We represent transitions from susceptible, infectious and recovered individuals in a multi-host community with a generalised directly transmitted microparasite using a system of differential equations (Dobson 2004):
where Si, Ii and Ri represent the densities of susceptible, infectious and recovered individuals of species i (other parameters defined in Table 1). Recovered individuals are assumed to have life-long protective immunity. We assumed allometric scaling of host vital rates, disease-free equilibrium densities and infection induced death rates following established methods (De Leo & Dobson 1996; Dobson 2004; Bolzoni et al. 2008). Details for each parameter are provided in Table 1.
Table 1. Model parameters, definitions, and values
| w i ||Body mass of species i (kg)|| a w min i−1 |
| w min ||Minimum species body mass||1 kg|
| a ||Weight scaling parameter||Varied from 1 to 5|
| b i ||Per capita birth rate||0.6 wi−0.27 + 0.4 wi−0.26|
| d i ||Per capita death rate||0.4 wi−0.26|
| K i ||Carrying capacity||(bi − di)/δ|
| v i ||Infection induced mortality||(m − 1) di|
| m ||Virulence term||1.5|
|σi||Recovery rate|| r d i |
| r ||Recovery scaling term||10|
| R 0i ||Intraspecific pathogen reproductive ratio||Gamma(k, θ) (0, 10)|
| k ||Gamma shape parameter||Varied from 0.1 to 1|
|θ||Gamma scale parameter||Varied from 1 to 2|
|βii||Intraspecific transmission rate|| |
|βij||Interspecific transmission rate|| |
| c ij ||Interspecific transmission scaling parameter||Varied from 0.01 to 0.99|
Host competence was represented by the probability of transmission given a contact between a susceptible and infectious individual, which is proportional to the intraspecific transmission rate βii (Keesing et al. 2006). Intraspecific transmission rates were calculated based on the potential for an epidemic in a naïve and isolated population (intraspecific R0), recovery rates σi, pathogen-induced death rates vi, background mortality rates di and host densities at carrying capacity Ki (Table 1). Interspecific R0 values were drawn from a right-skewed truncated gamma distribution so that populations of most species would not be readily invaded by the pathogen on their own, but some species would experience severe epidemics in the absence of other species, consistent with empirical evidence for multi-host pathogens (Komar et al. 2003; Ostfeld & LoGiudice 2003; Huang et al. 2013; Johnson et al. 2013). With strict allometric scaling, intraspecific transmission rates (βii) were linearly and positively correlated with intraspecific R0, and negatively related to body mass, so that small-bodied species were more competent than large-bodied species (Fig. S1; Johnson et al. 2012; Huang et al. 2013). Interspecific transmission between species j and species i (βij) was assumed to be symmetrical so that βij = βji, and was calculated as a scaled average of intraspecific transmission rates (Table 1; Dobson 2004).
We used a two-step approach to explore the degree to which host competence-extirpation risk relationships affect changes in disease risk during community disassembly. First, we produced a gradient of communities spanning from the strict assumption of small-bodied species being most competent to the reverse assumption of small species being least competent. To achieve this, we first assumed reverse rank-ordering between body size and intraspecific R0, so that smaller bodied species received the highest values of R0 (Roche et al. 2012, 2013). We then permuted the values of R0 to produce all possible species-R0 pairings, producing N! unique communities all consisting of the same N species, but with different species-R0 pairings. This created a gradient in the relationship between host competence and extirpation risk. On one extreme, smaller species always have the highest generated values of R0 (i.e. a strong, negative correlation between host competence and body size). On the other extreme, smaller species always have lowest values of R0. Intermediate permutations represent a range of randomness (i.e. strength of covariance) from one extreme to the other, with maximum randomness occurring at the median number of inversions (Fig. S2).
Second, we explored the consequences of competence-extirpation risk relationships in the context of community disassembly by sequentially removing individual host species from each of the N! communities generated in the previous step. We varied the extirpation ‘rules’ to always, often or only rarely extirpate larger bodied species, gradually relaxing the empirically supported assumption that large-bodied (i.e. slow-paced) species have high extirpation risk. Specifically, we assumed (1) deterministic extirpations of largest-bodied hosts, (2) semi-random extirpations with extirpation risk proportional to 1 − wi−1 and (3) completely random extirpations, irrespective of body size. The latter two methods represent partial and complete decoupling of extirpation risk and host competence. A range of initial host community richness values were explored, but our results were not sensitive to initial richness, and computational cost increased rapidly as host species were added. Thus, we present all model results with initial communities consisting of five species (N = 5).
We considered both compensatory and subtractive extirpations, corresponding to cases where persisting species do or do not increase in abundance following the extirpation of co-occurring species, respectively. With density compensation, following an extirpation, remaining species increased in density in proportion to their original relative abundances so that community density remained fixed and independent of species richness (MacArthur et al. 1972). Compensatory extirpations imply strong species interactions that result in host density regulation – one mechanism for dilution (Keesing et al. 2006). Finally, in the interest of generality we considered both density-dependent transmission and frequency-dependent transmission, where host contact rates increase with or are independent of host densities, respectively. Density-dependent transmission is often seen with directly transmitted diseases, and frequency-dependent transmission is common with vector-transmitted diseases (McCallum et al. 2001). To capture general patterns emerging from the processes of randomly drawing intraspecific R0 values from a distribution, and when applicable extirpation stochasticity, we iteratively produced and disassembled N! communities 1000 times for each extirpation scenario, assuming either density-dependent transmission or frequency-dependent transmission.
For each community, we estimated disease risk as community-level R0, which is calculated as the dominant eigenvalue of the next-generation matrix (Dobson 2004). This matrix incorporates within-species and between-species transmission, as well as host species traits related to survival and reproduction. Community R0 is positively related to the maximum prevalence of infection in the host community. Thus, using community R0 as a measure of disease risk, we are able to incorporate both the ability of the pathogen to invade the host community, and the intensity of the resulting epidemic if invasion is possible. We evaluate the effect of community disassembly by quantifying changes in disease risk resulting from species extirpations: the difference in community R0 before and after a species has been extirpated (ΔR0). When a species is removed and ΔR0 is positive, removing that particular species increases community-level disease risk; the opposite would be true if ΔR0 is negative. Simulations were conducted with R version 3.0.1 (R Core Team 2013).