Theoretical Ecology

, Volume 5, Issue 2, pp 211–217 | Cite as

Local transmission processes and disease-driven host extinctions

  • Alex Best
  • Steve Webb
  • Janis Antonovics
  • Mike Boots
Original paper

Abstract

Classic infectious disease theory assumes that transmission depends on either the global density of the parasite (for directly transmitted diseases) or its global frequency (for sexually transmitted diseases). One important implication of this dichotomy is that parasite-driven host extinction is only predicted under frequency-dependent transmission. However, transmission is fundamentally a local process between individuals that is determined by their and/or their vector’s behaviour. We examine the implications of local transmission processes to the likelihood of disease-driven host extinction. Local density-dependent transmission can lead to parasite-driven extinction, but extinction is more likely under local frequency-dependent transmission and much more likely when there is active local searching behaviour. Density-dependent directly transmitted diseases spread locally can therefore lead to deterministic host extinction, but locally frequency-dependent passive vector-borne diseases are more likely to cause extinctions. However, it is active searching behaviour either by a vector or between sexual partners that is most likely to cause the host to go extinct. Our work emphasises that local processes are essential in determining parasite-driven extinctions, and the role of parasites in the extinction of rare species may have been underplayed due to the classic assumption of global density-dependent transmission.

Keywords

Transmission Spatial structure Host–parasite Extinction 

Introduction

Parasites have been implicated in the extinction of a number of host species (McCallum and Dobson 1995; Pedersen et al. 2007; Antonovics 2009). A key question in disease ecology is therefore to understand under what conditions parasites may cause deterministic host extinction. In theoretical models, this depends crucially on the assumptions that are made about the transmission process (Getz and Pickering 1983). Clearly, transmission of disease is ultimately an individual level process, with disease passing from infected to susceptible host individuals through direct contact, environmental free-living stages or via a vector. The question is how we theoretically capture this individual process at a population level in order to understand the effect that transmission mechanisms may have on the potential for parasite-driven host extinctions.

Most commonly, transmission is modelled as a global, ‘mass-action’ process with the overall level of transmission related to the prevalence of infection across the entire population. When disease is spread through direct contact, transmission is typically assumed to be density-dependent (Anderson and May 1979; May and Anderson 1979; Levin and Pimental 1981; Bremermann and Pickering 1983; Bremermann and Thieme 1989). Any increase in the density of hosts is expected to increase the overall level of transmission since susceptible hosts will naturally make more contacts with infecteds. When disease is sexually transmitted or vector-borne, transmission is considered more likely to be frequency-dependent (Getz and Pickering 1983; Antonovics et al. 1995; Rudolf and Antonovics 2005; Ryder et al. 2007). In these cases, a host’s number of contacts is expected to stay relatively constant despite any changes in population density, as individuals will have a fairly fixed number of sexual contacts and vectors may search for a fixed number of hosts. Importantly, when transmission is frequency-dependent, it is possible for the parasite to drive the host population to extinction because the transmission rate does not drop towards zero as the population density decreases, as is the case when transmission is density-dependent (Getz and Pickering 1983). However, these global transmission assumptions do not explicitly address the fact that transmission often occurs locally between near neighbours or within social groups rather than across the entire population (Sato et al. 1994; Rand et al. 1995; Boots and Sasaki 1999; Roy and Pascual 2006). In such populations, an individual is more likely to be infected by a neighbour or member of their social group than by those spatially or socially more distant. Those host–parasite models that do include spatial structure and local interactions make predictions that are very different from their global counterparts (Sato et al. 1994; Rand et al. 1995; Keeling 1999; Boots and Sasaki 2000, 2002; Webb et al. 2007a, 2007b). In particular, extinction may occur in spatially structured models that assume local density-dependent transmission (Sato et al. 1994; Boots and Sasaki 2002). This is because, although the global population density may be low, the local density may be high and transmission remains at a high level locally even as the population density falls globally.

It is clearly important that we understand the implication of different transmission processes to disease risk and population extinction. Both frequency-dependent transmission (Getz and Pickering 1983; Antonovics et al. 1995; Ryder et al. 2007) and density-dependent local interactions (Boots and Sasaki 2000; Webb et al. 2007a, 2007b) have been implicated in causing parasite-driven extinctions. However, there has been little investigation into the properties of frequency-dependent transmission or, more generally, sexual and vector-borne transmission, when it occurs at a spatially local level (but, see Turner et al. 2003). Here, we investigate the population dynamics that result from three different local transmission mechanisms: density-dependent, frequency-dependent and active host searching. Our key result is that, when transmission is locally frequency-dependent, extinction is less likely than when it is globally frequency-dependent. However, when hosts actively search for contacts, the conditions for population extinction approach those of global frequency-dependence.

Model

We here give an overview of our model, but full details are given in the Online Resource. We assume an SIS framework (Kermack and McKendrick 1927; Anderson and May 1979), where S denotes the proportion of susceptible hosts, and I infected hosts. A model with global, mean-field transmission and birth processes takes the following form,
$$ \begin{array}{*{20}{c}} {\dot{S} = a\left( {S + fI} \right)\left( {1 - \left( {S + I} \right)} \right) - bS - \beta SI + \gamma I} \\{\dot{I} = \beta SI - \left( {\alpha + b + \gamma } \right)I} \\\end{array} $$
where dots denote derivatives with respect to time. Reproduction occurs at rate a (which is reduced for infected hosts by a factor f), and the natural death rate is b. Transmission occurs at rate β, and in the above formulation, the rate at which susceptible hosts become infected depends upon the density of infected hosts in the population. There is an additional mortality, or virulence, α, for infected hosts, and they can recover at rate γ. There is also disease-independent population regulation on birth, represented by the logistic term [1−(S + I)].
We choose to include a form of extreme spatial structure by assuming that static hosts exist on a regular, square lattice, where each individual has four neighbouring sites in a von Neumann neighbourhood, as shown in Fig. 1. Each site may be occupied by one host individual (S or I) or it may be empty (0). In Fig. 1, each site is shown by a circle, where white is an empty site (0), grey a susceptible host (S) and black an infected host (I), with a focal host shown as a square. The von Neumann neighbourhood of the focal host is highlighted by the dashed lines. We assume that hosts may only reproduce into neighbouring empty sites, whilst disease may be transmitted according to the infection process chosen. We investigate three alternative local transmission mechanisms. First, we assume transmission depends on the local density of infected hosts, as addressed in previous studies (e.g. Webb et al. 2007b). Referring to Fig. 1a, two of the four neighbouring sites of the focal host contain infecteds; this gives a transmission probability to the focal host per unit time as β/2. Second, we alternatively assume transmission depends on the local frequency of infected hosts. In Fig. 1a, the focal host has three neighbouring contacts, two of which are infected, thus giving a transmission probability of 2β/3. Third, we again assume that transmission depends on the frequency of infected hosts, but that hosts search the lattice for contacts beyond their immediate neighbourhood. In particular, as the lattice becomes sparse and the local population density decreases, hosts will search further for contacts, increasing the relative importance of distant contacts. Thus, in Fig. 1b, there are no individuals in the local neighbourhood, but the focal host is not constrained to interacting with its four immediate neighbours; thus, it may be infected from four more distant (but nearest) neighbours in the lattice. For details on model implementation of this, see Online Resource 1.
Fig. 1

Schematic representations of the lattice. A white circle denotes an empty site (0), a grey circle a susceptible host (S) and a black circle an infected host (I); with the grey square denoting the focal susceptible. The solid lines show the local, neighbour-to-neighbour interactions, with the dashed lines highlighting the focal host’s near (von Neumann) neighbours. In b, we show that in the searching model, when the focal host has no near neighbours, it may search in to further neighbourhoods to find contacts

In this paper, we explore the long-term consequences of these three transmission mechanisms in a spatial context. The spatial framework is incorporated using pair approximations, following the techniques of Matsuda et al. (1992) and Sato et al. (1994), details of which are given in the Online Resource 1. Using the AUTO continuation package (Doedel et al. 1997), we explore the bifurcation points of each system, particularly focusing on parasite-driven extinction. We also implement fully spatially explicit stochastic simulations of the corresponding systems to confirm our findings from the pair framework (see Online Resource 1).

Results

Local density-dependent transmission

We first summarise the results for when transmission depends on the local density of infected hosts, as this has been considered in previous studies (Boots and Sasaki 2000; Webb et al. 2007a, 2007b). In all the models, we assume initially that the pathogen causes complete sterility and a small increase in the disease-induced mortality. We plot the epidemiological behaviour predicted from the model for varying reproduction and transmission in Fig. 2b (we also show the results for the global model in 2a). With local, as opposed to global, interactions, two new dynamical behaviours arise: endemic cycles (these in fact occur just outside our parameter range) and parasite-driven extinction. For intermediate transmission, the outcome is an endemic equilibrium (host and parasite coexistence), and at high reproduction and transmission, the host and parasite exhibit cycles. Parasite-driven extinction occurs when host reproduction is low and parasite transmission high. This occurs because although the global population density of infecteds decreases, the local density and therefore transmission may remain relatively high; because of low disease-induced mortality, infected individuals continue to be infectious; yet, because they are sterilised, they do not contribute new susceptibles that permit host persistence. However, the dynamics are extremely sensitive to recovery and reproduction of infecteds, such that parasite-driven extinction becomes much less likely as either process is introduced (Webb et al. 2007a). We also use shading to show the results from stochastic simulations of the system in Fig. 2b, which broadly agree with the results from the pair approximation (PA).
Fig. 2

Equilibria classification diagrams from model with a global density-dependent and b local density-dependent transmission as transmission and reproduction are varied. Parameter values: b = 0.5, f = 0, θ = 0.25, ε = 0.8093, α = 0.1, γ = 0. We also show stochastic simulation results (shaded areas) for the host–parasite system. Parameter combinations shaded light grey indicate neither host nor parasite survived to 20,000 generations, dark grey that only the host survived and black that the host and parasite coexist

Local frequency-dependent transmission

For comparison, we first show the classification diagram for the globally frequency-dependent model in Fig. 3a (Getz and Pickering 1983). Provided the transmission rate exceeds the death rate, an endemic equilibrium can exist due to the density-dependence on host reproduction. However, for even moderately high levels of transmission relative to reproduction, the parasite drives the host to extinction. When transmission is instead locally frequency-dependent as described above (Fig. 3b), extinction becomes less likely. Also, in the local frequency-dependent model, population cycles are now possible. As in the density-dependent model, introducing small amounts of either reproduction or recovery of the infected individuals substantially reduces the potential for parasite-driven extinction and population cycles (see Online Resource 1).
Fig. 3

Analytic predictions from the model with a global frequency-dependent transmission and b local frequency-dependent transmission (white lines). Again, we also show stochastic simulation results (shaded areas) for the host–parasite system with local frequency-dependent transmission. Parameter combinations shaded light grey indicate neither host nor parasite survived to 20,000 generations, dark grey that only the host survived and black that the host and parasite coexist. Parameter values are as those in Fig. 2

In the frequency-dependent model, there is general agreement between the pair approximation analytic results and those of the spatially explicit stochastic simulations, at all but the lowest reproduction probabilities (Fig. 3b). When reproduction is low, the simulations suggest that the host is more likely to persist than the analytic results predict. Extinction does not occur here, as predicted from the pair approximation, because the lattice is very sparsely populated and often the last uninfected individual is not next to an infected individual, preventing host extinction, and allowing the host to persist once the last infected host dies. Host–pathogen persistence is also predicted in the region of high transmission and low reproduction (Fig. 3b), but over longer time spans and on larger lattices, the dynamics in this region tend to result in host-only survival.

Transmission through searching

In our final model, we assume that hosts can search the lattice for contacts, such that they search further as the local density drops; however, their total number of contacts remains constant, making transmission implicitly frequency-dependent. When the local density is high, transmission is dominated by local contacts, but as the lattice becomes sparse, transmission from distant infecteds becomes increasingly important (see Online Resource 1). Here, extinction occurs for low transmission (Fig. 4) because even when the local density is low, hosts will still search the lattice to find contacts, keeping the force of infection high. This effect is particularly noticeable at low reproduction where, unlike our previous two models, regions of host and/or parasite persistence are not present at high rates of transmission and low reproduction rates.
Fig. 4

Analytic predictions from the pair approximation model (white lines) and simulation results (shaded areas) for a host–parasite system with local reproduction and transmission where hosts search the lattice in increasing neighbourhood sizes for contacts. Parameter combinations shaded light grey indicate neither host nor parasite survived to 20,000 generations, dark grey that only the host survived and black that the host and parasite coexist. Parameter values are as those in Fig. 2

In stochastic simulations of the system, extinction is not as widespread as predicted in the pair approximation, and the extinction threshold in these simulations in fact resembles that in the global frequency-dependent model (see Fig 3a). To implement the pair approximation, it is necessary to assume that a host is able to sample all possible contacts on the lattice, whereas in the simulations, a host searches for exactly four contacts. This subtle difference means that, when the lattice empties, our pair approximation is likely to predict more transmission, and thus a higher likelihood of extinction, than that seen in the simulations. We conducted further simulations where hosts search for a greater number of contacts (eight) and found that the extinction boundary indeed moves down towards the analytic prediction. However, in both the pair approximation and the stochastic simulations, parasite-driven extinction is still much more likely when there is a searching mechanism than when there is purely local frequency-dependent transmission.

Discussion

We have investigated how local transmission processes may lead to parasite-driven extinction. In particular, since previous theoretical work has implicated both frequency-dependent transmission and local interactions as mechanisms that may result in extinction (Getz and Pickering 1983; Sato et al. 1994; Boots and Sasaki 2002), we have investigated the effects of different transmission mechanisms when transmission is localised between near neighbours. In spatially structured host–parasite systems, parasite-driven extinction can occur whether there is local density or frequency-dependence. Local transmission processes increase the likelihood of extinction when there is density-dependent transmission, but decrease the likelihood of extinction when there is frequency-dependent transmission. Extinction appears most likely to occur when transmission is frequency-dependent and especially when hosts search for contacts regardless of population density. The local transmission process therefore has a substantial impact on the potential for parasite-driven extinction.

The result that parasite-driven extinction is more likely when transmission is locally frequency-dependent rather than locally density-dependent, reflects the results from mean-field models (e.g. Anderson and May 1979; Getz and Pickering 1983). When transmission is globally density-dependent a feedback with disease prevalence prevents the parasite from driving the host to extinction since overall transmission will drop towards zero as the susceptible density falls (Anderson and May 1979). However, where transmission is frequency-dependent, no such feedback exists, meaning that transmission remains high even as the number of susceptibles drops towards zero, leading to the possibility of parasite-driven extinction (Getz and Pickering 1983). Similarly, in our spatially structured model, even when the local density of infected hosts is low, the frequency may remain high if the lattice is dominated by infecteds, keeping the transmission rate high. This effect is even stronger when hosts do not rely purely on local contacts but instead search the lattice and thus come in to contact with more distant infecteds. The predictions for the searching model closely match those of global frequency-dependent transmission with regards to parasite-driven extinction (Getz and Pickering 1983). As the extinction threshold is approached and the lattice empties, the searching model almost exactly coincides with the global frequency-dependent model.

One of the key questions when attempting to investigate spatially structured populations is how to incorporate that spatial structure. Many methods exist, including partial differential equations (White et al. 1999), perturbation expansions (Ovaskainen and Cornell 2006), contact networks (Meyers et al. 2005) or moment closures (Matsuda et al. 1992). Here, we have chosen to use the pair approximation on a lattice (Matsuda et al. 1992; Sato et al. 1994), a moment closure that assumes a rigid contact network where individuals are fixed in space and may only be contacted by neighbouring individuals. Whilst the searching model implicitly assumes some movement of individuals, the other local models are most appropriate to plants and other relatively sessile organisms. However, it is useful to understand the role of spatial structure by making extreme assumptions, and our results and insights are likely to apply more broadly (Dieckmann et al. 2000). Furthermore, host–parasite models that examine a range of spatial structures (Boots and Sasaki 1999; Webb et al. 2007b) show that fully spatial effects are not generally artefacts of the rigid lattice assumptions. For example, Webb et al. (2007b) showed clearly that extinction boundaries move smoothly between the two extremes of fully global and fully local interactions, suggesting that the patterns we have predicted here between the different transmission mechanisms are likely to hold for more realistic mixes of local and global interactions. They are therefore likely to give insights to a wide range of real biological systems.

Both the transmission mode and the degree of spatial structuring of this transmission are important factors in determining the dynamics in particular host–parasite systems. Particular host–parasite systems are likely to be characterised by different forms of transmission mode. Local density-dependent transmission reflects diseases spread through direct contact, since any increase in the local density of infected hosts will naturally lead to a greater exposure to disease and greater overall transmission. Local frequency-dependent transmission may reflect sexual and vector-borne infections without active searching. With sexual transmission, if an organism has a set dispersal range yet only has a limited number of sexual contacts, it is likely to conform to a local frequency-dependent model. Most obviously, this is likely to be true of many animal populations with territorial males. However, it may also occur in non-sexually transmitted diseases where individuals are organised into social groups whose structure and size remain relatively independent of overall density. As such, it may apply to a wide variety of diseases such that, for example, a local frequency-dependent transmission model, particularly if more detailed social contact networks are assumed, may provide an excellent fit to local measles epidemics in the UK, because transmission is in school classes of relatively fixed size regardless of whether schools are in high-density urban or low-density rural areas. The searching model, which most closely resembles the global frequency-dependent model, may best reflect sexually transmitted diseases, where when the local density drops hosts will search further afield to continue to make contacts. This is well known in humans, where distances over which people travel to marry has been shown to be negatively correlated with population density (Cavalli-Sforza 1958).

Vector-transmitted diseases are likely to conform to a very wide range of transmission models, depending on vector behaviour (Antonovics et al. 1995). Where vectors are passive and poor dispersers (e.g. aphids vectoring plant viruses), the transmission process is likely to be locally density-dependent. At very high host population densities, and if there is no or only a delayed numerical response on the part of the vectors, transmission may actually fall because the per-host contact rate declines. Where vectors actively search for hosts, such as in pollinator transmitted diseases, they are likely to travel further when hosts are further apart. They are therefore expected to conform (at least over a broad range of population densities) to the searching model. Global frequency-dependent transmission in such systems has been confirmed experimentally (Antonovics et al. 1995) and in field studies (Antonovics 2009) of anther-smut, a plant pollinator transmitted disease. Frequency-dependent transmission without active searching (local frequency-dependence) may occur where vectors are limited in both their dispersal distances (relative to host spacing) and the number of bites (or contacts per host). We would argue that this is likely to be the case in mosquito-borne diseases and indeed many models of malaria transmission assume frequency-dependent transmission of some type (Anderson 1981). Our work emphasises that, if transmission is localised and not active, the assumption of global frequency-dependence over estimates the likelihood of disease-driven host extinction.

The potential for parasite-driven extinction, as well as for population cycles, shrinks considerably when infected individuals can either reproduce (and produce healthy offspring) or recover from infection. By maintaining a supply of susceptible hosts, both of these processes not only prevent the host population from being driven to zero, but also generally stabilise the host–parasite system (Webb et al. 2007b). There are a large number of pathogens that either sterilise their hosts (Kuris 1974; Baudoin 1975; Jaenike 1992; Lafferty and Kuris 2009; Clay 1991; Wennstrom and Ericson 2003; Lockhart et al. 1996) or that are obligate killers (Kuris 1974; Hassell 1978, 2000; Boots and Begon 1994; Boots 2004). Our results suggest that these parasites may be more prone to causing host extinction than more conventional diseases that cause mortality and are transmitted in a density-dependent fashion (Antonovics 2009). As a consequence, we suggest that they deserve increased attention in conservation, disease management, and biological control (Lafferty and Kuris 1996).

The results from our analysis based on pair approximation closely matched the stochastic simulations. The fact that the general patterns we see in the PA and the full stochastic simulations are qualitatively similar gives us some further insight into the processes that underpin the results. In particular, higher-order spatial effects (triples, etc.) are not crucial to the general pattern, and the insights that we have gained hold under the deterministic large population assumption of the PA. There was some discrepancy in regions where the population densities were very low. This is not surprising as it is the higher-order effects that are likely to be important at low densities where the explicit spatial structures are likely to be important in determining contact rates. At low reproduction rates, and thus when the lattice is sparse, the analytic models tend to predict greater extinctions than witnessed in the simulations, as in the simulations the last susceptible host may not be next to an infected, preventing the extinction from occurring. The searching model in particular tended to over-predict the potential for extinction at all reproduction rates compared with the simulations, but we still found in the simulations that extinction was more likely with active searching than passive frequency-dependence. In the active searching model, hosts will always find a contact when the population density is very low, making transmission of disease and extinction more likely.

We have compared the predictions of local and global assumptions of density- and frequency-dependence in disease models and highlighted the importance that local infection processes may play. Knowledge of the local behaviour of hosts and their vectors is necessary to choose the most appropriate transmission model and, in particular, it is important to distinguish between active searching behaviour and passive local frequency-dependence. Taken as a whole, our results emphasise that even in one-host/one-pathogen systems, infectious disease may often lead to host extinction and that this is a fundamental consequence of the combination of sterilising effects of disease, the functional form of transmission dynamics and, in particular, the scale over which those dynamics operate.

Supplementary material

12080_2011_111_MOESM1_ESM.pdf (102 kb)
ESM 1(PDF 101 kb)

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  • Alex Best
    • 1
  • Steve Webb
    • 2
  • Janis Antonovics
    • 3
  • Mike Boots
    • 1
  1. 1.Department of Animal and Plant SciencesUniversity of SheffieldSheffieldUK
  2. 2.Department of Mathematics and StatisticsUniversity of StrathclydeGlasgowUK
  3. 3.Department of BiologyUniversity of VirginiaCharlottesvilleUSA

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