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Characterizing reservoirs of infection and the maintenance of pathogens in ecosystems

M. G. Roberts

M. G. Roberts

School of Natural and Computational Sciences, New Zealand Institute for Advanced Study and the Infectious Disease Research Centre, Massey University, Private Bag 102 904, North Shore Mail Centre, Auckland, New Zealand

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J. A. P. Heesterbeek

J. A. P. Heesterbeek

Department of Farm Animal Health, Faculty of Veterinary Medicine, University of Utrecht, Yalelaan 7, 3584 CL Utrecht, The Netherlands

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    Abstract

    We use a previously published compartmental model of the dynamics of pathogens in ecosystems to define and explore the concepts of maintenance host, maintenance community and reservoir of infection in a full ecological context of interacting host and non-host species. We show that, contrary to their current use in the literature, these concepts can only be characterized relative to the ecosystem in which the host species are embedded, and are not ‘life-history traits’ of (groups of) species. We give a number of examples to demonstrate that the same (group of) host species can lose or gain maintenance or reservoir capability as a result of a changing ecosystem context, even when these changes primarily affect non-hosts. One therefore has to be careful in designating host species as either maintenance or reservoir in absolute terms.

    1. Introduction

    All infectious disease agents rely on host species for their persistence. This ranges from single host species for highly specialized pathogens and parasites to diverse collections involving many host species at different ecological (trophic) levels. Where there are multiple host species for a given pathogen, these species may not all be equally important for its persistence [13]. In epidemiology, various terms are used to denote collections of species capable of sustaining a pathogen indefinitely, and their interpretation and use depends on whether the point of view is that of the pathogen or a particular host species. Terms in use include maintenance host, maintenance community, reservoir host/species and reservoir community, where the first two relate directly to the persistence of the pathogen (i.e. indefinite presence). The latter two terms relate to the ability of the pathogen to persist and to act as a sustained source of transmission (reservoir) of that pathogen to a specified (target) host species of concern (for example, humans, farm animals, agricultural crops, plant or animal species of conservation interest). The target species itself may or may not be essential for the life cycle or the persistence of the pathogen.

    There are at least three relevant points of view. If we take the point of view of the pathogen, it is entirely sensible to define persistence without a particular target host species playing a role. What counts is that the pathogen can rely on a species or group of species to sustain it indefinitely. This (group of) species acts as an ecological community that allows the pathogen to persist. Such a (group of) species is a maintenance host (community). A maintenance host/community is essential for the persistence of a pathogen, whether or not such a host/community can also transmit the pathogen to a specified target species of concern. Despite progress in clearly defining the terms for persistence (e.g. [4,5]), the terms remain imprecise in several ways that influence their use [6] as well as being challenging to quantify in practice [7,8]. While this is inherent in any attempt to capture complex multi-dimensional phenomena in simple words, it does pay to be aware of what is lost in the assumptions behind the language and the subsequent quantitative characterization of the ideas from models [1].

    From the point of view of a target host species, it makes no sense to call a (group of) species a reservoir for a pathogen if there is no way this (group of) species can transmit the pathogen to the target. Hence, to meet the definition of a reservoir of infection a maintenance host or community must have an infection pathway to the target, either directly or via additional host species (e.g. bridge hosts as introduced by Caron et al. [6]).

    There is a third point of view—that of the ecosystem—that highlights an aspect that is hitherto unaddressed to our knowledge. An ecosystem is a (local) complex network of species where individuals interact both ecologically and epidemiologically intra- and interspecifically and with their abiotic environment. An ecosystem can harbour and sustain a diverse range of pathogens, and it is the ecosystem species and their interactions that allow a pathogen to persist. For the pathogen this means that its host species and the other (non-host) species in the ecosystem, on all trophic levels, influence the ability of the pathogen to persist—for example, by collectively determining the dynamics of the population densities of all species.

    Our main argument is that, regardless of the point of view, whether a (group of) species has maintenance/reservoir capacity depends on the ecological context, i.e. the rest of the ecosystem this (group of) species lives in and interacts with ecologically and epidemiologically. Ecological changes can alter a maintenance/reservoir status, even without any changes in epidemiological parameters, and even with changes only related to non-host species of the ecosystem. Being a maintenance species or community or reservoir is never an absolute characteristic (i.e. it is not a life-history trait), but is dependent on the ecosystem context. So far, research into maintenance groups and reservoirs has concentrated on the host species. A relevant question is whether the maintenance capability of a group of host species can depend, in an essential way, on specific non-host species being present in the community. For example, can it be that two host species do not form a reservoir unless certain non-host species are also present, because the non-host species influence the ecological balance and therefore their epidemiological status? In other words, can the minimal set of species needed for that set to have maintenance/reservoir status contain non-host species?

    It is important to keep the above questions in mind whenever the terms maintenance host/community and reservoir are used, as these terms are too imprecise to characterize any specific situation. Indeed, one can never say that a (group of) species maintains a pathogen without describing the context for this maintenance, nor can one speak of a reservoir without the specifics of the target host and its ecological and epidemiological context. The terms are ambiguous and relative, yet the words are used in the literature and in characterizing persistence of pathogens as if they are unambiguous and absolute.

    In this paper, we show that the ecosystem context matters by exploring how, for a given pathogen, the maintenance/reservoir status of a community of species changes with changing ecological circumstances, including changes to non-host species and changes in predator–prey and competitive relationships. We employ a modelling framework for eco-epidemiological dynamics we used earlier to characterize R0 for pathogens in ecosystems [9], and to explore the interpretation and quantification of the dilution effect [10]. We restrict ourselves, for the purpose of exposition and computational tractability, to low-dimensional model ecosystems. In §2, we summarize the model used for our analysis. In §3, we present the philosophical basis of our argument, making a distinction between the static and dynamic perspectives. Then in §4, we present three theoretical examples of increasing complexity as illustrations. Further technical details may be found in the electronic supplementary material.

    2. Eco-epidemiological dynamics

    In this section, we describe the models used in the examples presented to illustrate our arguments. We first describe the ecological dynamics in the absence of pathogens, then extend our description of the ecosystems focusing on pathogens.

    2.1. Ecological dynamics

    Assume that an ecosystem consists of a number of species, labelled iΩ. Assume that the dynamics of the population size Ni of species i can be described by

    dNidt=νiNiμiNiNijΩϕijNj,2.1
    for iΩ. We take the birth rate of species i to be νijΩ(1aij)ϕijNj, and the death rate of species i to be μi+jΩaijϕijNj, where 0≤aij≤1. Equation (2.1) is a simplified version of the description used in [9], where more general density dependence was allowed.

    For each species in the ecosystem Ω, we define three subsets of Ω. Species i competes for resources with species j when jNi, is eaten by species j when jPi, or eats species j when jQi. We assume that NiPi=, NiQi= and PiQi= for all i. In effect, this means that if two species both compete for resources and one is a predator of the other, then one of the interactions can be neglected, and that cannibalism is not included. If jNiPi, then the per capita growth rate of species i is reduced by ϕijNj > 0. If jQi, then ϕij < 0 and the per capita growth rate of species i is increased by −ϕijNj; furthermore ϕij = −ejiϕji, where eji is the conversion rate of biomass from species j to species i.

    Equation (2.1) can be written in vector form

    dndt=(νμΦn)n,
    where the vectors n, ν and μ have components Ni, νi and μi, respectively; the matrix Φ has entries ϕij; and ° signifies the Hadamard product, νn has components νiNi. The steady-state population densities of the component species may be represented by a vector n*, where Φn=νμ. If Φ is non-singular and n=Φ1(νμ) has only positive components, then n* is a steady state (equilibrium) with all species present of the ecosystem for which the dynamics are specified by equation (2.1). If such a state exists, we call the system ecologically feasible. Although an ecosystem may be ecologically feasible, it need not be ecologically stable. Ecological stability is governed by the eigenvalues of the Jacobian or community matrix C=diag(n)Φ, where diag(n) is the matrix with Ni on the diagonals and zero entries off-diagonal. The condition for an ecosystem to be ecologically stable is that all the eigenvalues of C have negative real part. For ease of analysis, we can rescale parameters so that n* = 1, i.e. each component of n* is equal to one unit of biomass. The condition for ecological stability is then that all the eigenvalues of Φ have positive real part.

    2.2. Epidemiological dynamics

    Suppose that the ecosystem is infected by a single pathogen species of interest. The presence of the pathogen modifies the population dynamics of the host species, and equation (2.1) becomes

    dNidt=νiNiμiNiNijΩϕijNjαiIi,2.2
    for iΩ, where Ii/Ni is the proportion of the population of species i that is infected and αi is the increase in the rate of mortality due to infection. The dynamics of the infected population are expressed by
    dIidt=Λi(NiIi)(μi+αi+jΩaijϕijNj)Ii,2.3
    where the force of infection is
    Λi=γiIiNi+jΩβijIj+jQicjiϕjiIj.
    The model allows for three modes of transmission: frequency-dependent transmission within species at rate γi; density-dependent transmission within and between species at rate βij; and transmission from prey to predator while feeding. We assume susceptible–infected-type dynamics as in [9], but this is in no way essential. We also ignore potential ecological changes in the interactions between species due to the pathogen.

    The Jacobian matrix at the infection-free steady state n = n*, Ii = 0 for all i, can be written as

    J=(CD0H),
    where the matrix C=diag(n)Φ as before. The matrix H may be decomposed, H=T+Σ, where T is the transmission matrix and Σ is the transition matrix [9,11]. The transmission matrix T has diagonal components Tii=γi+βiiNi with off-diagonal components
    Tij=βijNi|jΩ+cjiϕjiNi|jQi,
    and can be written T=Tf+diag(n)Td. The transition matrix is diagonal with components
    Σii=μiαijΩaijϕijNj,
    and can be written Σ=Σf+diag(Σdn). The subscripts f and d denote the components owing to frequency and density dependence, respectively.

    The basic reproduction number is the largest eigenvalue of K, R0=ρ(K), where the next-generation matrix (NGM) K=TΣ1. An ecosystem is epidemiologically stable if R0<1. If that condition holds, then the pathogen cannot persist in the ecosystem [9]. If the ecosystem Ω includes non-host species i, then for these species Ii ≡ 0 by definition. The non-host species may be included in the transmission matrix by adding an appropriate row and column of zeros. The matrix then calculated is the NGM of large domain, KL=TΣ1, and we still have R0=ρ(KL). Alternatively, the NGM is calculated from K = EKLE, where E is the matrix whose columns consist of unit vectors relating to non-zero rows of T only and E is its transpose [11].

    3. The characterization of maintenance communities and reservoirs of infection

    Let there be an ecosystem Ω consisting of hosts and non-hosts of a particular pathogen P. Let ΩnΩ be a subset of Ω consisting of n host types of P, which we identify by number i = 1 … n. We now address the concepts of maintenance community and reservoir of infection. In doing so, we separate our discussion into static and dynamic perspectives. These concepts are illustrated in figure 1 for a theoretical ecosystem consisting initially of 45 species, 27 of which are hosts of a particular pathogen.

    Figure 1.

    Figure 1. An illustration of the possible effects of ecosystem changes on the epidemiology of a pathogen. (a) All host types belong to the maximal maintenance community. There are three minimal maintenance communities outlined with rectangles. There are two sources of infection for the target. (b) Two non-host and two host species have been removed from the ecosystem. Some host species no longer support the pathogen, and there is just one source of infection for the target. Red, infected host species; magenta, uninfected host species; green, non-hosts; yellow (t), target; red (s), source. Thin lines indicate ecological interactions. Black and grey arrows indicate active and inactive transmission pathways, respectively. Blue rectangles enclose minimal maintenance communities. (Online version in colour.)

    Figure 1a portrays the static perspective, showing all host species making up a maximal maintenance community, three minimal maintenance communities, two of which are overlapping, a target host and two sources of infection. Any maintenance community could be regarded as a reservoir of infection for the target, as there is always an infection pathway to the target. The transition from figure 1a,b illustrates the difference between the static and dynamic perspectives. The transition involves the removal of two host species and two non-host species from the ecosystem. As a consequence, it may be assumed that the population density of most, if not all, of the species has changed. What was a minimal maintenance community comprising six host species (top right in figure 1a) has been reduced to five species at a density no longer able to support the pathogen. In addition, there is no longer an infection pathway between this group of species and the remaining maintenance community for the pathogen. The host species shown in the centre of the figures also has no infection pathway to the maintenance community in figure 1b, and is no longer a source of infection for the target. The minimal maintenance community to which this species belongs in a static sense is not a maintenance community in the dynamic sense. Hence while there are three minimal maintenance communities in the static sense, there is only one minimal maintenance community in the dynamic sense.

    3.1. The static perspective

    The perspective in this section is static, in that, although we discuss removing host types from the ecosystem, we do not account for the consequential changes in the population dynamics of the remaining host types if this were to occur. We define a set of host types Ωn to be a maintenance community for the pathogen P if Ωn is connected with respect to P and the pathogen persists in Ωn. Let Kij be the expected number of infected hosts of type i that would result from a typical infected host of type j in an otherwise uninfected and susceptible ecosystem Ω. The set Ωn is connected with respect to P if the n × n matrix KΩn, with entries Kij for i, j = 1 … n, is irreducible ([12]; see also electronic supplementary material). This means that there is an infection pathway from i to j for all i and j in Ωn. Let RΩn=ρ(KΩn) be the spectral radius (largest eigenvalue) of the matrix KΩn. The pathogen P is said to persist in Ωn if RΩn>1. This means that if the set of host types Ωn were initially uninfected, and one or more of the hosts in Ωn were exposed to the pathogen P, then the pathogen would become endemic in Ωn. KΩn is the NGM for P in Ωn, and RΩn is the basic reproduction number for P in Ωn. We define Ωn to be a maximal maintenance community for the pathogen P if the set of host types Ωn is closed with respect to P. In other words, Ki = 0 if iΩn and ΩΩn. This means that if i is a host type in Ωn and ℓ is a host type not in Ωn, then there is no transmission of pathogen P from type ℓ to type i. Note that this does not preclude transmission from type i to type ℓ. Also note that an ecosystem Ω can theoretically harbour more than one maximal maintenance community for the same pathogen, although this requires that two sets of hosts that are ecologically and epidemiologically separated exist within the same ecosystem.

    The condition RΩn>1 means that if the host types in Ωn could somehow maintain their population dynamics and contact rates with each other, but in isolation from the rest of the ecosystem, then the pathogen would persist within the community consisting of only that subset of host types. This is an artificial situation, as removing any species from the ecosystem, or even reducing its population density, would almost always have repercussions for the population dynamics of the other species, and hence lead to changes in the NGM and basic reproduction number. Another way of looking at this is to say that we define a subset ΩnΩ and quantify the transmission of pathogen P between the host types in Ωn while ignoring transmission that involves host types not in Ωn. An immediate consequence of this perspective is that if a non-host species is removed from the ecosystem (ignored), then the epidemiology of the pathogen remains unchanged.

    A maximal maintenance community Ωn could potentially contain many separate, intersecting or nested maintenance communities. Let Ωn be a maintenance community for P in Ω. Let ΩmΩn be a subset consisting of m of the host types in Ωn, m < n, identified by ℓkΩm for k = 1 … m, with ℓk < ℓk+1 for k = 1 … m − 1. Let E be the n × m matrix with entries Eij = 1 if i = ℓj and Eij = 0 otherwise. Define an m × m matrix KΩm=EKΩnE. The matrix KΩm has entries that are also entries of KΩn, and its eigenvalues are equal to the eigenvalues of the matrix formed by deleting the appropriate rows and columns of KΩn. According to our definition, the set of host types Ωm is a maintenance community for the pathogen P in the ecosystem Ω if Ωm is connected with respect to P (KΩm is irreducible) and RΩm=ρ(KΩm)>1.

    If a maintenance community Ωn has no subsets that are also maintenance communities, we can say that Ωn is a minimal maintenance community. A sufficient test for a minimal maintenance community is to define n matrices KΩni, i = 1 … n, by deleting the ith row and column of KΩn. If ρ(KΩni)<1 for i = 1 … n but ρ(KΩn)>1, then Ωn is a minimal maintenance community ([12]; see also electronic supplementary material).

    The above definitions allow us to analyse the maintenance capability of various subsets of Ωn. Specifically, we will regard nested series of subsets, where host types are consecutively ignored or added and where we calculate RΩm to see how the maintenance capability changes when moving up or down in the nested series (see the examples in §§4.14.3).

    We now consider the definition of a reservoir of infection. First define a host kΩ to be a target for the pathogen P. Let ΩmΩn be a maintenance community for P, where Ωn is a maximal maintenance community. If there exists a community of host types ΩrΩ such that ΩmΩr, kΩr and KΩr is irreducible, then Ωm is a reservoir of infection for k. Note that a target may be outside the maximal maintenance community Ωn, Kik = 0 for iΩn, for example, if the target is a dead-end host species. If this is not the case, i.e. if kΩn, it is theoretically possible that the target is part of one or more of its reservoirs. Also, if kΩn then every maintenance community ΩmΩn is a reservoir of infection for k. Finally, it is sometimes useful to define a source of infection for a target species. A source is a host type in Ωn that transmits infection to the target k. Let Ωm be a reservoir of infection for k. If there exists a community of host types ΩrΩ such that ΩmΩr, ℓ ∈ Ωr and KΩr is irreducible, then ℓ ∈ Ω is a source of infection for k if Kk ≠ 0 (there exists a direct infection link from ℓ to k). Note that the source ℓ may or may not be part of a reservoir of infection Ωm.

    As explained, when adopting the static perspective we do not actually remove, but rather we ignore, host types and essentially look at the system through a narrower lens. This reflects the reality that one can never view the entire ecosystem, even locally, and that there may well be host species that one does not know about in the system, or has too little information about. Despite ignoring species, we do know that they are all connected within the ecosystem and that changes in their population dynamics will influence the species we do take into account. In the next subsection, we present a dynamic perspective that does account for such changes.

    3.2. The dynamic perspective

    Let us assume that the n epidemiologically distinct host types of pathogen P that make up a maximal maintenance community Ωn have biomass represented by the n-dimensional vector n, and their dynamics obey equation (2.1). Suppose that a subset of species, ΩmΩn, form a maintenance community in the static sense. The biomass of the m host types in Ωm can be represented by the m-dimensional vector En* that is merely a projection of n* onto Rm. We are applying a lens to the ecosystem, or we could say that we are ignoring some interactions, but we are not changing anything. If, by some mechanism, a species were actually removed from the ecosystem there would be some consequential changes in the population densities, or even the viability of the other species. We refer to the analysis of an ecosystem responding to change as the dynamic perspective, We have three possible scenarios.

    1.

    The removed species was in ΩΩn. The removal is from the wider ecosystem Ω, not from Ωn the maximal maintenance community for pathogen P. The removed species may not be a host of P.

    2.

    Host type ℓ is removed, where ΩnΩm. The removal is from the maximal maintenance community for P, but not from the maintenance community Ωm.

    3.

    Host type ℓ is removed, where ℓ ∈ Ωm. The removal is from the maintenance community Ωm itself.

    Note that we have chosen to view the dynamics starting from a larger set of species and removing species from it, but that one could just as easily study the problem by starting from a small set and adding species that successfully invade. In our description below, we take the former view, but in the discussion of examples both views are considered.

    Under each scenario, the population balance of the wider ecosystem Ω would change, including changes in the population dynamics of Ωm and Ωn. Either the maximal maintenance community Ωn would reach a new equilibrium with n host types, or one or more host types would be driven to extinction. Under all three alternatives, and unless transmission of the pathogen is entirely frequency-dependent, there would be a change in the matrices KΩn and KΩm. This could alter the status of Ωn as a maximal maintenance community, and/or the status of Ωm as a maintenance community. An additional consequence is possible if one of the host types is driven to extinction. There is then the possibility that the matrix KΩn could become reducible, potentially leading to the creation of two separate maximal maintenance communities. Should this happen, a further consequence could be that the species that made up a reservoir of infection are now present in one particular maintenance community, and the former target is a member of the other. With the transmission pathway broken, the status of Ωm as a reservoir of infection for the target would be lost.

    In summary, from a dynamic perspective any change in the wider ecosystem Ω will result in a change in the sub-system Ωn, and in all the communities denoted by Ωm. These changes in population dynamics will be reflected in changes in the NGM. We write R0(Ωm)=ρ(K) when K is the NGM of Ωm. The community Ωm is a maintenance community in the dynamic sense if R0(Ωm)>1, and a minimal maintenance community in the dynamic sense if R0(Ωm)>1 and there exists no subset ΩkΩm such that R0(Ωk)>1. If Ωn is a maximal maintenance community, then R0(Ωn)=RΩn. Let ΩmΩn and ωΩ. The community Ωm is a reservoir of infection for ω in the dynamic sense if R0(Ωm)>1 and there is an infection pathway from Ωm to ω.

    The difference between the two perspectives is highlighted in the notation we employ. When we use the static perspective we do not change the NGM apart from replacing selected elements (entire rows and columns referring to the species we choose to ignore) with zeros, and we write RΩn to emphasize the difference from the true basic reproduction number R0(Ωn) for that system. When we use the dynamic perspective we do regard the actual NGM of a changed system, hence we write R0(Ωn) in that perspective.

    4. Examples

    We illustrate the concepts discussed in §3 by the way of three theoretical examples. The first is a system with two predator species and two prey species, all of which are assumed to be hosts for the same pathogen. All sub-systems are ecologically feasible and the example provides a demonstration of the difference between the static and dynamic perspectives. The second example has five host species in competition for resources. Not all sub-systems are feasible and the example exhibits some bistability in transition between sub-systems. The third example has five host and five non-host species competing for resources. This was introduced to demonstrate that the presence of non-host species can change host population dynamics, and therefore the dynamics of the pathogen. For each example parameter values were chosen for illustration; see the electronic supplementary material, where the numerical results shown in figures 24 may also be found.

    Figure 2.

    Figure 2. The prey–predator ecosystem Ω discussed in §4.1 and its 14 sub-systems. (a) The static perspective: those shown with a red border have RΩm>1; those shown with a green border have RΩm<1. (b) The dynamic perspective: those shown with a red border have R0(Ωm)>1; those shown with a green border have R0(Ωm)<1. (Online version in colour.)

    Figure 3.

    Figure 3. The ecosystem Ω discussed in §4.2 and its 20 ecologically feasible sub-systems. (a) The static perspective: those shown with a red border have RΩm>1; those shown with a green border have RΩm<1. (b) The dynamic perspective: those shown with a red border have R0(Ωm)>1; those shown with a green border have R0(Ωm)<1. The sub-system Ωm = {1, 2, 3, 5} is ecologically feasible, as all population densities are positive, but ecologically unstable. Hence it is shown with a magenta border. (Online version in colour.)

    Figure 4.

    Figure 4. (a) The ecosystem Ω discussed in §4.3 and its 537 ecologically stable feasible sub-systems. White, non-hosts only; green, RΩm<1 and R0(Ωm)<1; yellow, R0(Ωm)<1<RΩm; blue, RΩm<1<R0(Ωm); red, RΩm>1 and R0(Ωm)>1. (b) A detail showing the relationship between some selected sub-systems highlighted as square symbols. These are referred to in the text by: moving left to right at level 8 Ω8g, Ω8y, Ω8r1, Ω8r2; at level 7 Ω7y, Ω7b1, Ω7r, Ω7g, Ω7b2; at level 6 Ω6r1, Ω6y, Ω6g, Ω6r2, Ω6b; and at level 5 Ω5g1, Ω5b, Ω5g2, Ω5y. For more details, see the electronic supplementary material, table S3. Thick blue lines show changes due to adding or subtracting a non-host species; thick black lines show changes due to adding or subtracting a host species. Labels on the vertical axes indicate the number of species in the sub-system. Labels on the horizontal axes indicate the species number ((a) only). (Online version in colour.)

    4.1. Example 1: a prey–predator system

    Consider four host species with species 1 and 2 prey for both species 3 and 4. We assume competition for resources between species 1 and species 2, and between species 3 and species 4. Hence Ω = {1, 2, 3, 4} and

    N1=N2=Q3=Q4={1,2},N3=N4=P1=P2={3,4}andP3=P4=Q1=Q2=.
    We assume frequency- and density-dependent transmission of the pathogen within species, density-dependent transmission between species and transmission from prey to predator while feeding. The NGM is given by K=TΣ1, where the transmission matrix is
    T=(γ1+β11N1β12N1β13N1β14N1β21N2γ2+β22N2β23N2β24N2β31N3+c13ϕ13N3β32N3+c23ϕ23N3γ3+β33N3β34N3β41N4+c14ϕ14N4β42N4+c24ϕ24N4β43N4γ4+β44N4),
    and the transition matrix Σ is diagonal, with components Σii = νi for i = 1, 2 and Σii=νi+j=1,2eijϕjiNj for i = 3, 4.

    With our sample parameter values, we have R0(Ω)=1.40. All of the sub-systems of Ω are depicted in a sub-system tree in figure 2, where figure 2a shows the results from a static perspective and figure 2b the results from a dynamic perspective. From a static perspective, the sub-systems {2, 3, 4}, {1, 3, 4} and {1, 2, 4} are maintenance communities; and {2, 4} and {1, 4} are minimal maintenance communities. None of the species have population densities at which they could support the pathogen in the absence of the other species. Moving down the sub-system tree, there are transitions from RΩm>1 to RΩm<1, but not from RΩm<1 to RΩm>1. Hence moving up or down in the sub-system tree maintenance capability can only change once from a static perspective. If ΩΩm, then RΩm<1 implies RΩ<1, and RΩ>1 implies RΩm>1.

    Figure 2b shows how the outcome differs from a dynamic perspective. Here, we observe, even in a simple example with only four species, repeated switches in maintenance capability when moving up or down the sub-system tree owing to the consequent changes on the population densities of the host species. If species 1, 3 and 4 are removed from Ωn, we see that R0({2})>1, i.e. species 2 would be able to maintain the pathogen on its own and so {2} is a minimal maintenance community in the dynamic sense. If species 3 is then added to species 2, we have R0({2,3})<1 and the pathogen would not be maintained in {2, 3}, even though species 2 is present in that community. However, if species 1 or species 4 were added, then either of the pairs {2, 4} and {1, 2} would maintain the pathogen. Also, if species 4 is added to the non-maintenance community {2, 3}, then maintenance capability is restored with R0({2,3,4})>1.

    4.2. Example 2: five species types in competition

    A system motivated by the interaction between waterfowl and avian influenza was analysed in [13]. Rather than five species of host, five groups of host species were recognized—mallards, other dabbling ducks, diving ducks, geese and swans, and waders. The only ecological interactions were assumed to be competition for resources, and infection was assumed to increase mortality. Hence the transmission matrix has components Tii = γi + βiiNi on the diagonal and Tij = βijNi on the off-diagonal. The transition matrix is diagonal with components Σii = νi + αi. The NGM has diagonal and off-diagonal elements Kii = (γi + βiiNi)/(νi + αi) and Kij = (βijNi)/(νj + αj) for ij. To illustrate this type of ecosystem, parameter values were assigned so that the infection-free steady states have unit biomass, and with all five host types present R0 is just above 1. The parameter values are not based on data and are for illustration only. See the electronic supplementary material.

    The results are shown in the sub-system tree in figure 3, where for the ecosystem Ω = {1, 2, 3, 4, 5} we have Ni=1 and R0=1.078. For these parameter values, there are just two ecologically feasible sub-systems with four host types: {2, 3, 4, 5} and {1, 2, 3, 5}. The latter sub-system gives rise to a community matrix with four non-zero eigenvalues, three negative and one positive, and is therefore ecologically unstable. Numerical results have confirmed that the sub-system {1, 2, 3, 5} has an unstable steady state, with solutions tending to {2, 3, 5} or {1, 2, 5} depending on initial conditions. Hence this sub-system exhibits ecological bistability. Figure 3a also shows that only the sub-systems {2, 3, 4, 5} and {2, 3, 5} have RΩm>1 and from a static perspective constitute maintenance communities, with {2, 3, 5} being a minimal maintenance community.

    From a dynamic perspective (figure 3b) R0({3})>1, so species 3 can support the pathogen in the absence of the other species. Of the ecologically stable sub-systems with two host species, R0(Ω2)>1 if Ω2 = {2, 3} or {3, 5}, hence these two sub-systems are maintenance communities, but adding species 4 to species 3 leads to R0({3,4})<1 and maintenance capability is lost despite the continued presence of species 3. Figure 3b shows that, in a dynamic sense, {3} and {1, 2} are minimal maintenance communities.

    Consider now the characterization of the reservoir of infection. Figure 3a shows that R0(Ω)>1, RΩ4>1 when Ω4 = {2, 3, 4, 5} and RΩ3>1 when Ω3 = {2, 3, 5}. Hence, in a static sense, the sub-system {2, 3, 5} is a reservoir of infection for species 4 and the sub-system {2, 3, 4, 5} is a reservoir of infection for species 1. If we consider species 2, 3 or 5 as targets, then each species would require the other two species to be present and, at the population densities they have in the ecosystem Ω, to be reservoirs of infection. Here, the target would be part of the reservoir.

    Now consider the dynamic perspective as illustrated in figure 3b. According to our criteria, species 1 and 2 are reservoirs of infection for each other, as the pathogen persists when both species are present, but ceases to persist following removal of either species. Our argument that the ecosystem context matters to determine whether a species can be designated as a reservoir is illustrated by species 3. Species 3 is a reservoir of infection for species 2 as well as for species 5. However, species 3 is no longer a reservoir for either species 2 or species 5 if species 4 is present. The reservoir capability of species 3 for species 2 and 5 is restored if species 1 is present in addition to species 4.

    4.3. Example 3: five host species and five non-host species in competition

    The third example is similar to example 2, but with five host species (numbered 1 : 5) and five non-host species (6 : 10). Increased mortality due to infection is neglected, αi = 0. The ecosystem and its sub-systems are illustrated in figure 4a, where only ecologically stable sub-systems are shown. The 10 species form a feasible and stable ecosystem with R0(Ω10)=RΩ10>1. However, only five out of 10 possible sub-systems with nine species are ecologically feasible and stable. Four of these have RΩ9>1 and R0(Ω9)>1, while the other has RΩ9<1 and R0(Ω9)<1. The two left-most sub-systems Ω9 in figure 4a differ from Ω10 by the absence of one host species. One of the sub-systems is a maintenance community in both the static and dynamic senses, whereas the other is not a maintenance community in either sense. The other three sub-systems Ω9 differ from Ω10 by the absence of a non-host species, and are maintenance communities in both senses. Overall, of the 509 sub-systems containing host species, 208(40.9%) have RΩm<1 and R0(Ωm)<1, 12(2.3%) have RΩm>1 and R0(Ωm)<1, 163(32.0%) have RΩm<1 and R0(Ωm)>1, and 126(24.8%) have RΩm>1 and R0(Ωm)>1.

    Figure 4b illustrates that from a static perspective removing or ignoring non-host species does not change the NGM, maintaining RΩm>1 (yellow or red squares) or RΩm<1 (green or blue squares). From a dynamic perspective, adding a non-host species can change the population densities of host species and hence the NGM. All theoretically possible transitions, R0(Ωm)<1 to R0(Ωm+1)<1 (yellow squares), R0(Ωm)<1 to R0(Ωm+1)>1 (yellow to red and green to blue), R0(Ωm)>1 to R0(Ωm+1)<1 (blue to green and red to yellow) and R0(Ωm)>1 to R0(Ωm+1)>1 (red to red) occur in our example. From a static perspective, removing or ignoring a host species changes the NGM, either maintaining RΩm>1 (yellow or red squares) or maintaining RΩm<1 (green or blue squares), or changing from RΩm>1 to RΩm<1 (red to blue or yellow to green). From a dynamic perspective, adding a host species changes the NGM. All possible transitions, R0(Ωm)<1 to R0(Ωm+1)<1 (green or yellow squares), R0(Ωm)<1 to R0(Ωm+1)>1 (green or yellow to red or blue), R0(Ωm)>1 to R0(Ωm+1)<1 (red or blue to green or yellow) and R0(Ωm)>1 to R0(Ωm+1)>1, (red or blue) are illustrated.

    Figure 4a shows that R0(Ω10)>1 for the ecosystem defined by the 10 species. Removing species 5 results in a sub-system with RΩ9g<1 and R0(Ω9g)<1 (see electronic supplementary material, table S3). This could be taken to imply that species 5 is a reservoir of infection for the host species 1–4. This view is strengthened by observing that, with species 5 only present, R0(Ω1)>1. However, there are a number of sub-systems where species 5 is absent but the infection still persists. For example, if Ω7b1 = {1, 3, 4, 6, 8, 9, 10} then R0(Ω7b1)>1 and if Ω5b = {1, 3, 4, 8, 9} then R0(Ω5b)>1. A number of other examples of reservoirs can be observed in electronic supplementary material, figure S4 and table S3, where one particular host species could be regarded as a reservoir of infection for other host species present under particular circumstances, but changing the context can change that view. To take another example, if Ω8r2 = {1, 2, 3, 4, 5, 6, 9, 10} then R0(Ω8r2)>1, and removing species 1 leads to R0(Ω7y)<1, so species 1 seems essential to maintain the pathogen in that particular eight-species community. One is tempted to argue that species 1 is therefore a reservoir for host species {2, 3, 4, 5}. However, if species 4 is then removed from Ω7y we have R0(Ω6r1)>1 and the pathogen is therefore able to persist in the six-species community Ω6r1, which does not contain species 1.

    5. Discussion

    In nature, each pathogen and parasite may have one or multiple host species co-existing in (local) ecosystems in a common abiotic environment with potentially many species at different trophic levels that are not susceptible. Because the ecosystem is defined by a complex set of interactions collectively determining the population dynamics of its constituent species, it is evident that all species will influence the dynamics of pathogens through the effects of these interactions on the host species of each particular pathogen and the non-hosts. In a series of papers, we are studying this interaction between ecology and epidemiology. Using a model that allows us to take all ecological and epidemiological interactions into account (outlined in [9]), we explored the consequences for infection dynamics, compared with the view where the ecosystem context is not taken into account. We showed previously that the ability of a pathogen to invade an ecosystem [9] and the influence of biodiversity on infection dynamics [10] are intricately linked to the characteristics and dynamics of the ecosystem. The ecological context and the ecological interactions between host species and with non-host species has such a profound influence that epidemiological conclusions on key issues such as invasion potential and persistence (i.e. indefinite presence) are meaningless without specifying the context in which these conclusions were reached.

    In the present paper, we show that the concepts of maintenance/reservoir host or maintenance/reservoir community are influenced by the ecosystem context. We do this by providing several examples. These are not chosen to model any natural system but to illustrate our main points. We show that specification of the ecosystem context is necessary in order to conclude whether a (group of) host species is a maintenance community for a specific pathogen or a reservoir for that pathogen in relation to a target host of concern. We show this in highly simplified systems with only four species, all of them hosts to the same pathogen, but also in more complex systems of 10 interacting species, with only five of them hosts to the pathogen. We show that the same (group of) host species can be a maintenance community or a reservoir in one context, but not in another context that is only slightly different (for example, where one non-host species is removed from or added to the system). The consequence of this is that one can never say in absolute terms that species X is a maintenance host or a reservoir for pathogen P if one does not specify the ecological context in which X and P exist. Even specifying that the population density (or size) of species X would need to be above a certain critical value is not sufficient as interactions and pathogen transmission are mediated by the other species in the same system, including other host and non-host species. Moreover, many pathogens will have multiple host species in the same ecosystem and a group of hosts can maintain this pathogen even if none of these host species is above a theoretical critical community size. One therefore needs to be very cautious when making and using statements about maintenance and reservoir species and communities in natural systems.

    Our results can also be interpreted in the context of dilution/amplification effects, as a sequel to our previous analysis [10]. Moving up or down the sub-system trees for our examples can be interpreted as adding or reducing biodiversity. We have examples that show a dilution effect in the sense that a collection of species that is able to maintain the pathogen can lose that ability when a new species is added (i.e. when biodiversity increases). We also have examples where the opposite occurs (i.e. leading to amplification when biodiversity is increased). Our results add to the growing body of evidence that dilution is not a universal principle, but is dependent on the context. This has also been emphasized in a recent review [14].

    In interpreting our results, it can pay to think in terms of functional ecology and functional epidemiology. Recently, a framework for this was described [6], although only host species were considered. The ideas, however, are very useful here. A (group of) species can be characterized by their (combined) function in the ecosystem; for example, a maintenance function or a transmission function [6]. In essence, we show that there is a regulating community in the ecosystem, consisting of the non-host species that can regulate the capability of the maintenance community to give persistence for the pathogen. These species therefore have a regulating function on persistence by influencing the densities at which the host species occur, their interactions and consequently pathogen transmission in the host community.

    The above discussion is not a disqualification of the concepts of maintenance and reservoir when describing infections in natural systems. Like other concepts, for example the basic reproduction number and the dilution effect, they are useful metaphors to guide our thinking about infection dynamics, despite the strong assumptions that lie beneath their definitions and the difficulty in characterizing and interpreting them in absolute terms for what are essentially complex systems. One can never take the full ecosystem dynamics into account, even locally. Species at all trophic levels influence and react to each other and to (changes in) the (abiotic) environment continuously. However, exploring eco-epidemiological models in more detail in case studies may help structure how ecosystem context can be usefully described to allow a more precise characterization of the concepts of maintenance and reservoir. Detailed analyses of well-defined systems are needed (such as [3,8,1519]), but they need to incorporate the wider ecosystem in which the multiple host species reside. Only then can we get a grasp on the question of what additional information needs to be specified if one is to state that a group of species is a maintenance community or reservoir community for a given pathogen. The theoretical examples in this paper show that additional information is needed on the ecological interactions between species, both host species and non-host species, if one is to more accurately gauge maintenance/reservoir capabilities of species and communities. In particular, and using the language introduced in [6], it is important to determine whether there are key non-host species that perform the regulatory function for maintenance. How exactly this needs to be characterized is an open and important problem for which it is essential that ecology and epidemiology are fully combined. The diverse range of approaches introduced over the years to study ecological stability (recently reviewed in [20,21]) and frameworks to study ecological questions [2224] could yield useful measures when combined with epidemiological stability (as in [9]) and characterizations of persistence from epidemiological models.

    Data accessibility

    This article has no additional data.

    Authors' contributions

    The authors participated equally in the design of the study and drafted the manuscript. Both authors gave final approval for publication.

    Competing interests

    We declare we have no competing interest.

    Funding

    The authors acknowledge financial support from the Marsden Fund under contract MAU1718. This research was partially supported by the Simons Foundation and by the Mathematisches Forschungsinstitut Oberwolfach.

    Acknowledgements

    The authors wish to acknowledge two anonymous referees whose comments contributed to the improvement of this manuscript.

    Footnotes

    Electronic supplementary material is available online at https://doi.org/10.6084/m9.figshare.c.4798329.

    Published by the Royal Society. All rights reserved.

    References