since hunting lowers the densities to the level that immigration/emigration virtually cease and the disease does not have a chance to spread. Our results show however, that if the hunting death rate is lowered to between one-half and one-quarter of its current level, the effect on the fox population is small during the course of the disease.

If the hunting death rate were reduced to half its current level, the mean steady-state population level would drop from 79,000 foxes to 56,000, a decline of about 30 percent. Twenty five years after the introduction of rabies, the long-term mean population with this level of hunting is within 15 percent of the un-hunted mean (see Figure 8). Thus hunting at this reduced level has little effect on the population level of the rabies-infected fox population. The reduced diseased rate due to the reduced density caused by hunting are nearly offset by the increased hunting death rate. The hunting pressure at any rate, is not selective for rabid foxes and therefore the only effect of hunting on the spread of the disease is through the reduction of the contact rate of rabid and healthy foxes. Fox death due to the disease is replaced by death due to hunting.

The patterns of morbidity are considerably different during the early phases of the disease. Hunting deaths are assumed to be distributed across the state on the basis of the local fox population. The deaths due to the disease spread in ever-broader waves, emanating from the initiation points. Eventually (after about 15 years) the disease becomes endemic and the time and spatial patterns of both death rates become identical (Figure 10).



Figure 10. The total Illinois fox population with the introduction of three rabid foxes across the Indiana-Illinois border (at time = 0), without hunting and with hunting at one-half the current level.

4. Controlling the disease

Control of wildlife disease is often expensive. Evaluation of control measures in a dynamic modeling facilitates decision making and gives policy makers additional information and insight into the effects of the control measures tested. Use of dynamic models in the study of wildlife diseases also identifies areas where information is lacking.

Control attempts without clear knowledge of the spatial qualities of the studied population or disease can produce less than optimal results . Bögel suggested that methods used in advance of a rabies epizootic would not be effective in controlling the spread of the disease once the epizootic has begun . Additional work by Bögel, resulted in a proposed method to evaluate populations and rabies control in the wild .

Anderson, et al. (1981) evaluated culling, vaccination and culling combined with vaccination as control measures for vulpine rabies. Their model revealed that population dynamics, most notably reproduction, limited the effectiveness of culling alone . Vaccination of foxes is a commonly chosen control method, but again, has had limited success, particularly in areas of high animal density and good habitat . It is estimated that nearly 100% vaccination is required in areas of good habitat and greater than 15 foxes per square kilometer .

So vaccine programs (the current preferred method of control) are expensive and have not been proven completely effective. The cost-effectiveness of an air-drop, vaccinated bait program could be increased dramatically by more precise knowledge of the location and rate of spread of the disease. A modeling process such as ours should be of great help in such in program given the proper monitoring of the current status of the disease. The model presented can describe a reasonable estimate of the location and rate of spread of the disease front. The model would need to include some additional temporal variability: the inevitable lag in the reporting of the appearance of rabid foxes, the probable lags in the initiation of the vaccine drop, in the effective life time of the bait and the probable lag time between drop and discovery by the foxes. Our model could be expanded to become effective in this arena if appropriate data could be furnished.

The current fox population in the State of Illinois is believed to be rabies-free but the disease appears to be spreading westward from the eastern regions of the country. The public health concern is that rabies could spread from foxes to pets to humans. We have developed a model of the spatial spread of the rabies disease through the Illinois fox population. We have calibrated the model to the best possible extent by altering model parameters such as the death rate (to match historic population estimates) and the biting coefficient (the parameter that modulates the rate of spread of rabies between rabid and healthy foxes). Historic record and model results indicate that vaccination and hunting seem to have some effect on the presence of the disease. Therefore there exists a governmental need to understand the spatial dynamic of the disease and the effects of control strategy scenarios.

Our model is a start in the development of such a public policy tool. We have shown that the disease would likely spread among the native healthy fox population from East to West at about 24 miles per year. The disease would oscillate in waves radiating from the point of introduction, in ever-broader and lower waves, superimposed on the annual cyclical swings in the normal fox population. (It is important to note here that we calibrated our model with total Illinois fox population estimates for the previous 30 year period by fitting our expected populations to the curve of the previous found populations. This helps to validate the modeled rate of spread of the disease.) The disease appears to become endemic across the State in approximately 15 years. We also found that while current hunting pressures would wipe out the fox in the State, hunting at about half this level has an effect indistinguishable from that of the disease itself. Half-current hunting death rates are substituted for the reductions in the death rate of the disease itself. This half-current level of hunting would reduce the fox population in the State to about 70 percent of the expected healthy, un-hunted level. We found that the disease in the absence of hunting would reduce the mean steady state fox population by 72 percent. We suspect that our model could be augmented with currently unavailable data to make it useful in guiding a fox rabies vaccination program.

The main weakness in the collaborative spatial modeling approach is the necessary process of simplification and aggregation. Modeling by definition is a process of simplification, of breaking down complex issues into their core components and looking for causal relationships. The Spatial Modeling Environment is a tool that is able to capture a wide amount of variables and effectively synthesize for output. But it may be this synthesizing or aggregation process into one computational time step, even though varying time steps may be needed to accurately reflect the data, that over-simplifies the outcome. In our approach we model the fox density rather than the individual, so the complex interactions between individuals become statistical probabilities with aggregations of probable outcomes.

We believe the SME approach has no more weaknesses overall than any other computer modeling approach. The key modeling problems lie in how the model is structured, which puts the responsibility on the modelers and modeling team. The SME approach allows the efficient involvement of a team of experts (due to its easily understandable iconographic front-end and spatially explicit output). This collaborative environment more than makes up for any computational inefficiencies. The resulting model is more believable and more supported by the participating experts, than if it were done by a single modeling consulting with expert input.

We are grateful to Laura Hungerford of the University of Nebraska, and Ron Larkin from the University of Illinois Department of Natural Resources and Environmental Sciences, Gail Scherba of the University of Illinois Department of Veterinary Pathobiology, Tim Vandeelan from the Illinois Natural History Survey, Steve Harper from the Geographic Modeling Systems Lab and the University of Miami, Ohio and George Hubert of the Division of Wildlife Resources, Illinois Department of Natural Resources for their valuable input to the collaborative modeling process.