Interactions in a host plant-virus–vector–parasitoid system: Modelling the consequences for virus transmission and disease dynamics

Virus Research 159 (2011) 183–193

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Interactions in a host plant-virus–vector–parasitoid system: Modelling the consequences for virus transmission and disease dynamics M.J. Jeger a,∗ , Z. Chen a , G. Powell b , S. Hodge b , F. van den Bosch c a

Division of Biology, Imperial College London, Silwood Park, Ascot SL5 7PY, UK Division of Biology, Imperial College London, South Kensington Campus, London SW7 2AZ, UK c Rothamsted Research, Harpenden, Herts AL5 2JQ, UK b

a r t i c l e

i n f o

Article history: Available online 18 May 2011 Keywords: Plant virus epidemiology Transmission Natural enemies and disease dynamics Biological control Mathematical model

a b s t r a c t A full understanding of plant virus epidemiology requires studies at different scales of integration: from within-plant cell processes to vector population dynamics, behaviour and broader ecological interactions. Vectors respond to cues derived from plants (both healthy and virus-infected), from natural enemies and from other environmental influences, and these directly affect the temporal and spatial patterns of disease development. The key element in linking these scales is the transmission process and the determining factors involved. We use a mathematical model to show how the presence of natural enemies, by increasing virus transmission, can increase the rate of virus disease development while at the same time reducing vector population size, supporting recent empirical evidence obtained in microcosm studies. The implication of this work is that biological control of arthropod pests, which are also virus vectors, using parasitoid wasps, may have unanticipated and negative effects in terms of increased incidence of virus disease. © 2011 Elsevier B.V. All rights reserved.

1. Introduction The spread of insect-transmitted plant viruses is highly dependent on the performance and dispersal patterns of their vector populations (e.g. Fiebig et al., 2004; Jeger et al., 2004; Reynolds et al., 2006). The various options for the control of insect-transmitted plant viruses (e.g. genetic resistance, cultural methods and pesticides) have been assessed for their long-term effects (Jeger et al., 2004; van den Bosch et al., 2007), but the effects of natural enemies of the vector population have not been considered in any detail. Biological control of arthropods has long been established as a major component of pest management programmes (Luck et al., 1988; Zehnder et al., 2007). In tri-trophic interactions, beneficial arthropods have been shown to greatly reduce levels of herbivore infestation and feeding intensity, potentially benefitting the host plant in terms of overall fitness (e.g. Dicke and van Loon, 2000; Tooker and Hanks, 2006; but see Coleman et al., 1999). If an insect-transmitted plant pathogen is added to the system, there is potential for predators and parasitoids to facilitate host plant performance, not only by reducing herbivore pressure but also by reducing vector longevity and decreasing vector numbers (e.g. Landis and van der Werf, 1997; Schroder and Basedow, 1999).

∗ Corresponding author. Tel.: +44 2075942428. E-mail address: [email protected] (M.J. Jeger). 0168-1702/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.virusres.2011.04.027

Knowledge of the effect of parasitism on development, reproduction and population growth of aphids at different (st)ages is critical to the success of biological control of aphids (He et al., 2005). Parasitoids can have complex effects on virus epidemiology, potentially affecting acquisition period, latent period and inoculation efficiency (Osler et al., 1999). There is potential for the developing endoparasitoid larvae to affect the efficiency of the aphid as a virus vector (prior to vector mortality), although pea aphids were able to transmit Pea enation mosaic virus (PEMV) right up until mummification was observed and the aphid was finally killed by the pupating Aphidius ervi larvae (Hodge and Powell, 2008a). Conversely, some plant viruses may have a negative impact on endoparasitoid larvae: development of A. ervi was delayed when the aphid host (Sitobion avenae) had acquired Barley yellow dwarf virus (BYDV) (Christiansen-Weniger et al., 1998). However, the effect of beneficial insects on virus dynamics is not always one of the simplistic inhibition. The presence of arthropod natural enemies can induce vector movement from infected source plants – small scale ‘vegetative’ movements rather than dispersal over larger spatial scales – but which nevertheless result in increased incidence of plant disease (Bailey et al., 1995; Losey and Denno, 1998; Jeger, 1999; Smyrnioudis et al., 2001; Reynolds et al., 2006). There are many instances of parasitism inducing dispersal of the host insect and several species of parasitized aphid move from their feeding site as death becomes imminent. It is generally believed this behaviour is primarily of benefit to the parasitoid by reducing the likelihood of hyperparasitism or predation, although

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it has also been suggested it could be an evolutionary response of the aphid to distance the emergent parasitoid from the extant aphid colony (McAllister and Roitberg, 1987; Moore and Gotelli, 1996; Chow and MacKauer, 1999). Mummification sites are found near preferred feeding sites of pea aphid (Acyrthosiphon pisum) parasitized by A. ervi but away from these sites when parasitized by Ephedrum californicus (Chow and MacKauer, 1999). However limited these movements are, it has been demonstrated that this dispersal of parasitized aphids can subsequently lead to an increase in virus incidence within a plant population (Weber et al., 1996; Hodge and Powell, 2008a). Hodge and Powell (2008a) showed the increased dissemination of PEMV by A. pisum in the presence of adult A. ervi was caused by the typical ‘drop-and-move’ escape response of the aphid vector to foraging natural enemies (see also Losey and Denno, 1998; Braendle and Weisser, 2001). However, in a similar study, Smyrnioudis et al. (2001) found that adults of the parasitoid Aphidius rhopalosiphi did not increase the incidence of BYDV in trays of wheat seedlings containing the aphid virus vector Rhopalosiphum padi. Predators and parasitoids may have differing effects on spread of vector-borne viruses: the spread of BYDV was greater with plants exposed to predators (Coccinella septempunctata) and lower with parasitoids (A. rhopalosiphi) (Smyrnioudis et al., 2001). Thus, natural enemyinduced dispersal of an aphid-vectored plant pathogen might be specific to certain aphid–host plant-virus combinations, and is primarily dependent upon the escape response of the aphid vector. Aphid alarm pheromones play an important role in mediating such escape responses (Pickett and Glinwood, 2007). Similarly, the performance of parasitoids can be modified by the presence of the plant virus, leading to a plant host–vector–virus–parasitoid interaction. When A. ervi is introduced into the bean (Vicia faba)–PEMV–A. pisum system, the benefits to the host plant due to reduced numbers of the vector species and associated herbivory must be balanced against the consequences of increased aphid dispersal and virus spread (Hodge and Powell, 2008a). Thus, in modelling virus disease dynamics (Madden et al., 2000; Jeger et al., 2004) there is a need to include the effects of parasitoids through these multitrophic interactions, and apply this approach to systems, as described by Hodge and Powell (2008a). Models should then include parameters describing the effects of natural enemies on vector dispersal and transmission to portray the consequences of higher trophic levels on epidemic development. This paper is the first attempt to model these multitrophic interactions explicitly. We first develop a model for a parasitoid introduced into a plant host–vector–parasitoid–virus system. We then establish criteria for the virus to invade where previously absent and similarly analyse different introduction strategies for the parasitoid. This model is then modified to consider the effect of a pheromone alarm signal in conditioning an escape response and increasing vector dispersal.

2. Basic model – without alarm signal The following assumptions are made:

i. A constant plant population size with new plantings making up losses due to natural (h) or disease-induced () mortality. ii. No plant mortality due to the vector. iii. Acquisition of the virus (ˇ, ˇP ) is proportional to numbers of ‘healthy’ vectors × fraction of hosts infected. Inoculation of the virus (˛, ˛P ) is proportional to ‘viruliferous’ vectors × fraction of hosts healthy. In both cases the suffix p refers to parasitized vectors.

Table 1 Variables in model: as host plant population size is a constant K variables are scaled per plant in Eq. (1). Initial values are for scaled variables in time plots. Initial values (scaled) H I X Y Xp Yp P V

Healthy host plants (H + I = constant host population size K) Infected host plants Non-parasitized non-viruliferous vectors Non-parasitized viruliferous vectors Parasitized non-viruliferous vectors Parasitized viruliferous vectors Number of parasitoids Total number of vectors (=X + Y + Xp + Yp )

0.90 0.10 10 0.1 1 0.01 0.2 11.11

iv. Vector births are given by a logistic function with intrinsic rate of increase () and carrying capacity (M). There is no transovarial transmission of virus. Parasitized vectors do not give birth. v. Vector death rates (, P ) are linear, with suffix p referring to parasitized vectors. We assume that P ≥ ; there may be additional mortality of parasitized vectors in addition to that caused directly by the parasitoid (e.g. because of increased exposure to other predators). vi. The parasitoid attack rate () is proportional to parasitoid numbers, the probability of encountering a vector, and the probability that a vector is non-parasitized. vii. Parasitoid emergence-rate () from parasitized vectors is linear. A constant (n) is necessary to transform parasitized vectors to parasitoids. In the following it is assumed that n = 1 (indicating that the parasitoid is a solitary endosymbiont; for gregarious parasitoids the assumption would be n > 1). viii. Parasitoid mortality rate (ω) is either linear or increases with vector population size and is then linear only at vector carrying capacity. ix. Parasitoids may be added to the system up to a maximum addition rate (a = ) that gives the same emergence rate as would occur if all non-parasitized vectors were parasitized; a special case of this assumption is where there is no external addition (a = 0). Variables and parameters used in the model are defined in Tables 1 and 2 respectively. In assumption (viii) there are two alternative forms of parasitoid mortality to be considered. The special case (˛ = 0) in assumption (ix) allows for the case where there is no augmentative or inundative release of parasitoids. With these assumptions, the basic equations are: dI H H = ˛Y + ˛p Yp − (h + )I K K dt where K = H + I



dX V = (X + Y ) 1 − M dt



− X − ˇX

X I − P K M

where X/M = (V/M) × (X/V) dY I Y = −Y + ˇX −  P K M dt dXp I X = −p Xp − ˇp Xp +  P − Xp K M dt dYp Y I = −p Yp + ˇp Xp +  P − Yp K M dt dP = (Xp + Yp ) − ωP + a(X + Y ) dt

M.J. Jeger et al. / Virus Research 159 (2011) 183–193 Table 2 Rate parameters and constants (V = vectors, P = parasitoids, T = time).

˛, ˇ

˛p , ˇp

˛min , ˇmin

˛max , ˇmax

k h   , p  (n)

ω a M

Virus inoculation/acquisition rate (unparasitized vectors) Virus inoculation/acquisition rate (parasitized vectors) Minimum inoculation/acquisition rate (unparasitized vectors when alarmed) Maximum inoculation/acquisition rate (unparasitized vectors when alarmed) Strength of the alarm signal Natural host mortality rate (≡harvesting) Disease-induced mortality rate (≡roguing) Vector birth rate Vector death rate Parasitoid attack rate Parasitoid emergence rate (nb. n (= 1) parasitoids/vector converts vectors to parasitoids) Parasitoid mortality rate Parasitoid addition rate Vector carrying capacity per plant

Units

Default (range)

V−1 T−1

0.01

values of the total vector population V, the unparasitized component Vs = X + Y, and the parasitized component Vp = Xp + Yp , with V = Vs + Vp (Appendix 1). 2.2. Invasion of the virus

0.015 (0.01–0.10)

0.02, 0.05 (0.01–0.10)

−1

0.02

T−1 T−1 VP−1 T−1 T−1 PV−1 T−1

0.1–1.2 0.05 0.10 0.20

T−1 PV−1 T−1 V

0.05 0.20 50

dY Y ˆ −  Pˆ = −Y + ˇXI M dt dYp Y = −p Yp + ˇp Xˆ p I +  Pˆ − Yp M dt

0–0.5 0.02

T

ˆ ˆ Xˆ p , P. ˆ Introducing The disease-free steady state is H(= 1), X, infinitesimally small amounts of I, Y and Yp to the system, gives dI ˆ + ˛p Yp H ˆ − (h + )I = ˛Y H dt

0.01

T−1

Eigenvalues of this system are calculated from

    −(h + ) −  ˛  ˛p      Pˆ   ˇXˆ − + − 0 =0  M     Pˆ  ˇp Xˆ p  −(p + ) −   M

from which is derived the basic reproductive number R0 =

ˇp Xˆ p ˛p ˛ ˇXˆ · · + h+ p +  h +  ˆ  +  P/M ˆ ˛p  P/M ˇXˆ · · p +   + P/M h + 

+ where 0 ≤ a ≤  and parasitoid mortality is assumed to be linear, alternatively dP V = (Xp + Yp ) − ωP + a(X + Y ) M dt

dI = ˛Y (1 − I) + ˛p Yp (1 − I) − (h + )I dt

(1a)

dX = (X + Y ) 1 − M dt

(1b)

 V

X − X − ˇXI −  P M

dY Y = −Y + ˇXI −  P M dt

(1c)

dXp X = −p Xp − ˇp Xp I +  P − Xp M dt

(1d)

dYp Y = −p Yp + ˇp Xp I +  P − Yp M dt

(1e)

dP = (Xp + Yp ) − ωP + a(X + Y ) dt

(1f)

X 1 −

X + Xp M

−p Xp +  XP − ωP



− X − 

X P=0 M

X P − Xp = 0 M

X + XP + aX = 0 M

where the non-linear form is used for parasitoid mortality. If a is equal to , to give the maximum introduction rate of parasitoids, then

emergence of parasitoids

θM Pˆ = ω

carrying capacity of vectors death of parasitoids

and

Alternatively dP V = (Xp + Yp ) − ωP + a(X + Y ) M dt

(2)

where for the virus to invade, R0 > 1. ˆ Xˆ P , P) ˆ in the disease-free system are The steady state values (X, calculated from:



where parasitoid mortality is assumed to be proportional to vector population size. Rescaling variables I¯ = I/K, . . . , P/K and dropping overbars:



185

(1g)

where M is now the vector carrying capacity per plant.

Xˆ =

ω(p + ) Xˆ p 

and



1 − (1/)( + (/ω)) 1 + (ω(p + )/)



2.1. Steady-state analysis

Xˆ p = M

The internal steady-state values I*, X*, Y*, Xp∗ , Yp∗ and P cannot in general be solved analytically. In the case where parasitoid mortality increases with vector population size further insight was obtained by considering the dynamics and steady state

with the constraint /( + /ω) > 1 for Xˆ p to be positive. If the parasitoid attack rate  is high and the parasitoid mortality ω low such that no additions have been made (a = 0), then Xˆ P  Xˆ and Pˆ ≈ M/ω as previously.

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Table 3 ˆ in the absence of virus in Values of the steady-state parasitoid population size (P) relation to assumptions made on parasitoid mortality and addition rate. Symbols as in Tables 1 and 2: steady state values for the vector population categories are obtained under the assumption made in the four cases. Addition rate

Parasitoid mortality

a=0 a=

Linear ωP

Non-linear ωP(V/M)

 ˆ X ω p  ˆ (X ω

 Xˆ p ω X+Xp  M ω

+ Xˆ p )

3. Introducing an alarm response In the presence of a parasitoid-induced alarm signal we make ˛ and ˇ functions of parasitoid numbers and assume that the alarm signal only affects the non-parasitized vectors. ˛ = ˛max



ˆ Xˆ and Xˆ P into Eq. (2) gives the parameter value Substituting P, combinations that must be satisfied to reduce R0 < 1. Depending on the alternative assumptions made on parasitoid mortality (linear or non-linear) and addition rate (whether there is no release or parasitoids are added at a constant rate), the expression for Pˆ varies as shown in Table 3. The values of Xˆ and Xˆ p also vary according to the assumptions made.

ˇ = ˇmax

 1−

ˇmax − ˇmin ˇmax

dI = ˛max dt



1−

˛



(4a)

exp(−kP)



− ˛min

max

 (4b)

exp(−kP)



exp(−kP)

˛max

Y (1 − I)

+ ˛P YP (1 − I) − (h + )I





× exp(−kP)

dY = −Y + ˇmax dt



XI − 



(5a)

− X − ˇmax



− X

⇒ Xˆ = M(1 − (/)) provided  < ; i.e. the vector birth rate is greater than the death rate. Steady state values in the absence of the parasitoid: ˆI , Xˆ and Yˆ are calculated from



˛max

V dX = (X + Y ) 1 − M dt

ˆ = 1): In the absence of disease and parasitoid (H



− ˛min

where k is the ‘strength’ of the alarm signal; ˛max and ˇmax are the maximum inoculation and acquisition rates possible; ˛min = ˛, ˇmin = ˇ in the system without an alarm system; and accordingly ˛p ≥ ˛min , ˇp ≥ ˇmin . The parasitized vector inoculation and acquisition rates remain constant as in the case with no alarm signal. The equations are then (scaled):

2.3. Invasion of the parasitoid



max



M

a: addition rate, ω: parasitoid mortality rate, : parasitoid emergence rate, V: total vector population size, X: steady-state non-parasitized vector population size in the absence of virus, Xp : steady-state parasitized vector population size in the absence of virus and M: vector carrying capacity (per plant).

X dX = X 1 − M dt

˛

1−

 1−

 1−

ˇmax − ˇmin ˇmax



X P M

(5b)

ˇmax − ˇmin ˇmax



 XI − 

exp(−kP)

Y P M (5c)

˛Y (1 − I) − (h + )I = 0



X +Y (X + Y ) 1 − M



− X − ˇXI = 0

−Y + ˇXI = 0 Solving for Xˆ and Yˆ gives Xˆ =

(h + ) ; ˇ˛(1 − I)

Yˆ =

(5d)

dYP Y = −P YP + ˇP XP I +  P − YP M dt

(5e)

dP = (XP + YP ) − ωP + a(X + Y ) dt

(5f)

or

(h + )I ˛(1 − I)

dP V = (XP + YP ) − ωP + a(X + Y ) M dt

For I to have reached an endemic steady state, i.e. 0 < ˆI < 1, following the introduction of virus into the parasitoid- and disease-free ˆ X), ˆ then M(1 − (/)) > ((h + )/˛)(/ˇ) must hold, i.e. system (H,

1 1 φ⎞ ⎛ >1 M ⎜ 1 − ⎟ ∗ β ∗ ∗α ∗ τ φ μ + h ⎝ ⎠

vector acquisition lifetime steady-state of virus per of vector (Xˆ ) in unit time absence of disease

X dXP = −P XP − ˇP XP I +  P − XP M dt

with 0 ≤ a ≤ . ˆ ˆ Xˆ P , P) ˆ At the disease-free steady state (H(= 1), X, dI = ˛max dt



1−

˛

dY = −Y + ˇmax dt

inoculation lifetime of of plants plant per unit time

(5g)

max

− ˛min

˛max



 1−





ˆ exp(−kP)

ˇmax − ˇmin ˇmax



Y + ˛P YP − (h + )I

 ˆ exp(−kP)

ˆ − XI

Y Pˆ M

Y dYP = −P YP + ˇP Xˆ P I +  Pˆ − YP M dt (3)

ˆ = 1 − ˆI , X, ˆ Yˆ . Invasion of the The parasitoid-free steady-state is H parasitoid is ensured for any addition rate a, where 0 < a ≤ .

Eigenvalues are then determined from this new system and give the same form of the invasion criterion as in Eq. (2), with ˛ (Eq. (4a)) and ˇ (Eq. (4b)) evaluated at Pˆ replacing ˛ and ˇ. ˆ Xˆ P , P) ˆ in the absence of disease are The steady state values (X, as given previously.

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187

Fig. 1. Dynamics of the system with no alarm signal (Eq. (1)) where parasitoid mortality is linear and no additions are made (a = 0): (a) disease incidence (proportion of infected plants) with vector birth rate  = 0.10; (b) numbers of vectors (blue X, cyano-blue Xp , red Y, pink Yp ) and parasitoids (green) with vector birth rate  = 0.10; (c) disease incidence (proportion of infected plants) with vector birth rate  = 0.15; (d) numbers of vectors (colour coded as in (b)) with vector birth rate  = 0.15.

4. Results Time plots of Eqs. (1) and (5) are shown in Figs. 1–4. In each case there is no addition of parasitoids to the system (a = 0). Initial conditions and parameter default values are as given in Tables 1 and 2. Fig. 1 illustrates the dynamics of infected plants (a, c) and vectors and parasitoids (b, d) for two values of the vector birth rate () such that disease does not establish (a, b) or does (c, d), in the case where there is no alarm signal (Eq. (1)) and where parasitoid mortality is assumed to be linear. In Fig. 2 the same plots are shown for the case where there is no alarm signal and parasitoid mortality depends on vector population size. Comparing Figs. 1a and 2a ( = 0.10), there is a similar monotonic decrease in number of infected plants up to about 40 days. The main difference between the plots in Figs. 1b and 2b is the more marked reduction in healthy unparasitized vectors and increased parasitoid numbers where parasitoid mortality is dependent on vector numbers. There are marginally more parasitized vectors as a consequence. The viruliferous vector population also goes to extinction. Comparing Figs. 1c and 2c ( = 0.15), in both cases there is an initial decrease in diseased plants but in this case subsequently disease is able to invade; even though the viruliferous vector population remains relatively low it does not go to extinction. The differences seen in Figs. 1b and 2b can

also be seen when comparing the vector and parasitoid dynamics in Figs. 1d and 2d. Fig. 3 shows the dynamics of infected plants (a, c) and vectors and parasitoids (b, d) for the same two values of the vector birth rate (), but with the alarm signal introduced (Eq. (5), ˛max = ˇmax = 0.05) and where parasitoid mortality is assumed to be linear. In Fig. 4 the plots are shown for the case where the alarm signal is introduced and parasitoid mortality depends on vector population size. Comparing Figs. 3a and 4a with Figs. 1a and 2a, and similarly Figs. 3c and 4c with Figs. 1c and 2c, the effect of increasing the transmission parameters due to the alarm signal is apparent, with disease increasing rapidly to >80% infection irrespective of the assumption made on parasitoid mortality (note the different scales on the x-axis in Figs. 3 and 4 compared with Figs. 1 and 2). The difference arising from the two different assumptions then becomes apparent. Where parasitoid mortality is dependent on vector numbers, there is a subsequent decrease in diseased plants (Fig. 4a and c) to a lower asymptote; whereas with the linear mortality term there is no such decrease (Fig. 3a and c). Figs. 3b, d and 4b, d show the higher numbers of viruliferous vectors as a consequence of the increased transmission due to the alarm signal. The same trends in unparasitized vectors and parasitoids when comparing the two assumptions on

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Fig. 2. Dynamics of the system with no alarm signal (Eq. (1)) where parasitoid mortality is proportional to vector population size and no additions are made (a = 0): (a) disease incidence (proportion of infected plants) with vector birth rate  = 0.10; (b) number of vectors (colour coded as in Fig. 1b) with vector birth rate  = 0.10; (c) disease incidence (proportion of infected plants) with vector birth rate  = 0.15; (d) number of vectors (colour coded as in Fig. 1b) with vector birth rate  = 0.15.

parasitoid mortality can be seen as in the case without an alarm signal. The numbers of unparasitized vectors decreases and the numbers of parasitoids increases where parasitoid mortality is dependent on vector numbers (Fig. 4b and d) compared with the linear case (Fig. 3b and d). There is also the marked reduction in viruliferous non-parasitized vectors (Fig. 4b and d) that matches the reduction in diseased plants (Fig. 4a and c) where parasitoid mortality is positively related to vector numbers. The effect of reducing the strength of the alarm signal k from the value of 0.5–0.2 was minimal for the default values used in the numerical plots. The effect of varying the transmission rates can be seen by analysis of the expression for R0 given in Eq. (2) and its variant for the case where an alarm signal is introduced. Calculations were made of R0 -values for the default parameter values (Table 2) and plotted against the parasitized transmission parameter values in the case where there was no alarm signal and against the maximum unparasitized transmission parameter values when the alarm signal was introduced. In both cases there was a greater than linear increase in R0 as then transmission parameter increased. Over the range tested for the transmission rates (0.01–0.10), the effect on R0 is marginally greater when varying the maximum unparasitized transmission parameter (with an alarm signal), than varying the parasitized transmission parameter (no alarm signal).

The influence of the alarm signal coefficient was further investigated by calculating the basic reproductive number as a function of the steady state parasitoid numbers at the time of introduction of the virus (Fig. 5). At low steady state P values the basic reproductive number is greater than 1 for some values of the alarm coefficient, hence the epidemic will proceed; however at large steady state P values the R0 value decreases to less than 1, indicating that the virus dies out (Fig. 5a). However, it is also clear that when the basic reproductive number is less than 1 in the absence of parasitoids, their introduction can lead to values greater than 1 (Fig. 5b) provided their steady state numbers remain low. 5. Discussion The model analysis presented in this paper clearly shows that increased acquisition and inoculation arising from an alarm pheromone produced by vectors in the presence of parasitoids can lead to increased levels of disease in a plant population. Such an effect has been shown experimentally to arise from increased vector movement. Thus the benefits of biological control, in reducing vector numbers, may be offset in some circumstances by an increase in disease. The ecological situation modelled is complex and some assumptions have been necessary, notably on parasitoid and vector life history parameters, and how these impact on the

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Fig. 3. Dynamics of the system with alarm signal (Eq. (5)) where parasitoid mortality is linear and no additions are made (a = 0): (a) disease incidence (proportion of infected plants – note difference in scale to Fig. 1(a)) with vector birth rate  = 0.10; (b) number of vectors (colour coded as in Fig. 1b) with vector birth rate  = 0.10; (c) disease incidence (proportion of infected plants – note difference in scale to Fig. 1(c)) with vector birth rate  = 0.15; (d) number of vectors (colour coded as in Fig. 1b) with vector birth rate  = 0.15.

host plant population. These assumptions enabled us to build on the epidemiological framework previously developed for virus transmission and epidemic development (Jeger et al., 1998; Madden et al., 2000). The justifications for these assumptions, possible alternatives, the implications for virus transmission, and epidemic development are now discussed.

on the balance between parasitoid emergence rate and mortality, and a measure of the vector population size. The expression used in the plots involves M, the vector carrying capacity per plant as the measure of the vector population size and hence gives the maximum possible value for P*. 5.2. Parasitoid attack

5.1. Parasitoid introduction The model assumes that parasitoids only enter the system through emergence from parasitized vectors within the system, or through releases made for biological control. There is no natural immigration from outside the system. In the numerical plots (Figs. 1–4) we show the case with no releases made (a = 0). If we add parasitoids to the system at a rate equivalent to the emergence that would have occurred from non-parasitized vectors, had they been parasitized, then this would correspond to the maximum release rate that could be considered rational in biological control programmes. Release at this rate then gives a particularly simple expression for P* where there is non-linear vector mortality (Table 3), and this expression was used in the basic reproductive number plots. Corresponding plots can be made for the other steady state values of P* shown in Table 3, according to the expressions obtained for the different assumptions made. In all cases P* depends

Only females oviposit but no attention is given to sex ratio in parasitoid populations, or sex allocation in oviposition – an aspect that may prove important in the efficiency of mass-rearing for biological control (Pandey and Singh, 1997; He and Wang, 2008). The model also assumes a linear relationship between host parasitism and host abundance, whereas other functional responses have been observed (e.g. Sagarra et al., 2000). A host plant related factor that affects parasitoid movement and attack is the so-called “cry-forhelp” signal from the plant, which is a blend of volatiles that attracts parasitoids to the site where pest damage is occurring (Dicke and van Loon, 2000). This signal, which in principle could reduce vector numbers, is not included in the model. 5.3. Parasitoid mortality There are many published reports dealing with larval and pupal mortality of parasitoids (e.g. Bell et al., 2005), but rela-

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Fig. 4. Dynamics of the system with alarm signal (Eq. (5)) where parasitoid mortality is proportional to vector population size and no additions are made (a = 0): (a) disease incidence (proportion of infected plants – note difference in scale to Fig. 2(a)) with vector birth rate  = 0.10; (b) number of vectors (colour coded as in Fig. 1b) with vector birth rate  = 0.10; (c) disease incidence (proportion of infected plants – note difference in scale to Fig. 2(c)) with vector birth rate  = 0.15; (d) number of vectors (colour coded as in Fig. 1b) with vector birth rate  = 0.15.

tively few of adult mortality rates. In the first assumption made on parasitoid mortality, we assume that there is a simple linear relationship between mortality rate and parasitoid population size (as found by Mishra and Singh, 1991). As an alternative, we assume a non-linear relationship in which parasitoid mortality is lower (and life-span longer) at low vector population numbers and increases to a constant term as the vector population approaches its carrying capacity. In the latter case there is some conflicting evidence depending on the host–parasitoid system considered. The pre-oviposition phase can be extended at low host densities (Lauziere et al., 1999), though this was for an insect pest where host feeding also occurred. Again in a host feeding behavioural model (Chan and Godfray, 1993) different formulations of parasitoid mortality were explored, including constant mortality and one in which resource-depleted individuals had a higher mortality rate. The inclusion of host feeding by parasitoids to obtain resources for egg maturation or for maintenance energy adds further complexities (Kidd and Jervis, 1991; Burger et al., 2005) not considered in our model. However, irrespective of the assumptions made on parasitoid mortality, the time trajectories of the model are similar except that with the non-linear assumption, disease increases to a maximum and then reduces to a lower asymptote (Fig. 4) compared with the monotonic dynamics with the linear assumption (Fig. 3). Also, in the non-linear case the parasitoid abun-

dance approaches a higher steady state value than in the linear case. 5.4. Vector birth and death rates In the model we allow for differential death rates (natural mortality) for parasitized and non-parasitized vectors, but within each category of vector the death rates for viruliferous and non-viruliferous vectors are the same. Births only occur in non-parasitized hosts, but again no distinction is made between viruliferous and non-viruliferous hosts. In populations without parasitism, the assumptions made about birth and death rates, based on whether vectors are viruliferous or non-viruliferous, can certainly affect vector population size and hence disease incidence (Sisterson, 2009). In our previous work we have generally specified models in which vector population size was constrained to be constant. Here, with parasitism, we cannot impose this constraint, but it would be interesting to extend the analysis with differential birth and death rates depending on virus status, but at some cost in the qualitative analysis that is possible. 5.5. Vector movement In the basic transmission model (Jeger et al., 1998), virus acquisition and inoculation were related to vector movement and vector

M.J. Jeger et al. / Virus Research 159 (2011) 183–193

4

k=0.12 R 0 without effect

k=0.10

of parasitoid on

k=0.10

Basic reproduction number,R 0

Basic reproduction number,R0

k=0.06

k=0.04

2

k=0.12

2

vector dispersal

k=0.08

3

Virus extinction

k=0.08 k=0.06 Virus extinction threshold

k=0.02

k=0.0

0 2

R0 without effect of parasitoid on vector dispersal

k=0.04

1

threshold

1

k=0.02

k=0.0

0 2

Population density

Population density

191

parasitized vectors unparasitized vectors

1

0 0

20

40

60

80

100

Parasitoid density

parasitized vectors unparasitized vectors

1

0 0

20

40

60

80

100

Parasitoid density

Fig. 5. The basic reproductive number of the system with alarm signal (Eq. (2), with Eqs. (4a) and (4b) inserted) as a function of the steady-state parasitoid density in the absence of the virus for values of k ranging from 0.0 to 0.12. In (a) the inoculation and acquisition rates are at the lower endpoint of the range in values, and R0 > 1 in the absence of the parasitoid; in (b) at the upper endpoint, and R0 < 1 in the absence of the parasitoid. Other parameters take on default values.

probing/feeding on host plants. That underlies the basic premise for incorporating the alarm signal as a means of increasing the transmission rate in the model. Such movement undoubtedly occurs in many host–parasitoid systems, including the pea aphid (Tamaki et al., 1970), where harassment of aphid colonies by parasitoids led to reductions in colony size and movement to the nearest plant. Some 50% of aphids affected by the alarm signal relocated to distant plants, compared with only 25% when mechanically dislodged (Phelan et al., 1976). Even when movement to new host plants occurs through aphids dropping to the ground, the dropped aphids soon relocate to new host plants in the absence of ground predators (Gish and Inbar, 2006). Similarly, the distance moved by aphid nymphs disturbed by coccinellid beetles rather than parasitoids was positively correlated with the density of aphids on the leaves on which the disturbance occurred (Roitberg et al., 1979). If such a relationship was found in response to a parasitoid then host density could be incorporated into the model. Currently the strength of the alarm signal is related only to parasitoid population density. As well as the alarm signal response, other plant-related factors undoubtedly influence vector movement, including vector preference for diseased or healthy plants (Sisterson, 2008; Hodge and Powell, 2008b; Hodge et al., 2011) and reduction in or cessation of host plant growth (Hodgson, 1991). An additional factor that affects the parasitoid rather than vector movement is the ‘cry-for-help’ signal from the plant that attracts parasitoids to the site where pest damage is occurring. Models which include the effects of cry-for-help as well as the alarm signal are currently being developed (Chen, 2009). So it seems the outcome in a host plant-virus–vector–parasitoid system might be influenced by three factors: the preference for healthy or diseased plant tissue shown by vectors; the plant response to pest damage caused by vectors in

attracting parasitoids; and the vector response to the presence of parasitoids. 5.6. Vector mortality due to parasitism In the model, death of vectors occurs through natural mortality (considered above) and through parasitism. The longevity and fecundity of parasitized aphids can be strongly correlated with the age at which parasitism occurs (Tsai and Wang, 2002). Our model allows for differential mortality of parasitized and non-parasitized vectors, but we assume only non-parasitized hosts produce offspring. Tsai and Wang (2002) found that 1st and 2nd instars of the brown citrus aphid did not reach adulthood, nor contribute to population growth, when parasitized. Net reproduction was markedly reduced when parasitism occurred in the 3rd and 4th instars, and in the adult stages, so in this case our assumption would be reasonable. 5.7. Virus transmission and epidemic development We have also made assumptions on virus transmission in relation to whether vectors are parasitized or non-parasitized. We include differential inoculation and acquisition rates according to whether the vector is parasitized. If parasitism increases the transmission rate then we have investigated the effect of this on the basic reproductive number: whether of the effect of the parasitoid on the parasitized vector if movement is increased; or the effect of the alarm signal more generally on the non-parasitized vector population. We assume that transmission increases according to parasitoid abundance. We have not shown the joint effect of increased transmission for both categories of vector or looked for

192

M.J. Jeger et al. / Virus Research 159 (2011) 183–193



 P∗ 

areas of parameter space where one or other of the two effects dominates. The relative balance of the two effects would probably depend on the proportions of parasitized and non-parasitized vectors in the population. Such information is vital if we are to make predictions on the likely consequences for epidemic development. This can readily be seen from comparing the time plots in Figs. 1 and 2 with those in Figs. 3 and 4. In the case of no alarm signal, Figs. 1 and 2 show how the model dynamics change in relation to changes in the vector birth rate. In Figs. 1a and 2a the effect is to reduce the basic reproductive number below 1 and disease dies out. In Figs. 1c and 2c, with higher vector birth rates it is greater than 1, the disease epidemic eventually takes off and approaches an asymptote. In the equivalent plots in Figs. 3 and 4, where the alarm signal is introduced the effect is a basic reproductive number greater than 1 for both vector birth rates, an effect that is still present when the alarm signal has a less dramatic effect on maximum transmission.

But

6. Concluding comments

V ∗ = Vs∗ + Vp∗ = 1 +

The model considered here has represented the interactions between host plant, virus, insect vector and parasitoid, and investigated how an alarm signal in response to parasitoid presence can under some circumstances lead to increased transmission of the virus. Although not modelled explicitly, this effect, as indicated in experimental studies, can be attributed to increased vector movement. This situation is complex, but in reality other interactions, both direct and indirect (Muller and Godfray, 1999), will affect these responses in ways not considered here. These factors include: interactions with host plant species (Raymond et al., 2000) and cultivars (Ronquim et al., 2004); with coccinelid predators with possible negative associations between parasitoid and predator abundance (Raymond et al., 2000); and with fungal entomopathogens (Furlong and Pell, 2000), where these may affect adult parasitoid survival and fecundity as well as larval mortality in parasitized hosts. Despite these limitations it is clear that biological control of insect pests such as whiteflies and aphids using parasitoids may have unanticipated and negative effects in cases where these pest species are also vectors of plant viruses. None of this detracts from the overall benefit of using biological control to control pest species; simply that as shown in this analysis the steady state parasitoid population density, as achieved through inundative or augmentative releases, may be critical in determining whether virus transmission is increased and with it the risk of a disease epidemic developing even though vector numbers may be reduced.

Internal steady states for the dynamics of the total, parasitized and non-parasitized vector populations in the case where parasitoid mortality is given by the term ωP(V/M). For a = Â: P* is readily obtained from 1 g as P * = (/ω)M. Adding Eqs. (1b) and (1c) and setting to zero, gives



V M



−−

from which



1 V∗ = M 1 − 

 ω



 =0 ω

 + ω

 −  (the net increase of unparasitized vectors)

M

or equivalently  >1  + /ω Adding Eqs. (1d) and (1e) and setting to zero, gives −p Vp + Vs

 − Vp = 0 ω

From which Vs∗ =

ω(p + ) ∗ Vp 





ω(p + ) Vp∗ 

Equating with Eq. (A.1) Vp∗ =

M[1 − (1/)( + (/ω))] 1 + (ω(p + )/)

(A.2)

For a = 0: P* is obtained from 1 g as (/ω)(Vp /V)M, which gives the expression for a =  as the proportion of vectors parasitized approaches 1. Adding Eqs. (1b) and (1c) and setting to zero, gives



 1−

V M



−−

 Vp =0 ω V

Adding Eqs. (1d) and (1e) and setting to zero, gives −p + 

Vs  − =0 V ω

From which ( + p )ω Vs∗ = V  But Vp∗ V

=1−

Hence



( + p )ω Vs∗ =1− V 



1  1− +  ω



( + p )ω 1− 

and Vs∗ , Vp∗ and P* follow. For V* > 0: −>

 ω



1−

( + p )ω 



=

P∗ M

  (A.3)



with (/( + p )ω) > 1 to ensure P* > 0. For a sufficiently high parasitoid attack rate  and sufficiently low parasitoid mortality ω, Vp /V approaches 1 and P * ≈ (/ω)M as in the case for a = . References



Hence for V* > 0

=

(the rate at which vectors are parasitized at carrying capacity)

V∗ = M

Appendix 1.

 1−

>

(A.1)

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