Non-linear transmission and simple models for bovine tuberculosis

Abstract

1. A new model is presented for a possum–tuberculosis (TB) system (Trichosurus vulpecula–Mycobacterium bovis) that is both realistic and parsimonious. The model includes a phenomenological treatment of heterogeneity of risk for susceptible hosts, similar to that used in insect host–parasitoid systems.

2. Parameter values for the model reflect current knowledge and differ significantly from those in other recent models of this system. Associated with these structural and parametric changes are substantially different predictions for the dynamics and control of TB in possums.

3. The model predictions include (i) only limited host suppression due to the disease (< 10%, cf. several earlier simple models for TB in both possums and badgers); (ii) asymptotically stable disease dynamics (cf. homogeneous-mixing models that predict either extremely weak stability such that disease fails to recover when host density is temporarily reduced, or oscillatory behaviour and potential elimination of disease following such a perturbation); (iii) TB that is harder to control than in the homogeneous-mixing model equivalents, in line with practical experience; and (iv) a threshold host density for disease elimination that differs substantially from the host equilibrium density in the presence of disease.

4. Homogeneous-mixing models are unable to reproduce this behaviour, whatever parameter values are chosen. Heterogeneous-mixing models with non-linear transmission may therefore be worth consideration in other endemic wildlife disease systems, as is now commonplace for insect–parasitoid and insect–pathogen ones.

Introduction

There is a growing interest in models for bovine tuberculosis (Mycobacterium bovis; TB) in wildlife, because of the challenge it poses both to understanding and to management (Anderson & Trewhella 1985; Barlow 1991a,b, 1993, 1994, 2000; Roberts 1992, 1995, 1996; Bentil & Murray 1993; Louie, Roberts & Wake 1993; Pfeiffer 1994; Barlow & Kean 1995; Kalmakoff, Mackintosh & Griffin 1995; White & Harris 1995a,b; Smith et al. 1995; Ruxton 1996; Smith, Cheeseman & Clifton-Hadley 1997; Swinton et al. 1997). This paper relates especially to the simpler models, and its purpose is threefold. First, it presents an updated parsimonious model for TB in possums, which differs from that of Roberts (1996) by significant changes in both structure and parameter values. The structural change involves the inclusion of heterogeneous mixing and non-linear transmission, while parameter values reflect best current knowledge of the biology. Secondly, it shows that these changes are more than cosmetic: they have significant effects on both dynamics and control, and the inclusion of non-linear transmission is necessary for the model to behave realistically. Thirdly, it suggests that there may be a need to consider heterogeneous mixing and non-linear transmission more widely in wildlife disease models, along the lines now commonplace for insect–parasitoid and insect–disease ones.

Model

The following model is proposed as the simplest ‘reasonable possum–TB system’ (Roberts 1996), including both heterogeneous mixing and the most realistic parameter values consistent with current knowledge (Table 1):

Table 1.  Parameters for a possum–TB SI model (model 1) and the best current estimates for their values, compared with those for homogeneous-mixing models 2 and 3 (model 3 = Roberts 1996)

Parameter	Value	Description	Model 2 value	Model 3 value
b	0·3	Maximum birth rate (year^−1)	0·3	0·267
d	0·1	Minimum death rate (year^−1)	0·1	0·167
r	0·2	Intrinsic rate of increase (year^−1)	0·2	0·1
θ	3	Density-dependence shape parameter	3	3
δ	0·5	Proportion of density-dependence affecting births	0·5	0·5
α	1	Disease mortality rate (year^−1)	1	3
β	2·5	Disease transmission coefficient (year^−1)	1·3	5
k	0·05	Disease aggregation parameter (year^−1)	∞	∞
ε	0·5	Disease contact rate–density parameter	0·5	0·5
p	0·25	Proportion of births giving disease transmission (pseudo-vertical transmission)	0·25	1

eqn 1

eqn 2

where N= total possums, S= susceptibles, I= infecteds (N = S + I), all expressed relative to carrying capacity; B(N) and D(N) are density-dependent birth and death rates; C(N) is the contact rate function; and F(S,I) the disease transmission term. These functions are:

• B(N) = b - δrN^θ

• D(N) = d + (1 - δ)rN^θ

formmu1

• 

a heterogeneous-mixing transmission term.

The function F(S,I) replaces the usual mass action term βSI, where β is the transmission coefficient, and introduces a further parameter k, the (time-dependent) index of aggregation. The smaller k the greater the degree of aggregation. The effect is to reduce the mean number of infecteds per susceptible, and the transmission term is equivalent to the continuous-time form of a negative binomial distribution of new infections across susceptible hosts. As such, it encompasses not only spatial patchiness but any ‘heterogeneity of risk’ for a susceptible host.

The biological basis for the parameter values is fully described in Barlow (1991a) but the present model includes several changes. For convenience, it scales host density to carrying capacity, as in Roberts (1996), thereby implying that the maximum disease contact rate and the threshold for maintenance of disease are proportional to the carrying capacity, K. This was also assumed by Barlow (1991a, 1993). Following Roberts (1996), the model assumes density-dependence in both recruitment and death rates and a non-linear contact rate–host density relationship, these being marginally more likely than the density-dependent death and linear contact rate–density relationship in Barlow (1991a,b). Although there is no direct evidence for a non-linear contact rate function, work is in progress using proximity recorders to establish this (M.N. Clout, personal communication) and a priori it seems a more realistic assumption than linearity, especially as mating is a likely route of much transmission. Recent data (Caley, Hickling & Cowan 1995; see below) showed that TB is eliminated at a possum density of 0·2K, so the threshold must be at least 0·2K. This corresponds to an ε value in the contact rate–density relationship (Roberts 1996; see below) of less than 0·63 (0 corresponding to linearity). This value is obtained by setting (dI/dt)/I = 0 in equation 2, letting S → N and I → 0, then determining the value of ε which makes N = 0. In the present model it is also assumed that 100% of births from infectious females result in pseudo-vertical transmission rather than the 50% assumed in earlier SEI (susceptible-exposed/latent-infectious) models (Barlow 1991a,b; Roberts 1992). However, only a proportion of the infected animals in the present susceptible–infected (SI) model are infectious, assumed to be around 0·25, corresponding to a latent period three times the length of the infectious period (Barlow 1991a). Therefore p = 0·25 (cf. Roberts 1996 in which p = 1).

As in Roberts (1996), the model omits an explicit latent class, for simplicity and because the observed disease behaviour can be reproduced without it. Although experimental infections (cited in Roberts 1996) suggest short latent and infectious stages, evidence from the field suggests that the time from infection to death may be as long as 1 year (D. Pfeiffer & R. Jackson, personal communication; Jackson et al. 1995). Because the simplified SI model defines the I as infecteds rather than infectious (as in an SEI model that has an additional ‘exposed’ or latent class), both the disease mortality rate α and the disease transmission coefficient β are reduced to the SI equivalents of those in the earlier SEI models (Barlow 1991a). They are therefore considerably smaller than those in Roberts (1996), who used similar ones for both SI and SEI models. Specifically α was assumed to be 1 in the present model, giving a time from infection to death of 1/(α + d) = 9 months. Disease transmission coefficients are notoriously hard to measure in the field, and the most robust way of estimating them is probably by tuning models with repeated trial values to mimic observed disease behaviour. Both β and k can be fitted in this way. In the possum–TB case, however, the information on disease patchiness (Fig. 1) was used in conjunction with the observed prevalence and equilibrium host density to provide slightly more rigorous estimates, assuming the aggregation parameter, k, to be independent of density (see the Appendix). This gave β = 2·5 and k = 0·05. With the possible exception of the degree of density-dependence associated with recruitment, the possum parameters are reasonably well established (Barlow 1991a). The greatest uncertainty lies in the disease parameters β, k, p and ε.

Figure 1. TB prevalence (% possums with gross lesions) per 0·5 km (25 traps) along a 66-km trap line in the Hauhungaroa Ranges, central North Island, New Zealand (from Barlow 1991a; with permission from the Journal of Applied Ecology).

The model (model 1) is compared with two homogeneous-mixing ones. Model 2 is the equivalent to model 1, giving the same equilibrium prevalence and host suppression. To achieve this, the disease transmission coefficient is reduced (1·3 compared with 2·5) but the other parameters are left as in model 1 (Table 1). Model 3 is that of Roberts (1996), which also has homogeneous mixing but has substantial parameter differences from model 1 as well, notably much higher values for the disease parameters α and β (3 and 5, compared with 1 and 2·5) and a much lower intrinsic rate of host increase (0·1 compared with 0·2; Table 1).

Results

Given that β and k are the least well established of the parameters in model 1, Table 2 presents a sensitivity analysis of model behaviour to changes in their values. Both prevalence and host suppression increase with the product βk, so in this respect one trades off against the other. In terms of the disease dynamics, however, high β and high k values (reduced aggregation) both promote oscillatory behaviour, and following a single reduction in host density the disease recovers most rapidly for intermediate values of β and low values of k (high aggregation). The basic reproductive rate of the disease, R₀, depends only on β (see below).

Table 2.  Sensitivity analysis of model 1 results to changes in the transmission parameters β (transmission coefficient) and k (aggregation parameter). Results are expressed in terms of the equilibrium host density (% of unit carrying capacity), the prevalence of infected possums (%), and the disease recovery time (time taken for the density of infected possums to return to 90% of the equilibrium level following a 75% single reduction in total possums). k = 100 corresponds effectively to no aggregation and homogeneous mixing; * denotes oscillatory behaviour

Transmission coefficient (β year^−1)
	1·5	2·5	5	1·5	2·5	5	1·5	2·5	5
Aggregation parameter, k year^−1	Equilibrium host density (%)			Prevalence of infecteds (%)			Disease recovery time (years)
0·025	99	96	91	0·8	2·6	4·8	26	12	11
0·05	98	91	81	1·5	4·6	9·6	26	12	15
0·10	95	81	50	2·7	9·4	18·0	27	13	20
100·0	73	35	15	12·3	19·2	20·0	26*	15*	14*

The TB dynamics predicted by the three models are summarized in Table 3, which shows that the heterogeneous-mixing model (model 1) gives results differing substantially from those of both the homogeneous alternatives (models 2 and 3).

Table 3.  Summary of the dynamics of possum TB, as predicted by the three models (model 3 = Roberts 1996). Disease r_m is the maximum specific rate of increase of diseased possums per year (when introduced into a susceptible host population), the equilibrium relative host density is that in the presence of disease, scaled to unit carrying capacity, the threshold under culling is the relative host density at which disease is eradicated under culling, the proportion vaccinated is that required to eliminate disease, and the culling and vaccination rates are those required yearly to achieve disease eradication

Model	Features	R0	Disease rm	Equilibrium relative possum density	Equilibrium prevalence (%)	Threshold under culling	Proportion vaccinated	Culling rate	Vaccination rate
1	Heterogeneous mixing, high β, high host r	2·13	1·35	0·91	4·6	0·32	0·54	0·16	0·23
2	Homogeneous mixing, low β, high host r	1·13	0·15	0·91	4·8	0·83	0·12	0·08	0·03
3	Homogeneous mixing, high β, low host r	1·59	2·00	0·43	3·0	0·43	0·40	0·09	0·15

Specifically, all three models reproduce realistic average prevalences of possums with gross TB lesions (around 5%; Hickling 1995; Table 3). Models 1 and 2 also predict that there is little suppression of possum density (9%) due to the presence of TB (Table 3 and Fig. 2a,b). In contrast, model 3 predicts a 50% reduction in possum density (Table 3 and Fig. 2c).

Figure 2. Predicted responses of total possums (dotted line) and TB-infected possums (solid line) to a 75% single reduction in possum density. (a) Model 1 (heterogeneous mixing β = 2·5). (b) Model 2 (homogeneous mixing with parameters the same as in model 1 except that β= 1·3). (c) Model 3 (Roberts 1996; homogeneous mixing β = 5).

In terms of their dynamic behaviour, both homogeneous-mixing models (models 2 and 3) generate oscillatory behaviour and predict that TB levels fail to recover after a single 75% reduction in possum density (Fig. 2b,c). Roberts' (1996)‘null model’ with his default parameter values (model 3) generates particularly high-amplitude oscillations and predicts that TB effectively dies out following a single 75% reduction in host density (Fig. 2c). In contrast, model 1, with heterogeneous transmission, generates asymptotically stable dynamics and a recovery of TB in about 10 years after a 75% reduction in possum density (Fig. 2a).

If host density is reduced then kept low, the models predict different rates of decline of the disease, although this is a less sensitive comparison than a single reduction in host density. Model 1 predicts a 98% decline in density of infected possums in 5 years, whereas model 3 predicts a very much more rapid decline of 98% within the first year.

The differences in dynamic behaviour of the models are captured by the respective maximum specific rates of disease increase (r_m for the disease = dI/Idt as I → 0 and N → 1). These equate to the difference (β – α), as opposed to the basic reproductive rate of the disease which varies as the ratio (β/α). As Table 3 shows, the basic reproductive rate for model 1 is higher than that for model 3 but the specific rate of disease increase is lower, because of the lower absolute values of α and β. Thus, model 1 predicts less rapid disease dynamics than model 3, and that changes in disease levels are less sensitive to host density. Model 2 generates very slow changes in disease levels because β is so small.

The equations and formulae for host thresholds, control rates for disease elimination and R₀ are the same as those in the Appendix of Roberts (1996), with βSI and γVI in his equation A2 replaced by F(S,I) and F(V,I), respectively, and corresponding changes made to equations A5 and A6 (V= relative density of vaccinated possums).

The significance of the difference between the present model and that of Roberts (1996) is particularly clear when considering his criterion of reducing TB prevalence by 90% of its original value within 5 years. In the model presented here, this requires a culling rate of 80% per year or a vaccination rate of 70% per year, and the virtual elimination of susceptible possums in both cases. These are very different from the 12% and 13%, respectively, in Roberts (1996).

Discussion

The possum tb model

The possum TB model presented here updates the homogeneous mixing ones of Roberts (1992, 1996) and the earlier heterogeneous mixing ones of Barlow (1991a,b, 1993). The latter, and that of Roberts (1992), were SEI ones, while Barlow's (1993) model was more detailed and included further complications like stochasticity, a birth pulse, local immigration and juvenile dispersal in a spatially explicit, individual-based, coupled map lattice, still with heterogeneous local transmission. Neither demographic stochasticity nor discrete births in this model gave significantly different local behaviour from that of the deterministic, continuous-time, model of Barlow (1991a,b), which itself gave behaviour indistinguishable from that of the present, considerably simpler, SI model. The different results for vaccination in Table 3, compared with those in Barlow (1991b), are largely because of the assumption in the present model that density-dependence applies to both recruitment and mortality rather than just mortality. This has a substantial effect on vaccination strategies (Barlow 1996).

The present model also removes several unrealistic features of the Roberts (1996) one (model 3). First, that model did not include heterogeneous transmission. Secondly, both the disease mortality rate and the extent of pseudo-vertical transmission were substantially higher than observed, given that they applied in the SI model to the total infected rather than just infectious animals. Thirdly, the host intrinsic rate of increase was about half the accepted value. The best published estimate and the one used in earlier models (Barlow 1991a,b; Roberts 1992) is 0·2 year⁻¹ (Hickling & Pekelharing 1989). In some habitats it is likely to approach 0·3 year⁻¹ (Hickling & Pekelharing 1989; Barlow 1991a), and even higher values than this have been published but largely discounted as arising partly from immigration (Hickling & Pekelharing 1989). The low value of 0·1 year⁻¹ in Roberts (1996) appears to have been derived from observed age distributions that included a component of density-dependence in the estimates of both minimum mortality rate and birth rate. Although Roberts (1996) tested the effects of varying most of these parameters in a sensitivity analysis, this was one-dimensional and an unrealistic default value for one parameter (r) can give substantially misleading indications of the significance of others. For example, the vaccination rates required to eliminate disease depend on whether density-dependence acts on mortality or recruitment, and the difference in the two rates depends on r (Barlow 1996).

These features lead to behaviour that is also unrealistic, namely a high level of host suppression by the disease (Fig. 2c); elimination of TB following a single reduction in possum density (Fig. 2c); and very rapid disappearance of TB when host density is kept low (Fig. 3). The model also predicts a 100% increase in host density under a vaccination policy (Roberts 1996), which would have extremely serious consequences for management if it were true. It would increase densities of possums over much of New Zealand, where they are pests in their own right because of their high numbers (up to 10 ha⁻¹ and approximately 70 million in New Zealand) and significant impact on native forests. The low r-value assumed in the model also results in predictions of exceptionally low culling rates necessary to eliminate disease (Table 3).

Figure 3. Predicted and observed (solid points) responses of total (dotted line) and TB-infected possums (solid line, model 1; dashed line, model 3), to a sustained 80% reduction in possum density at Hohotaka, North Island, New Zealand. Observed data from Caley et al. (1995) and P. Caley, personal communication.

Why non-linear transmission?

There are two main arguments for heterogeneous mixing and non-linear transmission in TB models, which may well apply to other wildlife disease systems. First, at least one type of heterogeneity evidently exists: TB in both possums and badgers is patchy in space (e.g. Fig. 1; Cheeseman et al. 1988; Cheeseman, Wilesmith & Stuart 1989; Barlow 1991a, 1994; Pfeiffer 1994; Hickling 1995; Smith et al. 1995; White & Harris 1995a,b), so an assumption of homogeneous mixing is at best a simplification. Spatial patchiness does not preclude the existence of other sources of heterogeneous risk, but even the spatial effect is difficult to explain. The most likely possibility consistent with the assumptions of the phenomenological model discussed here, is local heterogeneity in possum carrying capacity, K. This could give patches of disease where host density is above the threshold for persistence, separated by areas of subthreshold density, and with stochastic extinction of disease from a high-density patch balanced by reinvasion of infected juveniles. In support of this scenario, there is some evidence for a positive association between possum density and TB prevalence on a local scale (Hickling 1995; J. MacKenzie, personal communication).

The second argument for heterogeneous mixing is that the classic homogeneous-mixing epidemiological models described in Bentil & Murray (1993), Roberts (1996) and Ruxton (1996) do not appear able to reproduce the main features of observed disease behaviour.

For example, in a homogeneous-mixing model the threshold host density for disease elimination is similar (equal in the case of 100% pseudo-vertical transmission) to the steady state density in the presence of disease (Roberts 1996). So, if disease parameters α and β are chosen that allow for little host suppression and a host density close to the disease-free carrying capacity (as is the case in both possums and badgers, and as the homogeneous-mixing model 2 recreates), then the threshold must also be close to the carrying capacity and only small reductions in the density of susceptible hosts would cause elimination of disease (model 2; Table 3). This is not the case in practice. The homogeneous-mixing models cannot give both a low host threshold for disease elimination and a high equilibrium density in the presence of disease, whereas the heterogeneous-mixing one can.

Homogeneous-mixing models also tend to give oscillatory behaviour. If α and β are large, as in model 3, such behaviour results in a high level of sensitivity to host density. Thus, TB fades out after a single temporary reduction in host density (Fig. 2c). The model actually predicts lightly damped oscillations with a period of 18–20 years, but densities of TB possums reach around 8 × 10⁻⁸ × K, which implies extinction. If, on the other hand α and β are small, the oscillations are much less pronounced but TB levels effectively fail to recover after a single reduction in host density (model 2; Fig. 2b). Neither of these scenarios is realistic: not only does TB not die out after a single possum cull, but it recovers, as shown in Fig. 4a. Unfortunately there are no data showing the behaviour of TB possum densities after control operations, so those in Fig. 4a are for cattle reactor rates. However, there is a strong correlation between the two and reactor rates appear to be a good index of the density of infected possums [Fig. 4b; (relative density of TB possums) = 0·062 (cattle reactors per year (%)); R² = 0·72; d.f. 1,9; P < 0·001; intercept NS].

Figure 4. (a) Reduction then recovery of TB (% cattle reactors) following single possum control operations in a farm near Taupo, North Island, New Zealand (data from Batcheler & Cowan 1988). Arrows denote possum control operations. (b) Relationship between the relative density of TB possums (trap catch index × percentage with gross lesions per 6 km length of trap line) and the cattle reactor incidence (% reactors) on adjacent farms in 1981–82 at Taupo (the regression line is y = 0·062x, P < 0·001, and the data are from K. Patterson, personal communication). (c) Decline in cattle reactor incidence (% per year) in Buller, South Island, New Zealand, when possum densities are reduced then kept low by ‘maintenance control’ (data from Batcheler & Cowan 1988).

Another outcome of the oscillatory behaviour of model 3 is the almost immediate disappearance of disease if host density is kept low (Fig. 3). In the experiment shown in Fig. 3, and in large-scale control operations where possums are culled then kept low by ‘maintenance control’ (Fig. 4c), the decline in TB is much more gradual. If either of the homogeneous-mixing models 2 or 3 were realistic, then given their predicted disease behaviour shown in Figs 2 and 3 it would be a simple matter in practice to eliminate TB with two consecutive culls of possums.

Only the heterogeneous-mixing model (model 1) yields a more stable disease with relatively rapid recovery after a single reduction in host density (Fig. 2a) and gradual decline following a sustained reduction (Fig. 3). This is a special case of the well-known general effect that non-linearities in transmission rates stabilize oscillatory dynamics in host–pathogen models (Liu, Levin & Iwasa 1986; Hochberg 1991; Briggs & Godfray 1995, 1996). On the other hand, the badger TB models of Bentil & Murray (1993) and Ruxton (1996) were motivated partly by Cheeseman et al.'s (1989) suggestion that TB in badgers may be cyclic. There was some indication of this from their data but the suggestion was clearly tentative, and the effect is not apparent in a later presentation of the same data (Fig. 3 in Swinton et al. 1997). Neither is there evidence that TB cycles in possums. In one apparently exceptional case, a population was found with very high widespread prevalence (50%), following reduction in density of a possum population with moderate TB prevalence (7%; Coleman et al. 1994). This could indicate at least one oscillation, as it is highly unlikely that the 50% prevalence will be sustained. Such an effect is accommodated phenomenologically in the above model by a greatly reduced degree of disease aggregation (high k) at this site.

The main reason why the heterogeneous-mixing model gives generally more realistic behaviour than the homogeneous ones is that it allows a higher value for the transmission coefficient, while retaining the same realistic equilibrium disease prevalence and host density in the presence of disease (Table 3; compare models 1 and 2). The higher transmission coefficient allows a higher R₀ and r_m for the disease, hence realistic rapid dynamics and reasonable responses to control in accord with those so far observed. The formulae for disease elimination using culling or vaccination (Roberts 1996) are the same as for homogeneous-mixing models, as they depend on the basic reproductive rate of the disease (R₀). This is the same for both models, because the transmission terms become equivalent as prevalence tends to zero (note that R₀ does differ in the two cases if heterogeneity reflects spatial variation in parameters such as β). However, the results of the formulae are likely to be different because heterogeneous transmission leads to higher estimates for the transmission coefficient, hence R₀, where these are derived from observed prevalences and disease behaviour.

The problem of host suppression

A further problem with many TB models is an unrealistically high level of host suppression (50% in Anderson & Trewhella 1985; 57% in Roberts 1996, see Model 3 in Fig. 2c; and 93% in Bentil & Murray 1993; see Fig. 1b in Ruxton 1996). This does not happen in practice: there is no negative effect of TB prevalence on trap–catch indices of possum abundance in areas of apparently similar carrying capacity (Barlow 1991a; Pfeiffer et al. 1995); indeed, at a local scale of individual traps, the relationship between TB prevalence and number of possums caught is positive (Hickling 1995). Nor is there an apparent effect of TB prevalence on badger densities (Cheeseman et al. 1988, 1989).

The host suppression in these models is not so much a consequence of homogeneous mixing as of high disease prevalence (around 15%) in the case of badgers, or a high disease mortality rate in the case of Roberts' (1996) possum model (model 3 in Table 3; cf. model 2) and some of the badger TB models (Anderson & Trewhella 1985; Bentil & Murray 1993; White & Harris 1995a,b; Ruxton 1996). In other badger TB models the problem does not arise because the disease mortality rate is extremely low (Smith et al. 1995, 1997; Swinton et al. 1997). As equation 1 suggests, when dN/dt = 0 the equilibrium value for N depends on the disease mortality rate (α), prevalence (I/N) and the shape of the density-dependence curve (θ): asymmetric, rightward-peaked growth curves with θ > 1 cause less reduction in equilibrium density for a given additional mortality than does an ordinary symmetric logistic with θ = 1. Only in the earlier model of Roberts (1992) does homogeneous mixing directly cause unrealistic host suppression, because the same disease parameter values were used as in Barlow's (1991a) heterogeneous-mixing model, and this had the effect of both increasing prevalence and increasing host suppression. Assuming that these parameter values are realistic (cf. Roberts 1996), it might be argued that the homogeneous-mixing model is a useful metaphor for local disease behaviour within the disease ‘hot spots’. However, at least in the case of possums, any tendency for high local TB prevalences to give local suppression of population density in these disease foci would be partly offset by local immigration of susceptibles and the model would need to take this into account. This is effectively what the heterogeneous-mixing model does, as described in Barlow (1991a, 1994).

Current models for non-linear transmission

To accommodate heterogeneous transmission in a phenomenological way, three alternative terms have been used in the past. The negative binomial one in the present model was used by Godfray & Hassell (1989) in a continuous-time host–parasitoid model, and the discrete-time equivalent is widely applied in the insect parasitoid literature (May 1978). Briggs & Godfray (1995, 1996) also adopted this expression for insect disease–host models, with I replaced by W to represent the density of free-living infectious particles. In fact, the same force of infection term with k = 1 [i.e. ln(1 +βW)] could realistically be considered as an equally parsimonious and more appropriate default than βW for host–pathogen models with free-living pathogens, as their distributions are usually extremely aggregated, densities typically range through several orders of magnitude, and responses (proportions infected) often relate to log dose (density of infectious particles; T.A. Jackson & R. Hails, personal communication).

The negative binomial transmission term reduces the force of infection or effective number of infecteds per susceptible. Barlow (1991a) used a different approach that reduces the number of effective susceptibles per infected. This replaced βSI = β(N − I)I in the equivalent SI model (the original model was an SEI one) with β(N – I/q)I, where q is an aggregation parameter related to mean crowding and equal to (mean infecteds)²/(mean square infecteds) per unit area, or proportion of the habitat population containing disease (Barlow 1991a). The smaller q, the greater the degree of aggregation. For possum TB, Barlow (1991a) estimated q to be about 0·1 from field data.

The third option is a term of the form βS^pI^q (Severo 1969; Liu et al. 1986; Hochberg 1991), which allows a variety of non-linear responses depending on the values of p and q. Knell, Begon & Thompson (1998) showed that both this and the negative binomial term gave good fits to experimental data on the transmission of granulosis virus in the Indian meal moth Plodia interpunctella.

Conclusion

Comparing the badger–TB and possum–TB systems, the main difference appears to be that prevalence is higher in badgers but the disease-induced mortality is lower. Both host populations seem to be little affected by disease, and TB in both cases occurs in persistent spatial clusters. These features are reflected in the more realistic respective models. In the case of possums, there is additional information on the dynamics of disease following reductions in host density, which does not appear to be available for badgers. This imposes an additional criterion for realism on the possum–TB models, which is not addressed in those for badger–TB. The new possum–TB model presented here builds on and updates previous ones, and predicts radically different dynamics to some of the most recent alternatives. In the process, two general points emerge that are worthy of emphasis.

The first is the importance of realistic parameter values. The qualitative (data-free) behaviour of simple disease–host models is reasonably well understood, so the raison d'être for new models of this kind rests in their ability to mimic something found in nature, and thence to draw robust semi-quantitative conclusions about dynamics and control. The fact that models typically have several parameters based on poor or non-existent data reinforces the importance of realistic values for those parameters that can be quantified. However, more data on TB dynamics are badly needed, along with a greater emphasis on comparison of models with the data that do exist (such as Fig. 3 in Swinton et al. 1997).

The second point is the significance of model structure. In the case of possums, but possibly also for badger–TB and other wildlife disease systems, there is growing evidence that the simplest ‘classic’ models with linear contact rate–density relationships and homogeneous mixing are inappropriate. For possum–TB, the contact rate–density relationship is still uncertain but the evidence presented here for non-linear transmission, resulting from heterogeneity of risk among susceptible hosts, and the substantial difference in model predictions between the homogeneous- and heterogeneous-mixing versions, suggest that the latter deserves serious consideration as a default model structure. As such, it would bring wildlife disease models into line with both insect–disease and insect–parasitoid ones, where the applicability of non-linear transmission is becoming increasingly well recognized.

Acknowledgements

I am grateful to Peter Caley, Rosie Hails, Trevor Jackson and Joanna MacKenzie for helpful discussions, Charles Godfray and John Kean for comments on the draft manuscript, and two anonymous referees for their useful suggestions. The work was funded by the Foundation for Research, Science and Technology.

Received 12 May 1999; revision received 7 February 2000

Appendix

Estimation of the disease transmission parameters β and k

The mean field approximation to the transmission term [β × mean product of S and I] is:

where summation is over all patches. Assuming that k is constant, this is set equal to the transmission term in the model,

• kS¯ln[1+βI¯/k],

thus:

This enables the ratio β/k to be obtained from the observed pattern of S and I between patches. In the possum–TB case, because there is no association between TB prevalence and local possum density (N = S + I) at the scale represented in Fig. 1 (Barlow 1991a), it is more convenient to re-express the above equality in terms of prevalence

• p = I/N

thus:

with summation across patches (in this case lines of 100 traps). Hence:

As  ≈ 0·05,  ≈ 0·9 (scaled to unit host carrying capacity) and

(Barlow 1991a; Hickling 1995), this gives (β/k) ≈ 50. The quantity

is equivalent to the (mean square infecteds per patch–mean infecteds per patch squared) = (1/proportion of the habitat effectively occupied by disease) = (ratio of local prevalence to global prevalence). Given α = 1, the model was run iteratively with trial values of β and k satisfying β = 50 k until the correct steady-state prevalence and possum densities were achieved. This yielded β = 2·5, k = 0·05.