Shirli Bar-David, James O. Lloyd-Smith, and Wayne M. Getz. 2006. Dynamics and management of infectious disease in colonizing populations. Ecology 87:1215–1224.

Appendix A. A description of the simulation model.

The Deer Model

The model simulates the colonization process of a female Persian fallow deer population reintroduced over time into a conservation area in northern Israel. It links dispersal and home range establishment to landscape templates that represent the study area: a 639 km² rectangular section of the Galilee region (between 32°54’–33°05’N and 35°09’–35°28’E). Model parameter values are based on empirical data obtained during the first three years of the reintroduction program (1996–1999). The model was validated by comparing its predictions with the actual spatial distribution of the system over five years, two years beyond the time period on which parameters were based (Bar-David et al. 2005).

The model is a stochastic simulation from which we have extracted average spatial distributions. It includes the following elements (see Bar-David et al. 2005 for details): (1) a landscape matrix defined by a 213´300 1-hectare pixel grid overlaying the study area, where each pixel has a score that represents the quality of the habitat in the area surrounding it; (2) annual release of female deer from a habituation enclosure (10 females per year for the first five years, and five females per year during the following five years—see Saltz 1998); (3) age-dependent survival and reproduction; (4) dispersal of individuals—i.e., deer, newly released or wild-born, move through the landscape matrix evaluating potential home ranges by habitat quality and the presence of conspecifics, either establishing a home range or dying after a defined number of movements; (5) individual deer have home ranges of a fixed size (17 by 17 pixels) that can overlap with home ranges of up to eight other deer; (6) females born in the wild leave their mother and search for a new home range during their second year of life; (7) released females can shift their home range to improve its habitat quality during the first three years after release.

Incorporating disease

We modeled an SEI (Susceptible, Exposed but not infectious, Infectious) disease process (Hudson et al. 2001; Getz and Lloyd-Smith 2006), with characteristics of BTB, assuming no effect on host recruitment, behavior, or survival (McCarty and Miller 1998) Transmission was a stochastic, spatially-structured horizontal process, as elaborated below. In addition fawns born to infectious females could become infected via maternal (vertical or pseudo-vertical) transmission during their first year of life. Once a susceptible individual became infected, we assumed a one-year incubation period before the individual was infectious (Wahlström et al. 1998). Within each time step, fawns were recruited, maternal and horizontal transmission occurred, individuals were removed by natural mortality, and disease progressed in incubating animals. We assumed no immunity or recovery from BTB once deer were infected (McCarty and Miller 1998).

The model focuses on females (as in the base model). Males may play a major role in disease transmission (O’Brien et al. 2002) (e.g., as mixing agents due to possible life-long movements). Because the model describes a colonizing population, however, all individuals are mixing and dispersing due to the population expansion process (Perelberg et al. 2003). Thus we assume that the relative importance of males will be diminished, as compared with an established population where males may account for a greater proportion of dispersal.

Simulations ran for 20 years, covering roughly three deer generations (earliest age of reproduction is 2–3 years, with individuals living as long as 10–15 years), a reasonable time interval to consider short to mid-term disease management scenarios. All simulations started with one infectious individual released at the first time step.

Transmission

We incorporated the spatial structure in Persian fallow deer populations (Dolev et al. 2002) into the disease process by assuming that the infection rate for each susceptible individual is determined from contacts with infectious deer only in its immediate surroundings. In particular we assumed that the infection rate between two deer is proportional to the degree of overlap of their home ranges, an assumption derived from the observed high positive correlation between the degree of home range overlap and the level of interaction among individuals (Perelberg 2000). We also assumed that each individual’s total contact rate with other deer was density-independent (increase in population density in the wild did not impact significantly the degree of overlap between individuals’ home ranges, i.e., the growing population spreads to new territories so that the population density remains roughly constant, Bar-David et al. 2005), so that the risk of infection to each susceptible individual is proportional to the fraction of potential contacts that are infectious. Specifically, if W_ij(t) and V_ik(t) are the areas of overlap of each susceptible individual i with other susceptible (or vaccinated) individuals ( ) and infectious individuals ( ), respectively, then the force of infection for individual i is calculated as

,

(A.1)

where the transmission parameter β is discussed below. Assuming infection is a constant hazard process within each time step, the probability that individual i is infected in any given time step is now given by the expression

(A.2)

In addition, newborn fawns were assumed not to be infected by neighboring deer before establishing their own home ranges, but could be infected with probability b_mat if their mother was infectious.

Transmission coefficient (β)

We chose b to reflect measured rates of BTB transmission in herds of farmed fallow deer (Wahlström et al. 1998). That study used BTB incidence data in seven herds to estimate that the number of effective contacts (k) made by an individual during one year was in the range 0.07–0.88, assuming an incubation period of one year. The parameter k is approximately equal to our transmission coefficient b (see below), so we used values of b in the range 0.1–1.0 to represent settings with low (0.1), moderate (0.5) and high (1.0) disease transmissibility. Maternal transmission (β_mat) was set to 0.25 in most runs (McCarty and Miller, 1998); we tested sensitivity of model outputs to maternal transmission by setting β_mat=0.

The relationship between Wahlström et al.’s parameter k, from a Reed-Frost chain binomial model, and our transmission coefficient b can be evaluated as follows. If S and I are the number of susceptible and infectious individuals in contact with one another, then the number of new infections in the Reed-Frost model of Wahlström et al. is   When , which is true in this deer system where k<1 and the prevalence I/N<0.5, this is approximated by . Because total numbers of individuals are replaced by home range overlaps in our spatially-structured model, the latter expression is equivalent to Equations (1) and (2) above, with . Therefore we chose a range of values for b with reference to Wahlström et al.’s estimates of k.

Model Outputs

For each parameter combination 250 simulations were performed. Specifically, we randomly chose five spatial population scenarios (each depicting the spatial expansion of the population for 20 years) to serve as templates for disease simulations. For each of the five spatial scenarios, 50 repetitions of each disease scenario were performed.

For each parameter combination, at each time step, the following outputs were examined: (1) the proportion of model runs in which the disease went extinct, (2) the number of infectives, (3) the average spatial distribution of the population (a matrix that represents the overall occupancy of the study area with each grid pixel representing the average number of females that occupy the pixel as part of their home range), (4) the average spatial distribution of infectives (a matrix that represents the infectives’ use of the study area), (5) the average range distance of the population, and (6) the average range distance of infectives. The range distance was calculated as the square root of the area occupied divided by . This calculation yields the radius of a circle with area equal to the occupied region (Shigesada and Kawasaki 1997).

Sensitivity to structured landscape

To explore the influence of landscape structure on population and disease expansion, we ran the model over an unstructured landscape. We simulated population expansion for each of 10 random landscape templates (i.e. in which the habitat quality score of each pixel was generated as a random number from a uniform distribution on the same range as the real scores). For each of these spatial population scenarios, 25 repetitions of each set of disease parameters were performed. Average model outputs were compared to those based on the original, spatially explicit landscape (total of 250 repetitions in each case).

LITERATURE CITED

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Wahlstrom, H., L. Englund, T. Carpenter, U. Emanuelson, A. Engvall, and I. Vagsholm. 1998. A Reed-Frost model of the spread of tuberculosis within seven Swedish extensive farmed fallow deer herds. Preventive Veterinary Medicine 35:181–193.