Multihost pathogens are likely to exhibit spatiotemporal dynamics different from those of pathogens that infect only a single host species. From one perspective, multiple hosts could be considered an additional form of heterogeneity that divides the total host population into subpopulations, between which transmission occurs at a different rate from that within each subpopulation. Single-species ‘subpopulation’ approaches (with multiple scales of mixing) have been successfully developed to examine disease transmission between sexes in the case of sexually transmitted diseases (May & Anderson 1987; Anderson 1991); between children of different ages (measles, mumps, rubella) (Anderson & May 1985); people living in regions, cities and villages of different sizes (measles, influenza) (May & Anderson 1984; Grenfell & Bolker 1998; Grenfell, Bjornstad & Kappey 2001; Viboud et al. 2006); and hosts living as a metapopulation in different patches of habitat (Swinton et al. 1998; McCallum & Dobson 2002; McCallum & Dobson 2006).
However, using subpopulation approaches on multihost pathogens is not as straightforward as it seems; different host species might vary in their response to infection, have varying contact patterns based on social behaviour, and have different spatial distributions across the landscape (Dobson 2004). Due to these complexities, previous work on multihost models has made simplifying assumptions and assumed that each host population is well mixed, and specifically ignored heterogeneities due to social organization (Dobson 2004; Fenton & Pedersen 2005; McCallum & Dobson 2006). We have, therefore, developed a general stochastic, spatial model of a disease outbreak in two and three host-species communities with widely ranging social structures. Our model structure is based on a 1994 outbreak of canine distemper virus (CDV) in the Serengeti ecosystem that killed one-third of the lion population (Panthera leo) (Roelke-Parker et al. 1996; Kock et al. 1998; Packer et al. 1999). CDV is a contagious multihost virus spread by aerosol inhalation, which affects all carnivore families. Infected animals either die or obtain lifelong immunity (Appel 1987; Williams 2001).
Because lions are territorial, and most opportunities for disease transmission between social groups involve immediate neighbours (M.E.C., unpublished data), the erratic and discontinuous spatial pattern of CDV spread in the 1994 epidemic seems unlikely to have resulted solely from lion-to-lion transmission (Fig. 1). During the 1994 outbreak, the same CDV variant was responsible for deaths in spotted hyenas (Crocuta crocuta) (Haas et al. 1996; Roelke-Parker et al. 1996; Carpenter et al. 1998), while jackals (Canis adustus, Canis aureus, Canis mesomelas) also showed CDV-like symptoms and subsequently tested positive for CDV antibodies (Alexander et al. 1994; Roelke-Parker et al. 1996).
Hyenas and jackals had the potential to transmit CDV to lions, as the two species are more abundant than lions (Campbell & Borner 1986), and frequently interact with lions at kills (Schaller 1972; Cleaveland et al. 2008). While lions, hyenas, jackals, bat-eared foxes (Ototcyon megalotis) and potentially many other carnivore species (e.g. leopards, Panthera pardus) were affected by the 1994 CDV outbreak (Roelke-Parker et al. 1996), our most detailed data come from the long-term monitoring of the Serengeti lions (Packer et al. 2005). We therefore treat lions as the sentinel species when comparing the observed pattern of infection in the 1994 lion population with the model's CDV spatial spread.
We developed a stochastic simulation model to capture the general spatial and temporal patterns observed in the 1994 CDV outbreak. Although the model is based on the lion outbreak, it has been developed to provide more general insights into disease outbreaks in other communities, where multiple host species are susceptible to infection by the same pathogen. In particular, we ask whether differences in territorial social structure affect the spatial and temporal pattern of disease outbreaks, and if the time course of the epidemic is sensitive to different rates of within- vs. between-species interaction. Social organization due to territorial behaviour divides intraspecific transmission into two major components: within and between groups. Within-group transmission can occur during normal social interactions (feeding, grooming), whereas between-group transmission can occur during fights over food and territory, or during immigration events. Interspecific transmission occurs when multiple species feed together or during intraguild predation events.
We performed a set of simulations that examine the epidemic dynamics of a directly transmitted pathogen involving multiple host species with contrasting social organizations (e.g. isolated vs. well connected territorial structures), characterized by different within- and between-group transmission rates. After exploring the epidemic dynamics for each species in isolation, we examine the consequences of coexistence between pairs of species using high and low rates of interspecific transmission. Finally, we ask whether the coexistence of three hosts differs in any substantive way from any two-species scenario.
We use the simulation to ask:
How do within- and between-group contact patterns affect the incidence, rate of spread, probability and spatial pattern of infection in multiple hosts with coexisting pathogens?
How do the model results compare with the observed outbreak?
The model describes the spatial and temporal dynamics of a pathogen in a spatially structured, multihost community. The habitat is divided into a two-dimensional grid of 625 patches, with each patch containing a local population of each species. Because of the natural boundaries of the Serengeti ecosystem, we chose not to wrap the edges of the simulated habitat. Infection is spread within local populations, between different species occupying the same patch, and between any populations/species occupying the eight neighbouring patches. The pathogen is modelled in a stochastic, density-dependent, susceptible–infected–recovered (SIR) framework. The model was programmed in C.
The importance of group size to pathogen persistence is well known (Swinton et al. 2001; Park, Gubbins & Gilligan 2002; McCallum & Dobson 2006), so we held group size constant across species and across social groups in order to isolate the effect of social organization. Each patch begins with 10 individuals of each species. An individual may be categorized in one of three states: S (susceptible), I (infected) or R (recovered). All individuals, except an initially infected source, begin the simulation in state S. Transitions occur from S I (infection) and from I R (recovery). During each time-step, we determine the probability of a susceptible individual becoming infected, pS I, and of an infected individual recovering (either dying or obtaining lifelong immunity), pI R. The number of actual transitions is drawn from a binomial distribution, B(n, p). For the infection transition, n is the number of susceptible individuals in the group, while for the recovery transition, n is the number of infected individuals.
The probability that a susceptible individual i will be infected depends on the number of infections in its own social group, interspecific transmission within the same patch, and intra- and interspecific transmission from neighbouring patches. Two ‘who acquires infection from whom’ matrices (WAIFW; Anderson & May 1991) characterize the force of infection between individuals of each group; let bw,ij represent within-patch transmission and bB,ij represent between-patch transmission. The total probability of infection is given by:
where SL is the set of groups sharing the local patch and SN represents the groups in neighbouring patches and Ij is the number of infected individuals in group j. Each infected individual has a fixed probability, µ, of recovering.
Interspecific β values are taken as a weighted average of the intraspecific values so that
where c describes the level of interspecific interactions (or coupling). We used two different values of c, designated ‘high’ and ‘low’ (0·2, 0·01, respectively) for the multispecies simulations.
The value of the average reproductive rate of the pathogen is defined as R0. In general, a pathogen can persist only when R0 is >1 (when each infected individual infects at least one other individual). Species’ within- and between-patch transmission rates were chosen so that the R0 values in a single-species habitat equalled 2·2. CDV is closely related to phocine distemper virus, for which the empirically estimated R0 is 2·8 (Swinton et al. 1998). Different social systems were modelled by choosing different relative rates of within- and between-group transmission (Table 1).
Table 1. Relative rates of within- and between-group transmission
|Resembles||R0 within-group||R0 between-group|
|Lion||>1 (1·9)||<1 (0·3)|
|Hyena||>1 (1·1)||>1 (1·1)|
|Jackal||>1 (1·5)||<1 (0·7)|
In the Serengeti, the African lion lives in territorial social groups (prides) consisting of related females and their dependent offspring. Before the 1994 epidemic, average pride sizes (excluding cubs <3 months) were 10 individuals (M.E.C., unpublished data) defending territories ranging from 15 to 150 km2 (Mosser 2008). Lions form fission–fusion groups where pridemates are in frequent physical contact, but only occasionally contact their neighbours during territorial defence or fights over food (Schaller 1972; M.E.C., unpublished data). Thus the within-patch (or within-pride) transmission rate for lions will be far higher (R0 > 1) than between-patch transmission (R0 < 1).
The spotted hyena lives in social groups (clans) averaging about 45 individuals per clan (Hofer & East 1995). These hierarchical clans consist of related females and immigrant males who defend exclusive group territories (16–55 km2) and encounter their neighbours during territorial clashes, or when feeding at the same carcass (Hofer & East 1993a). Additionally, Serengeti hyenas have a unique feeding adaptation where they commute to migratory prey and associate with non-clan members at waterholes and resting sites (Hofer & East 1993b). Thus hyenas are expected to have high within-patch transmission (but contact each other less than lions), as well as high between-patch transmission.
Jackals live in small family groups of two to four, who are in close contact with each other (Moehlman 1983). Serengeti golden and black-backed jackals actively defend discrete territories (≈2–4 km2) from neighbours; they also make extraterritorial forays to water sources and large mammalian kills (Moehlman 1983). We therefore consider each ‘patch’ of 10 individuals to consist of two to five loosely connected groups of jackals. Although they interact with each other less frequently than pridemates, jackals contact individuals from neighbouring patches more frequently than lions.
Infections were introduced in a single individual at the edge of the grid to mimic a pathogen introduced from domestic dogs at the edge of the park (Cleaveland et al. 2000). We ran 150 simulations for each combination of species. To check whether changes in disease dynamics were due to social structure rather than to a simple increase in overall population size, we ran controls where the same species was coupled with itself within separate partitions of the same patch. Each simulation ran until all infections disappeared. For each species, we also varied the within- and between-group transmission rates to confirm that the results presented here were representative of the overall range of possible outcomes.
We used the package ncf (Bjornstad & Falck 2001) for r (R Development Core Team, 2006) to evaluate the spatial pattern in both simulated and observed outbreaks. For each time-step (day) in the simulated outbreaks, we entered the number of active infections per grid square (pride) into the nonparametric correlation function (ncf). Because of the coarse-grained resolution of within-pride mortality in 1994, we constructed within-pride epidemic curves from the simulated outbreaks by aligning the simulated start dates, averaging the number of infections at each time-step, and rounding the values into discrete integers. We combined these simulated within-pride epidemic curves with the observed first death date per pride and spatial location, to create a complete time-series for the observed outbreak.