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Journal of Theoretical Biology 309 (2012) 47–57 Contents lists available at SciVerse ScienceDirect 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 c 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 48 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. References Acthman, M., 2008. Evolution, population structure, and phylogeography of genetically monomorphic bacterial pathogens. Annu. Rev. Microbiol. 62, 53–70. Amarasekare, P., 2004. Spatial variation and density-dependent dispersal in competitive coexistence. Proc. R. Soc. Lond. B Biol. Sci. 271, 1497–1506. Antonovics, J., et al., 1995. A generalized model of parasitoid, venereal, and vectorbased transmission processes. Am. Nat. 145, 661–675. Arneberg, P., et al., 1998. Host densities as determinants of abundance in parasite communities. Proc. R. Soc. Lond. B Biol. Sci. 265, 1283–1289. Blaustein, L., Chase, J., 2007. Interactions between mosquito larvae and species that share the same trophic level. Annu. Rev. Entomol. 52, 489–507. Bryant, J.E., et al., 2007. Out of Africa: a molecular perspective on the introduction of yellow fever virus into the Americas. PLoS Pathog. 3, 668–673. Chitnis, N., Hyman, J.M., Cushing, J.M., 2008. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol. 70, 1272–1296. DeBach, P., Rosen, D., 1991. Biological Control by Natural Enemies, 2nd ed. Cambridge University Press. de Castro, F., Bolker, B., 2005. Mechanisms of disease-induced extinction. Ecol. Lett. 8, 117–126. Doberski, J.W., 1981. Studies on Entomogenous Fungi in Relation to the Control of the Dutch Elm Disease Vector Scolytus scolytus. Ph.D. Thesis. University of Cambridge. Ewald, P.W., de Leo, G., 2005. Alternative transmission modes and the evolution of virulence. In: Dieckmann, U., et al. (Eds.), Adaptive Dynamics of Infectious Diseases. Cambridge University Press, pp. 10–26. Fagan, W.F., Lewis, M.A., Neubert, M.G., van den Driessche, P., 2002. Invasion theory and biological control. Ecol. Lett. 5, 148–157. Faria, M., Wraight, S.P., 2001. Biological control of Bemisia tabaci with fungi. Crop Protect. 20, 767–778. Fitzgibbon, W.E., et al., 1995. Diffusion epidemic models with incubation and crisscross dynamics. Math. Biosci. 128, 131–155. Gardiner, M.M., Landis, D.A., Gratton, C., DiFonzo, C.D., O’Neal, M., Chacon, J.M., Wayo, M., Schmidt, N.P., Mueller, E.E., Heimpel, G.E., 2009. Landscape diversity enhances biological control of an introduced crop pest in the north-central USA. Ecol. Appl. 19, 143–154. Gerling, D., et al., 2001. Biological control of Bemisia tabaci using predators and parasitoids. Crop Prot. 20, 779–799. K.W. Okamoto, P. Amarasekare / Journal of Theoretical Biology 309 (2012) 47–57 Gourley, S.A., et al., 2007. Eradicating vector-borne diseases via age-structured culling. J. Math. Biol. 54, 309–335. Greer, A.L., Briggs, C., Collins, J.P., 2008. Testing a key assumption of host-pathogen theory: density and disease transmission. Oikos 117, 1667–1673. Gurney, W.S.C., Nisbet, R., 1998. Ecological Dynamics. Oxford University Press. Hajek, A., 2004. Natural Enemies: An Introduction to Biological Control. Cambridge University Press. Hartelt, K., et al., 2008. Biological control of the tick Ixodes ricinus with entomopathogenic fungi and nematodes: preliminary results from laboratory experiments. Int. J. Med. Microbiol. 298, 314–320. Hassell, M.P., 1978. The dynamics of arthropod predator–prey systems. Monogr. Popul. Biol., 1–237. Hawkins, B.A., Cornell, H.V. (Eds.), 1999. Theoretical Approaches to Biological Control. Cambridge University Press. Hemingway, J., Ranson, H., 2000. Insecticide resistance in insect vectors of human disease. Annu. Rev. Entomol. 45, 371–391. Herman, C.M., 1953. Regarding myxomatosis in rabbits. Am. Rabbit J. 23. Hochberg, M.E., 1989. The potential role of pathogens in biological-control. Nature 337, 262–265. Hoddle, M.S., et al., 1998. Discovery and utilization of Bemisia argentifolii patches by Eretmocerus ermicus and Encarsia formosa (Beltsville strain) in greenhouses. Entomol. Exp. Appl. 87, 15–28. Holt, R.D., Hochberg, M.E., 1997. When is biological control evolutionarily stable (or is it)? Ecology 78, 1673–1683. Holt, J., et al., 1997. An epidemiological model incorporating vector population dynamics applied to African cassava mosaic virus disease. J. Appl. Ecol. 34, 793–806. Hosack, G.R., Rossignol, P.A., van den Driessche, P., 2008. The control of vectorborne disease epidemics. J. Theor. Biol. 255, 16–25. Houle, C., et al., 1987. Infectivity of 8 species of entomogenous fungi to the larvae of the elm bark beetle, Scolytus multistriatus (Marsham). J. N. Y. Entomol. Soc. 95, 14–18. Inbar, M., Gerling, D., 2008. Plant-mediated interactions between whiteflies, herbivores and natural enemies. Annu. Rev. Entomol. 53, 431–448. Jeger, M.J., et al., 1998. A model for analysing plant-virus transmission characteristics and epidemic development. IMA J. Math. Appl. Med. Biol. 15, 1–18. Jeger, M.J., et al., 2004. Epidemiology of insect-transmitted plant viruses: modelling disease dynamics and control interventions. Physiol. Entomol. 29, 291–304. Jervis, M.A., 2005. Insects as Natural Enemies. Springer. Kaaya, G.P., Hassan, S., 2000. Entomogenous fungi as promising biopesticides for tick control. Exp. Appl. Acarol. 24, 913–926. Kaaya, G.P., Munyinyi, D.M., 1995. Biocontrol potential of the entomogenous fungi Beauveria-bassiana and Metarhizium anisopliae for tsetse-flies (Glossina spp) at developmental sites. J. Invertebr. Pathol. 66, 237–241. Kaaya, G.P., Okech, M.A., 1990. Horizontal transmission of mycotic infection in adult tsetse. Glossina morsitans morsitans. Entomophaga 35, 589–600. Kanzok, S.M., Jacobs-Lorena, M., 2006. Entomopathogenic fungi as biological insecticides to control malaria. Trends Parasitol. 22, 49–51. Klempner, M.S., et al., 2007. Focus on research: taking a bite out of vectortransmitted infectious diseases. N. Engl. J. Med. 356, 2567–2569. Knell, R.J., Webberley, K.M., 2004. Sexually transmitted diseases of insects: distribution, evolution, ecology and host behaviour. Biol. Rev. 79, 557–581. Lecuona, R.E., et al., 2001. Evaluation of Beauveria bassiana (Hyphomycetes) strains as potential agents for control of Triatoma infestans (Hemiptera: Reduviidae). J. Med. Entomol. 38, 172–179. 57 Levins, R., 1969. Some demographic and genetic consequences of environmental heterogeneity for biological control. Bull. Entomol. Soc. Am. 15, 237–240. Lounibos, L.P., 2007. Competitive displacement and reduction. In: Floore, T.E., Becnel J. (Eds.), Biorational Control of Mosquitoes. Bulletin No.7 of the American Mosquito Control Association, vol. 23 (Suppl. 2), pp. 276–282. Luz, C., et al., 1998. Beauveria bassiana (Hyphomycetes) as a possible agent for biological control of chagas disease vectors. J. Med. Entomol. 35, 977–979. May, R.M., Anderson, R.M., 1979. Population biology of infectious-diseases 2. Nature 280, 455–461. Moon, R.D., 1980. Biological-control through interspecific competition. Environ. Entomol. 9, 723. Moore, S.M., et al., 2010. Predators indirectly control vector-borne disease: linking predator-prey and host-pathogen models. J. R. Soc. Interface 7, 161–176. Murdoch, W.W., et al., 2003. Consumer-Resource Dynamics. Princeton University Press. Nelson, X.J., Jackson, R.R., 2006. A predator from east Africa that chooses malaria vectors as preferred prey. PLoS One 1. Oscherov, E.B., et al., 2004. Competition between vectors of chagas disease, Triatoma infestans and T. sordida: effects on fecundity and mortality. Med. Vet. Entomol. 18, 323–328. Ostfeld, R.S., et al., 2006. Controlling ticks and tick-borne zoonoses with biological and chemical agents. Bioscience 56, 383–394. Otim, M., et al., 2006. Population dynamics of Bemisia tabaci (Homoptera: Aleyrodidae) parasitoids on cassava mosaic disease-resistant and susceptible varieties. Biocontrol Sci. Technol. 16, 205–214. Pherez, F.M., 2007. Factors affecting the emergence and prevalence of vector borne infections (vbi) and the role of vertical transmission (vt). J. Vector Borne Dis. 44, 157–163. Pickett, C.H., Bugg, R.L., 1998. Enhancing Biological Control. University of California Press. Roderick, G.K., Navajas, M., 2003. Genes in new environments: genetics and evolution in biological control. Nat. Rev. Genet. 4, 889–899. Samish, M., Řeháček, J., 1999. Pathogens and predators of ticks and their potential in biological control. Annu. Rev. Entomol. 44, 159–182. Schelfer, R.J., et al., 2008. Biological control of dutch elm disease. Plant Dis. 92, 192–200. Sun, X.L., et al., 2006. Modelling biological control with wild-type and genetically modified baculoviruses in the Helicouerpa armigera-cotton system. Ecol. Modelling 198, 387–398. Tang, S., Xiao, Y., Chen, L., Cheke, R.A., 2005. Integrated pest management models and their dynamical behaviour. Bull. Math. Biol. 67, 115–135. van Driesche, R.G., Bellows, T.S., 1996. Biological Control. Chapman and Hall. Walter, D.E., Campbell, N.J.H., 2003. Exotic vs endemic biocontrol agents: would the real Stratiolaelaps miles (Berlese)(Acari: Mesostigmata:Laelapidae), please stand up? Biol. Control 26, 253–269. Weiser, J., 1991. Biological control of vectors: manual for collecting, field determination and handling of biofactors for control of vectors. Wiley. Weisser, W.W., Hassell, M.P., 1996. Animals ‘on the move’ stabilize host-parasitoid systems. Proc.R. Soc. Lond. Biol. Sci. 263, 749–754. Wonham, M.J., Lewis, M.A., Renc"awowicz, J., van den Driessche, P., 2006. Transmission assumptions generate conflicting predictions in host-vector disease models: a case study in West Nile virus. Ecol. Lett. 9, 706–725. Woolhouse, M.E.J., et al., 2001. Population biology of multihost pathogens. Science 292, 1109–1112.