Stochastic models of foot and mouth disease in feral pigs in the Australian semi-arid rangelands

Authors


Present address and correspondence: N. Dexter, Booderee National Park, Village Rd, Jervis Bay, Jervis Bay Territory, 2540, Australia (e-mail nick.dexter@ea.gov.au).

Summary

  • 1Foot and mouth disease (FMD) is one of the world's most important livestock diseases and pigs Sus scrofa are highly susceptible. In countries with significant feral pig populations, such as Australia, the possibility that FMD may become established in these populations is a cause of considerable concern.
  • 2Current models of FMD in feral pigs in Australia are based on deterministic population and behavioural parameters. However, the population dynamics of feral pigs in the semi-arid regions of Australia vary stochastically, in concert with the biomass of rainfall-driven pasture.
  • 3This study explored how stochastic variation in the population dynamics of feral pigs in semi-arid rangelands affected the probability of persistence of an FMD epizootic, and its impact on the density of feral pigs. A stochastic model of feral pig population dynamics, with death rate linked to vegetation and rainfall, was linked to a deterministic model of FMD in feral pigs.
  • 4Unlike the fully deterministic model, the stochastic model predicted inevitable extinction of the disease. When the transmission coefficient for FMD (β) was set at the mean value of 9 km2 pig−1 day−1, the mean persistence time of the epizootic (in 1000 simulations) was 3338 days, with a maximum persistence time of 8439 days.
  • 5On average the FMD epizootic reduced the population density of feral pigs, compared with the density of uninfected pig populations, by between 54% and 43% for transmission coefficients (β) equal to the estimated likely mean and minimum values (9·0 km2 pig−1 day−1 and 2·5 km2 pig−1 day−1), respectively. This compares with a suppression of 27% in density for the equivalent deterministic model.
  • 6Further stochasticity was introduced to the linked population and disease model by making the transmission coefficient β stochastic with values based on radio-telemetry of a population of feral pigs in the semi-arid rangelands over an 18-month period.
  • 7The addition of a stochastically varying β lowered the mean persistence time of the simulated epizootic to 1037 days in 1000 simulations and reduced the maximum persistence time to 3907 days. Pig density was reduced by 45% compared with uninfected populations.
  • 8Synthesis and applications. These simulations suggest that any control programme to suppress a FMD outbreak in feral pigs in the semi-arid rangelands of Australia should take account of prevailing environmental conditions when aiming to reduce feral pig populations beneath a threshold density (NT) below which the disease cannot persist. As the NT is higher than for the equivalent deterministic model, the disease may be easier to control than such models suggest.

Introduction

Foot and mouth disease (FMD) is one of the world's most important livestock disease that can devastate the livestock industries of whole countries, such as the 2001 FMD outbreak in the UK (Keeling et al. 2001). The danger to Australia's livestock industry posed by feral pigs Sus scrofa (L.) as potential vectors of exotic diseases is well recognized (O’Brien 1989; Wilson & O’Brien 1989). FMD, because of its potential to damage Australia's livestock export industry severely, has received particular attention. The situation is critical in the semi-arid rangelands where feral pigs coexist, often in close contact, with large numbers of domestic cattle and sheep (O’Brien 1989).

In the event of a FMD outbreak in feral pigs, current policy relies on ‘… establishing the limits of the identified zone (on a concentric ring basis starting from 20 to 30 km away from the point of disease detection); creating an infected depopulation zone to include 20–30 km beyond the limits of the spread of the disease, if possible using natural boundaries; reducing the population density within the infected zone to a level of less than 0·5 pigs km−2 within 30 days’ (AUSVETPLAN 1996). However, Hone & Pech (1990) estimated that the probability of detecting FMD in feral pigs is very low because these animals occupy large areas of Australia remote from human settlement that receive only sporadic veterinary surveillance, and that up to 3000 cases may occur before detection. Further problems occur because feral pigs are shy, nocturnal and cryptic so that even in the best case scenarios only 80% could be killed by helicopter shooting (Saunders & Bryant 1987). An alternative may be to force the abundance of feral pigs below the threshold abundance, NT, below which the disease cannot persist (May 1983).

A complementary approach has been to develop models that predict the level of feral pig control required to eliminate FMD (Pech & Hone 1988) and the rate at which an FMD outbreak would spread through feral pig populations (Pech & McIlroy 1990). Demographic and behavioural parameters for these models have been derived from studies of feral pigs, with disease parameters inferred from veterinary literature on epizootics of FMD in domestic pigs. Both available models were derived from deterministic fox rabies models (Anderson et al. 1981), which assume that populations of hosts grow logistically, demographic rates being a direct function of prevailing population density.

As host population density is a critical parameter in any non-sexually transmitted disease model (Anderson & May 1991), the unpredictable nature of semi-arid grazing systems implies that disease models with deterministic population parameters for the herbivore (host) component of the system are unlikely to be realistic. Despite eating some carrion and live animals, pigs are mainly herbivorous in the semi-arid rangelands (Giles 1980). Semi-arid grazing systems in Australia are characterized by low and erratic rainfall, high summer temperatures and an evaporation rate that exceeds rainfall (Robertson 1987). In these environments, the rate of change in herbivore abundance is heavily dependent upon prevailing vegetation biomass (Bayliss 1985), which is in turn largely dependent on unpredictable rainfall fluctuations (Noy-Meir 1973). Therefore, feral pig populations in semi-arid environments do not grow logistically (Giles 1980; Choquenot 1998). Variation in demographic rates and consequent direction and rate of change in population abundance are more related to prevailing environmental conditions (primarily food supply), than to population density, although intraspecific competition will increase in intensity when food becomes scarce. In the case of feral pigs in the semi-arid rangelands of Australia, prevailing environmental conditions influence population dynamics through their effect on mortality rate rather than fecundity, although higher birth rates occur during times of abundant food (Choquenot 1998).

Another critical parameter in many disease models is the transmission coefficient β. This parameter describes the rate at which the disease is transmitted between individuals, and is partially determined by the contact rate between individuals and the infectiousness of the disease. β is critical to the probability of establishment and velocity of spread in models of epidemic diseases such as rabies (Anderson & May 1979), classical swine fever (Hone, Pech & Yip 1992) and FMD (Pech & Hone 1988). Because contact rate and hence β (in part) is a behavioural parameter, it is likely to be influenced by a complex of environmental and social factors that vary over time. Although Pech & Hone (1988) suggest a range of contact rates and hence β, neither they nor Pech & McIlroy (1990) allow for stochastic or systematic variation in β.

The models described in this study differ from the models of Pech & Hone (1988) and Pech & McIlroy (1990) in that the assumption of logistic population growth in feral pigs is removed and replaced with a series of functions that link variable rainfall to demographic rates through food supply. The models use data collected in a study of the population dynamics of feral pigs in the semi-arid rangelands of north-west New South Wales (Choquenot 1998). This stochastic model of feral pig population dynamics was linked to the compartmental disease model of Pech & McIlroy (1990) to give a model of a FMD epizootic (designated SMI) in which the population parameters of the host vary in accordance with environmental conditions. A second stochastic model (designated SMII) was developed in which β also varies stochastically in accordance with the variation in β inferred from radio-telemetry data on a feral pig population in western New South Wales.

Populations subject to high levels of environmental stochasticity may be driven to extinction by adverse environmental conditions (Levins 1969). Anderson & May (1991) discussed the possibility of such a phenomenon for disease epidemics and called it ‘endemic fadeout’. To explore this possibility for FMD, 1000 simulations each of SMI with two different levels of β and 1000 simulations of SMII were conducted to determine the influence of stochastically varying parameters on the disease's probability of persistence and mean population density of feral pigs.

Methods

radio-telemetry

Estimates of two important disease model parameters (feral pig density and β) were derived from data collected by trapping and radio-telemetry from a feral pig popula-tion at a single site on the Paroo River at Nocoleche Nature Reserve, north-western New South Wales (144°8′E, 29°50′S), Australia. Feral pigs were monitored by radio-telemetry on seven occasions from November 1991 to July 1993: November 1991, February 1992, April/May 1992, July 1992, November 1992, April 1993 and July 1993. Each radio-tracking session lasted from 5 to 9 days. The tracking sessions covered a period of intense drought that eased following heavy rains in late 1992.

Prior to each tracking session, steel mesh traps were set up at several locations in Nocoleche and baited with soaked wheat (Choquenot, Kilgour & Lukins 1993). Captured feral pigs were subdued with a combination of xylazine hydrochloride and ketamine hydrochloride. Adult pigs were fitted with radio-collars (150–151 MHz; Titley Electronics, Ballina, Australia) weighing approximately 300 g and released. At least 2 days were allowed between the end of trapping and the commencement of tracking to allow pigs that had been collared to adjust to the collars. However, the pigs did not appear to vary their behaviour much in response to being collared and were frequently retrapped the day after the collars had been fitted.

The number of pigs tracked varied between tracking sessions from seven to 22. This was due to equipment failure, death of the collared pigs between tracking sessions during the prolonged drought, and to animals moving in and out of the study area. To replace radio-collared pigs that moved out of the study area or died, collars were fitted to new pigs during the course of the study. Tagged pigs moved up to 50 km from their point of capture over the 2 years of this study.

For the duration of the tracking session the position of each pig was recorded hourly, 24 h a day, from three 12-m radio-telemetry masts fitted with twin seven-element yagis. The masts were positioned approximately 3·5 km apart. At the beginning of every hour each person assigned to a tower would scan for a set sequence of pigs on a Telonics TR-4 Receiver (Mesa, AZ), and record to the nearest degree the direction from which the signal strength was strongest. Trackers kept in radio-contact to synchronize readings. The position of each pig was estimated by triangulation using the programme locate II as the centre point of a 95% error ellipse (Nams 1990). Using the Lenth estimator available in locate II, the mean size of the 95% error ellipse around all locations was 0·7 km2.

density estimates

Feral pig density was required for both the population and epidemiological components of the model developed in this study. Density in the trapping session prior to each tracking session was estimated using the Jolly–Seber mark–recapture technique for open populations (Caughley 1977). This technique was used for the first five trapping sessions but recaptures were too few for the tracking sessions in April 1993 and July 1993. Therefore the sum of all pigs caught, seen and known to be in the study area by radio-telemetry were used as a minimum estimate of density for those two tracking sessions. As the Jolly–Seber method makes daily estimates of abundance, the average daily estimate of abundance for the whole trapping session was calculated. To estimate pig density the area of the minimum convex polygon described by all radio-telemetry locations for each tracking session was divided by the mean abundance estimated for the preceding trapping session (Table 1). This minimum convex polygon was used because it represents the combined area covered by all the radio-collared pigs in their day-to-day movements. Incidental long-distance movements and occasional fixes on pigs that had left the study area were not used to calculate the minimum convex polygon.

Table 1.  Density of feral pigs at Nocoleche Nature Reserve, New South Wales, Australia, by Jolly–Seber technique
Trapping sessionJolly–Seber estimate ± 1 SEArea of minimum convex polygon in km2Density km−2
  • *

    Minimum number of pigs known to be in the area.

November 199139·9 ± 7·9 460·85
February 199239·4 ± 16·0 391·18
April 199225·0 ± 10·3 580·41
July 199226·7 ± 5·21030·26
November 199229·8 ± 5·3 730·41
April 199316* 570·28
July 199317* 780·21

stochastic population model

The model of pig population dynamics used in this study was an extension of a stochastic herbivore model derived empirically by Caughley (1987) to describe the interactive grazing system comprising red kangaroos Macropus rufus (Desmarest 1822) and the chenopod shrubland pastures of Kinchega National Park in south-western New South Wales (350 km south of Nocoleche but still in the semi-arid rangelands). To this Choquenot (1998) added the population dynamics of feral pigs through their numerical and functional responses to pasture biomass.

The three components of this system are change in vegetation biomass, change in kangaroo abundance, and change in pig abundance. The system changes quarterly (90 days) approximated by the functions:

ΔV = F(V,R) – C(K,P,V)eqn 1
ΔK = rk(V)K eqn 2
ΔP = rp(V)Peqn 3

The full model was constructed in an EXCEL® spreadsheet. The driving variable for this model R was drawn from the relevant quarterly rainfall distribution, with mean and variance equal to the 100-year seasonal average for Wanaaring (mean 193 mm, standard deviation 90 mm), 20 km north of the study site. For each quarter of a 100-year simulation, R was a random draw from this distribution. As V, P and K depended on R, they also changed quarterly. In these equations V is current vegetation biomass expressed in kg ha−1. F is a function describing change in vegetation biomass expressed in kg ha−1 per quarter, in the absence of grazing. The function C describes the quarterly removal of vegetation, expressed in kg ha−1 per quarter, by kangaroos K and pigs P. In equations 2 and 3, rk and rp are the annual exponential rates of increase for kangaroos and pigs, respectively. The function F takes the form:

F = –55·12 – 0·01535V – 0·00056V2 + 4·9Reqn 4

This function was derived empirically by Caughley (1987) for pastures similar to those at Nocoleche and describes the influence of rainfall and existing vegetation biomass on change in vegetation biomass. Biomass increases with increasing rainfall but the rate of increase decreases with increasing biomass as there is density-dependent limitation to plant growth. As with Caughley's (1987) model, the growth increment not accounted for by V and R provided a standard deviation around the regression of 52 kg ha−1 quarter−1. The function C takes the form:

C = K[86(1 – eV/34)] + P[58(1 – e–−(V –92)/302)](eqn 5)

This function describes the amount of vegetation biomass eaten as the sum of the functional responses of pigs and kangaroos. The kangaroos’ functional response was estimated by Short (1985) for kangaroos of a uniform 35 kg in trials at Kinchega National Park, while the functional response of pigs was estimated by Choquenot (1998) for pigs of a uniform 40 kg in trials at Trangie Agricultural Research Station, New South Wales. The numerical response equations describing the yearly change in herbivore abundance (rk for kangaroos and rp for pigs) were:

rk = –1·6 + 2(1 – e−0·007V)(eqn 6)
rp = –2·045 + 2·78(1 – e−0·0055V)(eqn 7)

In both equations r increases asymptotically with increasing vegetation abundance. Equation 6 was estimated by Bayliss (1985) for kangaroos at Kinchega National Park while equation 7 was estimated by Choquenot (1998) for pigs over the entire Paroo-Cuttaburra Creek System, with Nocoleche forming one of his study sites. Annual increments in population size for both pigs and kangaroos were converted to quarterly increments by dividing equations 2 and 3 by four.

The model was interactive (Caughley & Lawton 1981), with vegetation biomass strongly affecting herbivore abundance but with herbivore abundance also lowering the long-term vegetation biomass by deepening the troughs in vegetation biomass during drought, and cropping off the peaks in vegetation biomass during flushes in vegetation growth following abundant rains. The chief characteristic of this system is that it is centripetal. That is, there is a negative feedback loop such that the system tends to return towards some equilibrium that is never achieved because of the perturbing effects of erratic rainfall.

To aid in comparing the outputs of this model with the earlier work of Caughley (1987), the initial vegetation biomass estimates of 295 kg ha−1 and initial kangaroo abundance estimates of 45 km−2 were taken from Caughley (1987). Pig abundance was set at 0·51 km−2 from the average density of pigs from the seven estimates of density in Table 1 (Fig. 1). The models produce results that are in accord with measured changes in the frequency and amplitude of population fluctuations of kangaroos and pigs in the semi-arid rangelands of Australia (Giles 1980; Bayliss 1985; Choquenot 1998). The coexistence of the two herbivores on the shared resource, without competitive exclusion, is explicable in terms of the storage effect whereby species-specific responses to environmental fluctuation promote the coexistence of potentially competing species (Chesson 1994). In this modelled system pigs were at a competitive disadvantage because they were inefficient grazers compared with kangaroos but were able to persist because their higher intrinsic rate of increase allowed them to exploit times of abundant pasture biomass more effectively than kangaroos (Choquenot 1998).

Figure 1.

An example of simulated 100 years of rainfall for Nocoleche, New South Wales, Australia, with the associated changes in abundance of vegetation, kangaroos and feral pigs.

deterministic disease model

To describe the progress of an outbreak of FMD in a feral pig population, Pech & Hone (1988) adapted a fox rabies model of Anderson et al. (1981) in which individual pigs move through four categories at a daily rate, according to their disease status: susceptibles (X), latents (I), infectives (Y) and immunes (Z). Susceptibles have never been exposed to FMD or have lost their immunity, latents have acquired the disease but are not excreting the virus, infectives are excreting the virus and immunes have recovered from the disease and have temporary immunity. The model is as follows (see Table 2):

Table 2.  Parameters used in the Pech & Hone (1988) deterministic FMD feral pig model
X= density of susceptibles
I= density of latents
Y= density of infectives
Z= density of immunes
N= density of pig population
a= birth rate in absence of FMD = 0·0025 day−1
b= death rate in absence of FMD = 0·00089 day−1
r= rate of increase in absence of FMD (= a − b) = 0·0016 day−1
Nmax= carrying capacity 0·47 pigs km−2
g= density dependent death rate in the absence of FMD = 0·00011 day−1
α= death rate due to FMD alone = 0·0064 day−1
β = FMD transmission coefficient = 9·0 pigs 1 km−2 day−1
σ= rate of change from latent to infective = 0·5 day−1
υ= rate of recovery = 0·17 day−1
ω= rate of loss of immunity = 0·011 day−1
dX/dt = (a – b – gN)X – βXY + ωZ(eqn 8)
dI/dt = βXY – (b + gN + σ)I(eqn 9)
dY/dt = σI – (b + α + gN + υ)Y(eqn 10)
dZ/dt = υY + (a – b – gN – ω)Z(eqn 11)
dN/dt = a(X + Z) – ( b + gN)N – αY(eqn 12)

This disease model assumed logistic population growth with an equilibrium pig density, Nmax. Per capita birth rate, a, was fixed and death rate in the absence of FMD, b, constant. The parameter gN represents the density-dependent component of mortality where Nmax =r/g.

Before developing the disease model and adding it to the model of pig population dynamics, the transmission coefficient β (expressed in km2 pig−1 day−1) was calculated from the radio-telemetry data gathered by the methods described earlier. Ideally β should be gathered from data from a real epidemic of FMD in feral pigs but this was not practical. Therefore β in this study was estimated using the method described by Pech & McIlroy (1990) and modified by Caley (1993; see the Appendix). Estimates of β calculated for each tracking session are contained in Table 3.

Table 3.  Estimates of the transmission coefficient β for each tracking session
Tracking sessionβ
November 1991 3·78 km2 pig−1 day−1
February 1992 2·52 km2 pig−1 day−1
April/May 1992 4·48 km2 pig−1 day−1
July 199227·72 km2 pig−1 day−1
November 1992 9·18 km2 pig−1 day−1
April 199313·12 km2 pig−1 day−1
July 1993 2·5 km2 pig−1 day−1

To test the suitability of Pech & Hone's (1988) model as a basis for a stochastic model of FMD, their deterministic model was tested with data obtained in this study. Pech & Hone (1988) showed that at a density of 15 pigs km−2 and a β of 0·026 km2 pig−1 day−1 the disease became endemic after initial oscillations in abundance of pigs within disease classes. The model was adapted by setting Nmax at the mean density of pigs recorded at Nocoleche, 0·51 pigs km−2, and the mean β was set at 9·0 km2 pig−1 day−1 (the mean value for β calculated from Table 3). Equations 8 to 12 describe the daily changes in density of pigs in the disease categories. The model was constructed in an EXCEL spreadsheet with the change in density of pigs in the disease categories calculated daily. In this derivation of Pech & Hone's (1988) model the differential equations 8–12 were solved by the Euler method so that daily changes in the abundance of infectives were calculated as Xt + 1 = Xt+ (dX/dt). This is a less precise method than the first order Runge–Kutta equations used in the original model (R. Pech, personal communication) but it produced results identical to the original model when using Pech & Hone's (1988) original data.

Despite a much higher β (9·0 km2 pig−1 day−1 compared with 0·026 km2 pig−1 day−1) and a much lower overall density of feral pigs (0·51 km−2 compared with 15 km−2), the dynamics of the FMD epizootic were similar to those in Pech & Hone's (1988) model. The disease rapidly approached a stable equilibrium after initial fluctuations in the abundance of the four disease classes, with a density of 12·9 pigs km−2, equivalent to a 14% reduction in abundance after 3 years in Pech & Hone's (1988) model, and a density of 0·37 pigs km−2, equivalent to a 27% reduction in abundance, after 3 years in this model (Fig. 2). This indicates that Pech & Hone's (1988) model provided a robust framework for exploring FMD further in feral pigs in the rangelands, as the qualitative outcomes were the same.

Figure 2.

Change in densities of total pig population (–), susceptibles (–), latents, infectives and immunes following the introduction of FMD for the deterministic disease model.

stochastic model i

The first stochastic model was constructed by combining the full kangaroo, pig, vegetation model described in equations 1–7 with the deterministic model of FMD described in equations 8–12. As the dynamics of the FMD epizootic component of the model were calculated on a daily time scale while equations 1–3 describe quarterly and annual change, increments in V, equation 1, were divided by 90 (quarters rounded to 90 days) while annual increments in K, equation 2, and P, equation 3, were converted to daily increments by dividing by 360 (years rounded to 360 days). The values for the daily increments in V, K and P were calculated at the beginning of each quarter and held constant for the duration of the quarter. To remove the assumption of logistic growth in the host population and the requirement for an equilibrium pig density from the model of Pech & Hone (1988), the density-related component of death rate, gN, in equations 8–12 was removed and the constant death rate, b, was replaced with a variable death rate, b′. This death rate fluctuates according to available pasture biomass drawn from the interactive plant–herbivore model. Values of b′ were estimated at the beginning of each quarter as the difference between the prevailing rate of population increase, r, estimated from the numerical response of pigs to current pasture biomass equation, and the instantaneous daily birth rate, a. In this way, while instantaneous birth rate remains constant, mortality rate varies with prevailing pasture biomass, increasing when food availability is low and decreasing asymptotically when it is high. Thus the equations describing the progression of the FMD epizootic become:

dX/dt = (a – b′)X – βXY + ωZ(eqn 13)
dI/dt = βXY – (b′ + σ)I(eqn 14)
dY/dt = σI – (b′ + α + υ)Y(eqn 15)
dZ/dt = υY + (a – b′ – ω)Z(eqn 16)
dN/dt = a(X + Z) – bN – αY(eqn 17)

where b′, like the density of pigs in various compartments, is recalculated daily.

In this stochastic model I (SMI), β is deliberately kept constant at two alternative values: the mean (9·0) and the lowest recorded (2·5) used in the simulations. The lower limit of β was included as a sensitivity analysis because the arbitrary definition of a contact being less than 200 m meant that the true contact rate could be substantially over-estimated. Furthermore, FMD is known to cause lameness in pigs (Callis 1984), which may reduce contact rates. To assess the influence of the varying death rate, b′, on the behaviour of FMD in pig populations, the stochastic FMD model was run 1000 times each for the average and minimum value of β. Initial densities of pigs, N0, were drawn randomly from a normal distribution of densities taken from Table 1 with a mean of 0·51 pigs km−2 and a standard deviation of 0·56 pigs km−2 (truncated at 0·1 so as to avoid random draws of 0 density). The initial density of infectives Y0 was set at 0·1 pigs km−2 (the same value as the Pech & Hone 1988 model), and initial density of susceptibles X0 was set at X0 = N0 − Y0. The model was run until the epizootic was deemed extinct when the density of latents fell below 0·0005 km−2 (the same criterion used by Pech & Hone 1988). For each iteration, the duration of the epizootic was recorded in days. The impact of the epizootic on mean pig population abundance was estimated by comparing mean abundance of feral pigs between simulations of SMI and simulations of the plant herbivore model with identical rainfall but no FMD for the 1000 simulations with each of the two values of β.

NT is an abundance of susceptibles below which an epizootic will not persist if an infected individual is introduced to a susceptible population, that is when the rate of change in latents and infectives = 0. Setting dI/dt = 0 in equation 14, and dY/dt = 0 in equation 15, gives:

βXY – (b′ + σ)I = 0 σI – (b′ + α + υ)Y = 0∴ X = [(b′ + σ)(b′ + α + υ)]/σβ = NT(eqn 18)

Therefore NT will be sensitive to variation in death rate b′ as well as β and so will be variable over the course of the epizootic.

stochastic model ii

As β is partially a behavioural parameter it may be influenced by the same environmental parameters that influence other aspects of pig behaviour, such as temperature and pasture biomass (Dexter 1998, 1999). To determine whether β was influenced by these parameters the seven measured values of β were entered as the dependent variable into a stepwise multiple regression with temperature, feral pig density and pasture biomass as the independent variables. Specifically, temperature was mean maximum temperature during each tracking session while pasture biomass was mean ln kg ha−1 available during each tracking session. Pasture biomass was estimated by the comparative yield method (Haydock & Shaw 1975) for the whole minimum convex polygon described under Density estimates. Explicit descriptions of these methods are contained in Dexter (1998). Despite the influence of these variables on other behavioural parameters, there was no significant effect of temperature, density or total pasture biomass on β (F3,3 = 0·27, R2 = 0·2, P < 0·84). Therefore, in this model β becomes a random variable β′, which is drawn weekly from a normal distribution fitted to the seven values of β estimated from the seven tracking sessions. As it is assumed that there will always be some contact between pigs, to prevent β′ equalling zero the distribution of β was truncated at 0·25, the lowest estimated value of β. β′ was redrawn at weekly intervals because that approximates the length of a tracking session, the interval over which β′ was measured in this study.

As with SMI, this model was run for 1000 simulations with the persistence time (measured as the time in days till density of latents fell below 0·0005 km−2) recorded for each iteration. The impact of the epizootic on feral pig abundance was measured in the same way as for SMI but using an identical random draw of β′ as the infected population.

Results

stochastic model i

In contrast to the deterministic disease model, SMI never approached a stable equilibrium. For simulations with β set at the mean value of 9 the longest persistence time was 8439 days with a mean of 3338 days; for simulations with β set at the minimum value of 2·5 the longest persistence time was 9574 days with a mean of 2270 days (Fig. 3).

Figure 3.

Frequency distribution of persistence times for SMI: (a) 1000 simulations of β = 9·0; (b) 1000 simulations of β = 2·5; (c) 1000 simulations of SMII.

An example of a simulation can be seen in Figs 4 and 5, in which an epizootic went extinct after 2424 days. As can be seen when comparing the abundance of susceptibles in Fig. 5 with NT in Fig. 4, NT rose sharply above the abundance of susceptibles at about this time. This rise in NT was due to a large increase in b′ as pasture biomass plummeted following a drought (Fig. 4).

Figure 4.

Example of simulated change in pasture biomass and associated change in density of a pig population infected with FMD and NT for SMI (–) and density of an uninfected pig population with identical rainfall and pasture biomass (–).

Figure 5.

Change in abundance of susceptibles, latents, infectives and immunes, for the simulation in Fig. 4.

In Fig. 4 the overall population density was lower for the population infected with FMD than for the uninfected population subject to the same simulated rainfall. This pattern is reflected in an overall reduction in population density for simulations with FMD compared with simulations without FMD. When β was set at the mean value of 9 over 1000 simulations, the population density for the infected population was 0·29 pigs km−2 while the mean density for the uninfected population was 0·63 pigs km−2, a reduction of 54%. For the minimum value of β (2·5) over 1000 simulations, the mean population size was 0·35 pigs km−2, a reduction of 43%.

stochastic disease model ii

The major effect of making the transmission coefficient stochastic was to reduce the mean persistence time of the epidemic and lower the overall suppressive effect of the epizootic. The mean persistence time for SMII was 1037 days, the longest persistence time was 3907 days (Fig. 3).

The addition of a stochastic transmission coefficient influenced the behaviour of the epizootic (Figs 6 and 7). This example of SMII has an identical draw of rainfall to the example of SMI illustrated in Figs 4 and 5. The density of pigs in all disease classes fluctuated considerably more than for SMI and the persistence time was lower at 2224 days. This high frequency oscillation is also apparent in NT, with the amplitude of oscillation in NT being very much greater than for SMI. This was because β′ had a much greater influence over NT than b′. NT was inversely proportional to β′ in equation 16 but varied little with b′, as its value was small compared with σ and υ.

Figure 6.

Example of simulated change in pasture biomass and associated change in density of a pig population infected with FMD and NT for stochastic model II (–) and density of an uninfected pig population with identical rainfall and pasture biomass (–).

Figure 7.

Change in abundance of susceptibles, latents, infectives and immunes for the simulation shown in Fig. 6.

Adding stochasticity to β reduced the suppressive effect of the FMD epizootic on population density compared with SMI. The mean population density for pig populations suffering the epizootic in SMII was 0·34 pigs km−2, which was 45% lower than the uninfected pig population.

Discussion

Heterogeneity in behaviour, distribution and abund-ance of hosts strongly influences the likely prevalence and persistence of diseases (Anderson & May 1991). This study extends the concept of heterogeneity to include temporal heterogeneity in the population dynamics and behaviour of pigs. The most important result to flow from the small alterations made to the compartmental models of Pech & Hone (1988) and Pech & McIlroy (1990) to accommodate temporal heterogeneity was the eventual extinction of the epizootic in all of the 3000 simulations conducted. The reason for this is that the density of feral pigs drops below NT during drought for sufficient time for the disease to be driven to extinction.

In this model there are some untested assumptions that could substantially change the behaviour of the disease, in particular that of a linear relationship between the contact rate C (Appendix) and density N. If contact rate does not increase at the same rate as density then the dynamics of the disease may be more stable with an increased mean persistence time for the epizootic and a less severe impact on population density.

The imposition of further stochasticity in SMII imparts several differences on the course of the FMD epizootic compared with SMI. β′ varies with higher amplitude than b′ and NT is more responsive to β′ than to b′. Consequently NT will fluctuate much more in relation to X than for SMI, and NT will exceed X more often and to a greater extent. The disease will therefore be more prone to extinction. However, more estimates of β may have more tightly constrained the variance in β′ and imposed less variability on the course of the epizootic.

The estimates of β are higher than those estimated by Caley (1993) at 0·091–0·307 for tropical woodland, and 0·119 for Pech & McIlroy's (1990) recalculated values in temperate montane forest. This is quite surprising as both areas had higher densities of pigs, 2·5–3·5 km−2 for Caley's (1993) site and 1·4 km−2 for Pech & McIlroy's (1990) site, which should lead to higher contact rates There was considerable variation in β in this study, from 2·5 to 27. There is no reason why there should be an upper limit to contact rate and hence β, set by spacing behaviour. Pigs are non-territorial and frequently exhibit close physical contact with one another (Frädrich 1974). The high values of β estimated in this study may be due to the concentration of pigs for much of the time into a relatively small area of riverine woodland during dry weather (Dexter 1998). It is suspected that similar dry-season aggregations of impala Aepyceros melampus (Lichtenstein, 1812) in South Africa help to maintain FMD epizootics in that species (Plowright 1988).

In SMI increasing β lowered the mean population abundance of the pigs. This is due to the increasing infection rate with increasing β and hence more infected pigs susceptible to disease related mortality. There is a density-dependent effect of the epizootic on the pig population because the term controlling the transmission rate between latents and infectives βXY will increase with increasing density of infectives and latents. The result of this can be seen in Fig. 4, where there is a pronounced dampening in the amplitude of the feral pig population fluctuations compared with an uninfected population.

In SMI, with β set at 2·5, and for SMII the distribution of persistence times is skewed to short persistence times for two reasons. First, for SMI and SMII the X0 drawn in the simulation will be below NT in many instances, and hence the disease will rapidly decline to extinction. This is likely to happen in the wild when an infectious individual enters a population below NT. Secondly, for SMII, the disease may disappear through ‘epidemic fadeout’ (Anderson & May 1991), when the initial density of susceptibles falls so low after the initial intense epidemic that stochastic events, such as a decrease in β forcing NT above X, may cause the disease to disappear.

There are several important practical consequences of these simulations for controlling FMD. Hone & Pech (1990) suggest ranking areas of Australia at risk from FMD in feral pigs, according to criteria such as proximity to ports, feral pig density and the likelihood of detection. To this could be added seasonal conditions, with good conditions increasing an area's risk ranking and poor seasonal conditions lowering an area's risk ranking. These results qualify Hone & Pech's (1990) suggestion that areas with a feral pig population density below NT should attract a low risk ranking. As demonstrated in these models, NT can fluctuate depending on b′ or on β′. Furthermore, the relatively high number of simulations in which the FMD epizootic rapidly goes extinct, due to epidemic fadeout or X0 being below NT, indicates that a number of outbreaks could never be detected.

While it is desirable to eradicate FMD as swiftly as possible and it is also likely that enormous financial resources will be available for the control of an FMD epizootic (Wilson & O’Brien 1989), total eradication of the feral pig population may not be practical on a regional or even local scale. This study reinforces the advice of Hone & Pech (1990) in targeting NT as the goal for disease eradication rather than total feral pig eradication. However, NT has become a moving target because NT is so dependent on β, which is very difficult to assess in the field. The safest course would be to assume β is at its highest and aim to reduce pig density to below NT for the combination of the highest possible β and a b that can be deduced from the pig's numerical response to prevailing pasture biomass, which can be measured relatively easily.

Given the assumptions inherent in this model and the small sample sizes used to derive the parameters, and the uncertainty inherent in representing a continuous process as a discrete process (Gurney & Nisbet 1998), the results of these simulations should be regarded as indicative rather than prescriptive. Furthermore, the results of the simulation should be seen in context with ecological factors not covered by these models, such as proximity to other feral animals and stock. However, this study highlights the need for considering the prevailing environmental conditions when planning control strategies to eliminate an exotic disease from a wild population of animals.

Acknowledgements

I thank David Choquenot for assistance in developing the disease models in this paper. I also thank B. Lukins, D. Mula, P. Boglio, R. Pizel and D. Tolces for help with trapping the pigs and the many volunteer students from the University of New England, University of Canberra, and Dookie Agricultural College for their assistance with the radio-telemetry. I thank New South Wales National Parks and Wildlife Service for permission to use the study site. Funding for this project was provided by the Wildlife and Exotic Disease Preparedness Programme.

Appendix

For each tracking session the hourly locations of all k radio-collared pigs were recorded. However, many locations were not recorded because signals were not received. From this the total number of possible distances, s, between pairs of pigs was calculated by k!/(k − 2)!2!. Using the same criterion as Pech & McIlroy (1990), a contact c was recorded when the distance between a pair of pigs was < 200 m. Thus c/s contacts as a proportion of pair-wise distances occurred. To estimate contact rate C and from this β the following equations are taken from Caley (1993). With d recordings per day and assuming k pigs are randomly mixing and representative of the total population then the total contacts day−1 km−2 would be:

(c/s)(dN!)/[(N – 2)!2!] = (c/s)(dN)(N – 1)/2  = cdN(N – 1)2s

and contacts km−2 day−1 would be cdN(N − 1)2sA, where A is the area (km2) over which pigs were recorded.

To determine the change in number of susceptibles (X) and infectives (Y), the number of pig to pig contacts involving a susceptible and a infective pig needs to be calculated. For any contact between pig A and pig B in a population of N pigs, the probability that pig A is infective and pig B is susceptible is (Y/N)(X(N − 1) while the probability that B is infective and A is susceptible is (X/N)(Y/N − 1), Therefore the probability that a contact results in disease transmission is:

(Y/N)[X/(N – 1)] + (X/N)[Y/(N – 1)] = 2XY/N(N –1)] 

Hence the number of infectious contacts per unit area per day:

[cd(N – 1)N]/2sA × 2XY/[N/(N – 1)] = βA(X/A)(Y/A) ∴ βAcdA/s km2 pig−1 day−1

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