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Journal of Theoretical Biology 309 (2012) 47–57
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Journal of Theoretical Biology
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
c
c
c
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We compare how different vector biological control agents can reduce infections.
In general, parasitoids with high attack rates can suppress disease incidence.
Virulent pathogen biocontrol agents require a high transmission rate to be effective.
Disease incidence can be reduced permanently even if vector populations recover.
Inundating the system with a large number of competitors can reduce disease incidence.
a r t i c l e i n f o
abstract
Article history:
Received 31 October 2011
Received in revised form
3 April 2012
Accepted 21 May 2012
Available online 31 May 2012
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.
Keywords:
Infectious diseases
Indirect interactions
Vector-borne diseases
1. Introduction
Many infectious diseases are spread between hosts via an
intermediary carrier (vector). The transmission rate of such a
disease is intimately linked to the number of encounters between
vectors 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 of
great economic and public health concern. One approach of
controlling vector-borne diseases is to introduce biological
n
Corresponding author.
E-mail addresses: [email protected] (K.W. Okamoto),
[email protected] (P. Amarasekare).
1
Present address: Department of Entomology, North Carolina State University, Raleigh, NC 27695, USA.
0022-5193/$ - see front matter & 2012 Published by Elsevier Ltd.
http://dx.doi.org/10.1016/j.jtbi.2012.05.020
enemies (biocontrol agents) of the vector. Biological control of
vectors is increasingly becoming recognized as a promising tool in
controlling a variety of disease pathogens, including well-known
human diseases such as malaria, chagas, trypanosomiasis and
Lyme disease (e.g., Kaaya and Munyinyi, 1995; Kaaya and Hassan,
2000; Nelson and Jackson, 2006; Ostfeld et al., 2006; Samish and
Řeháček, 1999), crop diseases such as the tomato leaf curl virus in
India and the cassava mosaic virus in sub-Saharan Africa (e.g.,
Jeger et al., 2004; Otim et al., 2006), and diseases in natural
systems such as the dutch elm disease in North American forests
(e.g., Schelfer et al., 2008). Potential biocontrol agents of disease
vectors include predators (e.g., Nelson and Jackson, 2006), noninfective competitors (e.g., Blaustein and Chase, 2007 and references therein, and Moon, 1980), and infective pathogens of the
vectors (e.g., Lecuona et al., 2001; Luz et al., 1998; Ostfeld et al.,
2006). Moreover, vector control efforts based entirely on chemical
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K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57
insecticide have often exacerbated disease incidence by selecting
for insecticide resistant vectors (e.g., Hemingway and Ranson,
2000). Thus, the biological control of disease vectors offers an
environmentally safe alternative to pesticide use in managing
costly or deadly vector-borne diseases.
Although biological control of vectors provides a potentially
important tool for controlling vector-borne diseases, a successful
control program requires a thorough understanding of the interactions between the host–vector disease community and the
biological control agents that attack the vector. Biological control
of herbivore pests by directly reducing pest density is extremely
well-studied from both theoretical (e.g., Levins, 1969; Hassell,
1978; Hawkins and Cornell, 1999; Fagan et al., 2002; Murdoch
et al., 2003) and empirical (e.g., DeBach and Rosen, 1991; van
Driesche and Bellows, 1996; Pickett and Bugg, 1998; Hajek, 2004;
Jervis, 2005) perspectives. In contrast, biological control of vectors
which is aimed at decreasing disease incidence in the host, rather
than pest density per se, is much less well studied (e.g., Weiser,
1991; Hartelt et al., 2008). There are several reasons why the
conclusions about the biological control of herbivorous pests may
not readily apply when the objective is reducing disease incidence
in the host. First, biological pest control focuses on reducing pest
density through the direct action of a natural enemy on the pest.
Biological vector control, however, involves reducing disease
incidence through the indirect action of a natural enemy that
attacks the vector rather than the disease organism. Given the
indirect interactions inherent in vector-borne disease systems
and the resulting non-linear feedbacks, predictions about the
efficacy of biological control based on direct pest-enemy interactions may not readily apply to host–vector disease interactions. In
particular, reduced vector densities, rather than outright eradication, can suffice to eradicate a disease and not a vector, so the
standard of comparison is different. Indeed, how the range of
dynamics between different control agents and vectors ultimately
translate into disease prevalance in the host is hard to predict
without exploring the consequences of the indirect effects inherent in the host–vector disease interactions.
Previous theory on vector-borne diseases highlights the
importance of the indirect interactions between the vector’s
population 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 has
focused on the dynamics of the host–vector disease system
without considering the dynamics of natural enemies of the
vector. Two studies have investigated the use of a single biocontrol agent of the vector to reduce disease incidence. Moore et al.
(2010) examined how a predator of the vector affects the
prevalence of a vector-borne disease in the absence of predator
satiation; Gourley et al. (2007) investigated how pulsed application of biological larvicides or chemical insecticides on different
life stages affected disease prevalence. Both studies focus on one
type of biological control agent and how the vector–agent interaction affects infection rates in the primary host. No study to date
has compared whether different types of biocontrol agents (e.g.,
predators, parasitoids, pathogens) can exert differential effects on
the same host–vector interaction. Here, we develop a comparative
framework to ask how a range of antagonistic interactions
between the vector and a potential control agent can indirectly
reduce the prevalence of a vector-borne disease in the host
population.
A comparative analysis of different types of biocontrol agents
is key to elucidating the consequences of indirect interactions
that are characteristic of vector-borne disease systems. It can also
help inform ongoing efforts to control vector-borne diseases.
Different vector-borne disease systems involve diverse natural
enemies of the vectors, and thus the appropriate control
strategies could vary across systems (e.g., Weiser, 1991). Moreover, by highlighting how different forms of ecological interactions can indirectly affect the vector-host–pathogen system, a
comparative framework can help elucidate the role of such
indirect effects in community ecology more generally.
Here we use a common mathematical framework to compare
how different types of biological control agents reduce the incidence of vector-borne diseases. The novel aspect of this work is the
comparative approach, which allows us to make predictions about
the efficacy of different biological control agents (predator/parasitoid, competitor, or pathogen of the vector) to reduce disease
incidence. We discuss how these predictions can inform ongoing
and 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 expected
to predominate in agricultural vector-borne diseases of plants
(e.g., Jeger et al., 2004; Otim et al., 2006), as well as non-zoonotic
vector-borne diseases of humans (i.e., those diseases that cannot
be spread from non-human animals to humans), such as the
dominant malaria pathogen Plasmodium falciparum (Woolhouse
et al., 2001). Our goal is to understand the impact of introducing
different biological control agents into such a vector-borne disease system.
The basic dynamics of a single vector–single pathogen–single
host 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 as
a starting point. Their continuous time model analyzed the
dynamics of the African cassava mosaic virus, a pathogen of
cassava (Manihot esculenta) transmitted by the whitefly (Bemisia
tabaci Gennadius). Unlike many earlier models of vector-borne
infectious 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–vector
disease system are given by:
dS
SþI
¼ rðS þIÞ 1
þ yIfVSdS,
ð1aÞ
dt
K
dI
¼ fVSðy þd þaÞI,
dt
ð1bÞ
dU
U þV
¼ FðU þVÞ 1
cIUoU,
dt
mðS þ IÞ
ð1cÞ
dV
¼ cIUoV,
dt
ð1dÞ
where S,I,U and V denote the densities of susceptible hosts,
infected hosts, susceptible vectors, and infected vectors, respectively. In what follows, we will refer to the susceptible and
infected host populations (S and I, respectively) as ‘‘hosts.’’ In
the absence of the disease, the host population grows at percapita rate rð1S=KÞd, where r is the host’s unconstrained percapita birth rate, K is the carrying capacity of the host, and d is the
host’s per-capita death rate. The disease increases the host’s percapita 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 to
uninfected hosts at a rate f. Holt et al. (1997) assumed densitydependent transmission, which we retain because it allows us to
compare our results with previous studies (e.g., Gourley et al.,
2007; Moore et al., 2010). Focusing on density-dependent transmission 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
tractable while allowing the transmission rate to depend on host
and vector availability.
Following Holt et al. (1997) we assume that the disease
pathogen can multiply sufficiently quickly so that hosts and
vectors that contract the disease become infectious immediately.
Uninfected vectors encounter and contract the disease
from infectious hosts at a rate c. The disease transmits from
infected hosts to susceptible vectors at rate cIU, and thus the
transmission dynamics of the disease depend critically on the
vector density.
The vector’s per-capita population growth rate consists of F,
their unconstrained per-capita birth rate, m, the extent to which
vectors are limited by the host population, and o, the vector
natural death rate. Thus, the vector’s population growth rate is a
function of both total vector density and total host density. The
model assumes the disease has no impact on the vector, and that
the parameters are scalar constants (Table 1).
Holt et al. (1997) found that increasing the host carrying
capacity K and unconstrained growth rate r increased disease
incidence. They showed that increased transmission between
hosts to vectors (analogous to c and f) increased disease
incidence only when the disease’s virulence (a) was moderate
or high. They also found that allowing infectious hosts to
reproduce or recover from infection stabilizes model dynamics
(1). Indeed, extensive numerical integration we conducted across
the parameter space found no indication that Eq. (1) exhibit
persistent limit cycles. When the model did not converge to an
interior, disease-endemic equilibrium, the fraction of infected in
our numerical simulations always decreased to zero or the
primary host went extinct. Thus, we expect the long-term
behavior of Eq. (1) to be stable over a wide range of parameter
values.
Holt et al. (1997) assumed that infected individuals either did
not reproduce or did not give birth to infected offspring. The main
difference in our model is that we allow infectious hosts to give
birth to susceptible offspring. We ignore vertical transmission
both for analytical tractability, and based on the fact that its
relative importance in the epidemiology of most vector-borne
diseases remains unclear (e.g., Pherez, 2007).
49
Appendix B gives the equilibria of Eq. (1). The two important
equilibria are the disease-free boundary equilibrium (S ,0,U ,0)
that allow us to specify the conditions for disease invasion in
Eq. (1), and the interior-equilibrium (S , I , U , V ) that constitutes
the resident community which is invaded by the biocontrol agent.
We use Eq. (1) as a starting point for investigating the effects
of different biocontrol agents of the vectors on disease suppression. We consider three different types of biocontrol agents: a
predator/parasitoid of the vector, a non-infective competitor of
the vector, and an infectious pathogen of the vector.
%
%
%
%
%
%
2.1. Parasitoid (or predator) biocontrol agent
Biological control frequently involves introducing a specialist
parasitoid or predator that consumes the pest (e.g., Murdoch
et al., 2003). Although there are important biological differences
between predator and parasitoid biocontrol agents (e.g., Hassell,
1978) the dynamics of their interactions with the host–vector
disease system is qualitatively similar. Hence, we consider both
under a common modification of Eq. (1) involving an increase in
the vector’s per-capita mortality rate driven by the biological
control agent (e.g., Weisser and Hassell, 1996). For brevity, we
refer to the predator or parasitoid biocontrol agent as ‘‘the
parasitoid’’. The dynamics of the host–vector disease-parasitoid
system is given by:
dS
ðS þ IÞ
¼ rðS þIÞ 1
þ yIfðV 1 þ V 2 ÞSdS,
ð2aÞ
dt
K
dI
¼ fðV 1 þ V 2 ÞSðy þd þaÞI,
dt
dU
U þV
U
¼ FðU þ VÞ 1
cIUoUPqðU þ VÞ
,
dt
mðS þ IÞ
U þV
ð2bÞ
ð2cÞ
dV
V
¼ cIUoVPqðU þ VÞ
,
dt
U þV
ð2dÞ
dP
¼ ce PqðU þ VÞmP P,
dt
ð2eÞ
Table 1
Parameters of models (1)–(4).
Parameter
d
Interpretation
Units
Per-capita natural mortality of host
Per-capita natural mortality of the vector
Infected host mortality rate (i.e., disease virulence)
Rate at which an individual vector contacts an infected host
Rate at which an individual vector gets infected from biting an infected host
Host per-capita birth rate
Host recovery probability
Vector self-limitation terma
Vector intrinsic per-capita birth rate
o
a
f
c
r
y
m
F
1
day
day 1
day 1
day 1
day 1
day 1
day 1
vectors host 1
day 1
Value in Holt et al. (1997)
0.003
0.12
0.003
0.008
0.008
0.05
0.003
500
0.2
Parameter
Interpretation
Units
Range examined
q1
q2
R
m2
Effect of the competitor on the vector’s per-capita growth rate
Scaled effect of the vector on the competitor’s per-capita growth rate
The competitor’s unconstrained birth rate
The competitor’s self-limitation term
The competitor’s death rate
The per-capita attack rate of the parasitoid
The parasitoid’s conversion efficiency
The parasitoid’s handling time
The parasitoid’s death rate
Recovery of rate of vectors infected with the vector-specific pathogen
The virulence of the vector-specific pathogen
Transmission rate of the vector-control pathogen
Competitor 1
Competitor 1
day 1
Competitor 1
day 1
day 1
Vector attacked 1
parasitoid 1
day 1
day 1
day 1
vector 1 under density dependent transmission
Varied
Varied
Varied
Varied
Varied
Varied
Varied
Varied
Varied
Varied
Varied
Varied
mC
A
ce
h
mP
yv
a
b
a
Holt et al. (1997) give the units for m as (host 1) instead of (vectors host 1), but this would render their Eq. (10) to no longer be in units of (vectors time 1).
50
K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57
where P is the density of the parasitoid biocontrol agent. The
function qðÞ describes the per-parasitoid consumption rate (functional response) of the vector. We consider both Type I (qðxÞ ¼ Ax)
and Type II (qðxÞ ¼ Ax=ð1 þ AhxÞ) functional responses, where A and
h are the attack rate and handling times of the parasitoid,
respectively. Type I functional responses describe the behavior
of voracious parasitoids with a very high vector saturation
density. Type II functional responses are appropriate when the
number of vectors consumed saturates relatively quickly to 1=h as
vector density increases. Because a Type II functional response
can introduce a delayed feedback in the parasitoid population
(e.g., Gurney and Nisbet, 1998), comparing the two types of
functional responses can highlight how parasitoid biocontrol
agents induce delayed negative density dependence in the vector
population, and the effect of such delays on disease suppression.
The parameter ce describes the conversion efficiency of the
consumed vectors into individual predators or parasitoids, while
mP 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 the
vector (but not the host) and reduces the vector’s per-capita
growth rate. Infectious pathogens have a well-established history
as biocontrol agents (e.g. Herman, 1953; Hochberg, 1989; Sun
et al., 2006), and the potential for vector pathogens to control
vector-borne diseases has begun receiving closer attention (e.g.,
Doberski, 1981; Houle et al., 1987; Lecuona et al., 2001; Kanzok
and Jacobs-Lorena, 2006; Schelfer et al., 2008). Here, we investigate when and how pathogens of the vector can help suppress a
disease. We refer to the pathogen biocontrol agent as the ‘‘control
pathogen’’ or simply the ‘‘pathogen’’, to distinguish it from the
disease infecting the host that is being targeted for suppression
(referred to as ‘‘the disease’’). The dynamics of a host–vector
disease pathogen system is given by:
dS
ðS þ IÞ
¼ rðS þ IÞ 1
þ yIfðV 1 þV 2 ÞSdS,
ð3aÞ
dt
K
dI
¼ fðV 1 þ V 2 ÞSðy þ d þ aÞI,
dt
dU 1
U 1 þV 1 þ U 2 þ V 2
¼ FðU 1 þ V 1 þ U 2 þV 2 Þ 1
dt
mðS þ IÞ
cIU 1 oU 1 þ yv U 2 U 1 bðU 1 ,U 2 ,V 1 ,V 2 Þ,
ð3bÞ
ð3cÞ
dU 2
¼ cIU 2 ðo þ aÞU 2 yv U 2 þ U 1 bðU 1 ,U 2 ,V 1 ,V 2 Þ,
dt
ð3dÞ
dV 1
¼ cIU 1 oV 1 þ yv V 2 V 1 bðU 1 ,U 2 ,V 1 ,V 2 Þ,
dt
ð3eÞ
dV 2
¼ cIU 2 ðo þ aÞV 2 yv V 2 þ V 1 bðU 1 ,U 2 ,V 1 ,V 2 Þ:
dt
ð3fÞ
Vectors uninfected with the disease are partitioned into two
groups—vectors (U1) that are neither infected with the disease nor
the biocontrol pathogen, and vectors (U2) infected with the
biocontrol pathogen but not the disease. Similarly, vectors infected
with the disease are partitioned into vectors (V1) infected with the
disease but not with the biocontrol pathogen, and vectors (V2)
infected with both the disease and the biocontrol pathogen. We
assume that the vector contracts the control pathogen from other
infected vectors which contract the control pathogen at rate bðÞ.
We consider both density-dependent (i.e., bðU 1 ,U 2 ,V 1 ,V 2 Þ ¼
bD ðU 2 þV 2 Þ) and frequency-dependent (bðU 1 ,U 2 ,V 1 ,V 2 Þ ¼ bF U 2 þ
V 2 =U 1 þU 2 þV 1 þ V 2 ) transmission. Examples of density-dependent 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; Knell
and Webberley, 2004) typically spread via frequency-dependent
transmission. Once infected, the vector recovers from its pathogen at
rate yv and suffers an increased per-capita mortality rate a due to
the control pathogen’s virulence.
2.3. Competitor biocontrol agent
Finally, we ask whether a competitor biocontrol agent of the
vector can reduce disease incidence in the host. Ideally, such an
agent should be unable to transmit the disease. Although much
rarer than parasitoids or pathogen biocontrol agents, observed
reductions in vector populations in the presence of inter-specific
competitors have led some investigators to examine the potential
of such competitors as biocontrol agents. For example, in a review
of responses of mosquito populations to competitors, Lounibos
(2007) attributed larval resource competition from less effective
vector species as the key mechanism leading to a reduction in
vector population densities in several case studies. In some
instances, whole-scale local displacement of vectors may occur.
For example, the chagas disease vector Triatoma infestans requires
a host-blood meal to reproduce. T. infestans competes with a less
effective vector, the hemophagous congener T. sordida, and the
two species are known to segregate spatially (Oscherov et al.,
2004). However, the efficacy of such competitors as biocontrol
agents and their impact on host disease incidence has not been
previously studied. Here we analyze the epidemiological consequences of introducing a non-vectoring competitor of the vector.
The dynamics of a host–vector–competitor disease system are
given by
dS
ðS þ IÞ
¼ rðS þIÞ 1
þ yIfðV 1 þ V 2 ÞSdS,
ð4aÞ
dt
K
dI
¼ fðV 1 þ V 2 ÞSðy þd þaÞI,
dt
ð4bÞ
dU
U þ V þ q1 C
¼ FðU þVÞ 1
cIUoU,
dt
mðS þ IÞ
ð4cÞ
dV
¼ cIUoV,
dt
dC
C þq2 ðU þ VÞ
¼ RC 1
mC C,
dt
m2 ðS þIÞ
ð4dÞ
ð4eÞ
where C is the competitor’s density, R and mC are, respectively, its
per-capita exponential growth rate and death rate, and q1 ,q2 are
the per-capita, interspecific competitive effects of the competitor
on the vector and vice versa. Because the competitor cannot itself
act as a secondary vector, as would a parasitoid biocontrol agent
(model (2)), the primary host’s dynamics remain unchanged from
model (1). Host availability can differentially affect the self
limitation 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 competition between the vector and its competitor. We assume that
the hosts (or their by-products, e.g., household water containers
in which mosquito larvae can grow) are the limiting resource for
both the vector and its competitor, and that competition is both
intra- and inter-specific.
3. Model analyses and results
An ideal biological control agent should exhibit three key
attributes. First, it should be able to establish when rare. Second,
K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57
it should be able to suppress recurring outbreaks of a vectorborne disease. Third, it should reduce the proportion of infected
individuals in the host population when the disease is already
established. We applied a combination of analytical and numerical methods to non-dimensionalized versions (Appendix A) of the
models (1)–(4) to predict and compare the efficacy of the different
biological control strategists according to these three criteria.
We derived invasion criteria for when each type of control
agent and the conditions under which competitors and parasitoids exhibiting a Type I functional response can prevent disease
recurrence. Numerical analyses were required to assess when the
disease-free equilibrium is locally stable when parasitoids with a
Type II functional response or pathogens are used as control
agents. We also used numerical analyses to determine the ability
of 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. This
system is an excellent case study for several reasons. First, the
vector-borne disease occurs in a relatively simple agricultural
setting, which allows us to focus on a single vector and a single
host species. Second, the disease is endemic in much of its range,
and in the absence of biocontrol agents, models of this system
exhibit a stable interior equilibrium. Third, the white fly (Bemisia
tabaci), the mosaic virus vector, is a common agricultural pest
with many studies on its biological control. A range of organisms,
including predators, parasitoids (Gerling et al., 2001) and pathogenic fungi (Faria and Wraight, 2001) have been proposed as
control agents. Research also exists on competition between
Bemisia tabaci and other herbivorous insects such as leafminers
(Diptera: Agromyzidae), a confamilial whitefly (Trialeurodes
vaporariorum), the cabbage looper (Trichoplusia ni), as well as
spider mites (reviewed in Inbar and Gerling, 2008). Thus, the
cassava–whitefly–cassava mosaic virus system presents an
51
attractive model system to compare the impact of different
biocontrol 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 control
agent are released initially because of logistical challenges and
the cost involved (e.g., Walter and Campbell, 2003). In such
situations, demographic stochasticity and Allee effects can prevent the control agent from becoming established. We assessed
the establishment success of a biocontrol agent by identifying
the conditions under which it can increase from initially small
numbers (see Appendix C for details). The key results are given
in Table 2, and the derivations are given in Appendix C. An
important point to appreciate is that the equilibrium host
density determines the successful establishment of a competitor
biocontrol agent but not that of a predator or pathogen
control 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) are
more likely to become established. By contrast, highly virulent
pathogens with moderate or low transmission rates quickly kill
infected vectors and hence fail to become established. If the
biocontrol pathogen is spread through frequency-dependent
transmission, it can establish only if its transmission rate exceeds
the vector’s recovery rate and the biocontrol pathogen’s virulence.
If the biocontrol pathogen is spread through density-dependent
transmission, even biocontrol pathogens with low transmission
rates can still become established provided the susceptible vector
population is sufficiently high.
Table 2
Predictions for the successful establishment of the biocontrol agent.
Biocontrol agent
Target system
Parasitoid with Type 1 functional response Vector population in the absence of control agent very abundant
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 1 functional response, as well as:
Handling time large relative to lifetime of control agent
Pathogen with frequency-dependent
transmission
Pathogen with density-dependent
transmission
Competitor of the vector
Expected results
Parasitoid successfully
established
Parasitoid successfully
established
Parasitoid cannot become
established
Parasitoid cannot become
established
Parasitoid successfully
established
Parasitoid cannot become
established
Handling time small relative to lifetime of control agent
Parasitoid successfully
established
Recovery of vectors from control pathogen is lower than infection rates
Pathogen successfully
established
Mortality of vector or control pathogen virulence greater than infection rate
Pathogen cannot become
established
Mortality of vector and control pathogen virulence, as well as recovery rate, less than Pathogen successfully
infection rate
established
Similar results as a pathogen spread through frequency-dependent transmission, as
well as:
Uninfected vectors highly abundant
Pathogen successfully
established
Uninfected vectors rare
Pathogen cannot become
established
The ratio of hosts to vectors is large
Competitor successfully
established
Competitor has high birth rate or low mortality rate
Competitor successfully
established
Vector has a strong per-capita effect on the competitor’s growth rate
Competitor cannot become
established
52
K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57
In the next section, we ask when a vector borne disease can
invade a disease-free vector–host–biocontrol agent system. We
use analytical and numerical results based on the African cassava
mosaic virus disease.
3.2. When does a biocontrol agent prevent a vector-borne disease
from becoming endemic?
An effective biological control agent ideally maintains disease
incidence at low levels and prevents the vector-borne disease
from becoming endemic in the face of reintroduction of the
disease (due to, for example, immigration of infectious vectors
Bryant et al., 2007). We consider a biocontrol agent as successfully preventing a disease from becoming endemic if the diseasefree equilibrium density of the vectors, hosts, and biocontrol
agent is feasible and locally stable.
Factors that influence the stability of the disease-free boundary equilibrium of Eq. (1) are key to understanding the conditions
for successful disease suppression. Provided the disease-free
equilibrium is feasible, it is locally unstable (R0 41) if
m4 F oðd þ a þ yÞ=frðd1Þ2 ðFoÞc (Appendix B). The vector’s selflimitation is key because it determines (i) the configuration of the
host–vector disease system in the absence of a biocontrol agent
(e.g., Holt et al., 1997) and (ii) vector density, which in turn
determines the success of natural enemies such as pathogens and
parasitoids (e.g., Arneberg et al., 1998; Murdoch et al., 2003).
Indeed, when the vector’s self-limitation (m) is strong, vector
densities are low and the disease may not readily spread or
emerge even in the absence of a control agent. We therefore
investigated the conditions under which the biocontrol agent
could prevent disease emergence when the vector’s self-limitation is weak using parameter values from previous studies of the
cassava mosaic virus disease.
For parasitoid control agents exhibiting a Type I functional
response, a threshold conversion efficiency exists below which
the parasitoid consistently fails to prevent the disease from
becoming endemic. We found that even parasitoids with conversion efficiencies above this threshold may fail when the vector’s
self-limitation is sufficiently strong. In these cases, the vector
population at the disease-free equilibrium is too small to support
Fig. 1. Stability diagrams for the disease-free equilibrium in the presence of a biocontrol agent. The black regions describe regions of the parameter space leading to stable
disease-free equilibria, white regions depict unstable disease-free equilibria (i.e., the disease can invade), and the light or dark grey regions mean the boundary equilibrium
is not feasible. The parameter values have been rescaled (Appendix A) to facilitate comparison. Other parameter values not depicted in a given panel are the same as those
in Holt et al. (1997) (Table 1), as well as: (A—competitors) mc =r ¼ 1,R=F ¼ 1, (B—parasitoids with a Type II functional response) m=r ¼ 5,c ¼ 1, and (C—pathogens with
density-dependent transmission) yv =r ¼ 0:3. High vector self-limitation improves the ability of competitors to keep the disease from becoming endemic (1A; the re-scaled
vector’s effect on its competitor is normalized by the vector’s self-limitation term). When the parasitoid exhibits a Type II functional response (B), suppression is only
possible when the parasitoid’s scaled handling time is low and vector’s self limitation is strong. If the control pathogen is spread through density-dependent transmission
(C), as long as the vector’s self limitation is above a certain threshold the pathogen can prevent the disease from becoming endemic, while if the control pathogen spreads
through frequency-dependent transmission, a higher transmission rate can stabilize the 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
a large parasitoid population. Simililarly, when the vector’s selflimitation is strong, weak competitors of the vector could also fail
to prevent the disease from becoming endemic (Appendix D).
Numerical analyses show that parasitoid biocontrol agents
with a Type II functional response require very short handling
times when vector self-limitation is weak in order to keep the
disease from becoming endemic (Fig. 1B). Also, biocontrol pathogens can prevent a disease from becoming endemic as long as the
vector’s self limitation is above a certain threshold for the case of
density-dependent pathogens (Fig. 1C). By contrast, we found
frequency-dependent pathogens can prevent the disease from
becoming endemic with a high transmission rate even if the
vector 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
a large number of biocontrol agents are released (same level as
the vector population density).
^ as
We quantified the degree of disease suppression (q)
q^ ¼
Ic =ðSc þ Ic Þ
,
I0 =ðS0 þ I0 Þ
%
%
%
%
%
%
ð5Þ
where the numerator and denominator are, respectively, the
proportion of infected hosts in the presence of and in the absence
of the biocontrol agent. We calculated q^ based on host population
densities after 15,000 time steps. The integrations were periodically checked to see if either the scaled density of vectors (U þV)
or the biocontrol agent was below 2E or E (where E is the machine
epsilon), respectively, or the density of infected individuals in
both the host and vector populations (i.e., I þ V) was below 2E.
This allowed for the possibility of extinction at low density due to
demographic stochasticity. If any of these conditions held, the
disease (or the biocontrol agent) was considered extinct and the
simulations terminated. We decided to use a conservative measure of extinction because complete eradication is neither logistically or biologically likely (Tang et al., 2005), and because under
some parameter ranges the nonlinearities in models (1)–(4) can
potentially allow a population to recover from low densities. All
numerical integrations were carried out using the NDSolve
routine in Mathematica 8 with E ¼ 252 .
We found that eradicating the whitefly vector was not a
prerequisite for reducing disease incidence in the cassava plant
(Figs. 2–4). For instance, parasitoids exhibiting a Type II functional
response could still reduce infection rates in the cassava plant if
Conversion efficiency
Conversion efficiency
Perhaps the strongest impetus for the biological control of a
vector occurs when a vector-borne disease has already become
endemic in the host–vector disease system. To quantify a control
agent’s effect on disease incidence, ideally one would compare
equilibrium disease incidence in the host before and after the
introduction of a biocontrol agent. We did this by numerically
integrating (2)–(4). We considered two different implementation
regimes: (i) when a small number of biocontrol agents (1% of the
vector density) are released initially into the system and (ii) when
53
Scaled parasitoid mortality rate
Scaled handling time
Conversion efficiency
Conversion efficiency
Scaled
vector density
Scaled parasitoid mortality rate
Scaled handling time
Fig. 2. The ratio q^ of disease incidence with and without the biocontrol agent for parasitoids with (A) a Type I functional response and (B) a Type II functional response, as
well as the vector density for parasitoids with (A) a Type I functional response. 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 mP ¼ 3 in panels (B,D). For all types of parasitoids, strong disease reduction is facilitated by highly efficient predators with low mortalities
(A–D), but does not require vector extinction. The scatter of points in panels (B,D) result from the deterministic fluctuations in vector density.
54
K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57
Transmission rate
Transmission rate
^q
Virulence
Virulence
Transmission rate
Transmission rate
Scaled
vector
density
Virulence
Virulence
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) frequencydependent 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 =r ¼ 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.
the handling time was sufficiently small, despite high mortality
rates (lower-left hand corner of Fig. 2D). This occurred because
the non-linear functional response causes oscillations in vector
and parasitoid abundances, which in turn cause infection rates in
the host plant population to decline as vector population densities
cycle through troughs. However, if the parasitoid has a short
handling time and high conversion efficiency, it can go extinct
before the vector does.
The biocontrol pathogen’s mode of transmission affects its
effects on the cassava mosaic virus prevelance in the host. If the
biocontrol pathogen spreads through density-dependent transmission, it can drive the cassava mosaic virus extinct without causing
vector (whitefly) extinction (Fig. 3A and B). If the biocontrol
pathogen spreads through frequency-dependent transmission, its
growth rate is no longer strongly coupled to vector density.
Substantial reductions in disease incidence in cassava therefore
occur concomitantly with substantial reductions in whitefly populations (Fig. 3C and D). Competitors could also cause substantial
reductions in cassava mosaic virus prevalence in the cassava
population without causing vector extinction (Fig. 4A and B).
When the biocontrol agent is a parasitoid or a pathogen, these
results are unaffected by whether one inundates the system with a
control agent or releases a small number. When the biocontrol
agent is a competitor of the whitefly, the initial biocontrol agent
density strongly affects the ultimate success or failure of biological
control. Inundating the system with competitors could successfully
reduce disease incidence among cassava hosts as long as such
competitor’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 its
numerical advantage, can exclude the competitor and reach
equilibrium, thus preventing a reduction in infection rates. However, this numerical advantage diminishes if the competitor also
is at high density and has a strong negative effect on the whitefly.
4. Discussion
While reducing the encounter rate between hosts and vectors
by altering vector or host behavior can lower the transmission
rates of vector-borne diseases (e.g., Jeger et al., 2004), reducing
vector densities also has the potential to lower the transmission
rate, and, ultimately, the incidence of the disease (Klempner et al.,
2007). Increasingly, the use of natural enemies of the vector is
seen as a possible strategy to reduce vector densities, and thereby
suppress the disease (e.g., Blaustein and Chase, 2007; Kaaya and
Hassan, 2000; Lecuona et al., 2001; Luz et al., 1998; Nelson and
Jackson, 2006; Ostfeld et al., 2006).
A great deal is known about the use of biological control to
suppress herbivorous pests, but relatively little is known about
using biological control to suppress pathogens causing vector-borne
diseases. Here we have developed a theoretical framework to
compare the efficacy of three types of biological control agents (a
competitor, a predator or a parasitoid, and an infectious pathogen)
in controlling vector-borne infectious diseases. We have identified
the conditions under which different types of biocontrol agents
K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57
q^
55
Effect of competitor on vector relative
to vector’s self limitation
Effect of competitor on vector relative
to vector’s self limitation
^q
Effect of vector on competitor
Effect of vector on competitor
Effect of competitor on vector relative
to vector’s self limitation
Scaled vector
density
Effect of vector on competitor
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 ¼ o.
could successfully become established, prevent disease endemicity,
and reduce disease incidence. We believe our work can serve
as a point of departure for researchers to develop system-specific
models aimed at assessing the efficacy of specific biocontrol
scenarios.
Existing theory on the biological control of herbivorous pests
suggests that the stability of the pest-natural enemy interaction
comes at the cost of higher pest density (the ‘‘stability-suppression trade off’’ - e.g., Murdoch et al., 2003). This suggests that a
trade off between suppression and stability may not be as
important for the biological control of vectors as it is for the
biological control of herbivorous pests. Parasitoid biocontrol
agents with short handling time, high fecundity, and short life
span are an important exception. When vector abundances are
low, such agents can go extinct before the vector population has
time to recover.
Previous work has also recognized that the biological control
of vectors can reduce disease incidence in primary hosts without
causing vector eradication (Gourley et al., 2007; Moore et al.,
2010). However, these studies focus on only one type of biological
control agent (e.g., biological larvicide in Gourley et al., 2007 or
predators with a Type I functional response in Moore et al., 2010).
The novelty of our approach is the comparative analysis of
different types of biocontrol agents, thus identifying the conditions under which predators, competitors or pathogens of the
vector can successfully control a disease. Our key result is that the
ability of the biocontrol agent to reduce the fraction of infectious
hosts without reducing the vector’s long-term density depends
crucially on the nature of the interaction between the vector and
the biocontrol agent. For some potential biocontrol agents, even
temporarily or moderately reducing vector density dramatically
reduced disease incidence (e.g., parasitoids and pathogens with
density-dependent transmission). Yet for other potential biocontrol agents (e.g., pathogens with frequency-dependent transmission), substantially reducing long-term vector density was
required to reduce disease incidence in the host. Our comparative
framework thus allowed us to identify the conditions when
biocontrol agents could reduce disease incidence in the host
without strongly reducing vector density.
Our results have important implications for the ongoing efforts
to reduce disease incidence via the biological control of vectors.
For instance, Hoddle et al. (1998) show how different types of
functional responses influence the biological control of a disease
vector. The hymenopteran parasitoid Encarsia formosa Gahan
parasitizes the whitefly (Bemisia argentifoli Bellows and Perring¼ Bemisia tabaci Gennadius strain B) in greenhouses growing
ornamental plants. Bemisia argentifoli is closely related to the
whitefly vector of the African cassava mosaic disease studied by
Holt et al. (1997). Hoddle et al. (1998) demonstrated that the
functional response of E. formosa was saturating (Type II) in small
canopies of plants, but approximately linear (Type I) in large
canopies. If parasitoids such as E. formosa are used to control the
whitefly and thus reduce incidence of the African cassava mosaic
virus, then whether a cassava plot is characterized by large or
small canopies, which in turn determines whether the functional
response of E. formosa is linear or saturating, affects what
additional steps planters could take to facilitate suppressing the
virus. For example, if the plot has large canopies the functional
response of the parasitoid is likely to be linear, and therefore a
parasitoid control agent with low conversion efficiency and
mortality can become established regardless of the whitefly’s
self-limitation. In plots with smaller canopies, the functional
response of the parasitoid to B. tabaci may saturate, in which
case effects of the parastioid’s handling time and conversion
56
K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57
efficiency on disease suppression will depend crucially on the
vector’s self-limitation.
Our findings also have implications for the role of pathogens in
controlling vector populations. While competitors and parasitoid
biocontrol agents that directly reduce the vector’s per-capita
growth rate can suppress disease incidence in the host, highly
virulent pathogens must have a correspondingly high transmission rate to be effective biocontrol agents. For instance, both the
entomopathogenic fungi Beauveria bassiana and Metarhizium
anisopliae are transmitted through physical contact between
individual tsetse flies (Glossina morsitans), vectors of trypanosomiasis (Kaaya and Okech, 1990). Conducting experiments to
determine whether transmission of M. anisopliae and B. bassiana
is frequency- or density-dependent (e.g., Greer et al., 2008 and
references therein), and quantifying the transmission rate, will
help guide how much public health officials should conjoin
biological control with additional measures. For instance, if
transmission of these entomopathogens is density-dependent
highly virulent control agents may actually be less successful at
preventing endemic disease than less virulent control agents.
However, if transmission is frequency-dependent, then control
pathogens with higher virulence are likely to be more successful
at disease suppression.
Our work shows that releasing competitors in large numbers
can potentially compensate for their reduced per-capita effects on
vectors. For instance, even when the biocontrol agent is competitively inferior to the vector, releasing a sufficiently large number
of such competitors may achieve satisfactory control.
In conclusion, we have presented a comparative analysis of
biological control agents, with testable predictions about the
conditions under which different types of agents can successfully
control a vector-borne disease. Our mathematical framework
lends itself to modifications that allow for investigating vector
control in specific host–vector disease systems. For instance, we
have assumed density-dependent transmission both to enable
comparisons with previous studies and to ensure analytical
tractability. Transmission between the vector and the host may
not be strictly density-dependent in many systems (e.g., Wonham
et al., 2006). Departures from density-dependent transmission
can affect the dynamics of model (1). For example, in vector–host
disease 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 and
host abundance have additive effects on R0. Investigating the
effects of different modes of transmission on vector biological
control is an important future direction.
Following previous studies (e.g., Gourley et al., 2007; Holt
et al., 1997; Moore et al., 2010), we have also considered
instantaneous infectivity of host and vectors. The presence of
exposed but latent hosts and vectors as well as susceptible and
infectious individuals is quite common in many vector-borne
diseases, especially when the incubation period can be long
compared to the lifespan of the vector (e.g., Hosack et al., 2008).
The presence of non-infectious disease carriers could affect the
epidemiological dynamics in the absence of the vector-control
agent. For example, Chitnis et al. (2008) illustrate that decreasing
the rate at which vectors progress from initial exposure to
becoming infectious can reduce the endemic equilibrium and
disease prevelance in the host. Incorporating an incubation period
can potentially alter properties such as the transitory duration of
a disease (Hosack et al., 2008). These factors need to be considered when the model is extended to include periodic immigration of disease organisms from outside the community or the
existence of reservoir host populations. Infectious immigrants
could potentially cause future outbreaks even if the biocontrol
agent help maintain the local stability of the disease-free
equilibrium, and whether such outbreaks occur can depend on
the time-lag between exposure of the vector to the disease and
the vector becoming infectious (Hosack et al., 2008).
An important future direction is to incorporate greater biological realism into the framework developed here. These include
spatial heterogeneity within sites that could allow for vectornatural 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 different
types of biocontrol agents, and genetic variation in vector traits
that could lead to the evolution of resistance against biocontrol
agents (e.g., Holt and Hochberg, 1997; Roderick and Navajas,
2003). These processes have been thoroughly investigated in
studies of biologically controlling herbivorous pests, and they
hold similar promise for further investigations of the biological
control of disease vectors.
Acknowledgments
This research was funded by grants from the Systems and
Integrative Biology Training Grant from the National Institute of
Health in the United States to the Department of Biomathematics
at the University of California, Los Angeles, as well as a Chair’s
Fellowship from the Department of Ecology and Evolutionary
Biology at the University of California, Los Angeles. We would like
to thank R. Vance, G. Grether, and two anonymous reviewers for
their valuable comments on earlier versions of this manuscript.
Appendix A. Supplementary data
Supplementary data associated with this article can be found in
the online version at http://dx.doi.org.10.1016/j.jtbi.2012.05.020.
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