Keywords: Contact rate; density-dependent transmission; frequency-dependent transmission; heterogeneity; mass action; nonlinearities; pseudo-mass action

Introduction

A central aim of disease ecology is to understand how pathogens spread within host populations (Brandell et al. 2020). Implicit to this is the elucidation of a pathogen’s transmission dynamics (Smith et al. 2009). For directly transmitted pathogens, transmission is effectively the product of the contact rate between hosts, the proportion of contacts that are between susceptible and infectious hosts, and the proportion of effective contacts that result in infection (McCallum, Barlow & Hone 2001). In models of infectious diseases, transmission mechanisms are typically combined into a single term, the transmission coefficient (β), and modelled with one of two simplified functions that describe how infectious contacts scale with population size: that scaling is linear (‘density-dependent’) or independent (‘frequency-dependent’) (McCallum, Barlow & Hone 2001; Begon et al. 2002; McCallum et al. 2017). Selecting a function that provides a useful approximation for modelling transmission has been debated extensively (e.g. McCallum, Barlow & Hone 2001; Begonet al. 2002; Lloyd-Smith et al. 2005; Cross et al.2013), although in practice, the model is often based on the transmission route of the pathogen: density-dependent for direct transmission, and frequency-dependent for vector-borne or sexual transmission. While this approach may accurately reflect transmission at the local scale where contacts happen (though see Ryder et al.2005), there is growing evidence to suggest this may not scale correctly to describe transmission in the population as a whole (the ‘global’ scale) (Smith et al. 2009; Ferrari et al. 2011; Cross, Caillaud & Heisey 2013). For example, transmission may be density-dependent at the local scale, but appear more consistent with frequency-dependent transmission at the global scale (Ferrari et al. 2011; Cross, Caillaud & Heisey 2013). Two interrelated questions arising from this paradox need to be answered to more thoroughly consider the nature of transmission in wildlife populations: how should density in natural populations be defined and measured, and what spatial scales are appropriate for understanding transmission in particular host pathogen systems (De Jong 1995; McCallum, Barlow & Hone 2001; De Jong 2002). Understanding how transmission scales with density is especially important for wildlife populations, in comparison with human infections, where population size may change by orders of magnitude over relatively short periods.
Quantifying animal density can be complex, however. Animal behaviour and heterogeneity in the environment can create aggregate distributions of animals that are not adequately represented in estimates when divided by total inhabited area (Krebs 1999). Defining and measuring density is additionally complicated if animals are distributed in three dimensions (i.e. across horizontal and vertical space). Finally, aggregative distributions of animals may also deviate from the random-mixing assumption that underlies density-dependent and frequency-dependent transmission models, if contacts between neighbours are more frequent than contacts between distantly spread animals (McCallum, Barlow & Hone 2001). An imperative question, therefore, lies in determining the appropriate ‘local’ scale at which transmission occurs, and where contacts may be more homogenous (McCallum, Barlow & Hone 2001). In addition, for highly aggregative species, where local groups form within the global population, processes that drive transmission within groups may not match processes that drive transmission between groups. In such species, transmission within groups may be driven by local group size, while transmission between groups by the structure of the population and the connectivity between local groups (Jong, Diekmann & Heesterbeek 1995; Ferrari et al. 2011).
These issues are prevalent across most wildlife disease systems. Indeed, it has been suggested that seal colonies are one of the few natural situations where transmission can be adequately modelled with population abundance rather than animal density (possibly owing to uniform distancing between individuals) (De Koeijer, Diekmann & Reijnders 1998; McCallum et al. 2017). Issues surrounding the definition and estimation of density are particularly problematic in models of zoonotic pathogens that have bat reservoirs. Bats are among the most gregarious of all mammals – a high proportion of species are social, with some forming the largest aggregations of resting mammals known (Kerth 2008). Bats typically gather together during inactive periods of the diurnal cycle, either in natural habitat (e.g. tree foliage, tree hollows and caves) or anthropogenic structures (e.g buildings, mines and bat boxes) (Kerth 2008). Some species switch roosts frequently (e.g. Rhodes 2007) while others regularly return to the same roost space, or even to specific locations within the roost (Nelson 1965; Lewis 1995; Markus & Blackshaw 2002). This can also be variable among individuals within species (e.g. Welbergen 2005; Welbergen et al. 2020). Spatio-temporal changes to roost structure and organisation are often observed in response to ecological factors like season, mating and gestation, food availability, thermoregulation, parasite accumulation, or site disturbance (Lewis 1995; Kerth 2008). By altering rates of contact, spatial and temporal changes in bat aggregations can contribute to spatio-temporal dynamics of transmission, infection, and risk of disease spillover (Altizeret al. 2006). Framing transmission in ecologically relevant contexts will therefore be important for accurate infection modelling in these species, where population size often changes dramatically.
In models of bat viral transmission where contact rate is assumed to be density-dependent, and where pathogen transmission occurs within the roost (generally the case if individuals forage independently), the transmission coefficient is often parameterised with total roost abundance (George et al. 2011; Plowright et al. 2011; Wanget al. 2013; Hayman 2015; Jeong et al. 2017; Colombiet al. 2019; Epstein et al. 2020). Likewise in statistical models, population size is often fit with total abundance (Serra-Coboet al. 2013; Giles et al. 2016; Páez et al. 2017). If the population size is modelled to be constant (e.g., Plowrightet al. 2011; Wang et al. 2013) it is irrelevant how transmission scales with population size. If population size is variable however, parameterisation with total abundance implicitly assumes that the area occupied remains constant with increasing population size (so that roost abundance scales linearly with bat density), and that this is consistent across scales. Whether this occurs in reality is not routinely evaluated. Indeed, changes to density may be multifaceted and hard to predict. In tree-roosting Pteropus , for example, new individuals arriving into the roost could be accommodated by an expansion of the total roost area, by increasing the number of trees occupied within the roost perimeter, or by crowding more animals into occupied individual trees. These processes may all occur simultaneously.
Over what spatial scale transmission can reasonably be expected to occur within roosts is also a critical question, and one that will define the scale at which density is ecologically relevant to infection dynamics. This is likely to depend on: (i) the mode of pathogen transmission, (ii) animal behaviour and (iii) the structural heterogeneity in the roost environment. First, the mode of transmission will determine the distance over which transmission can occur and will frame the scale relevant for population measures – transmission via aerosols or droplets, or indirect contact with infectious excretions has a greater potential for spread over large distances compared with transmission via direct contact, meaning that a larger scale of density estimation may be warranted. Second, animal behaviour, specifically range of movement, site fidelity, and tendency to aggregate, will influence the spatial extent of contact throughout populations, and so also the scale relevant for population measures. Animals that constantly move through their environment will be likely to contact other animals over a greater area than animals that are more sedentary, for example. Finally, environmental heterogeneity can influence the probability and rate of movement between groups in highly aggregative populations, and so influence the rate and extent of spread over space. Currently, there is little empirical evidence available to understand how viral transmission depends on density in bat populations, or the spatial scale on which density might be relevant. Moreover, there is little empirical support for traditional density-dependent viral transmission in bats, which may reflect complexity in transmission underpinned by ecological processes across different scales (Plowright et al. 2015; McCallum et al. 2017).
Here, we investigate the roosting characteristics of AustralianPteropus bats. Australian Pteropodid bats are the reservoir hosts for Hendra virus (HeV), an emerged paramyxovirus in the genusHenipavirus that causes lethal disease in horses and humans in eastern Australia (Plowright et al. 2015). We present a 13-month dataset of roosting Pteropus spp. from 2,522 spatially referenced trees across eight roost sites, to compare estimates of density across scales (roost-level, subplot-level, and tree-level). We focus on tree-level measures of abundance and density to then evaluate whether roost features at the different scales relate to these local dynamics. We focus our analyses on tree-level measures of abundance and density, as this is the scale at which the majority of contacts are likely to occur, given the nature of viral transmission between bats, and aspects of bat behaviour – i.e. while vertical transmission of Hendra virus has been documented (Halpin et al. 2000), transmission between bats is assumed to be primarily horizontal through contact with infectious urine, either through close contacts with individuals, contacts through the vertical tree column (e.g. with excretion from bats roosting above), or exposure to clouds of aerosolised urine over small distances (Fieldet al. 2001; Plowright et al. 2015). In addition, flying-fox activity within roosts is limited - bats rarely move from their roosting position after they return at dawn, and diurnal activities primarily consist of roosting, sleeping and grooming (Markus & Blackshaw 2002). Moreover the roosting positions of individuals can be highly consistent, with animals often returning to the same branch of a tree over many weeks or months (Markus 2002; Welbergen 2005). Considered together, it is plausible that tree-level measures of abundance or density will be the most relevant for understanding transmission in these species. Through these analyses we aim to provide data on ecologically relevant estimates of density for these species and highlight predictors of the local density most important for transmission of Hendra virus. Understanding gained on the nature of animal density will be important to give more realistic predictions of pathogen invasion and persistence within bat populations. To this end, we also propose a framework to help guide incorporation of heterogenous contact structures into bat infectious disease models more generally.

Methods

Data collection
We collected data on roosting structure of three species (black flying-fox: P. alecto , grey-headed flying-fox: P. poliocephalus and little red flying-fox:P. scapulatus ) from eight roost sites in south-east Queensland and north-east New South Wales, Australia (Fig. 1).P. alecto are believed to be the primary reservoir for Hendra virus in this study region (Goldspink et al. 2015), however a newly-identified Hendra virus variant has been detected in P. poliocephalus and P. scapulatus tissues (Veterinary Practitioners Board of New South Wales 2021). All sites were previously documented as having continuous occupation by at least one species of flying-fox (National Flying-Fox Monitoring Program 2017). Roosting surveys were repeated once a month for 13 months (August 2018 - August 2019).
Methodological details are described in Lunn et al. (2021). Briefly, we mapped the spatial arrangement of all overstory, canopy and midstory trees in a grid network of 10 stratified random subplots (20 x 20 meters each) using an ultrasound distance instrument (Vertex Hypsometer, Haglöf Sweden). Trees were mapped and tagged using tree survey methods described in the “Ausplots Forest Monitoring Network, Large Tree Survey Protocol” (Wood et al. 2015). This approach allowed for precise spatial mapping of trees, with locations of trees within subplots accurate to 10-30 cm. Tagged trees were revisited monthly, and the number of bats per tree was visually estimated and recorded per species using a quasi-logarithmic index: 0: no bats, 1: 1-5 bats, 2: 6-10 bats, 3: 11-20 bats, 4: 21-50 bats, 5: 51-100 bats, 6:101-199 bats and 7: 200+ bats. In total 2,522 trees were mapped across the eight sites. For a subset of trees (60 per site, consistent through time) absolute counts, minimum roosting height, and maximum roosting height of each species were recorded. The roost perimeter boundary (defined as the outermost perimeter delimitating occupied space, as per Clancy and Einoder (2004)) was mapped with GPS (accurate to 10 meters) immediately after the tree survey by walking directly underneath roosting flying-foxes. This track was used to calculate perimeter length and occupied roost area (QGIS 3.1). Total abundance at each roost was estimated with a census count of bats where feasible (i.e. where total abundance was predicted to be <5,000 individuals), or by counting bats as they emerge in the evening from their roosts (“fly-out”), as per Westcott et al.(2011). If these counts could not be conducted, population counts from local councils (conducted within ~a week of the bat surveys) were used, as total abundance of roosts are generally stable over short timeframes (Nelson 1965). Because roost estimates become more unreliable with increasing abundance, we converted the total estimated abundance into an index estimate, as per values used by the National Flying-Fox Monitoring Program (2017). Index categories were as follows: 1: 1-499 bats; 2: 500-2,499 bats; 3: 2,500-4,999 bats; 4: 5,000-9,999 bats; 5: 10,000-15,999 bats; 6: 16,000-49,999 bats; and 7: 50,000+ bats.
All observations were made from a distance to minimise potential disturbance to bats during the survey. In general, bats showed minimal response to the observers during the surveys, providing observers remained quiet, did not move quickly, and kept an appropriate distance, consistent with other studies on flying-foxes (Markus & Blackshaw 2002; Klose et al. 2009).