interspeciﬁc contact and competition may affect the strength … · 2015. 10. 9. ·...

Interspecific Contact and Competition May Affect the Strength

and Direction of Disease-Diversity Relationships

for Directly Transmitted Microparasites

Suzanne M. O’Regan,1,2,* John E. Vinson,1 and Andrew W. Park1,3

1. Odum School of Ecology, University of Georgia, Athens, Georgia 30602; 2. National Institute for Mathematical and Biological Synthesis,University of Tennessee, Knoxville, Tennessee 37996; 3. Department of Infectious Diseases, College of Veterinary Medicine, University ofGeorgia, Athens, Georgia 30602

Submitted July 22, 2014; Accepted June 3, 2015; Electronically published September 4, 2015

Online enhancements: appendix, scripts.

abstract: The frequency of opportunities for transmission is key tothe severity of directly transmitted disease outbreaks in multihostcommunities. Transmission opportunities for generalist micropara-sites often arise from competitive and trophic interactions. Addition-ally, contact heterogeneities within and between species either hinderor promote transmission. General theory incorporating competi-tion and contact heterogeneities for disease-diversity relationshipsis underdeveloped. Here, we present a formal framework to exploredisease-diversity relationships for directly transmitted parasites that in-fect multiple host species, including influenza viruses, rabies virus, dis-temper viruses, and hantaviruses. We explicitly include host regulationvia intra- and interspecific competition, where the latter can be depen-dent on or independent of interspecific contact rates (covering resourceutilization overlap, habitat selection preferences, and temporal nichepartitioning). We examine how these factors interact with frequency-and density-dependent transmission along with traits of the hosts inthe assemblage, culminating in the derivation of a relationship describ-ing the propensity for parasite fitness to decrease in species assemblagesrelative to that in single-host species. This relationship reveals that in-creases in biodiversity do not necessarily suppress frequency-dependentparasite transmission and that regulation of hosts via interspecific com-petition does not always lead to a reduction in parasite fitness. Our ap-proach explicitly shows that species identity and ecological interactionsbetween hosts together determine microparasite transmission outcomesin multispecies communities.

Keywords: biodiversity, dilution effect, disease ecology, epidemiology,host-parasite interactions, transmission.

Introduction

Directly transmitted microparasites often infect multiplehost species, and consequently the ecology of species in-teractions is crucial for determining parasite establishment

and persistence in multihost communities. The frequencyand magnitude of disease outbreaks is often determined bythe rate of transmission opportunities in the assemblage.Cross-species contact—achieved, for example, throughfeeding on or utilizing a common resource—may facilitateparasite spillover from one host species to another. Alter-natively, ecological interactions that regulate susceptiblehost species, such as intra- and interspecific competition,can control or possibly promote parasite transmission inmulti-host-species communities. For example, reducedhost species abundances resulting from interspecific com-petition may lead to a decreased number of cross-speciestransmission events. Additionally, community structuresthat result in contact heterogeneities within and betweenspecies will affect disease outcomes (Dalziel et al. 2014).Given that interspecific competition and contact may beeither correlated or independent of one another accordingto community contexts (fig. 1), it is unclear under whatconditions competitive interactions and contact betweenmultiple hosts combine to enhance or, alternatively, re-duce the risk of outbreaks of microparasitic infections inmultispecies communities.To quantify how community structure and ecological

context, including competition and contact heterogeneities,may affect outbreak tendency, it is possible to calculate thefitness of a parasite in a community of host species. Parasitefitness is typically measured by the basic reproductionnumber R0, the average number of secondary infectionscaused by one infectious individual (Anderson and May1991). Although this was originally defined to describetransmission in a susceptible single-species host popula-tion, the basic reproduction number of the parasite in amultihost community can be readily derived through calcu-lation of the dominant eigenvalue of the next-generationmatrix obtained from linearizing the infected-host sub-system at the disease-free equilibrium (Diekmann et al.

* Corresponding author; e-mail: [email protected].

Am. Nat. 2015. Vol. 186, pp. 480–494. q 2015 by The University of Chicago.0003-0147/2015/18604-55653$15.00. All rights reserved.DOI: 10.1086/682721

vol . 1 86 , no . 4 the amer ican natural i st october 20 1 5

2010). Theoretical studies have considered how parasite fit-ness in a multihost community (hereafter, community R0)is impacted by the composition and abundance of the hostcommunity (Norman et al. 1999; Roche et al. 2012; Mihal-jevic et al. 2014). Dobson (2004) explored the conditionsunder which species richness, in combination with trans-mission mode, increases or decreases the propensity foroutbreaks, which he quantified by calculating the basic re-production number acrossmultispecies communities. Rudolfand Antonovics (2005) described how reduction in parasiteincidence might arise from different assumptions abouttransmission in combination with regulation mechanisms,but their study investigated the effects of intraspecific com-petition and frequency-dependent transmission only. Neitherof the latter models included interspecific competition. Whilemodels for interspecific competition and disease transmis-sion are well developed, they are rarely examined jointly(but see Bowers and Turner 1997; Peixoto and Abramson2006). Recently, community epidemiology has made prom-ising advances in the study of parasite transmission in mul-tispecies host communities by invoking phenomenologicalassumptions linking species richness, abundance, and even-ness (Roche et al. 2012; Joseph et al. 2013; Mihaljevic et al.2014), but these studies do not include interspecific contactheterogeneities and host competition. Our approach intro-

duces these scenarios and asks under what conditions is par-asite fitness in a multi-host-species community reduced rel-ative to parasite fitness in a community consisting of asingle focal host species. This question represents a funda-mental knowledge gap concerning how broad assumptionsabout transmission mechanisms and strength of competi-tion contribute to the spread of directly transmitted para-sites in multispecies communities. This “missing piece” in-cludes the potential independence of contact rates andstrength of competition mediated by variable overlap inresource-utilization functions, habitat selection, temporalniche partitioning, and behavioral avoidance.Accordingly, we consider simple epidemiological mod-

els encompassing intra- and interspecific competition alongwith intra- and interspecific contact. We compare parasitefitness across communities of a single host species (the res-ident or focal host) and communities composed of a focaland alternative host species. Analytical results of the effectof host species richness on parasite fitness are achieved via aone- and two-host species comparison, and themain resultsare shown to hold in larger host species assemblages bymeans of numerical simulations (appendix, available on-line). The models are flexible in their assumptions aboutthe correlation between contact rates and strength of com-petition (fig. 1). Contrasting contact and regulatory regimes

Figure 1: Ecological contexts encompassing competition and interspecific contact structure that may affect microparasite transmission inmultispecies communities. Four separate cases are considered: (i) low interspecific contact and weak interspecific competition, (ii) highinterspecific contact and weak interspecific competition, (iii) low interspecific contact and strong interspecific competition, and (iv) highinterspecific contact and strong interspecific competition. The degree of susceptible host regulation that arises from interspecific competitionstrengthens along the interspecific competition axis. We use this framework to explore how susceptible host regulation through competitionand cross-species contact interact to drive parasite transmission.

Disease-Diversity Relationship Framework 481

include (i) low interspecific contact and weak interspecificcompetition (e.g., when host species are spatially separatedor select different habitats); (ii) high interspecific contactand weak interspecific competition (e.g., host species usedifferent resources, but the resources co-occur); (iii) lowinterspecific contact and strong interspecific competition(e.g., host species compete for a limiting resource but havelimited contact with each other, such as through temporalpartitioning or behavioral avoidance); and (iv) high contactand strong interspecific competition (e.g., when species arecompeting for a common resource).

In addition to regulatory forces, the transmission modealso influences parasite fitness in a community of host spe-cies. Per capita contact rates may increase with host den-sity, for example, if the parasite is transmitted primarilythrough random contact between individuals (density-dependent transmission). Alternatively, contact rates mayremain approximately constant across a range of densities,for example, if parasite transmission occurs through sex-ual contact or between members of populations that arestrongly socially structured (frequency-dependent trans-mission). We incorporate mechanistic host competitioninto susceptible-infectious-susceptible (SIS) mathematicalmodels spanning frequency- and density-dependent trans-mission. Using these models, we derive analytical relation-ships that describe the propensity for amplification and re-duction of disease transmission as a function of host speciesrichness by calculating basic reproduction numbers, whichmeasure parasite fitness (Dobson 2004; Roche et al. 2012).These expressions for the disease-diversity relationship canbe represented graphically in ecologically relevant parame-ter space. Our results demonstrate that disease-diversityrelationships are more complex than is often recognized,including the potential for disease amplification in host-parasite systems with frequency-dependent transmissionand decreased outbreak potential in communities exhibitingdensity-dependent transmission. Additionally, we demon-strate that elements of community composition, such as hostspecies traits and contact patterns between species, are keycomponents of disease-diversity relationships that shouldnot be neglected by predictive models. In “Discussion,” wesummarize caveats linked to our modeling approach. Fi-nally, we place our findings in the context of the dilution-effect hypothesis, which broadly posits that the net effectof increasing biodiversity is reduction in parasite trans-mission (quantified in this article as the propensity for de-creases in parasite fitness as species richness increases).

Quantifying Parasite Fitness across Communities

Here, we extend theoretical studies that have examined theeffects of increasing biodiversity—for example, throughchanging species richness or evenness—on parasite dy-

namics (Norman et al. 1999; Holt et al. 2003; Dobson2004; Rudolf and Antonovics 2005). To quantify parasitefitness across host communities, we use the basic repro-duction number of the parasite in a community of hosts,R0, which has been used to identify conditions for in-creased and decreased parasite transmission in multihostdisease systems (Norman et al. 1999; Dobson 2004; Rocheet al. 2012; Joseph et al. 2013; Mihaljevic et al. 2014). Tocalibrate the effect of increasing host species richness,we additionally calculate the basic reproduction numberof the parasite in a single host species, Rj0. The advantageof using basic reproduction numbers to quantify parasitefitness is that they are epidemiological properties corre-lated with parasite incidence and prevalence and may be ex-pressed analytically for single- and two-species communi-ties. Moreover, the basic reproduction number describesthe initial growth rate of the parasite population in an en-tirely susceptible mono- or heterospecific host community,and consequently it is a measure of the propensity for out-breaks across different communities.

Model Formulation and Analysis

We consider a focal host species (species 1) that is a per-manent community resident and the introduction of a sec-ond alternative host species (species 2) to the communitysuch that parasite transmission occurs between the resi-dent and additional host species. We use the SIS frame-work as a general model for parasite transmission. By set-ting parameters to 0, it is easy to generalize this model bynoting that the form of the next-generation matrix used tocalculate community R0 is the same for models that makedifferent assumptions regarding immunity and reinfec-tion, such as SI, SIR, and SIRS systems (see the appendix).The SIS model for a two-species community can be writ-ten as

dS1dt

p ½b01 2 b11(N1 1a12N2)�N1 2X2jp1

b1j f (.)IjS1 2 m1S1 1 g1I1,

dI1dt

pX2jp1

b1j f (.)IjS1 2G1I1,

dS2dt

p ½b02 2 b12(N2 1a21N1)�N2 2X2jp1

b2j f (.)IjS2 2 m2S2 1 g2I2,

dI2dt

pX2jp1

b2j f (.)IjS2 2G2I2.

(1)

Parameters and variables of the model are listed in table 1.The model encompasses single- and two-species commu-nities, with the single-species model being achieved by

482 The American Naturalist

Table 1: Model parameters and formulas

Symbol/formula Meaning

Sj Number of susceptible individuals ofspecies j

Ij Number of infectious individuals ofspecies j

Nj p Sj 1 Ij Population size of species jm Number of species in the communityb0j Per capita natural birth rate of species jb1j Per capita density-dependent reduction in

birth rate of species jmj Per capita natural mortality rate of

species jrj p b0j 2 mj Per capita natural growth rate of species jKj p rj=b1j Carrying capacity of species jsj Susceptibility of species jij Infectiousness of species jpjk p sjik Transmissibility of the parasite from

species k to species jcjk Per capita contact rate between members

of species j and kbjk p pjkcjk Per capita transmission rate from a

member of species k to a member ofspecies j

gj Per capita recovery rate of species jdj Per capita disease-induced mortality rate

of species jGj pgj 1 dj 1 mj Per capita removal rate from the Ij classajk Competition coefficient (relative compet-

itive effect of species k on species j)ajk p 0 Interspecific competition absent0

setting symbols with subscript 2 to 0. For analytical trac-tability and to allow ease of comparison of this model withother published models (e.g., Getz and Pickering 1983;Holt and Pickering 1985; Dobson 2004; Rudolf and An-tonovics 2005; McCormack and Allen 2007), we assume thatdensity dependence arises via reduction in birth rates withincreasing host density. Density- and frequency-dependenttransmission modes are characterized by the function f(.).We assume that both intra- and interspecific transmissionoccur in the two-species system. The per capita transmissionrate bjk from species k to species j is the product of transmis-sibility of the parasite pjk and the number of contacts perunit of time per infected host k with susceptible host j, cjk.If transmission is density dependent, the per capita contactrate between hosts of species j and k scales linearly withcommunity size,

cjk(Nj,Nk)p cjk(Nj 1Nk),

and we assume that the force of infection exerted by spe-cies k hosts on species j hosts is bjkIk. If transmission is fre-quency dependent, the per capita contact rate betweenhosts of species j and k is constant,

cjk(Nj,Nk)p cjk,

and the force of infection is bjkIk=(Nj 1Nk). Thus, the forceof infection is proportional to the frequency of infectiousindividuals of species k relative to the total communityabundance. Intuitively, the proportion of infectious indi-viduals of species j (probability of contacting species jhosts) is reduced when an alternative host species k isadded to the assemblage, which may reduce disease trans-mission (encounter reduction, sensu Keesing et al. 2006).

Our framework can be adapted to flexibly account for dif-ferent encounter probabilities (appendix).

Heterogeneities in Host Contact Ratesand Transmissibility

Contact patterns—for example, through foraging, sexual,or antagonistic encounters—are key to the transmissionprocess. To describe the interspecific mixing patterns infigure 1 mathematically, we use the who-acquires-infection-from-whom (WAIFW) matrix (Anderson and May 1991),whose entries are composed of the per capita transmissionrates bjk,

Wp

�b11 b12b21 b22

�p

�p11c11 p12c12p21c21 p22c22

�,

noting that the per capita contact rate cjk in each entry isconstant for frequency-dependent transmission and equalto cjk(N1 1N2) for density-dependent transmission. In ad-dition to mixing between species, the WAIFW matrix con-veniently describes elements of host competence. Competenceis central to studies of parasite transmission in multispecieshost communities and embodies the ways in which differenthost species contribute unequally to parasite fitness. Compo-nents of competence include behavioral exposure (Hawleyet al. 2011), grooming of ectoparasites and vectors (Keesinget al. 2009), and parasite replication, shedding, and immuneactivation (Komar et al. 2003).We allow variation in compe-tence to be manifested in susceptibility (sj), infectiousness(ij), or both (fig. 2A). Here, we assume that transmissibilitypjk is composed of the product of susceptibility of host j

1 2

1 2

β21=ci1s2

β12=ci2s1 β22=c22i2s2β11=c11i1s1

2β21

β11

p11c11 p12c

p21c p22c22WS I=

1s1c11 i2s2c

2s1c i2s2c =

11 0

21 0W = =

1i1c11 0

s2i1c21 0

A

B

Figure 2: Schematics representing transmission networks for species 1 and 2. Circles represent species 1 (resident host) and 2 (alternativehost species), and arrows represent transmission rates within and between species. The corresponding who-acquires-infection-from-whommatrix is shown for each network. A, Transmission rates are composed of host susceptibility, infectiousness, and per capita contact rates.B, Spillover from the resident host (circle 1) to an alternative dead-end host for transmission (circle 2). Arrows indicate transmission rateswithin and between members of each population. Since species 2 is a dead-end host, no transmission occurs between members of this species.


and the infectiousness of host k. For example, if compe-tence is driven by susceptibility and the infectiousness ofeach host species in the assemblage is equal, then each en-try of the WAIFW matrix is

Wjk p sjicjk.

On the other hand, if transmission is driven by infectious-ness ik and the susceptibility of each host is equal to s, theneach entry of the WAIFW matrix is

Wjk p sikcjk.

To summarize the contributions of inter- and specifictransmission to parasite fitness, we introduce the ratio ofinter- to intraspecific transmission coefficients,

bpb12b21

b11b22.

Interestingly, b depends only on the ratio of interspecific(c12 p c21 p c) to intraspecific (cjj) contact rates,

bps1i2c#s2i1c

s1i1c11#s2i2c22p

c2

c11c22.

Thus, under this framework, differences in host contact ratesare the key determinant of the interspecific-intraspecifictransmission ratio, not parasite transmissibility heteroge-neities. Consequently, we assume that b is a function ofcontact only and hereafter refer to it as a contact ratio.If interspecific contact is weak (b< 1), then spillover ofthe parasite between species may occur infrequently, whereasstrong interspecific contact (b > 1) will facilitate spillover.The ratio of inter- to intraspecific transmission rates hasbeen shown to determine parasite establishment and per-sistence in some contexts (e.g., Bowers and Turner 1997;Holt et al. 2003; Begon et al. 2008). Since b summarizesthe relative degree of mixing between species, we considerthe following subcases for the contact ratio representingdifferent contact networks: (1) bp 1 (e.g., high degreeof mixing due to shared resource utilization; panel ii infig. 1); (2) b > 1 (e.g., species are highly territorial and en-counter other species more frequently than members oftheir own species); and (3) b< 1 (e.g., species with high so-ciality within groups). We will show that b is a key contrib-utor to predicted parasite fitness outcomes in multispeciescommunities.

Criterion for Comparing Parasite Fitness acrossMono- and Heterospecific Communities

To compare parasite fitness across mono- and hetero-specific communities, we compare the basic reproductionnumber of the parasite in a resident host (R10) to community

R0. We use equation (1) to calculate single-host- and two-species-community basic reproduction numbers (see theappendix for details and table 1 for expressions). In a two-species assemblage, community R0 is expressed analyticallyin terms of the resident and alternative hosts’ basic repro-duction numbers (R10 and R20; see table 1). For example, as-suming that a second host species is added to an assem-blage at the same abundance as the resident host, with thetwo species having equal interspecific competitive effectson one another, and additionally assuming that transmis-sion is density dependent, the community R0 for the two-species assemblage is expressed as

R0 p12

R10

11a1

R2011a

1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�R10

11a1

R2011a

�22 4

R10R20(11a)2

(12 b)

s !,

where a is the symmetric interspecific competition coeffi-cient, b is the ratio of inter- to intraspecific transmission, and

Rj0 pbjjKjGj

,

jp 1, 2, is single-species R0. This expression is derived in thesection titled “Density-Dependent Transmission and Intra-and Interspecific Competition” in the appendix, whereas theassumption of symmetric competition is relaxed in the sec-tion titled “Density-Dependent Transmission and Intraspe-cific and Asymmetric Interspecific Competition” in the ap-pendix. If competition is asymmetric, the community R0expression is different (table 1).We posit that the net outcome of community interac-

tions and increase in biodiversity is reduction of fitnessof the parasite in a heterospecific community relative toits fitness in a monospecific resident host community (adilution effect) if the following inequality is satisfied:

R0 R10 as anamplification effect.

Results

Table 2 summarizes the analytical conditions on the modelparameters for which relation (2) will hold for a two-speciescommunity and predicted outcomes for parasite fitness (seethe appendix for mathematical details). A dilution effect isnot a ubiquitous outcome (fig. 3). In all combinations oftransmission mode and competition that we examined, in-creasing species richness through adding a second host spe-cies to the assemblage may result in enhanced parasite fit-ness (i.e., R0 > R10). Since the addition of a second hostincreases overall population abundance, the combinationof density-dependent transmission and regulation by intra-


specific competition cannot lead to a reduction in parasite fit-ness (e.g., as shown in Dobson 2004), but a dilution effect maymanifest through all other transmission-competition combi-nations. Importantly, neither frequency-dependent transmis-sion nor susceptible host regulation via interspecific competi-tion guarantees reduction of parasite transmission.

The analytical conditions in table 2 for reduction of par-asite fitness in a two-host-species community relative toparasite fitness in a community consisting of a single focalhost species are characterized by the degree of contact be-tween species and host traits. Specifically, conditions forR0 pR10 are represented by a linear configuration of the ra-tio of the single-host basic reproduction numbers (j) andthe ratio of inter- to intraspecific contact coefficients (b).In addition to intraspecific contact rates, the ratio of basicreproduction numbers j captures species-level properties,such as carrying capacity, life span, recovery, and disease-induced mortality rates, thereby quantifying relative differ-ences in life-history traits (carrying capacity, life span, trans-mission competence) as well as effects of parasites on hostsurvival and morbidity (virulence, infectious period). Onthe other hand, the interspecific-intraspecific contact ratio bcharacterizes the relative contact rates between and withinspecies only. For brevity, we refer to trait ratio and trait effectsto emphasize that the ratio of the single-host basic reproduc-tion numbers j includes intrinsic host-species properties andeffects of parasites on hosts, whereas b simply refers to ecolog-ical contact rates.

The linear configurations in table 2 divide contact-traitparameter space into disjoint regions for which dilutionand amplification of parasite fitness manifest (fig. 3), withthe slope and intercept of each line determining the orien-tation. Above and to the left of each line in figure 3 are pa-rameter pairs for which a dilution effect occurs, whereasbelow and to the right of each line are parameter pairsfor which amplification in parasite fitness is the outcomein a two-species community.

When transmission is density dependent and popula-tions of each species are regulated by interspecific compe-

tition, resulting in fewer available susceptible hosts, the ex-pression arising from the dilution-effect criterion is

j >(12a12)b

a12(12a12a21)1

12a2112a12a21

. (3)

The relative competitive effect of the alternative host (spe-cies 2) on the resident host (species 1), a12, determines theparameter space for which a dilution effect manifests be-cause it is the dominating parameter in the slope term.As a12 increases, the magnitude of the slope declines (asdoes, to a lesser extent, the intercept), and the parametercombinations for which an amplification effect may man-ifest declines in area (compare fig. 3C with 3D). This sug-gests that as competition strengthens, a dilution effect maymanifest even if the additional host species possesses prop-erties that make it a more competent host for the parasite(R20 > R10). On the other hand, high interspecific contact(b > 1) is predicted to overwhelm the diluting effect ofadding a less competent host to the assemblage, and weakcompetition facilitates this amplification effect. The analyt-ical results generalize to multispecies communities exhib-iting density-dependent transmission and interspecific com-petition (appendix, figs. A2, A3; figs. A1–A3 are availableonline). In conclusion, interspecific competition, manifestedas an increase in the competitive effect of the additional hostspecies on the resident, facilitates a reduction in parasitetransmission for the resident host species.In contrast, if transmission is frequency dependent and

species are regulated through intraspecific competition,the relative abundances of each species at carrying capac-ity feature in the relationship:

j >K1b

K1 1K21

K2K1 1K2

. (4)

The slope depends on the relative abundance of the resi-dent host species, and the intercept is a function of the rel-ative abundance of the additional host species. Since therelative abundances of each host species sum to 1, there

Table 2: Predicted parasite transmission outcomes for directly transmitted microparasites in a two-host species assemblage

Density-dependenttransmission Dilution-effect criterion

Frequency-dependenttransmission

Dilution-effectcriterion

Additive (regulation byintraspecific competi-tion)

Amplification/nochange

. . . Amplification/dilution j > K1bK1 1K2 1K2

K1 1K2

Substitutive (regulationby interspecificcompetition)

Amplification/dilution j > (12a12)ba12(12a12a21)

1 12a2112a12a21 Amplification/dilution j >K1(12a12)b1K2(12a21)K1(12a12)1K2(12a21)

Note: Rows describe the means of regulation, and columns represent transmission mode. A row-column combination represents the predicted outcomes formodels with those combinations (eq. [1]). Each relationship is linear in jpR10=R20, and bpb12b21=b11b22. The analytical relationships arise from the criterionR0

is an inverse relationship between the slope and the inter-cept (compare fig. 3A with fig. 3B). Notably, an amplifica-tion effect is possible if a host species with a single-host re-production number that is less than that of the resident(hereafter, a diluting host species) and a lower carrying ca-pacity than the resident host species is added to the assem-blage, provided cross-species contact is sufficiently high(R10 > R20, K1 > K2; fig. 3A). On the other hand, an ampli-fication effect is predicted if the additional host species has

properties that enhance parasite transmission and its carry-ing capacity is greater than that of the resident, even if inter-specific contact is low (b< 1, R20 > R10, K2 > K1; fig. 3B). Therelative disease-free equilibrium abundances additionallyfeature in the relationship for frequency-dependent transmis-sion and interspecific competition between host species (ta-ble 2). Finally, we note that the predictions for frequency-dependent transmission result from the assumption thatthe force of infection depends on the frequency of infec-

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

trait

ratioσ

DE

AE K1 K2

AE

DE

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

trait

ratioσ

contact ratio b

DE

DE

DE

α = 0.95

A

B

C

D

Figure 3: Analytical relationships for combinations of frequency-dependent transmission and intraspecific competition and density-dependent trans-mission and interspecific competition. The horizontal axis in eachfigure represents the ratio of inter- to intraspecific transmission (contact ratio, b), andthe vertical axis describes the ratio ofR10 to R20 (trait ratio, j). The black lines represent parameter sets (b, j) for which community R0 equals the basicreproduction number of the parasite in the resident host, R0 pR10. These relationships separate parameter pairs for which a dilution effectmanifests (R0 R10; gray region below each line).Symbols representing parameter pairs that result in dilution (DE) and amplification (AE) are discussed in detail in the text. A, The black lineseparating parameter sets that result in amplification or dilution is a linear configuration resulting from frequency-dependent transmission, in-traspecific competition, and no interspecific competition (eq. [4]) when K1 K2. Here, K1 p 90 andK2 p 10. C, The black line delineating parameter pairs that lead to amplification or dilution is a linear configuration arising from density-dependenttransmission and symmetric interspecific competition, jp ½1=(11a)�½(b=a)1 1�. Here, ap 0.5 (moderate competition). D, A linear configura-tion arising from density-dependent transmission and strong symmetric interspecific competition. Here, ap 0.95.


tious hosts relative to the community abundance Ik=(Nj 1Nk). We show in the section titled “Generalizing the Disease-Diversity Relationship for Frequency-Dependent Transmis-sion” in the appendix that reduction of parasite transmissionfor the focal host will not arise unless the frequency of infec-tious hosts is reduced when a second species is added.

Relating the Analytical Conditions Back to the EcologicalContext (Fig. 1): Interspecific Contact, Host Traits,and Parasite Effects as Drivers of Transmission

Outcomes in Communities

The derived relationships shown in table 2 indicate thatthe ratio of single-species reproduction numbers, j (whichencompasses host traits such as reservoir competence andlife span), and b (which determines the degree of parasitespillover mediated by interspecific contact) are both key tomanifestation of a dilution effect for a focal host when aparasite is directly transmitted. Parameter pairs with j > 1represent communities with a resident species that is amore competent host for the parasite than the speciesadded to the assemblage by providing a more optimal en-vironment for the parasite to grow and spread in the ab-sence of alternative host species (since R j0 represents para-site fitness in a population consisting of a single hostspecies). Intrinsic host traits that lead to enhanced R10 rel-ative to R20 include longer life span or infectious periodof the resident than the additional host species and greatercarrying capacity if transmission is density dependent.Parasites with relatively low virulence and resident hostswith high transmission competence also lead to larger R10relative to R20. Alternatively, parameter sets with j< 1 indi-cate that the additional host species has properties thatmake it a more competent host for the parasite than thefocal species. In particular, we are interested in the effectof introducing a second host species to the assemblage thatis either a diluting host (i.e., j > 1) or an amplifying host(j< 1) on parasite fitness in a community.

Examining where the relationships shown in table 2 liein parameter space enables us to distinguish between trait-based (species-specific intrinsic properties that determineparasite fitness in a single host) and contact-based (ecolog-ical) mechanisms that determine parasite transmissionoutcomes in different contexts. Figure 4 is a schematicrepresenting the possible outcomes for community R0depending on combinations of b and j. The slope and in-tercept of the line, which are determined by transmissionmode and the influence of interspecific competition, affectthe range of parameters for which each outcome can man-ifest. Vertical movement away from the horizontal jp 1line represents host species becoming less alike in terms ofcarrying capacity, life span, competence, infectious period,and virulence. Using figure 4 as a guide, we will describe

what outcomes occur for low and high interspecific contactcases separately (under ecological contexts depicted in panels iand iii and in panels ii and iv, respectively, of fig. 1). We sum-marize the suite of outcomes for frequency-dependent trans-mission in an assemblage where each species is regulated byintraspecific competition in table 3 and the outcomes fordensity-dependent transmission where each species is reg-ulated by interspecific competition in table 4.

Model Predictions Assuming Low Interspecific Contact(b ≤ 1; Panels i and iii in Fig. 1)

A dilution effect is predicted if interspecific contact is weak(b≪ 1), the transmission competence of the additional hostspecies is weak relative to the resident (j > 1), and trans-mission is frequency dependent (figs. 3, 4). We term this di-lution effect a “trait- and contact-driven dilution effect” be-cause host and parasite traits, along with low contact betweenspecies, combine to reduce community R0 relative to R10.Counterintuitively, if the additional host provides higherparasite fitness than the resident (R10

resident (compare the cross symbol in fig. 3C with its coun-terpart in fig. 3D).

Model Predictions Assuming High Interspecific Contact(b ≥ 1; Panels ii and iv of Fig. 1)

In cases of high interspecific contact (e.g., panels ii and ivof fig. 1) and assuming frequency-dependent transmis-

sion, the addition of a second host species will result incontact- and trait-driven amplification effects if R10 R20, although this does not guaranteethe effect.

Table 3: Summary of conditions and potential transmission outcomes in different interspecific contact contexts (fig. 1) given thattransmission is frequency dependent and species are regulated by intraspecific competition

Contact ratioRegulationregime Analytical result Potential outcomes

b > 1 ii R10 > R20 is not a sufficient condition for dilution Trait-driven dilution effectContact-driven amplification effect

bp 1 ii R10 > R20 is a necessary and sufficient conditionfor dilution

Trait- and contact-driven amplificationeffect

0< b< 1 i R10 > R20 is a sufficient condition for dilution Contact-driven dilution effectTrait-driven amplification effectTrait- and contact-driven dilution effect

Figure 4: Schematic based on linear relationship for frequency-dependent transmission and intraspecific competition (bold line). Thedashed vertical line at bp 1 indicates the parameter pairs for which per capita interspecific contact rates equal per capita intraspecific contactrates. The dashed horizontal line at jp 1 indicates the parameter pairs for which parasite fitness as propagated by the resident species equalsthat propagated by the additional host species (R10 pR20). The schematic is relevant for frequency-dependent transmission (since jp 1 at bp 1;eq. [4]) but is also representative of transmission outcomes when interspecific competition is a component of community interactions. Thebold and dashed lines divide parameter space into regions for which different kinds of amplification and dilution effects can manifest. Blueregions indicate intuitive outcomes for the disease-diversity relationship, and red regions suggest counterintuitive outcomes. For example, in-troducing an alternative host species with low transmission competence of the parasite relative to that of the focal host, together with highsociality within species and a low interspecific contact rate, leads to a dilution effect, as expected (trait- and contact-driven dilution effect,in blue), but this type of social regime, combined with an additional species with transmission competence higher than that of the resident,could also lead to dilution (contact-driven dilution effect, in red), contrary to expectations.


Again, the relative abundance of the alternative host spe-cies is important to transmission outcomes. Greater abun-dances of diluting alternative hosts may counteract the am-plifying effect of strong interspecific contact, thereby drivinga net dilution effect mediated by traits of the host and para-site (triangle in fig. 3A). Less obviously, if the abundance ofthese additional hosts is less than the resident abundance, acontact-driven amplification effect is predicted (triangle infig. 3B) because of the similarity of parasite fitness in each hostspecies (R10 is only slightly greater than R20 in this scenario).As the intercept of the line (eq. [4]) increases, the parameterspace for trait-driven dilution effects increases in area (e.g.,triangle in fig. 3A; fig. 4).

Additionally, trait-driven dilution and contact-drivenamplification are possible outcomes if transmission is den-sity dependent and species are regulated by interspecificcompetition. In this context, the competitive effect of theadditional species on the resident, along with relatively poorcompetency of the additional species, combine to overcomethe potential for spillover via high contact between speciesand thereby drive a dilution effect (compare the outcomesfor the systems represented by the squares in fig. 3D [dilu-tion] with those in fig. 3C [amplification]). In the appendix,we show that contact-driven amplification effects generalizeto multispecies communities exhibiting interspecific compe-tition (fig. A2).

Special Case: Spillover of the Parasite to Dead-End Hosts

Many species novel to a parasite are capable of becominginfected but do not play a role in onward transmission(e.g., humans are “dead-end” hosts for directly transmitteddiseases such as rabies and hantavirus). Figure 2B shows atransmission regime between the focal host and an addi-

tional dead-end host species. Assuming that the additionalhost is not infectious (i.e., i2 p 0), the WAIFW matrix is

Wp

�b11 0b21 0

�p

�s1i1c11 0s2i1c21 0

�.

To investigate whether the dilution effect occurs, we com-pared the expression for community R0 derived from thenext-generation matrix to the basic reproduction number ofthe parasite in the resident species R10 for all transmission-competition combinations previously considered (table 5).A dilution effect occurs if transmission is frequency de-pendent and if hosts are regulated by interspecific competi-tion when transmission is density dependent. Addition ofa dead-end species to the community does not lead to am-plification effects in these contexts since the expressionsfor community R0 are a fraction of R10 (table 5).

Discussion

We have presented a simple tractable model to link host-species diversity, abundance, and parasite transmission tocompare the propensity for infectious disease outbreaksacross mono- and multihost communities. Adding more spe-cies can either increase or decrease parasite fitness, dependingon ecological context and the traits of the species in an assem-blage.We demonstrate that disease transmission outcomes inmultihost communities are more complex than expected, in-cluding the potential for disease amplification in communi-ties exhibiting frequency-dependent transmission and di-lution of outbreak risk in density-dependent transmissionsystems. Our approach offers important insights to otherstudies that have exploited robust empirical or phenomeno-

Table 4: Summary of conditions and potential transmission outcomes in different interspecific contact contexts (fig. 1) given thattransmission is density dependent and species are regulated by interspecific competition

Contact ratio Regulation regime Analytical result Potential outcomes

bp 1 iv R10=R20 > 1=a12 (necessary andsufficient)

Trait-driven dilution effectContact-driven amplification effect

b > 1 iv: weak competitive effectof species 2 on species 1(a12 → 0)

R10=R20 →∞⇒R10≫R20 Contact-driven amplification effectTrait- and contact-driven amplification effect

iv: strong competitive effectof species 2 on species 1(a12 → 1)

R10=R20 → 1⇒R10 > R20 Trait-driven dilution effectContact-driven amplification effectTrait- and contact-driven amplification effect

0< b< 1 iii: weak competitive effectof species 2 on species 1(a12 → 0)

R10 > R20 is not a sufficientcondition for the dilution effect

Trait-driven amplification effectContact-driven amplification effectTrait- and contact-driven amplification effect

iii: strong competitive effectof species 2 on species 1(a12 → 1)

R10 > R20 is not a sufficientcondition for the dilution effect

Contact-driven dilution effectTrait-driven amplification effectTrait- and contact-driven dilution effect


logical patterns, including the predictability of host commu-nity changes (Johnson et al. 2013) and relationships betweenspecies richness and abundance (Roche et al. 2012; Mi-haljevic et al. 2014). Our findings emphasize the importanceof contact rates, competition, and relative interspecies dif-ferences in parasite fitness in developing theory describ-ing the relationship between diversity and microparasiticdisease outbreak propensity, thus corroborating empiricalstudies that have shown these elements to be importantfor disease-diversity relationships (e.g., Clay et al. 2009; Hallet al. 2009; Johnson et al. 2013). Incorporating competitionand contact heterogeneities into simple models changes ourpredictions of how increasing species richness alters com-munity R0 (e.g., Dobson 2004; Rudolf and Antonovics 2005).

Our findings may be interpreted in the context of thedilution-effect hypothesis, which is broadly stated as the re-duction of parasite transmission in increasingly diverse hostcommunities, acknowledging that some researchers some-times refer to the particular risk of infection in one species,for example, Lyme disease in humans (Ostfeld and Keesing2000a, 2000b; Schmidt and Ostfeld 2001). Many empiricalstudies have indicated that there is an association betweenhigh biodiversity and reduced infectious disease risk, for ex-ample, Lyme borreliosis in small mammals and ticks (Lo-Giudice et al. 2003), West Nile virus in wild birds (Ezenwaet al. 2006) and mosquito vectors (Allan et al. 2009), hanta-virus in rodents (Suzán et al. 2009), trematodes in amphibians(Johnson et al. 2013), fungal pathogens of rice crops (Zhu et al.2000), and yellow barley virus in plants (Lacroix et al. 2014).However, reduction of infectious disease risk in diverse com-munities may be idiosyncratic and more likely determined byecological interactions between host species (Salkeld et al.2013). Simple mathematical models have been used to de-scribe hypothetical mechanisms for a dilution effect, such asencounter reduction and susceptible host regulationmediatedby nonhost species (Keesing et al. 2006), but not to study howdiluting mechanisms might reduce parasite transmission in amultihost community under a general framework with flexi-bility in transmission mode and dominant competitive forces.Our tractable models bridge this gap in theory, in particularby providing a theoretical basis for the effects of communitycomposition on the propensity for disease outbreaks inmulti-host assemblages. Our results suggest that reduction of para-

site transmission in multispecies communities will occur ifinterspecific contact rates are sufficiently low and species inthe assemblage differ substantially in their parasite trans-mission potential. Importantly, the propensity for disease out-breaks may be enhanced on adding additional host specieswhen transmission is either frequency or density dependent,and this propensity is host, parasite, and ecological contextdependent.We have shown that reduction of parasite transmission

in multihost communities may manifest for all contact andcompetition regimes considered. However, the predictedoutcomes for parasite transmission are due to different mech-anisms. Dilution effects driven by life-history traits such astransmission competence tend to manifest in high interspe-cific contact scenarios, for example, if there is high overlap inspecies’ resource acquisition functions or if species interactwith each other through antagonistic encounters. In general,hosts in an assemblage that compete with each other for alimiting resource must differ substantially in their parasitefitness for a dilution effect to manifest. On the other hand,dilution effects driven by low contact rates between speciestend to occur if species exploit different niches or are com-petitors that avoid one another. These are new theoreticallines of inquiry that could potentially be tested empirically.Our work challenges some of the key assumptions of the

dilution-effect hypothesis (Ostfeld and Keesing 2012); weshow that adding a less competent host to the assemblagedoes not inevitably lead to a dilution effect and that addinga more competent host does not invariably lead to amplifi-cation of disease risk. Our analysis suggests that it is possiblefor host traits to negate the effect of contact on transmissionoutcomes and for interspecific contact to neutralize the in-fluence of relative parasite fitness afforded by different hostspecies. Our findings lend theoretical support to empiricalstudies that have shown that species identity is key to thedisease-diversity relationship (LoGiudice et al. 2008; Salkeldand Lane 2010; Venesky et al. 2014). For example, the pres-ence of a noncompetent host has been shown to drive a di-lution effect in many systems (Johnson et al. 2008; Hall et al.2009; Keesing et al. 2009). On the other hand, the presenceof competent hosts may enhance parasite transmission insome cases (Power and Mitchell 2004; Hamer et al. 2011).Our findings suggest potential new avenues for empirical ex-

Table 5: Conditions that guarantee a dilution effect when a dead-end host for the infection isadded to the assemblage, where K1=(K1 1K2) is the abundance of susceptible resident hosts rel-ative to community abundance

Transmission mode and regulation regime Community R0 Result

Density dependent, intraspecific competition R01 No change

Frequency dependent, intraspecific competition K1R10

K1 1K2Dilution effect

Density dependent, interspecific competition (12a12)R1012a12a21

Dilution effect


ploration that may clarify mechanisms behind dilution andamplification of parasite transmission and may help refinethe predictions of the effect of species identity on disease riskin communities.

The generality of the hypothesis that biodiversity is pro-tective against disease has been questioned (Randolph andDobson 2012; Wood and Lafferty 2013). Our results eluci-date conditions under which dilution and amplification ofdisease transmission with increased species richness can beexpected in simple models. We demonstrate that both dilu-tion and amplification effects are possible if a diluting hostis added to the assemblage (i.e., the single-host reproductionnumber of the additional species is less than that of the resi-dent). Whether dilution or amplification is observed dependson ecological context (degree of contact and strength of com-petition). For example, a greater relative abundance of lesscompetent additional hosts is not sufficient to guarantee a di-lution effect in frequency-dependent transmission systems ifthe degree of interspecific contact is high (e.g., in marine orfreshwater assemblages). Ecological dynamics as well astraits of the host species and parasite effects all combine todetermine the fate of parasite transmission, even in simplemodels of two-host-species assemblages.

Naturally, the simple analytical model developed here hassome limitations. We compare parasite transmission acrossmultihost communities that are monospecific and hetero-specific rather than comparing how increasing diversity af-fects parasite transmission in a focal host. Parasite fitness ina community is measured using R0, a quantity capturing par-asite dynamics in single and multispecies communities thatis correlated with parasite prevalence. Infectious disease riskhas been measured by various means in empirical studies, in-cluding, for example, prevalence in the focal host, rate ofchange of infected hosts, prevalence in the community, den-sity of infected vectors, and prevalence of infected vectors(e.g., Mitchell et al. 2002; Johnson et al. 2008, 2013; Lo-Giudice et al. 2008; Clay et al. 2009; Salkeld and Lane 2010;Searle et al. 2011). We use R0 as a measure of disease riskacross communities, acknowledging that it may not directlymeasure disease risk in the focal host when a second speciesis added to the assemblage and that it measures outbreakpotential in naive host populations only.

Community ecology is more complicated than the simpleLotka-Volterra models that we use here to calculate com-munity R0. We ignore trophic levels above and below thelevel of the host species, which may control host speciesabundance via top-down and/or bottom-up effects (Keesinget al. 2006). Additionally, the structure of contact networks,host age structure, and the effects of demographic stochas-ticity are elements that affect the propensity for outbreaks(Newman 2002; Meyers et al. 2005; Dalziel et al. 2014) thatare not included in our simple models. We do not examinethe impact of correlations between life-history traits and

the effects of parasites on hosts (Keesing et al. 2010). Allof these complexities will impact transmission outcomes,underscoring the importance of initially studying simplemodels, which here lead to more nuanced predictions fordilution and amplification effects.In conclusion, predictions for the level of microparasite

transmission in communities cannot be made simply fromknowledge of parasite transmission mode or host regulatorymechanism. Metrics of biodiversity such as species richnessand evenness may fail to capture predictable outcomes ofcommunity assembly and disassembly on parasite fitness.We demonstrate the importance of contact rates, competi-tion, and relative interspecies differences in R0 on the pro-pensity for disease outbreaks in multispecies communities.Moreover, our analysis demonstrates that even simple ca-nonical models predict that directly transmitted parasitetransmission outcomes in community settings are likely todepend on the context in which the ecological dynamicsplay out. We recommend that these elements be includedas components of more complicated predictive models ofdisease-diversity relationships.

Acknowledgments

We thank T. Dallas, B. Dalziel, J. Haven, the associate edi-tor, and three anonymous reviewers for valuable commentson the manuscript. We also thank K. Carter for assistancein preparing figure 1. This research was funded by grant220020193 from the James S. McDonnell Foundation.S.M.O. completed this work while a postdoctoral fellow atthe National Institute for Mathematical and Biological Syn-thesis, an institute sponsored by the National Science Foun-dation (NSF) through NSF award DBI-1300426, with addi-tional support from the University of Tennessee, Knoxville.

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Associate Editor: C. Jessica E. MetcalfEditor: Susan Kalisz

“Vastness of size is so generally, and it may almost be conceded, so naturally associated in the popular idea with the whales, that some mayscarcely be able to realize at first the fact that there are species no larger than ordinary porpoises; and yet which agree so closely in all themore essential elements of structure with some of the whales, that it is impossible, in a natural system, to separate them from their giganticrelatives.” Pictured: Two views of the skull of an adult Physeter macrocephalus. From “The Sperm Whales, Giant and Pygmy” by TheodoreGill (The American Naturalist, 1871, 4:725–743).


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