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Keywords:

  • contact heterogeneity;
  • contact networks;
  • epidemiological models;
  • extinction;
  • network model;
  • R0;
  • tasmanian devil facial tumour disease;
  • transmission;
  • wildlife disease

Summary

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Conclusions
  8. Acknowledgements
  9. References
  10. Supporting Information

1. Understanding the nature and characteristics of contact heterogeneities is crucial for predicting the epidemic behaviour of infectious diseases. Nonetheless, few studies include contact heterogeneities when modelling disease outbreaks in wildlife, which differ in their population impact from human diseases.

2. We use empirical estimates of contact heterogeneities and network metrics to simulate outbreaks of devil facial tumour disease (DFTD), an extinction-threatening infectious cancer. We incorporate tuneable algorithms, with a range of transmission rates and latent periods of DFTD, to grow devil population networks capable of reproducing observed aspects of devil ecology, demographic and seasonal-based mixing preferences. The outputs of the network model are compared with a stochastic mean-field model, in which every individual is equally likely to pass or acquire infection through time.

3. Our network model predicts a lower epidemic threshold for DFTD compared with the stochastic mean-field model. While host extinction probabilities are similar in both models, the network model predicts faster devil extinction and higher DFTD extinction probabilities, particularly for intermediate levels of transmissibility.

4. While the time taken to devil extinction increases with the longer estimate of latent period, probabilities of both, disease and devil extinction, are greater with the shorter latent period. Host–pathogen coexistence is strictly subject to the longest plausible estimate of latent period and low transmissibility.

5.Synthesis and applications. In the particular case of DFTD, incorporating observed host network structure has only a modest effect on the outcome of the host pathogen interaction. In general, however, non-random network structure may have major implications for the management of wildlife diseases. Our results suggest that this is particularly likely for pathogens in which the probability of transmission given a contact is intermediate. Our approach provides a template for using empirically obtained data on contact networks to develop models to explore the extent to which network structure influences R0, the probability of extinction and the mean time until extinction.


Introduction

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Conclusions
  8. Acknowledgements
  9. References
  10. Supporting Information

Most compartmental models of disease (e.g. Anderson & May 1979) make a mean-field assumption, which is that disease dynamics can adequately be modelled by assuming that individuals make contact with each other at random. Estimates of the basic reproductive number R0 derived from such models form the basis of most management strategies, because elimination of a disease requires driving R0 to below one. However, it has become increasingly clear that the structure of many contact networks for both human and wildlife diseases indicates that mean-field assumptions can lead to unreliable estimates of R0 (Meyers et al. 2005; Keeling 2005; Porphyre et al. 2008).

Modelling disease spread through contact networks, which can explicitly account for non-random mixing patterns of hosts, is becoming a powerful predictive epidemiological tool (Gomes-Gardeñes et al. 2008; Mosson et al. 2008; Eames, Read & Edmunds 2009). All models must make assumptions about contact patterns between individuals, either by incorporating certain heterogeneities in contact or by assuming random mixing. The use of social network analysis has been a major step towards assessing the validity of the assumption that the mean contact rate between individuals is adequate to describe transmission dynamics. The application of network theory in human epidemiology has proved to be extremely useful for estimating individual heterogeneities in contact patterns and examining their implications for disease transmission dynamics for both sexually transmitted (Liljeros et al. 2001; Gomes-Gardeñes et al. 2008) and non-sexually transmitted diseases (Anderson et al. 2004; Lloyd-Smith et al. 2005). Contact networks can be used not only to investigate epidemiological parameters of an infectious disease but also to identify key individuals or core groups (‘superspreaders’) in epidemic outbreaks, to predict their impact on disease dynamics and to design targeted control strategies (Christley et al. 2005; Böhm., Hutchins & White 2009).

Social networks often differ from many other types of networks in two important properties: high levels of clustering (termed ‘transitivity’, which is a network metric that estimates the density of triangles in a network) and a positive correlation between the degree (or the number of connections) of neighbouring nodes (Newman & Park 2003). This means that individuals are more likely to be connected in the network if (i) they share a mutual neighbour or (ii) if they have a similar number of contacts. Such network properties play important roles in determining the likelihood of an epidemic outbreak and in mediating infection dynamics.

The epidemiology of wildlife diseases is qualitatively different from that of most human diseases because wildlife diseases, particularly highly pathogenic ones such as devil facial tumour disease (DFTD), strongly affect population size. This may have a major influence on the structure of the network itself. Most existing network models assume that association between individuals remain fixed in time. Recent studies suggest, however, that including dynamic contact networks in epidemic models can drastically change the estimates of R0 produced by static networks (Volz & Meyers 2007, 2009). Difficulties in collecting data on dynamic contact networks in free ranging animals have hampered application of network approaches to understanding wildlife diseases. Recent advances in radiotelemetry technologies combined with the emerging interest in network theory have allowed the collection of more accurate information on animal contact networks in wildlife disease studies (Böhm., Hutchins & White 2009; Hamede et al. 2009).

A common assumption when modelling epidemics through networks is that all interactions have the same probability of causing infection. This is clearly not the case for transmission of many parasites and pathogens (Perkins, Ferrari & Hudson 2008; Drewe 2010). For instance, for diseases that require direct contact for transmission (such as DFTD), detailed information on the social setting of interactions is needed (Hamede et al. 2009). This is in contrast to diseases that can be spread through aerosol spray or fomites, such as bovine Tb, in which it may be reasonable to assume that animals sharing the same home range or trapping grid over a certain period are at greater risk of passing infection (Porphyre et al. 2008).

The Tasmanian devil Sarcophilus harrisii, the largest living carnivorous marsupial, is currently threatened with extinction by DFTD, a fatal infectious cancer (McCallum & Jones 2008; McCallum et al. 2009). DFTD is transmitted by direct inoculation of tumour cells between infected and uninfected hosts through biting (Pearse & Swift 2006), a process facilitated by very low host genetic diversity at the major histocompatibility complex (MHC) genes associated with tumour recognition (Siddle et al. 2007). Decreased survival in affected populations predicts that local extinctions are possible within 10 years of disease arrival (McCallum et al. 2007). We recently derived an empirical contact network for wild Tasmanian devils, obtained from proximity sensing radiocollars (Hamede et al. 2009), in which we characterized seasonal contact network dynamics in a distinct population. Our key question in this study is to determine how the structure of this observed contact network might affect the dynamics and epidemic behaviour of this disease in the wild and hence potential management strategies.

In this study, we simulate the epidemic spread of DFTD into naïve devil populations using weighted networks, which incorporate gender mixing properties and seasonal shifts in network metrics associated with mating behaviour (Hamede et al. 2009). We use tuneable algorithms for growing virtual networks capable of recreating our observed devil contact patterns into epidemic simulations. We run these epidemic simulations by including population and disease parameters into a susceptible-exposed-infected (SEI) network model to investigate seasonal, sex- and population-based network metrics on the spread of DFTD under different transmission rates and latent periods of the disease. Finally, we compare our results and estimations of R0 with those of a stochastic mean-field model, in which every individual is equally likely to pass or acquire infection through time. We have used DFTD as a case study, but the approaches we have used to investigate the effect of realistic contact network structure on the transmission of wildlife disease should be applicable to understanding and managing epidemics of other pathogens in dynamic host populations.

Materials and methods

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Conclusions
  8. Acknowledgements
  9. References
  10. Supporting Information

Model structure

The devil population is described by a basic age-structured stochastic population dynamic model, which determines the number of nodes (devils) at any point in time (see Supporting Information). Empirically determined seasonal and sex-mixing preferences (Hamede et al. 2009) were used as input for an algorithm to create networks with realistic metrics (mean degree and transitivity: see Table 1). To incorporate the effects of seasonal and sex-based contact patterns within this population, we use a formula for probabilistically joining a pair of nodes. Consider a square of side length inline image (hence yielding an area of K units) wrapped into a torus to avoid boundary effects, where K is the carrying capacity of the landscape. After assigning random uniform coordinates to each node within this area, the probability p12 that two nodes are joined by an edge depends on (i) dist, their separation in the landscape and (ii) the (empirically estimated) rate of association between their sex classes:

  • image(eqn1)

where inline image where A and B denote the sexes of the two animals. Here eAB represents the affinity for an animal of sex A for an animal of sex B. The Gaussian width parameter V can be thought of as representing either animal ranging or network clustering. Essentially for V∼0·003K, long-range contacts within a set of K animals (in this article, the population at carrying capacity) occur at a rate expected for a random collection of nodes (Badham, Abbass & Stocker 2008). For V values below this value (short range), clustering is preferred (Fig. S1a Supporting Information) and the network giant component (the connected subnetwork containing the majority of the nodes in the entire network) is likely to be small. As V increases beyond this threshold, any node is likely to be joined to any other node and the model approaches mean-field (hereafter ‘MF’) behaviour (Fig. S1b Supporting Information). This is borne out in (Fig. S2, Supporting Information), which shows the dependence of giant component and normalized mean degree upon V. Above the value V/K = 0·003, most nodes are contained in the giant component, whereas below this value, a randomly selected node is likely to have only a few connected neighbours. In this study, we set = 0·003 K, consistent with the assumption that neither ‘long range’ nor ‘short range’ contacts are preferred. Once a choice for V has been made, the affinities eAB are scaled so as to generate networks with mean degree and transitivity comparable to those observed empirically (Hamede et al. 2009).

Table 1.   Parameters and symbols used in network and mean field models
QuantitySymbolValue
Number of adult devilsN
Number of reproducing femalesnF
Carrying capacityK100 animals
Annual recruitment rateR1–2 female
Natural mortality ratem0·07–0·12 season
Gompertz parameterG1–2
Gaussian variance parameterV0·003K
Mating gender-mixing ratioseMM:eMF:eFF1:3·5:1·1
Non-mating mixing ratioseMM:eMF:eFF1·5:3:2·2
Rewiring interval 1 season
Mean mating season degree 14
Mean non-mating season degree 10
Mean mating season transitivity 0·47
Mean non-mating transitivity 0·39
Recruitment intervalδtrec4 seasons
TransmissibilityΤ0–1
Latency rateλ0·25–1 season
Induced mortalityμI0·25 season

As described in the Supporting Information, births, deaths and disease transmission are propagated in these simulated contact networks using the ‘next-reaction’ formalism of Gillespie (1977). ‘Reactions’ correspond to transitions between susceptible, exposed and infectious classes. If an edge exists in the network between an infected and a susceptible devil, the transmissibility T is the probability that the susceptible devil becomes exposed in the 3-month time step. The basic approach was modified to account for (i) spatial structure in terms of network connectivity and (ii) the need to include age-dependence in natural devil mortality. Parameter definitions and initial values used for the simulations are presented in Table 1, and Fig. 1 shows the basic model structure. All models were implemented using custom-written code in the R programming environment (R Development Core Team 2011).

image

Figure 1.  Compartmental model with the transitions of susceptible, exposed and infected classes. Ageing of each class is given by the Gompertz parameter G described in Table 1. The parameter ß (transmission rate) is replaced for T (transmissibility) in the network model.

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Devil population models

The stochastic algorithm we used to build dynamic contact networks is able to capture several aspects of devil ecology and DFTD transmission. Based on the length of the mating season and observed changes in sex-based mixing preferences (Hamede et al. 2009), we set the ‘rewiring’ period (the interval after which the network edges were regenerated from the algorithm described earlier) at 3 months (see Appendix S1 Supporting Information). This means that for every (annual) recruitment event, there are four seasonal rewiring events. Then, the ‘full network’ model consists of three consecutive seasons where non-mating season mixing parameters determine the patterns of contact followed by a fourth season where the annual recruits appear and mating season mixing quantities are used to grow the network. In addition, we examined the effects of three other network rewiring schemes: ‘season only’ (mating versus non-mating mixing preferences), ‘sex only’ (male versus female mixing preferences) and non-preferential mixing (‘rewiring effect only’). Because of the high pathogenicity of DFTD, the population size is expected to be driven by the host–pathogen interaction. Rewiring our network at discrete time intervals allows modelling the spread of DFTD through dynamic contact networks. Given that we have empirically estimated heterogeneities in contact patterns in sexually mature devils only (Hamede et al. 2009) and that prevalence in animals <2 years of age is very low (McCallum et al. 2009), recruitment into the network model occurs only once devils reach sexual maturity (at 2 years of age).

Finally, for comparison to the established approach in network modelling, we consider the MF model population. That is, network structure is ignored and all susceptible animals are equally likely to contract DFTD from an infected devil, regardless of location, season or sex.

Estimates of latent period

The latent period of DFTD is unknown but is likely to be variable depending on a number of factors such as the genotype and immunological response of the infected host, its overall individual fitness, the number of tumour cells transferred during contact and the location of the cells in the recipient host. Our estimates of latent period are inferred from the best information available based on field (R. Hamede, unpublished data) and experimental observations (Kreiss et al. 2010). There is one anecdotal case of a wild Tasmanian devil brought into captivity, which developed DFTD 10 months after its removal from the wild (Tasmanian Department of Primary Industries, Water and Environment, unpublished manuscript). Experimental inoculation of tumour cells in captivity has shown a much shorter but still highly variable latent period, although the dosage used in these trials was considerably higher than what could be expected in a ‘natural’ transmission scenario (Kreiss et al. 2010). Field observations suggest that tumours can increase tenfold in size in as little as 3 months and can cause death in 6–12 months (R. Hamede, unpublished data). We use four different estimates of latent period, ranging from 3 to 12 months.

Estimating R0

We used several methods of estimating R0 as a function of different contact patterns. For basic reference, we use a stochastic SEI compartmental model assuming MF behaviour. By analogy with a deterministic SEI model with frequency dependent transmission, this basic estimate is

  • image(eqn2)

where β represents the infection rate (transition from susceptible to exposed classes), λ is the (exponentially distributed) rate of transition through the exposed class to the infected class, μ is disease independent mortality and μI is the mortality rate of infected animals (for details see Appendix S1 Supporting Information). We estimate this MF R0 for our randomly generated networks by making the following substitution:

  • image

where the angled brackets represent the mean (infected) node degree per unit time and T is the probability per unit of time that a transmission event occurs, given two nodes are connected (here the factor of four arises in converting from contacts per 3-month season to an annual transmission rate).

Anderson et al. (1986) suggest that heterogeneity in contacts can be accounted for by adjusting the mean-field approximation of R0 as follows:

  • image(eqn3)

where as Newman (2002) suggests:

  • image(eqn4)

In both equations (3) and (4), the angled brackets denote globally averaged quantities and σd is the standard deviation of the degree distribution. The superscripts A and N differentiate the arguments used by and Anderson et al. (1986) and Newman (2002), respectively.

For each simulation, we record the life span of each animal and, when infection occurs, the time it spends in the exposed and infected classes. By extracting transition rates (μ, λ, μI) from each simulation, we are able to incorporate the effects of intraspecific regulation and ageing into our estimates of R0. At the beginning of each simulation, we assumed that 97% of the population was susceptible and 3% was infectious. On the basis on the field observations (R. Hamede, unpublished data), the value of induced mortality μI was kept fixed (6 months).

Results

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Conclusions
  8. Acknowledgements
  9. References
  10. Supporting Information

Three different outcomes after 50 years are possible: devil extinction, devil coexistence with disease and devil persistence with disease extinction. For each set of values for transmissibility (T) and latent period (λ), 500 simulations with identical conditions were used.

Estimation of R0

Figure 2a shows the differences in the value of R0 (and hence the epidemic threshold) for network and MF models for a latent period of 6 months. Incorporating empirically derived devil contact patterns increases the R0 of DFTD by between 5 and 15%, depending on whether the estimates of Newman (2002) or Anderson et al. (1986) are used. Accordingly, the epidemic threshold value of transmissibility is c. 10% lower than the MF estimate, Tmf∼ 0·44.

image

Figure 2.  Estimates of R0 (median) and epidemic threshold (R0 = 1) in the three methods using two different latent periods, (a) = 6 months, (b) = 9 months.

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Figure 2b shows analogous plots, but for a latent period of 9 months. Again, the network R0 values are 5–15% larger than the MF prediction. However, increasing the latent period has marginally raised the (MF) epidemic threshold, Tmf∼ 0·47.

Effect of contact patterns in disease simulation models

To demonstrate the effect of contact patterns on disease transmission and epidemic outcome, we ran simulations comparing MF models to the network model. Additionally, we considered scenarios where neither preferences (‘rewiring’ effect only), one of (either ‘season’ or ‘sex’ only) or both (‘full network’) sex and seasonal dependent association behaviours, were present. Predicted devil extinction was higher at intermediate values of transmission rate in the MF model, but similar between both models at the lower and upper values of transmissibility (Fig. 3a). The predicted time to host extinction is faster in the network model than in the MF model but this difference between models converges as transmissibility approaches one (Fig. 3b). The lower values of transmissibility, at which these large differences between models in devil extinction processes occur, produce estimates of R0 (see Fig. 2a) that are inconsistent with the observed estimates from field data (McCallum et al. 2009).

image

Figure 3.  Effect of network association preferences versus mean-field behaviour for (a) devil extinction probability and (b) median projected time for devil extinction with 95% confidence intervals (latent period = 12 months). Note: in (b), time of extinction was obtained only from cases where devil extinction occurred. For clarity, only the results for the mean-field (solid circles) and full network models are shown, but results of all network models were similar.

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Host–pathogen coexistence was only possible at low to intermediate values of transmissibility. Higher values of transmissibility led to the extinction of either host or pathogen (Figs 3a and 4b). There was a large difference between the MF and network predictions for the probability of host–pathogen coexistence (Fig. 4a). This result shows that the constraints imposed on connectivity between hosts in the network models directly increase the likelihood of DFTD extinction at the expense of devil-DFTD coexistence (Fig. 4b) while at the same time accelerating the time-scale of host extinction processes (Fig. 3b). For both probability of host–pathogen coexistence and probability of pathogen extinction, the difference between the models converges at higher transmissibility values (Fig. 4a, b).

image

Figure 4.  Effect of network association preferences versus mean-field behaviour for (a) devil-DFTD coexistence after 50 years, (b) DFTD extinction probability (latent period = 12 months). Note: coexistence probabilities were obtained only from cases in which both devil and DFTD survived after 50 years. DFTD extinction probabilities are conditional to devil survival.

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Effect of latent period

We ran simulations of our network model to estimate the effect of latent period on devil and DFTD extinction and coexistence probabilities (Fig. 5a–c) as well as predictions of time taken to drive either devil or DFTD to extinction (Fig. 6a, b) at different values of transmissibility. A shorter latent period for DFTD increased the probability of host extinction, particularly at intermediate values of transmissibility (Fig. 5a). The time taken to devil extinction also increased at longer latent periods (Fig. 6a). Probabilities of disease extinction were greater with the shorter latent period, increasing up to c. 15% at intermediate rates of transmissibility (Fig. 5b). Similarly, the time to disease extinction increased at intermediate levels of transmissibility, while the shorter latent period favours a faster rate of pathogen extinction (Fig. 6b).

image

Figure 5.  Effect of latent period on (a) devil and (b) DFTD extinction probabilities and (c) coexistence (after 50 years) in the full network model.

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image

Figure 6.  Effect of latent period on (a) devil and (b) DFTD extinction time (with 95% confidence intervals) in the full network model.

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Interestingly, host–pathogen coexistence was not possible with a short latent period regardless of the transmission rate (Fig. 5c). With a long latent period, coexistence was possible, reaching a maximum probability of c. 25% at intermediate transmissibility (Fig. 5c).

Discussion

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Conclusions
  8. Acknowledgements
  9. References
  10. Supporting Information

A limited but increasing number of studies are demonstrating the profound effect that the structure of a contact network can have on the dynamics of infectious diseases (Keeling 2005; Christley et al. 2005; Read, Eames & Edmunds 2008). Our network modelling approach allows us to make several inferences regarding the role of heterogeneities in contact patterns on the transmission dynamics of DFTD. The most immediate conclusion is that our network model, incorporating known heterogeneities in contact patterns with reproductive season and sex variation, predicts a lower transmissibility threshold for an epidemic to occur compared to the MF model (Fig. 2). Individuals with anomalously high number of contacts (such as occur in scale-free networks) are capable of decreasing the epidemic threshold of a disease (Pastor-Satorras & Vespignani 2001) compared to randomly mixed populations. The devil social networks used in this study included some highly connected individuals, in accordance with our empirical data on devil interactions (Hamede et al. 2009).

The values for R0 in the network model for the intermediate estimates of transmissibility range between 0·9 < R0 < 2·0. McCallum et al. (2009) derived estimates of R0 from increase in DFTD prevalence in 2- to 3-year-old devils in two different populations. These estimates ranged between 1·6 < R0 < 3·1 using a 6 months latent period estimate and 2·1 < R0 < 5·4 using a 9 months latent period. Applying those estimates to our network simulation model with identical latent periods suggests transmissibility values for DFTD of at least 0·8. More accurate estimates of R0 will depend on better estimation of the latent period, which at present is hindered by the lack of a diagnostic test for DFTD in the absence of clinical signs.

The latent period of DFTD influences the probability of extinction of both host and tumour and the potential for coexistence between them. Although devil extinction probability differs by only 10% depending on shorter (3 months) or longer (12 months) estimates of latent period at the intermediate values of transmissibility, a shorter latent period in the network model substantially hastens the rate of devil extinction. Similarly, the shortest latent period increases the probability of DFTD extinction by a 15%, most likely as a result of the pathogen dying out within a local patch before it can reach the whole network. Conversely, a longer latent period permits ‘exposed’ hosts spending more time in that state, allowing the pathogen to reach and spread across the entire network.

Incorporating real-world contact rates amongst devils into our network model also resulted in higher predicted probability of pathogen extinction, as a direct result of the greater seasonal and sex-based constraints on connectivity between hosts within the population. Conversely, under the MF model the chance of pathogen extinction is considerably less likely than in the network model, even when values for transmissibility approach zero. This suggests that homogeneous mixing models might tend to overestimate potential contagious contacts.

Our network models do not take into account contact patterns of juvenile devils (1-year old), which are only recruited in the model when they become sexually mature in the subsequent year. However, juvenile devils usually do not engage in mating, except in populations where older sexually mature devils have succumbed to DFTD, which results in precocial breeding (Jones et al. 2008). In addition, prevalence of DFTD in juveniles is very low in most populations (McCallum et al. 2009). Further research into the role of juvenile devils in the dynamics of DFTD, once the older age classes have disappeared because of DFTD mortality, is needed.

Network structure has implications for the likelihood of disease-induced devil extinction. Extinction is somewhat more likely (c. 10%) at intermediate values of transmissibility in the MF compared to the network model, possibly due to the higher connectivity in the MF model. Host extinction probability does not differ between the MF and the network models for transmissibility close to zero or one. This suggests that for diseases with either very high or very low transmissibility, contact heterogeneities may have little effect on the likelihood of disease-induced host extinction. However, transmission rates per contact are likely to be intermediate in many wildlife diseases. In the case of DFTD, it is likely that the probability of transmission from a single contact between an infected and a susceptible animal is low because direct inoculation of live tumour cells (and their subsequent establishment) is necessary to acquire infection. However, an edge in our empirical network may represent repeated contacts. Biting is the most likely route for transmission. While biting is very common in Tasmanian devils (Hamede, McCallum & Jones 2008), not all bites result in penetration of the dermal layer (Pemberton & Renouf 1993), with the potential to result in disease transmission. Likewise, not all contacts recorded by proximity sensing radiocollars will have resulted in a bite, but conversely, some very short contacts that may have resulted in bites may not have met the 10 s threshold we used to define a contact in our empirical devil network (Hamede et al. 2009). Given that our contact rate estimates were obtained from a disease-free population, further studies are needed to clarify whether DFTD is capable of changing contact patterns at individual or population level.

Rewiring the network creates further opportunities for disease transmission by connecting individuals that were previously not connected, increasing the likelihood of devil extinction. Previous studies have demonstrated that rewiring nodes in dynamic networks has direct consequences for the spread and persistence of infectious diseases (Gross, Dommar & Blasius 2006; Volz & Meyers 2007). Rewiring the entire network was performed at discrete intervals for computational tractability but is clearly only an approximation to the actual continual changes in network structure that would occur in a real population as seasons change, animals die and new individuals are recruited. We found rewiring had a greater impact on simulation outcomes than the specific forms of mixing preferences (sex and/or season). Although differences in individual mixing preferences within our network model are small at most values of T, they reach a 15–20% difference in devil extinction probabilities when only gender or seasonal mixing preferences are compared with the effect of rewiring.

There are substantial differences between network and MF models in the probability of coexistence of host and the pathogen (defined here as persistence of both for 50 years). The likelihood of coexistence is much higher in the stochastic MF model, where contact probabilities are homogeneous, than in the network model. However, this occurs at the lower values of transmissibility only, at the expense of underestimating DFTD extinction (Fig. 4a, b). Both models peak in the likelihood of coexistence at similar low values of transmissibility, but coexistence probabilities drop to zero as values for transmissibility approach one. Coexistence is possible in the network model only for the longest latent period estimate (12 months).

A limitation of this study is that our models assume constant values for key disease parameters such as transmissibility and disease-induced mortality with each generation of infection. Cancerous tumour cells are, however, usually under strong evolutionary pressure (Merlo et al. 2006) and so these parameters may change. The potential for co-evolution of DFTD and the devil is evident in the ongoing evolution of multiple strains of the tumour (A.M. Pearse, unpublished data), in geographical variation in MHC gene diversity involved in tumour recognition (Siddle et al. 2010) coupled with recent data indicating that devil populations in which MHC differs from the tumour have lower infection rates and longer survival periods with DFTD (Hamede et al. in press), as well as genetic and adaptive phenotypic changes in the devil (Jones et al. 2008; Lachish et al. 2011). Including tumour strain dynamics and heterogeneities in disease-induced mortality in devil-DFTD epidemic models should help to further investigate effects of virulence in the epidemiology of DFTD.

Implications for disease epidemiology and management

We draw several conclusions relevant to the epidemiology and thus management of DFTD and other infectious wildlife diseases. We predict local extinctions of Tasmanian devil populations affected by DFTD between 5 and 15 years (dependent on the value of transmissibility used) following disease arrival. This time period is similar to those predicted using both mark–recapture and stochastic-dynamic modelling (c. 10 years, McCallum et al. 2007). To date, no local extinctions from DFTD are known, although as yet the longest known time since disease arrival at a monitored field site is 15 years (at Mt. William National Park, where the disease was first detected in 1996, McCallum et al. 2007). The decrease in population size at this site has been estimated to be at least 90% (McCallum et al. 2007) and devils might have already become locally extinct without subsequent recolonization through immigration from adjacent populations. Once population size has been drastically reduced, other important threatening processes such as stochastic environmental and demographic processes or Allee effects can also hasten local extinctions. Given that only the north-west of Tasmania remains free of DFTD and the predicted short time until local extinction after disease arrival, establishing captive and wild insurance populations must form a critical component of any comprehensive management plan aiming to prevent devil extinction.

Selectively culling all infected individuals captured has failed to decrease the rate of disease prevalence or to reduce the population-level impacts (Lachish et al. 2010), and models suggest that no realistic capture rate would have eliminated the disease (Beeton & McCallum 2011). One explanation for the persistence of DFTD at very low population densities is that cyclic periods of high transmission, which could mimic a sexually transmitted disease, might allow DFTD persistence in affected populations. Indeed, age-structured deterministic models have shown that DFTD transmission is consistent with frequency rather than density dependence (McCallum et al. 2009), typical of sexually transmitted diseases.

The ability to obtain robust predictions from modelling of the likely epidemic outcome (extinction or persistence and coexistence of host and pathogen) is important in the management of any wildlife disease, but particularly for an emerging disease. Whether coexistence between DFTD and the devil is the outcome of our models depends to a great extent on the length of the latent period and its interaction with the disease generation time (latent + infectious periods). When the latent period is less than the infectious period (fixed at 6 months in our model), the disease can only persist for high values of transmissibility. On the other hand, when the latent period is greater than the infectious period, the disease can be sustained for a wider range of transmissibility: at high values of transmissibility, it brings the host to extinction and at low values of transmissibility achieves coexistence. Obtaining information on the variability of the infectious period across different genotypic populations and strains of DFTD may offer new insights for predicting the likelihood of coexistence.

Conclusions

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Conclusions
  8. Acknowledgements
  9. References
  10. Supporting Information

Determining the epidemic threshold above which disease outbreaks are possible depends not only on the transmission and recovery rates but also on social mixing parameters and the structure of the host population (Volz & Meyers 2009). Estimating the dynamics of social mixing in host–pathogen systems provides a step forward for understanding the role of contact heterogeneities in epidemic behaviour. This is particularly relevant to those diseases that cause long-lasting epidemics and substantial host mortality meaning that individual and population network structural properties are unlikely to remain static. Measuring contact heterogeneities and social networks in wild populations should become more practical as technological innovations surrounding data collection become available (Krause, Wilson & Croft 2011). In this particular case, the inclusion of observed network properties had a modest effect on the transmissibility threshold for R0 > 1, decreasing it by c. 15% compared to a stochastic model with MF assumptions. Given the other uncertainties in estimating R0 in these wild populations, it is probably unnecessary to obtain further estimates of contact heterogeneities or to include them in models for assessing alternative management strategies. This is unlikely to be the case for many other wildlife diseases (Tompkins et al. 2011). Quantifying heterogeneities in dynamic contact networks and using modelling approaches to investigate the implications of these heterogeneities for disease dynamics is an important step in developing approaches for managing emerging diseases in wildlife populations. We provide a template for modelling of these heterogeneities for diseases in dynamic networks.

Acknowledgements

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Conclusions
  8. Acknowledgements
  9. References
  10. Supporting Information

This research project was funded by an Australian Research Council Linkage grant to M.J. and H.M. (LP0561120), an Australian Research Council Discovery Grant (DP0772644) to H.M., and by an Eric Guiler Tasmanian Devil Research grant through The University of Tasmania and the Save the Tasmanian devil appeal to M.J and R.H. The Tasmanian Department of Economic Development sponsored R.H. under the State and Territory Nominated Independent Scheme Ref. No. STNA-ITAS200501. The research was carried out with approval from the University of Tasmania’s Animal Ethics Committee (A0010296).

References

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Conclusions
  8. Acknowledgements
  9. References
  10. Supporting Information

Supporting Information

  1. Top of page
  2. Summary
  3. Introduction
  4. Materials and methods
  5. Results
  6. Discussion
  7. Conclusions
  8. Acknowledgements
  9. References
  10. Supporting Information

Appendix S1. Stochastic model structure and algorithm.

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