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The effect of fragmented host distributions on the transmission dynamics of directly-transmitted pathogens was explored via stochastic automata simulation. Sixteen diverse population distributions varying in shape and density were used as a substrate for simulated outbreaks. Extended neighborhoods (80 cells), with probability of infection weighted by proximity to an infective source were used to define the overall probability of transitions from susceptible to infected. A static probability defined transitions from infected to recovered. The duration of active transmission as well as the proportion of each population infected per outbreak was averaged over a series of 30 simulations per parameter set. The level of aggregation for each population, measured in terms of the Moran Coefficient (MC) of spatial autocorrelation, was found to affect both the intensity of an outbreak and its length of persistence. Denser populations produced the most cases and lasted longer than those that were sparser. Elongated distributions, measured as the ratio between perimeter and area (PA) reversed some of the trends of increasing density. Long, narrow distributions produced fewer cases and were less persistent than populations composed of more compact clusters but with similar MC. Thus, both the shape and density of host distribution patterns affected the incidence rate, duration of epidemics and the percent of the population infected. Certain patterns of habitat fragmentation, thus, may put more hosts at risk of becoming infected than others.

Fragmented populations of endangered flora and fauna may be exposed to greater stresses than populations that are larger and more continuous [

For pathogens that rely on close contact for transmission, the configuration of populations in space may greatly affect the rate and stability of epidemic outbreaks. Such effects have been repeatedly noted in the natural documentation and simulation of measles outbreaks [8, 9].

Distributions of organisms that are organized into discontinuous clusters display especially complex and unpredictable dynamics. The manner in which such ensembles of populations or “metapopulations” are arranged can affect the persistence of certain populations both free-living or parasitic. In metapopulation models of free-living organisms, dispersal can either stabilize [10,11] or destabilize [

The mathematical modeling of infectious disease transmission still generally relies on non-spatial methods based on ordinary differential equations (ODEs) pioneered early in the 20th century [

These extensions generally rely on subdividing populations into smaller groups with shared characteristics. Such methods become very quickly limited in how much spatial structure they are able to represent before becoming computationally intractable.

Cellular-automata offer an opportunity to study the effect of host spatial distribution on the dynamics of infectious disease transmission in a manner that captures the full complexity of the spatial structure of natural population distributions [

Cellular automata are discrete mathematical systems that allow “bottom-up” simulations of complex systems by applying simple sets of local rules to a matrix of integers over time [

Automata have been widely applied to ecological systems including forest gap dynamics, forest fires [

Here we apply methods derived from stochastic automata to explore the effect of host organization in specific spatial pattern on the persistence and intensity of outbreaks of directly-transmitted pathogens.

Patchy population distributions were created by first seeding 100 cells randomly in a blank matrix then applying a cellular automata-based growth algorithm that used various rules governing the probability of an empty cell becoming occupied based on the number of neighboring cells that were already occupied. In some cases, probabilities were weighted more heavily along certain axes in order to create long, narrow aggregations. These population growth models were suspended once the population reached 10,000.

To produce more variation in patterns, differing numbers of initial seeds were used to initiate growth and were allowed to expand for different durations until the population size limit was reached. To produce more linear patterns, probabilities of growth were weighted to favor expansion along particular axes. The intensity of the weighting affected the length and width of the resulting pattern which is shown in

MC = 0 indicates total randomness, while MC = 1 indicates maximum spatial clustering; values in between indicate various degrees of aggregation. See

All simulations were encoded using the interpreted “J” programming language (Version 4.62, Iverson Software), which is optimized for array manipulation. Simulations were structured as follows:

Assumptions and rules:

1) The environment is represented by a square matrix (224 × 224) of 50,176 integers.

2) Population distributions of hosts are represented by 10,000 variously distributed cells.

Patchy population distributions were created by first seeding 100 cells randomly in a blank matrix then applying a cellular automata-based growth algorithm that used various rules governing the probability of an empty cell becoming occupied based on the number of neighboring cells that were already occupied. In some cases, probabilities were weighted more heavily along certain axes in order to create long, narrow aggregations. These population growth models were suspended once the population reached 10,000.

3) Codes representing the current infective state of individual cells are as follows:

0 = unoccupied

1 = susceptible and uninfected

2 = infected and infectious

3 = immune

4) Time (t) advances in discrete units of one week per time step.

5) During each time step (t), an infection can be transferred from a source (i) to adjacent or nearby contacts up to 4 cells away from the source, as described in

The basic structure of an automaton is as follows:

The future composition of a matrix is thus a function of the current state of that matrix and external factors such as treatment or vaccination that might be imposed. In the present simulation, only the internal state of the matrix is considered.

6) The probability of a set of contacts giving rise to an infection is calculated using a state transition probability function as in

where:

= the probability that an uninfected host becomes infected.

= an index of the relative transmissibility of a particular pathogen.

= the number of infective neighbors at each distance a-d.

= the relative effect of distance on relative transmissibility portraying a geometric decline.

7) For each time step, transitions from an uninfected state to infected are determined by comparing the probability of transmission to each uninfected cell to a set of randomly generated numbers between 0 and 1 using the Mersenne Twister algorithm.

8) Births and mortality from other causes are not considered due to the short term (up to several months) that was assumed for the duration of an outbreak.

Each simulation was terminated until extinction of the pathogen occurred and all infected individuals either became immune or dead, however long that required. For each of the eight distributions, simulations were replicated 10 times and averaged.

The spatial characteristics (shape, density) of host population distributions were found to greatly affect the dynamics of infectious disease outbreaks. Epidemic curves generated for each distribution (Figures 2(a)-(p)), were diverse in amplitude and period (

The mean number of infections occurring in an outbreak tended to be positively associated with the intensity of spatial autocorrelation (

Spatial autocorrelation was much less significantly associated (R2 = 21.4%) with the average duration of outbreaks (

Variation in the total number of infections occurring in an outbreak between simulations tended to be negatively associated (R2 = 61.4%) with the degree of spatial autocorrelation (

Spatial statistics that account for degree of aggregation and exposure did not always suffice to explain the variation in outcomes between simulated outbreaks. Other aspects of shape, such as degree of elongation and fenestration may have impacted outbreak dynamics. For ex-

ample, cumulative number of infections was lower in elongated patches (MC = 0.84) with a higher perimeterarea ratio (

The degree of fragmentation and the connectivity of population distributions contribute to the dynamics of the spread of infectious pathogens. The shape of population distributions can affect both the duration of persistence of an epidemic outbreak. While the intensity of contacts reflected by density was also a critical factor, differences in shape had the ability to negate trends that were otherwise apparent across a gradient of density values.

Shape modulates the intensity of infectious contacts in a population by altering the number of individuals exposed on the edges of a distribution. When the perimeter to area ratio is small, the population behaves more like a homogeneous distribution. For a given density, when the perimeter to area ratio is large, simulations tended to terminate more quickly, and left a greater number of individuals uninfected.

Habitat fragmentation may either protect populations or render them more vulnerable to disease outbreaks, depending on the nature of the pathogen and the arrangement and size of habitat patches. Protection may be

provided in certain cases by fragmentation by interrupting transmission between patches. In natural populations of a Swiss wetland plant (Primula farinosa), outbreaks of smut fungus (Urocystis pimulicola) were more likely to occur in larger, more contiguous habitats [

Conditions might be considered as well in which fragmentation allows a pathogen to persist by imposing asynchrony on its infection dynamics. While an active outbreak is occurring in one patch, other patches are recovering and increasing their immunologic vulnerability through the recruitment of young individuals. Certain pathogens, especially those which either kill or render hosts permanently immune, might be expected to cause their own extinction in host populations that are distributed uniformly and contiguously.

These results may be important in terms of evaluating trends in local and global biodiversity. Models similar to this may define more precisely the vulnerability of spatially discontinuous populations to pathogens, particularly populations of threatened flora and fauna that have become isolated through habitat fragmentation. Population viability analyses (PVAs) have not been reliable in defining these risks, partly because of the complexity that homogeneous models fail to capture but which can be highly relevant to outcomes. The manner in which hosts are distributed can fundamentally affect the dynamics of disease spread through a population.

This work was supported in part by a grant from the Robert Wood Johnson Foundation to Tamara AwerbuchFriedlander. Also we thank Hyung Park for helping edit the manuscript.