Journal of Theoretical Biology 309 (2012) 47–57

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Journal of Theoretical Biology

0022-51

http://d

n Corr

E-m

amarase1 Pr

sity, Ral

journal homepage: www.elsevier.com/locate/yjtbi

The biological control of disease vectors

Kenichi W. Okamoto n,1, Priyanga Amarasekare

Department of Ecology and Evolutionary Biology, University of California, Los Angeles, CA 90095, USA

H I G H L I G H T S

c We compare how different vector biological control agents can reduce infections.c In general, parasitoids with high attack rates can suppress disease incidence.c Virulent pathogen biocontrol agents require a high transmission rate to be effective.c Disease incidence can be reduced permanently even if vector populations recover.c Inundating the system with a large number of competitors can reduce disease incidence.

a r t i c l e i n f o

Article history:

Received 31 October 2011

Received in revised form

3 April 2012

Accepted 21 May 2012Available online 31 May 2012

Keywords:

Infectious diseases

Indirect interactions

Vector-borne diseases

93/$ - see front matter & 2012 Published by

x.doi.org/10.1016/j.jtbi.2012.05.020

esponding author.

ail addresses: kokamoto@ucla.edu (K.W. Okam

k@ucla.edu (P. Amarasekare).

esent address: Department of Entomology, N

eigh, NC 27695, USA.

a b s t r a c t

Vector-borne diseases are common in nature and can have a large impact on humans, livestock and

crops. Biological control of vectors using natural enemies or competitors can reduce vector density and

hence disease transmission. However, the indirect interactions inherent in host–vector disease systems

make it difficult to use traditional pest control theory to guide biological control of disease vectors. This

necessitates a conceptual framework that explicitly considers a range of indirect interactions between

the host–vector disease system and the vector’s biological control agent. Here we conduct a

comparative analysis of the efficacy of different types of biological control agents in controlling

vector-borne diseases. We report three key findings. First, highly efficient predators and parasitoids of

the vector prove to be effective biological control agents, but highly virulent pathogens of the vector

also require a high transmission rate to be effective. Second, biocontrol agents can successfully reduce

long-term host disease incidence even though they may fail to reduce long-term vector densities. Third,

inundating a host–vector disease system with a natural enemy of the vector has little or no effect on

reducing disease incidence, but inundating the system with a competitor of the vector has a large effect

on reducing disease incidence. The comparative framework yields predictions that are useful in

developing biological control strategies for vector-borne diseases. We discuss how these predictions

can inform ongoing biological control efforts for host–vector disease systems.

& 2012 Published by Elsevier Ltd.

1. Introduction

Many infectious diseases are spread between hosts via anintermediary carrier (vector). The transmission rate of such adisease is intimately linked to the number of encounters betweenvectors and hosts, which in turn depends on the density of vectors(e.g., Klempner et al., 2007). Vector-borne diseases affect humans,livestock and crops, and thus the eradication of such diseases is ofgreat economic and public health concern. One approach ofcontrolling vector-borne diseases is to introduce biological

Elsevier Ltd.

oto),

orth Carolina State Univer-

enemies (biocontrol agents) of the vector. Biological control ofvectors is increasingly becoming recognized as a promising tool incontrolling a variety of disease pathogens, including well-knownhuman diseases such as malaria, chagas, trypanosomiasis andLyme disease (e.g., Kaaya and Munyinyi, 1995; Kaaya and Hassan,2000; Nelson and Jackson, 2006; Ostfeld et al., 2006; Samish andRehacek, 1999), crop diseases such as the tomato leaf curl virus inIndia and the cassava mosaic virus in sub-Saharan Africa (e.g.,Jeger et al., 2004; Otim et al., 2006), and diseases in naturalsystems such as the dutch elm disease in North American forests(e.g., Schelfer et al., 2008). Potential biocontrol agents of diseasevectors include predators (e.g., Nelson and Jackson, 2006), non-infective competitors (e.g., Blaustein and Chase, 2007 and refer-ences therein, and Moon, 1980), and infective pathogens of thevectors (e.g., Lecuona et al., 2001; Luz et al., 1998; Ostfeld et al.,2006). Moreover, vector control efforts based entirely on chemical

K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–5748

insecticide have often exacerbated disease incidence by selectingfor insecticide resistant vectors (e.g., Hemingway and Ranson,2000). Thus, the biological control of disease vectors offers anenvironmentally safe alternative to pesticide use in managingcostly or deadly vector-borne diseases.

Although biological control of vectors provides a potentiallyimportant tool for controlling vector-borne diseases, a successfulcontrol program requires a thorough understanding of the inter-actions between the host–vector disease community and thebiological control agents that attack the vector. Biological controlof herbivore pests by directly reducing pest density is extremelywell-studied from both theoretical (e.g., Levins, 1969; Hassell,1978; Hawkins and Cornell, 1999; Fagan et al., 2002; Murdochet al., 2003) and empirical (e.g., DeBach and Rosen, 1991; vanDriesche and Bellows, 1996; Pickett and Bugg, 1998; Hajek, 2004;Jervis, 2005) perspectives. In contrast, biological control of vectorswhich is aimed at decreasing disease incidence in the host, ratherthan pest density per se, is much less well studied (e.g., Weiser,1991; Hartelt et al., 2008). There are several reasons why theconclusions about the biological control of herbivorous pests maynot readily apply when the objective is reducing disease incidencein the host. First, biological pest control focuses on reducing pestdensity through the direct action of a natural enemy on the pest.Biological vector control, however, involves reducing diseaseincidence through the indirect action of a natural enemy thatattacks the vector rather than the disease organism. Given theindirect interactions inherent in vector-borne disease systemsand the resulting non-linear feedbacks, predictions about theefficacy of biological control based on direct pest-enemy interac-tions may not readily apply to host–vector disease interactions. Inparticular, reduced vector densities, rather than outright eradica-tion, can suffice to eradicate a disease and not a vector, so thestandard of comparison is different. Indeed, how the range ofdynamics between different control agents and vectors ultimatelytranslate into disease prevalance in the host is hard to predictwithout exploring the consequences of the indirect effects inher-ent in the host–vector disease interactions.

Previous theory on vector-borne diseases highlights theimportance of the indirect interactions between the vector’spopulation dynamics and disease prevelance in the host (e.g.,Ewald and de Leo, 2008; Fitzgibbon et al., 1995; Gourley et al.,2007; Holt et al., 1997; Jeger et al., 2004). Most of this work hasfocused on the dynamics of the host–vector disease systemwithout considering the dynamics of natural enemies of thevector. Two studies have investigated the use of a single biocon-trol agent of the vector to reduce disease incidence. Moore et al.(2010) examined how a predator of the vector affects theprevalence of a vector-borne disease in the absence of predatorsatiation; Gourley et al. (2007) investigated how pulsed applica-tion of biological larvicides or chemical insecticides on differentlife stages affected disease prevalence. Both studies focus on onetype of biological control agent and how the vector–agent inter-action affects infection rates in the primary host. No study to datehas compared whether different types of biocontrol agents (e.g.,predators, parasitoids, pathogens) can exert differential effects onthe same host–vector interaction. Here, we develop a comparativeframework to ask how a range of antagonistic interactionsbetween the vector and a potential control agent can indirectlyreduce the prevalence of a vector-borne disease in the hostpopulation.

A comparative analysis of different types of biocontrol agentsis key to elucidating the consequences of indirect interactionsthat are characteristic of vector-borne disease systems. It can alsohelp inform ongoing efforts to control vector-borne diseases.Different vector-borne disease systems involve diverse naturalenemies of the vectors, and thus the appropriate control

strategies could vary across systems (e.g., Weiser, 1991). More-over, by highlighting how different forms of ecological interac-tions can indirectly affect the vector-host–pathogen system, acomparative framework can help elucidate the role of suchindirect effects in community ecology more generally.

Here we use a common mathematical framework to comparehow different types of biological control agents reduce the inci-dence of vector-borne diseases. The novel aspect of this work is thecomparative approach, which allows us to make predictions aboutthe efficacy of different biological control agents (predator/para-sitoid, competitor, or pathogen of the vector) to reduce diseaseincidence. We discuss how these predictions can inform ongoingand prospective efforts to biologically control disease vectors.

2. Models

We focus on the biological control of vectors in single vector–single pathogen–single host systems. Such systems are expectedto predominate in agricultural vector-borne diseases of plants(e.g., Jeger et al., 2004; Otim et al., 2006), as well as non-zoonoticvector-borne diseases of humans (i.e., those diseases that cannotbe spread from non-human animals to humans), such as thedominant malaria pathogen Plasmodium falciparum (Woolhouseet al., 2001). Our goal is to understand the impact of introducingdifferent biological control agents into such a vector-borne dis-ease system.

The basic dynamics of a single vector–single pathogen–singlehost system has been studied by Holt et al. (1997), among others(e.g., Jeger et al., 1998, 2004). We use Holt et al. (1997)’s model asa starting point. Their continuous time model analyzed thedynamics of the African cassava mosaic virus, a pathogen ofcassava (Manihot esculenta) transmitted by the whitefly (Bemisia

tabaci Gennadius). Unlike many earlier models of vector-borneinfectious diseases (e.g., May and Anderson, 1979), Holt et al.(1997) explicitly modelled the vector population dynamics.

Following Holt et al. (1997), the dynamics of the host–vectordisease system are given by:

dS

dt¼ rðSþ IÞ 1�

Sþ I

K

� �þyI�fVS�dS, ð1aÞ

dI

dt¼fVS�ðyþdþaÞI, ð1bÞ

dU

dt¼ FðUþVÞ 1�

UþV

mðSþ IÞ

� ��cIU�oU, ð1cÞ

dV

dt¼cIU�oV , ð1dÞ

where S,I,U and V denote the densities of susceptible hosts,infected hosts, susceptible vectors, and infected vectors, respec-tively. In what follows, we will refer to the susceptible andinfected host populations (S and I, respectively) as ‘‘hosts.’’ Inthe absence of the disease, the host population grows at per-capita rate rð1�S=KÞ�d, where r is the host’s unconstrained per-capita birth rate, K is the carrying capacity of the host, and d is thehost’s per-capita death rate. The disease increases the host’s per-capita death rate by a; however, infected hosts recover at rate y,after which they become susceptible to the disease again.

Infectious vectors encounter and transmit the disease touninfected hosts at a rate f. Holt et al. (1997) assumed density-dependent transmission, which we retain because it allows us tocompare our results with previous studies (e.g., Gourley et al.,2007; Moore et al., 2010). Focusing on density-dependent trans-mission is a reasonable first step that allows us to keep the model

K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57 49

tractable while allowing the transmission rate to depend on hostand vector availability.

Following Holt et al. (1997) we assume that the diseasepathogen can multiply sufficiently quickly so that hosts andvectors that contract the disease become infectious immediately.Uninfected vectors encounter and contract the diseasefrom infectious hosts at a rate c. The disease transmits frominfected hosts to susceptible vectors at rate cIU, and thus thetransmission dynamics of the disease depend critically on thevector density.

The vector’s per-capita population growth rate consists of F,their unconstrained per-capita birth rate, m, the extent to whichvectors are limited by the host population, and o, the vectornatural death rate. Thus, the vector’s population growth rate is afunction of both total vector density and total host density. Themodel assumes the disease has no impact on the vector, and thatthe parameters are scalar constants (Table 1).

Holt et al. (1997) found that increasing the host carryingcapacity K and unconstrained growth rate r increased diseaseincidence. They showed that increased transmission betweenhosts to vectors (analogous to c and f) increased diseaseincidence only when the disease’s virulence (a) was moderateor high. They also found that allowing infectious hosts toreproduce or recover from infection stabilizes model dynamics(1). Indeed, extensive numerical integration we conducted acrossthe parameter space found no indication that Eq. (1) exhibitpersistent limit cycles. When the model did not converge to aninterior, disease-endemic equilibrium, the fraction of infected inour numerical simulations always decreased to zero or theprimary host went extinct. Thus, we expect the long-termbehavior of Eq. (1) to be stable over a wide range of parametervalues.

Holt et al. (1997) assumed that infected individuals either didnot reproduce or did not give birth to infected offspring. The maindifference in our model is that we allow infectious hosts to givebirth to susceptible offspring. We ignore vertical transmissionboth for analytical tractability, and based on the fact that itsrelative importance in the epidemiology of most vector-bornediseases remains unclear (e.g., Pherez, 2007).

Table 1Parameters of models (1)–(4).

Parameter Interpretation

d Per-capita natural mortality of host

o Per-capita natural mortality of the vector

a Infected host mortality rate (i.e., disease virulence)

f Rate at which an individual vector contacts an infected host

c Rate at which an individual vector gets infected from biting an

r Host per-capita birth rate

y Host recovery probability

m Vector self-limitation terma

F Vector intrinsic per-capita birth rate

Parameter Interpretation

q1 Effect of the competitor on the vector’s per-capita growth rate

q2 Scaled effect of the vector on the competitor’s per-capita growth ra

R The competitor’s unconstrained birth rate

m2 The competitor’s self-limitation term

mC The competitor’s death rate

A The per-capita attack rate of the parasitoid

ce The parasitoid’s conversion efficiency

h The parasitoid’s handling time

mP The parasitoid’s death rate

yv Recovery of rate of vectors infected with the vector-specific pathog

a The virulence of the vector-specific pathogen

b Transmission rate of the vector-control pathogen

a Holt et al. (1997) give the units for m as (host�1) instead of (vectors � host�1), b

Appendix B gives the equilibria of Eq. (1). The two importantequilibria are the disease-free boundary equilibrium (S%,0,U%,0)that allow us to specify the conditions for disease invasion inEq. (1), and the interior-equilibrium (S%, I%, U%, V%) that constitutesthe resident community which is invaded by the biocontrol agent.

We use Eq. (1) as a starting point for investigating the effectsof different biocontrol agents of the vectors on disease suppres-sion. We consider three different types of biocontrol agents: apredator/parasitoid of the vector, a non-infective competitor ofthe vector, and an infectious pathogen of the vector.

2.1. Parasitoid (or predator) biocontrol agent

Biological control frequently involves introducing a specialistparasitoid or predator that consumes the pest (e.g., Murdochet al., 2003). Although there are important biological differencesbetween predator and parasitoid biocontrol agents (e.g., Hassell,1978) the dynamics of their interactions with the host–vectordisease system is qualitatively similar. Hence, we consider bothunder a common modification of Eq. (1) involving an increase inthe vector’s per-capita mortality rate driven by the biologicalcontrol agent (e.g., Weisser and Hassell, 1996). For brevity, werefer to the predator or parasitoid biocontrol agent as ‘‘theparasitoid’’. The dynamics of the host–vector disease-parasitoidsystem is given by:

dS

dt¼ rðSþ IÞ 1�

ðSþ IÞ

K

� �þyI�fðV1þV2ÞS�dS, ð2aÞ

dI

dt¼fðV1þV2ÞS�ðyþdþaÞI, ð2bÞ

dU

dt¼ FðUþVÞ 1�

UþV

mðSþ IÞ

� ��cIU�oU�PqðUþVÞ

U

UþV, ð2cÞ

dV

dt¼cIU�oV�PqðUþVÞ

V

UþV, ð2dÞ

dP

dt¼ cePqðUþVÞ�mPP, ð2eÞ

Units Value in Holt et al. (1997)

day�1 0.003

day�1 0.12

day�1 0.003

day�1 0.008

infected host day�1 0.008

day�1 0.05

day�1 0.003

vectors � host�1 500

day�1 0.2

Units Range examined

Competitor�1 Varied

te Competitor�1 Varied

day�1 Varied

Competitor�1 Varied

day�1 Varied

day�1 Varied

Vector attacked �1 Varied

parasitoid�1 Varied

day�1 Varied

en day�1 Varied

day�1 Varied

vector�1 under density dependent transmission Varied

ut this would render their Eq. (10) to no longer be in units of (vectors � time�1).

K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–5750

where P is the density of the parasitoid biocontrol agent. Thefunction qð�Þ describes the per-parasitoid consumption rate (func-tional response) of the vector. We consider both Type I (qðxÞ ¼ Ax)and Type II (qðxÞ ¼ Ax=ð1þAhxÞ) functional responses, where A andh are the attack rate and handling times of the parasitoid,respectively. Type I functional responses describe the behaviorof voracious parasitoids with a very high vector saturationdensity. Type II functional responses are appropriate when thenumber of vectors consumed saturates relatively quickly to 1=h asvector density increases. Because a Type II functional responsecan introduce a delayed feedback in the parasitoid population(e.g., Gurney and Nisbet, 1998), comparing the two types offunctional responses can highlight how parasitoid biocontrolagents induce delayed negative density dependence in the vectorpopulation, and the effect of such delays on disease suppression.The parameter ce describes the conversion efficiency of theconsumed vectors into individual predators or parasitoids, whilemP characterizes the per-capita mortality rate of the control agent.

2.2. Pathogen biocontrol agent

We modify Eq. (1) to introduce a pathogen that infects thevector (but not the host) and reduces the vector’s per-capitagrowth rate. Infectious pathogens have a well-established historyas biocontrol agents (e.g. Herman, 1953; Hochberg, 1989; Sunet al., 2006), and the potential for vector pathogens to controlvector-borne diseases has begun receiving closer attention (e.g.,Doberski, 1981; Houle et al., 1987; Lecuona et al., 2001; Kanzokand Jacobs-Lorena, 2006; Schelfer et al., 2008). Here, we investi-gate when and how pathogens of the vector can help suppress adisease. We refer to the pathogen biocontrol agent as the ‘‘controlpathogen’’ or simply the ‘‘pathogen’’, to distinguish it from thedisease infecting the host that is being targeted for suppression(referred to as ‘‘the disease’’). The dynamics of a host–vectordisease pathogen system is given by:

dS

dt¼ rðSþ IÞ 1�

ðSþ IÞ

K

� �þyI�fðV1þV2ÞS�dS, ð3aÞ

dI

dt¼fðV1þV2ÞS�ðyþdþaÞI, ð3bÞ

dU1

dt¼ FðU1þV1þU2þV2Þ 1�

U1þV1þU2þV2

mðSþ IÞ

� �

cIU1�oU1þyvU2�U1bðU1,U2,V1,V2Þ, ð3cÞ

dU2

dt¼�cIU2�ðoþaÞU2�yvU2þU1bðU1,U2,V1,V2Þ, ð3dÞ

dV1

dt¼cIU1�oV1þyvV2�V1bðU1,U2,V1,V2Þ, ð3eÞ

dV2

dt¼cIU2�ðoþaÞV2�yvV2þV1bðU1,U2,V1,V2Þ: ð3fÞ

Vectors uninfected with the disease are partitioned into twogroups—vectors (U1) that are neither infected with the disease northe biocontrol pathogen, and vectors (U2) infected with thebiocontrol pathogen but not the disease. Similarly, vectors infectedwith the disease are partitioned into vectors (V1) infected with thedisease but not with the biocontrol pathogen, and vectors (V2)infected with both the disease and the biocontrol pathogen. Weassume that the vector contracts the control pathogen from otherinfected vectors which contract the control pathogen at rate bð�Þ.We consider both density-dependent (i.e., bðU1,U2,V1,V2Þ ¼

bDðU2þV2Þ) and frequency-dependent (bðU1,U2,V1,V2Þ ¼ bFU2þ

V2=U1þU2þV1þV2) transmission. Examples of density-depen-dent transmission include airborne pathogens, in which the

transmission rate is directly related to vector density. By contrast,sexually transmitted pathogens (e.g., Antonovics et al., 1995; Knelland Webberley, 2004) typically spread via frequency-dependenttransmission. Once infected, the vector recovers from its pathogen atrate yv and suffers an increased per-capita mortality rate a due tothe control pathogen’s virulence.

2.3. Competitor biocontrol agent

Finally, we ask whether a competitor biocontrol agent of thevector can reduce disease incidence in the host. Ideally, such anagent should be unable to transmit the disease. Although muchrarer than parasitoids or pathogen biocontrol agents, observedreductions in vector populations in the presence of inter-specificcompetitors have led some investigators to examine the potentialof such competitors as biocontrol agents. For example, in a reviewof responses of mosquito populations to competitors, Lounibos(2007) attributed larval resource competition from less effectivevector species as the key mechanism leading to a reduction invector population densities in several case studies. In someinstances, whole-scale local displacement of vectors may occur.For example, the chagas disease vector Triatoma infestans requiresa host-blood meal to reproduce. T. infestans competes with a lesseffective vector, the hemophagous congener T. sordida, and thetwo species are known to segregate spatially (Oscherov et al.,2004). However, the efficacy of such competitors as biocontrolagents and their impact on host disease incidence has not beenpreviously studied. Here we analyze the epidemiological conse-quences of introducing a non-vectoring competitor of the vector.The dynamics of a host–vector–competitor disease system aregiven by

dS

dt¼ rðSþ IÞ 1�

ðSþ IÞ

K

� �þyI�fðV1þV2ÞS�dS, ð4aÞ

dI

dt¼fðV1þV2ÞS�ðyþdþaÞI, ð4bÞ

dU

dt¼ FðUþVÞ 1�

UþVþq1C

mðSþ IÞ

� ��cIU�oU, ð4cÞ

dV

dt¼cIU�oV , ð4dÞ

dC

dt¼ RC 1�

Cþq2ðUþVÞ

m2ðSþ IÞ

� ��mCC, ð4eÞ

where C is the competitor’s density, R and mC are, respectively, itsper-capita exponential growth rate and death rate, and q1,q2 arethe per-capita, interspecific competitive effects of the competitoron the vector and vice versa. Because the competitor cannot itselfact as a secondary vector, as would a parasitoid biocontrol agent(model (2)), the primary host’s dynamics remain unchanged frommodel (1). Host availability can differentially affect the selflimitation of the vector and its competitor through m1 and m2.This yields a conventional logistic competition model (e.g.,Amarasekare, 2004) that describes inter- and intra-specific com-petition between the vector and its competitor. We assume thatthe hosts (or their by-products, e.g., household water containersin which mosquito larvae can grow) are the limiting resource forboth the vector and its competitor, and that competition is bothintra- and inter-specific.

3. Model analyses and results

An ideal biological control agent should exhibit three keyattributes. First, it should be able to establish when rare. Second,

K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57 51

it should be able to suppress recurring outbreaks of a vector-borne disease. Third, it should reduce the proportion of infectedindividuals in the host population when the disease is alreadyestablished. We applied a combination of analytical and numer-ical methods to non-dimensionalized versions (Appendix A) of themodels (1)–(4) to predict and compare the efficacy of the differentbiological control strategists according to these three criteria.

We derived invasion criteria for when each type of controlagent and the conditions under which competitors and para-sitoids exhibiting a Type I functional response can prevent diseaserecurrence. Numerical analyses were required to assess when thedisease-free equilibrium is locally stable when parasitoids with aType II functional response or pathogens are used as controlagents. We also used numerical analyses to determine the abilityof each control agent to reduce host disease incidence.

Parameters for our numerical analyses came from Holt et al.(1997)’s study of the African cassava mosaic virus disease. Thissystem is an excellent case study for several reasons. First, thevector-borne disease occurs in a relatively simple agriculturalsetting, which allows us to focus on a single vector and a singlehost species. Second, the disease is endemic in much of its range,and in the absence of biocontrol agents, models of this systemexhibit a stable interior equilibrium. Third, the white fly (Bemisia

tabaci), the mosaic virus vector, is a common agricultural pestwith many studies on its biological control. A range of organisms,including predators, parasitoids (Gerling et al., 2001) and patho-genic fungi (Faria and Wraight, 2001) have been proposed ascontrol agents. Research also exists on competition betweenBemisia tabaci and other herbivorous insects such as leafminers(Diptera: Agromyzidae), a confamilial whitefly (Trialeurodes

vaporariorum), the cabbage looper (Trichoplusia ni), as well asspider mites (reviewed in Inbar and Gerling, 2008). Thus, thecassava–whitefly–cassava mosaic virus system presents an

Table 2Predictions for the successful establishment of the biocontrol agent.

Biocontrol agent Target system

Parasitoid with Type 1 functional response Vector population in the absence of cont

Control agent conversion efficiency high

Control agent conversion efficiency low

Control agent mortality high

Control agent mortality low

Parasitoid with Type II functional response Same results as a parasitoid with a Type

Handling time large relative to lifetime o

Handling time small relative to lifetime

Pathogen with frequency-dependent

transmission

Recovery of vectors from control pathog

Mortality of vector or control pathogen v

Mortality of vector and control pathogen

infection rate

Pathogen with density-dependent

transmission

Similar results as a pathogen spread thro

well as:

Uninfected vectors highly abundant

Uninfected vectors rare

Competitor of the vector The ratio of hosts to vectors is large

Competitor has high birth rate or low m

Vector has a strong per-capita effect on

attractive model system to compare the impact of differentbiocontrol agents on epidemiological dynamics.

3.1. When will a biological control agent of the vector become

established when rare?

In practice, only a small number of individuals of a controlagent are released initially because of logistical challenges andthe cost involved (e.g., Walter and Campbell, 2003). In suchsituations, demographic stochasticity and Allee effects can pre-vent the control agent from becoming established. We assessedthe establishment success of a biocontrol agent by identifyingthe conditions under which it can increase from initially smallnumbers (see Appendix C for details). The key results are givenin Table 2, and the derivations are given in Appendix C. Animportant point to appreciate is that the equilibrium hostdensity determines the successful establishment of a competitorbiocontrol agent but not that of a predator or pathogencontrol agent

Regardless of the functional response (Type I and Type II),parasitoids that are more effective at exploiting the vector (i.e.,through increased attack rates or shorter handling times) aremore likely to become established. By contrast, highly virulentpathogens with moderate or low transmission rates quickly killinfected vectors and hence fail to become established. If thebiocontrol pathogen is spread through frequency-dependenttransmission, it can establish only if its transmission rate exceedsthe vector’s recovery rate and the biocontrol pathogen’s virulence.If the biocontrol pathogen is spread through density-dependenttransmission, even biocontrol pathogens with low transmissionrates can still become established provided the susceptible vectorpopulation is sufficiently high.

Expected results

rol agent very abundant Parasitoid successfully

established

Parasitoid successfully

established

Parasitoid cannot become

established

Parasitoid cannot become

established

Parasitoid successfully

established

1 functional response, as well as:

f control agent Parasitoid cannot become

established

of control agent Parasitoid successfully

established

en is lower than infection rates Pathogen successfully

established

irulence greater than infection rate Pathogen cannot become

established

virulence, as well as recovery rate, less than Pathogen successfully

established

ugh frequency-dependent transmission, as

Pathogen successfully

established

Pathogen cannot become

established

Competitor successfully

established

ortality rate Competitor successfully

established

the competitor’s growth rate Competitor cannot become

established

K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–5752

In the next section, we ask when a vector borne disease caninvade a disease-free vector–host–biocontrol agent system. Weuse analytical and numerical results based on the African cassavamosaic virus disease.

3.2. When does a biocontrol agent prevent a vector-borne disease

from becoming endemic?

An effective biological control agent ideally maintains diseaseincidence at low levels and prevents the vector-borne diseasefrom becoming endemic in the face of reintroduction of thedisease (due to, for example, immigration of infectious vectorsBryant et al., 2007). We consider a biocontrol agent as success-fully preventing a disease from becoming endemic if the disease-free equilibrium density of the vectors, hosts, and biocontrolagent is feasible and locally stable.

Factors that influence the stability of the disease-free bound-ary equilibrium of Eq. (1) are key to understanding the conditionsfor successful disease suppression. Provided the disease-freeequilibrium is feasible, it is locally unstable (R041) if

Fig. 1. Stability diagrams for the disease-free equilibrium in the presence of a biocontro

disease-free equilibria, white regions depict unstable disease-free equilibria (i.e., the dis

is not feasible. The parameter values have been rescaled (Appendix A) to facilitate comp

in Holt et al. (1997) (Table 1), as well as: (A—competitors) mc=r¼ 1,R=F ¼ 1, (B—para

density-dependent transmission) yv=r ¼ 0:3. High vector self-limitation improves the ab

vector’s effect on its competitor is normalized by the vector’s self-limitation term). W

possible when the parasitoid’s scaled handling time is low and vector’s self limitation is

(C), as long as the vector’s self limitation is above a certain threshold the pathogen can

through frequency-dependent transmission, a higher transmission rate can stabilize th

m4FoðdþaþyÞ=frðd1Þ2ðF�oÞc (Appendix B). The vector’s self-limitation is key because it determines (i) the configuration of thehost–vector disease system in the absence of a biocontrol agent(e.g., Holt et al., 1997) and (ii) vector density, which in turndetermines the success of natural enemies such as pathogens andparasitoids (e.g., Arneberg et al., 1998; Murdoch et al., 2003).Indeed, when the vector’s self-limitation (m) is strong, vectordensities are low and the disease may not readily spread oremerge even in the absence of a control agent. We thereforeinvestigated the conditions under which the biocontrol agentcould prevent disease emergence when the vector’s self-limita-tion is weak using parameter values from previous studies of thecassava mosaic virus disease.

For parasitoid control agents exhibiting a Type I functionalresponse, a threshold conversion efficiency exists below whichthe parasitoid consistently fails to prevent the disease frombecoming endemic. We found that even parasitoids with conver-sion efficiencies above this threshold may fail when the vector’sself-limitation is sufficiently strong. In these cases, the vectorpopulation at the disease-free equilibrium is too small to support

l agent. The black regions describe regions of the parameter space leading to stable

ease can invade), and the light or dark grey regions mean the boundary equilibrium

arison. Other parameter values not depicted in a given panel are the same as those

sitoids with a Type II functional response) m=r¼ 5,c¼ 1, and (C—pathogens with

ility of competitors to keep the disease from becoming endemic (1A; the re-scaled

hen the parasitoid exhibits a Type II functional response (B), suppression is only

strong. If the control pathogen is spread through density-dependent transmission

prevent the disease from becoming endemic, while if the control pathogen spreads

e disease-free equilibrium even if the vector’s self limitation is weak.

K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57 53

a large parasitoid population. Simililarly, when the vector’s self-limitation is strong, weak competitors of the vector could also failto prevent the disease from becoming endemic (Appendix D).

Numerical analyses show that parasitoid biocontrol agentswith a Type II functional response require very short handlingtimes when vector self-limitation is weak in order to keep thedisease from becoming endemic (Fig. 1B). Also, biocontrol patho-gens can prevent a disease from becoming endemic as long as thevector’s self limitation is above a certain threshold for the case ofdensity-dependent pathogens (Fig. 1C). By contrast, we foundfrequency-dependent pathogens can prevent the disease frombecoming endemic with a high transmission rate even if thevector has very weak self-limitation (Fig. 1D).

3.3. How do different biocontrol agents reduce the incidence of an

endemic, vector-borne disease? A case study using the African

cassava mosaic virus disease

Perhaps the strongest impetus for the biological control of avector occurs when a vector-borne disease has already becomeendemic in the host–vector disease system. To quantify a controlagent’s effect on disease incidence, ideally one would compareequilibrium disease incidence in the host before and after theintroduction of a biocontrol agent. We did this by numericallyintegrating (2)–(4). We considered two different implementationregimes: (i) when a small number of biocontrol agents (1% of thevector density) are released initially into the system and (ii) when

Con

vers

ion

effic

ienc

y

Scaled parasitoid mortality rate

Con

vers

ion

effic

ienc

yC

onve

rsio

n ef

ficie

ncy

Con

vers

ion

effic

ienc

y

Scaled parasitoid mortality rate

Fig. 2. The ratio q of disease incidence with and without the biocontrol agent for parasi

well as the vector density for parasitoids with (A) a Type I functional response. Parame

et al. (1997) (Table 1) as well as mP ¼ 3 in panels (B,D). For all types of parasitoids, stron

(A–D), but does not require vector extinction. The scatter of points in panels (B,D) resu

a large number of biocontrol agents are released (same level asthe vector population density).

We quantified the degree of disease suppression (q) as

q ¼I%

c =ðS%

c þ I%

c Þ

I%

0=ðS%

0þ I%

0Þ, ð5Þ

where the numerator and denominator are, respectively, theproportion of infected hosts in the presence of and in the absenceof the biocontrol agent. We calculated q based on host populationdensities after 15,000 time steps. The integrations were periodi-cally checked to see if either the scaled density of vectors (UþV)or the biocontrol agent was below 2E or E (where E is the machineepsilon), respectively, or the density of infected individuals inboth the host and vector populations (i.e., I þ V) was below 2E.This allowed for the possibility of extinction at low density due todemographic stochasticity. If any of these conditions held, thedisease (or the biocontrol agent) was considered extinct and thesimulations terminated. We decided to use a conservative mea-sure of extinction because complete eradication is neither logis-tically or biologically likely (Tang et al., 2005), and because undersome parameter ranges the nonlinearities in models (1)–(4) canpotentially allow a population to recover from low densities. Allnumerical integrations were carried out using the NDSolveroutine in Mathematica 8 with E¼ 2�52.

We found that eradicating the whitefly vector was not aprerequisite for reducing disease incidence in the cassava plant(Figs. 2–4). For instance, parasitoids exhibiting a Type II functionalresponse could still reduce infection rates in the cassava plant if

Scaledvector density

Scaled handling time

Scaled handling time

toids with (A) a Type I functional response and (B) a Type II functional response, as

ter values have been re-scaled as in Fig. 1 and other parameters are based on Holt

g disease reduction is facilitated by highly efficient predators with low mortalities

lt from the deterministic fluctuations in vector density.

q

Scaledvector density

Virulence

Tran

smis

sion

rate

Tran

smis

sion

rate

Tran

smis

sion

rate

Tran

smis

sion

rate

Virulence

VirulenceVirulence

^

Fig. 3. q for pathogens with (A) density- and (B) frequency-dependent transmission, as well as the vector density for pathogens with (C) density- and (D) frequency-

dependent transmission. Parameter values have been re-scaled as in Fig. 1 and other parameters are based on Holt et al. (1997) (Table 1) as well as yv=r¼ 0:3. For both

frequency- and density-dependent biocontrol pathogens, when the scaled transmission rate b is low, pathogens with intermediate virulence more successfully eradicate

the vector-borne disease. However, pathogens with density-dependent transmission could cause substantial reductions in disease incidence without a correspondingly

large reduction in vector density, but pathogens which spread through frequency-dependent transmission, the vector’s density had to be reduced considerably before

disease incidence in the host also decreased.

K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–5754

the handling time was sufficiently small, despite high mortalityrates (lower-left hand corner of Fig. 2D). This occurred becausethe non-linear functional response causes oscillations in vectorand parasitoid abundances, which in turn cause infection rates inthe host plant population to decline as vector population densitiescycle through troughs. However, if the parasitoid has a shorthandling time and high conversion efficiency, it can go extinctbefore the vector does.

The biocontrol pathogen’s mode of transmission affects itseffects on the cassava mosaic virus prevelance in the host. If thebiocontrol pathogen spreads through density-dependent transmis-sion, it can drive the cassava mosaic virus extinct without causingvector (whitefly) extinction (Fig. 3A and B). If the biocontrolpathogen spreads through frequency-dependent transmission, itsgrowth rate is no longer strongly coupled to vector density.Substantial reductions in disease incidence in cassava thereforeoccur concomitantly with substantial reductions in whitefly popu-lations (Fig. 3C and D). Competitors could also cause substantialreductions in cassava mosaic virus prevalence in the cassavapopulation without causing vector extinction (Fig. 4A and B).

When the biocontrol agent is a parasitoid or a pathogen, theseresults are unaffected by whether one inundates the system with acontrol agent or releases a small number. When the biocontrolagent is a competitor of the whitefly, the initial biocontrol agentdensity strongly affects the ultimate success or failure of biologicalcontrol. Inundating the system with competitors could successfullyreduce disease incidence among cassava hosts as long as suchcompetitor’s negative effect on the vector was strong (Fig. 4A and C,

upper right corners). Indeed, when interspecific effects are strong,a priority effect occurs whereby the whitefly, by virtue of itsnumerical advantage, can exclude the competitor and reachequilibrium, thus preventing a reduction in infection rates. How-ever, this numerical advantage diminishes if the competitor alsois at high density and has a strong negative effect on the whitefly.

4. Discussion

While reducing the encounter rate between hosts and vectorsby altering vector or host behavior can lower the transmissionrates of vector-borne diseases (e.g., Jeger et al., 2004), reducingvector densities also has the potential to lower the transmissionrate, and, ultimately, the incidence of the disease (Klempner et al.,2007). Increasingly, the use of natural enemies of the vector isseen as a possible strategy to reduce vector densities, and therebysuppress the disease (e.g., Blaustein and Chase, 2007; Kaaya andHassan, 2000; Lecuona et al., 2001; Luz et al., 1998; Nelson andJackson, 2006; Ostfeld et al., 2006).

A great deal is known about the use of biological control tosuppress herbivorous pests, but relatively little is known aboutusing biological control to suppress pathogens causing vector-bornediseases. Here we have developed a theoretical framework tocompare the efficacy of three types of biological control agents (acompetitor, a predator or a parasitoid, and an infectious pathogen)in controlling vector-borne infectious diseases. We have identifiedthe conditions under which different types of biocontrol agents

Effect of vector on competitorEffect of vector on competitor

Effect of vector on competitor

Effe

ct o

f com

petit

or o

n ve

ctor

rela

tive

to v

ecto

r’s se

lf lim

itatio

n

Effe

ct o

f com

petit

or o

n ve

ctor

rela

tive

to v

ecto

r’s se

lf lim

itatio

n

Effe

ct o

f com

petit

or o

n ve

ctor

rela

tive

to v

ecto

r’s se

lf lim

itatio

n

q q

Scaled vectordensity

Fig. 4. q for competitors released at 1% of the vector’s density (A) and 100% of the vector’s density (B), as well as the vector density for the 1% release ratio (C). Parameter

values have been re-scaled as in Fig. 1 and other parameters are based on Holt et al. (1997) (Table 1) as well as F ¼ RC and mC ¼o.

K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57 55

could successfully become established, prevent disease endemicity,and reduce disease incidence. We believe our work can serveas a point of departure for researchers to develop system-specificmodels aimed at assessing the efficacy of specific biocontrolscenarios.

Existing theory on the biological control of herbivorous pestssuggests that the stability of the pest-natural enemy interactioncomes at the cost of higher pest density (the ‘‘stability-suppres-sion trade off’’ - e.g., Murdoch et al., 2003). This suggests that atrade off between suppression and stability may not be asimportant for the biological control of vectors as it is for thebiological control of herbivorous pests. Parasitoid biocontrolagents with short handling time, high fecundity, and short lifespan are an important exception. When vector abundances arelow, such agents can go extinct before the vector population hastime to recover.

Previous work has also recognized that the biological controlof vectors can reduce disease incidence in primary hosts withoutcausing vector eradication (Gourley et al., 2007; Moore et al.,2010). However, these studies focus on only one type of biologicalcontrol agent (e.g., biological larvicide in Gourley et al., 2007 orpredators with a Type I functional response in Moore et al., 2010).The novelty of our approach is the comparative analysis ofdifferent types of biocontrol agents, thus identifying the condi-tions under which predators, competitors or pathogens of thevector can successfully control a disease. Our key result is that theability of the biocontrol agent to reduce the fraction of infectioushosts without reducing the vector’s long-term density dependscrucially on the nature of the interaction between the vector andthe biocontrol agent. For some potential biocontrol agents, eventemporarily or moderately reducing vector density dramaticallyreduced disease incidence (e.g., parasitoids and pathogens with

density-dependent transmission). Yet for other potential biocon-trol agents (e.g., pathogens with frequency-dependent transmis-sion), substantially reducing long-term vector density wasrequired to reduce disease incidence in the host. Our comparativeframework thus allowed us to identify the conditions whenbiocontrol agents could reduce disease incidence in the hostwithout strongly reducing vector density.

Our results have important implications for the ongoing effortsto reduce disease incidence via the biological control of vectors.For instance, Hoddle et al. (1998) show how different types offunctional responses influence the biological control of a diseasevector. The hymenopteran parasitoid Encarsia formosa Gahanparasitizes the whitefly (Bemisia argentifoli Bellows and Per-ring¼Bemisia tabaci Gennadius strain B) in greenhouses growingornamental plants. Bemisia argentifoli is closely related to thewhitefly vector of the African cassava mosaic disease studied byHolt et al. (1997). Hoddle et al. (1998) demonstrated that thefunctional response of E. formosa was saturating (Type II) in smallcanopies of plants, but approximately linear (Type I) in largecanopies. If parasitoids such as E. formosa are used to control thewhitefly and thus reduce incidence of the African cassava mosaicvirus, then whether a cassava plot is characterized by large orsmall canopies, which in turn determines whether the functionalresponse of E. formosa is linear or saturating, affects whatadditional steps planters could take to facilitate suppressing thevirus. For example, if the plot has large canopies the functionalresponse of the parasitoid is likely to be linear, and therefore aparasitoid control agent with low conversion efficiency andmortality can become established regardless of the whitefly’sself-limitation. In plots with smaller canopies, the functionalresponse of the parasitoid to B. tabaci may saturate, in whichcase effects of the parastioid’s handling time and conversion

K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–5756

efficiency on disease suppression will depend crucially on thevector’s self-limitation.

Our findings also have implications for the role of pathogens incontrolling vector populations. While competitors and parasitoidbiocontrol agents that directly reduce the vector’s per-capitagrowth rate can suppress disease incidence in the host, highlyvirulent pathogens must have a correspondingly high transmis-sion rate to be effective biocontrol agents. For instance, both theentomopathogenic fungi Beauveria bassiana and Metarhizium

anisopliae are transmitted through physical contact betweenindividual tsetse flies (Glossina morsitans), vectors of trypanoso-miasis (Kaaya and Okech, 1990). Conducting experiments todetermine whether transmission of M. anisopliae and B. bassiana

is frequency- or density-dependent (e.g., Greer et al., 2008 andreferences therein), and quantifying the transmission rate, willhelp guide how much public health officials should conjoinbiological control with additional measures. For instance, iftransmission of these entomopathogens is density-dependenthighly virulent control agents may actually be less successful atpreventing endemic disease than less virulent control agents.However, if transmission is frequency-dependent, then controlpathogens with higher virulence are likely to be more successfulat disease suppression.

Our work shows that releasing competitors in large numberscan potentially compensate for their reduced per-capita effects onvectors. For instance, even when the biocontrol agent is competi-tively inferior to the vector, releasing a sufficiently large numberof such competitors may achieve satisfactory control.

In conclusion, we have presented a comparative analysis ofbiological control agents, with testable predictions about theconditions under which different types of agents can successfullycontrol a vector-borne disease. Our mathematical frameworklends itself to modifications that allow for investigating vectorcontrol in specific host–vector disease systems. For instance, wehave assumed density-dependent transmission both to enablecomparisons with previous studies and to ensure analyticaltractability. Transmission between the vector and the host maynot be strictly density-dependent in many systems (e.g., Wonhamet al., 2006). Departures from density-dependent transmissioncan affect the dynamics of model (1). For example, in vector–hostdisease models when transmission is frequency-dependent, R0

depends on the ratio of vectors to hosts (Wonham et al., 2006).When transmission is density-dependent vector abundance andhost abundance have additive effects on R0. Investigating theeffects of different modes of transmission on vector biologicalcontrol is an important future direction.

Following previous studies (e.g., Gourley et al., 2007; Holtet al., 1997; Moore et al., 2010), we have also consideredinstantaneous infectivity of host and vectors. The presence ofexposed but latent hosts and vectors as well as susceptible andinfectious individuals is quite common in many vector-bornediseases, especially when the incubation period can be longcompared to the lifespan of the vector (e.g., Hosack et al., 2008).The presence of non-infectious disease carriers could affect theepidemiological dynamics in the absence of the vector-controlagent. For example, Chitnis et al. (2008) illustrate that decreasingthe rate at which vectors progress from initial exposure tobecoming infectious can reduce the endemic equilibrium anddisease prevelance in the host. Incorporating an incubation periodcan potentially alter properties such as the transitory duration ofa disease (Hosack et al., 2008). These factors need to be con-sidered when the model is extended to include periodic immigra-tion of disease organisms from outside the community or theexistence of reservoir host populations. Infectious immigrantscould potentially cause future outbreaks even if the biocontrolagent help maintain the local stability of the disease-free

equilibrium, and whether such outbreaks occur can depend onthe time-lag between exposure of the vector to the disease andthe vector becoming infectious (Hosack et al., 2008).

An important future direction is to incorporate greater biolo-gical realism into the framework developed here. These includespatial heterogeneity within sites that could allow for vector-natural enemy coexistence (e.g., Levins, 1969; Hassell, 1978;Woolhouse et al., 2001; de Castro and Bolker, 2005; Acthman,2008; Gardiner et al., 2009), simultaneously releasing differenttypes of biocontrol agents, and genetic variation in vector traitsthat could lead to the evolution of resistance against biocontrolagents (e.g., Holt and Hochberg, 1997; Roderick and Navajas,2003). These processes have been thoroughly investigated instudies of biologically controlling herbivorous pests, and theyhold similar promise for further investigations of the biologicalcontrol of disease vectors.

Acknowledgments

This research was funded by grants from the Systems andIntegrative Biology Training Grant from the National Institute ofHealth in the United States to the Department of Biomathematicsat the University of California, Los Angeles, as well as a Chair’sFellowship from the Department of Ecology and EvolutionaryBiology at the University of California, Los Angeles. We would liketo thank R. Vance, G. Grether, and two anonymous reviewers fortheir valuable comments on earlier versions of this manuscript.

Appendix A. Supplementary data

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org.10.1016/j.jtbi.2012.05.020.

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