Risk Analysis, Vol. 35, No. 9, 2015
DOI: 10.1111/risa.12278
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity Sara Pasquali,1,∗ Gianni Gilioli,2 Dirk Janssen,3 and Stephan Winter4
The introduction of invasive species causes damages from the economic and ecological point of view. Interception of plant pests and eradication of the established populations are two management options to prevent or limit the risk posed by an invasive species. Management options generate costs related to the interception at the point of entry, and the detection and eradication of established field populations. Risk managers have to decide how to allocate resources between interception, field detection, containment, and eradication minimizing the expected total costs. In this work is considered an optimization problem aiming at determining the optimal allocation of resources to minimize the expected total costs of the introduction of Bemisia tabaci-transmitted viruses in Europe. The optimization problem takes into account a probabilistic model for the estimation of the percentage of viruliferous insect populations arriving through the trade of commodities, and a population dynamics model describing the process of the vector populations’ establishment and spread. The time of field detection of viruliferous insect populations is considered as a random variable. The solution of the optimization problem allows to determine the optimal allocation of the search effort between interception and detection/eradication. The behavior of the search effort as a function of efficacy or search in interception and in detection is then analyzed. The importance of the vector population growth rate and the probability of virus establishment are also considered in the analysis of the optimization problem. KEY WORDS: Bemisia tabaci; invasive species; optimization; population dynamics; probabilistic model
1. INTRODUCTION
morphocryptic species(3) some of these, Med and MEAM1, occur in Europe and are highly invasive. B. tabaci is included in the 100 World’s Worst Invasive Species list (http://www.issg.org). These insects typically exhibit resistance to certain insecticides, have high fecundity and spread capacity, and a broader plant host range.(4) B. tabaci also transmits globally over 200 plant virus species, belonging to five different genera: Begomovirus, Crinivirus, Ipomovirus, Torradovirus, and Carlavirus.(5) Many of these viruses cause serious diseases in crops, and hence crop production requires extensive protection measures. The importance of newly emerging plant viruses is then directly related to the adaptability of invasive B. tabaci to new agricultural habitats, in open fields, and to greenhouses
The whitefly Bemisia tabaci(1) is a polyphagous sap-sucking insect commonly found in tropical and subtropical regions(2) and it is a serious pest of agricultural crops and ornamental plants. It causes direct damage by its feeding activity, leading to reduced host growth and yield, and indirectly by transmitting viruses. B. tabaci occurs worldwide and it constitutes a complex of at least 28 indistinguishable 1 CNR-IMATI, 2 DMMT,
Via Bassini 15, 20133 Milano, Italy. University of Brescia, Viale Europa 11, 25123 Brescia,
Italy. 3 IFAPA,
Centro La Mojonera, Almer´ıa, Spain. Institut DSMZ, Plant Virus Department, Germany. ∗ Address correspondence to Sara Pasquali, CNR-IMATI, Via Bassini 15, 20133 Milano, Italy; [email protected]. 4 Leibniz
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C 2014 Society for Risk Analysis 0272-4332/15/0100-1663$22.00/1
1664 of temperate climates where much of the vegetable production occurs in northern regions of Europe. To date, only 10 of these emerging B. tabacitransmitted viruses have been found in Europe, such as tomato yellow leaf curl begomovirus, in tomato,(6) and cucurbit yellow stunting disorder crinivirus in cucurbits(7) and those are mostly confined to the Mediterranean region. Nevertheless, the introduction of new viruses currently not present in the European Union presents a constant risk to crop production. Three major processes characterize biological invasions of plant pests like B. tabaci and the viruses it transmits. During the entry phase, a nonindigenous species arrives in an area, begins to reproduce, and builds up a local population, leading to the establishment. Locally established populations can grow numerically and spread by invading new areas suitable for establishment, causing irreversible ecological and economic damage.(8) Hence, preventing the introduction of new species and eradicating populations at establishment are the only strategies to prevent the risk posed by B. tabaci and the viruses it transmits. In recent years, the management of invasive species has been widely considered, and methods and approaches to decision making during different phases of invasion have been proposed. EpanchinNiell and Hastings(9) give a complete review of studies on economically optimal control of established invasive species. Three aspects of prevention through quarantine, surveillance, and removal have been taken into account in optimal management problems.(10–12) A model developed by Mehta et al.(13) captures also the stochasticity of the detection time and combines the population density with the economic impact. The “density-impact curve” is treated also by Yokomizo et al.(14) Also, spatial aspects are considered in management of invasive species.(15–19) In this article, the problem of optimal allocation of the search and control efforts aiming at preventing introduction and establishment of a new virus vectored by an insect is considered. The insect and the virus can enter into new areas via the international trade of plant products and planting materials. Interception is the opportunity to prevent entry of the vector and the virus in infected plants at the point of entry. If the insect escapes interception it may establish viable populations and, provided suitable conditions, populations grow and subsequently spread. The objective of this study is to determine the optimal allocation of resources between interception, detection
Pasquali et al. in the field, and eradication. For the sake of simplicity, containment is not taken into account, also because the wide host range and the high spread capacity of the vector make this option not feasible in the long term. The main issue is to minimize the expected total costs, which are composed of interception costs, detection costs up to the time of detection, and eradication costs depending on the population abundance at the time of detection. The time of detection is considered stochastic(13) and it is supposed that once the detection takes place, eradication is successfully carried out. This simplifying assumption is justified in an initial phase of establishment when dispersal is only limited around the area of entry and control measures can be highly effective. The damage costs, useful to encompass the range of costs associated with large population,(15) are not taken into account here because detection and eradication are supposed in the early stage of invasion. If this is not the case and the virus escapes interception and eradication in the early stage of establishment or spread, it causes an economic damage far more important that the costs for the management of invasion, and the control of the vector is performed by integrated pest management (IPM) tactics and strategies. The optimization problem, aimed at determining the optimal allocation of resources to minimize the expected total cost, is presented in Section 2 and it takes into account a population growth modeled through a differential equation. The importance of considering the population dynamics in management problems has been pointed out by other authors.(13,14,17,20) An application in plant biosecurity is considered in Section 3. In particular, as a case study the arrival and establishment of new B. tabacitransmitted viruses in Europe is considered, taking into account the trade volume of commodities carrying the insect and the percentage of viruliferous insect populations estimated by means of a probabilistic model. Other works remark on the importance of taking into account the trade in management invasive species.(21–23) Results of the optimization problem are presented in Section 3.3 considering the variation in the parameters of cost functions. Changes in the growth rate of the population and in the virus probability establishment have been considered and the results are discussed in Section 4. 2. OPTIMIZATION PROBLEM This section is devoted to the analysis of the resource allocation problem. The optimization
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity problem aims to minimize the costs deriving from different allocation of the search effort between the vector interception at the point of entry and the detection/eradication of established vector populations. Eradication costs depend on population abundance, which in turn depends on the time of detection. Before setting the optimal control problem it is necessary to make some definitions and assumptions.
r A single new virus is considered and is supposed r
r r r
to be vectored by an insect species. A propagule population of the vector is defined as the population of abundance A0 corresponding to the minimum number of individuals surviving the arrival process and able to establish a local population. The population abundance of the propagules is supposed to be constant until reaching the area of potential establishment. An area of potential establishment is an area with suitable environmental conditions (e.g., temperature) and availability of host plants. As soon as the propagule arrives in a suitable area it establishes a local population (establishment phase). The local population growths and the population abundance N(t) (with N(0) = A0 ) follows the dynamics: dN(t) = a N(t). dt
(1)
Equation (1) accounts for two processes, the local population growth and the possibility that local populations give rise to other local populations by a discontinuous spread process (branching process). The global pattern of population growth, which takes into account both short - and long-distance dispersal, is not limited by local constraints and it is well approximated, in the first phase of colonization, by exponential growth.(24,25)
r The probability of transmitting the virus to the
r
host plants in the area of establishment is related to the prevalence of the infection in the vector population and population abundance, and it also changes over time. The control operations may occur in three phases: at the point of entry (interception, I), searching for establishing local populations (detection, D), and eliminating local populations (eradication, E). Detection and eradication occur at the same time.
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r The r r
costs of eradication depend on the total population abundance. Economic losses to producers due, for example, to consignments rejected at port of entry are not considered here. The optimization problem does not take into account budget constraints for interception, detection, and eradication.
Denoting by α the search effort for the interception phase measured in man-hours at time t, by θ the productivity per man-hour, and by SI the interception cost for a man-hour, the total interception cost is supposed to be quadratic in the search effort: α 2 θ SI . The quadratic cost reflects, for example, overtime pay for increased surveying for more intensive interception effort.(13,17) Let TVt be the trade volume at time t, and q the search sensitivity of the interception process related to the type of goods, measured in ton/man-hour. Parameter q represents a measure of the efficacy of an operator in intercepting a pest considering a unit of search effort (man-hour) and a pest density one (A0 for unit of trade volume). Then, exp(−qα/TV) is the probability that a propagule population survives the interception phase. For the sake of simplicity, the trade volume is supposed constant over time. Denote by SP = AI TV exp(−qα/TV) the number of propagule populations that survive the interception phase, where AI is the number of propagule populations per unit of trade volume. The number of propagule populations that survive the interception phase is equal to the number of establishing local populations SP of dimension A0 at the end of the process of arrival. Let PE ηSP be the number of local foci of virus infection, depending on the number of propagule populations SP establishing in suitable areas, on the probability of virus establishment PE , and on the percentage of viruliferous insect populations η. Denote by β the search effort for the detection of established populations at time t (in man-hours), that is, the number of man-hours allocated to search an establishing insect vector population, by SD the cost unit per man-hour for the detection, and by SE the eradication cost (including labor and the use of control techniques) per individual. Let τ denote the time of detection, which depends on the search effort for the detection β and on the efficacy of the detection process γ , that is, the number of viruliferous individuals found per man-hour. The time of detection is distributed according to an exponential
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distribution(13) and its probability density function is given by γβ exp(−γβτ ) for τ > 0. The expected discounted total cost composed of the costs of interception, detection, and eradication is given by: TC(α, β) = α 2 θ SI ∞ β 2 θ SD + 0
τ
qα
+SE PE η AI TVe
qα − TV
e−δτ N(τ ) γβe−γβτ dτ, (2)
where PE AI TVexp(−qα/TV)N(t) is the dimension of the establishing populations to be eradicated and δ is the discount rate. The objective is to find the optimal search strategy (α ∗ , β ∗ ) that minimize TC with constraints: α > 0,
β>0
Commodity C1 C2 C3
e−δt PE η AI TVe− TV N(t)dt
0
Table I. Notation for the Commodities and Macro-Areas from Which the Commodities Originate
A1 A2 A3 A4 A5 A6 A7 A8 A9
(3) Table II. Correspondence Between Values Assumed by the Variable “Arrival of Viruses with Bemisia” and Intervals Representing the Percentage of Infested Commodities
and Equation (1), that is: TC(α ∗ , β ∗ ) = min TC(α, β), α>0,β>0
(4)
subject to Equation (1). To solve the optimization problem (Equation (4)) subject to Equation (1), it is necessary to compute the first derivatives with respect to α and β and the determinant of the Hessian matrix (see the Appendix for details). Under the condition: δ < a < δ + γβ,
and α ∗ solution to: qα ∗
θ (β ∗ )2 SD +γβ ∗ SE = 0. δ+γβ ∗ − a
“Arrival of Viruses with Bemisia”
Percentage of Infested Commodities in the Interval
= very low = low = medium = high = very high
[0, 0.001] [0.001, 0.01] [0.01, 0.05] [0.05, 0.2] [0.2, 1]
H1 H2 H3 H4 H5
(5)
the first derivatives of TC with respect to α and β are null for: θ SD(a − δ) β∗ = θ γ SD 2 θ 2 (a − δ)2 SD + θ γ 2 (a − δ)SE SD + (6) θ γ SD 2α ∗ θ SI − q PE η AI e− TV N0
Plants for planting Cut flowers Fruits and leafy vegetables Macro-Area Africa North Africa SS Asia Australia Non - EU Europe Near East North America Oceania South America
(7)
Furthermore, it can be easily seen that the second derivative with respect to α is positive and the determinant of the Hessian matrix is positive (see the Appendix). Consequently, the minimum of TC(α, β) is reached in (α ∗ , β ∗ ) solution to Equtions (6)–(7). 3. APPLICATION TO RISK ASSESSMENT AND MANAGEMENT IN PLANT BIOSECURITY In this section, an application to the problem of preventing the invasion in Europe by B. tabaci-
transmitted viruses is considered. These viruses can enter into the risk assessment area with the vector insect or with the infected host plants. Because the focus is on a circulative virus, the vector insect plays a major role in the arrival of the virus; therefore, in this application only the contribution of B. tabaci to the virus arrival is considered. To estimate the value of η, the percentage of viruliferous individuals in the propagule population, a probabilistic model is applied. The model takes into account the trade volume and the estimation, based on expert judgment, of the outcome of the different processes involved in the arrival. This model is illustrated in Section 3.1. For further details on the probabilistic model of arrival, refer to the EFSA (European Food Safety Authority) Opinion on B. tabaci.(26) 3.1. Arrival Probability for a Virus by Infected B. tabaci The probability of arrival of a virus with B. tabaci is calculated using a formula combining expert judgment with data on the trade volumes of
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity
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Table III. Import Volumes in EU (in Tons) Relative to the Three Commodities, Summarized for Different Macro-Areas of Origin
C1 C2 C3 TOT.
A1
A2
A3
A4
A5
A6
A7
A8
A9
TOT.
3,671.97 1,387.73 1,641,167 1,646,227
6,932.57 1,558.27 1,969,635 1,978,126
30,621.73 1,476.03 1,207,587 1,239,685
812.07 254.5 77,175.73 78,242.3
5,370.43 1,967.43 1,655,379 1,662,717
3,656.8 8,216.63 692,715.3 704,588.7
18,221.97 38,532.73 1,633,474 1,690,229
92.47 0.7 400,276.8 400,370
28,475.03 19,440.13 7,020,722 7,068,637
97,855.03 72,834.17 16,298,132 16,468,821
different types of commodities. The model assumes that the probability of arrival depends on the type of commodity and on the geographical macro-area from which the commodities originate. Three different types of commodity (plants for planting, cut flowers, fruits, and leafy vegetables) have been considered and denoted by C j , j = 1, 2, 3. Nine different geographical macro-areas in the world are defined according to the homogeneity in the type and frequency of infestation of the commodities at the origin. The macro-areas are denoted by Ai , i = 1, 2, ..., 9. The arrival probability can be calculated for each different commodity separately or for all the commodities considered together. In the following, it is shown how to compute the probabilities for the latter case, the extension to the case of a single commodity is trivial. Moreover, it is possible to calculate the probabilities for the whole Europe and for the different EU countries. The notation reported in Table I will be used. The random variable “arrival of viruses with Bemisia” is supposed to assume five possible values: H1 = very low, H2 = low, H3 = medium, H4 = high, and H5 = very high. In Table II, a correspondence between these values and numerical intervals representing the percentage of infested commodities is given. The probability of arrival of viruses with B. tabaci can be calculated as:
=
P(arrival of viruses with Bemisia = Hk) 3 i=1 j=1 P(arrival of viruses with Bemisia = Hk|
9
arrival of C j from Ai )P(arrival of C j from Ai ), (8) where P(arrival of viruses with Bemisia = Hk|arrival of Cj from Ai ), that is, the probability that the arrival of viruses with Bemisia coming from the macro-area Ai with the commodity C j is Hk depends on expert evaluation. A panel of experts provided estimation of the probability of association of viruliferous vectors with the commodity at place of
origin and survival of the propagules during transport. The probability of arrival of C j coming from Ai , P(arrival of C j from Ai ), is given by “n. of plants C j coming from Ai /total n. of plants.” The values P(arrival of viruses with Bemisia = Hk) allows to obtain the probability distribution of the variable “arrival of viruses with Bemisia.” To estimate the value η representing the percentage of viruliferous individuals, the mean of the variable “arrival of viruses with Bemisia” is computed choosing as representative point of an interval its middle point. 3.2. Data To calculate the probabilities of arrival, it is necessary to know the trade matrix and the expert judgment on the probabilities of arrival with respect to a given commodity and macro-area of origin. The trade matrix reporting the trade volume in EU relative to different commodities and macro-areas is reported in Table III. Data on trade volume allow the computation of the second probability in Equation (8). An estimation of the arrival probabilities to be used in the computation of the first probability in Equation (8) is formulated by a panel of experts on B. tabaci and it is reported in Table IV.(26) These probabilities are used in combination with data on trade volume of commodities to estimate the probability of arrival of a virus with B. tabaci. The percentage η of viruliferous insect populations is calculated applying Equation (8). The probability distribution of the variable “arrival of viruses with Bemisia” is reported in Table V together with its mean used as an estimate of η. In the optimization problem, η is set equal to 7.2/1000. Other data necessary to solve the optimization problem are summarized in Table VI. It is supposed that the interception cost is smaller than the detection cost because the control in the interception phase is less heavy than the control
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Table IV. Probabilities of Arrival of a Virus with B. tabaci for Different Commodities and Macro-Areas A1
A2
A3
A4
A5
A6
A7
A8
A9
Very low
C1 C2 C3
0 0.7 0.8
0 0.5 0.9
0 0.5 0.8
0 1 1
0 1 1
0 0.6 0.8
0 1 1
0 1 1
0 0.7 0.9
Low
C1 C2 C3
0.2 0.3 0.2
0.2 0.5 0.1
0.2 0.5 0.2
0.2 0 0
0.2 0 0
0.2 0.4 0.2
0.2 0 0
0 0 0
0.2 0.3 0.1
Medium
C1 C2 C3
0.8 0 0
0.8 0 0
0.8 0 0
0.8 0 0
0.8 0 0
0.8 0 0
0.8 0 0
0.2 0 0
0.8 0 0
High
C1 C2 C3
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0.8 0 0
0 0 0
Very high
C1 C2 C3
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
Table V. Probability Distribution of the Variable “Arrival of Viruses with Bemisia” (Columns 1–5) for the Whole Europe and Mean of the Distribution (Column 6) Very Low 0.5336
Low
Medium
High
0.2822
0.01448
0.0393
Very High 0
Mean Value 7.2/1000
Table VI. Values of the Parameters Appearing in the Optimization Problem Parameter
Value
A0 = N0 a TV AI δ SI SD SE θ PE η
10 individuals 0.014/day 10 tons 2 individuals/ton 0.001/day 50€/man-hour 100€/man-hour 10/individual 1/man-hour 0.1 0.0072
relative to detection. The probability of virus establishment PE is supposed constant, but in the next section the model will be analyzed considering different values of PE and θ .
3.3. Results In this section, the solution of the optimization problem (Equation (4)) subject to Equation (1) is studied as a function of the parameters q and γ and for different values of the probability of establishment PE , and of the growth rate a. Different values of θ do not produce significant variations in the parameters α and β and in the total cost. For the sake of simplicity, here and in the following the superscript ∗ , for the optimal search efforts α and β, is omitted. In Fig. 1, the search efforts α and β and the total cost TC as a function of q and γ are represented using the parameter values indicated in Table VI. For small values of the efficacy of search and detection, the search efforts α and β are high and, consequently, also the total cost TC is high. On the contrary, high values of the efficacies q and γ require a smaller search effort and, consequently, a smaller cost. From the comparison between α and β it can be seen that for small values of the efficacy of detection γ (γ < 0.004) and large values of the efficacy of interception q (q > 0.45), it is more convenient to allocate more time to detection than to interception. In all the other cases, it is convenient to allocate more time to interception than to detection. The search effort for detection β does not depend on the efficacy of search in the interception process (Fig. 1). It follows that, for fixed γ , the time (in man-hours) spent in detection is the same for all values of the efficacy of interception q. Moreover, fixing q, it is possible to obtain a graphic representing the mean time of detection as a function of γ (Fig. 3). Increasing the growth rate a, it is necessary to spend more time for both interception and detection, and in particular for detection. For a = 0.04 (Fig. 2), the time spent in detection is more than two times that of the previous case, as the total cost. Moreover, for small values of the efficacy of detection γ , it is always convenient to spend more time in detection than in interception. For γ > 0.02, it is always convenient to spend more time in interception than in detection. In this case, the mean time of detection is less than in the previous case (Fig. 3), due to the fact that the population increases quickly and it is easier to find an infected insect than in the previous case. Increasing the probability of establishment PE there is again an increase in the time spent for control, in particular for interception (Fig. 4). For PE = 0.5, only in a few cases, for high q and small γ , it is more convenient to spend more time in detection than in interception. Moreover, the cost for γ small is approximately five
40
β (man−hours)
α (man−hours)
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity
20 0 1 0.5 q
0.1
40 20 0 1 0.5 q
0.05 0 0
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γ
0.1 0.05 0 0
γ
5
x 10
α β
40
TC
man−hours
5
0 1 0.5 q
0.1 0.05 0 0
γ
20 0 1 0.5 q
0.1 0.05 0 0
γ
Fig. 1. Search effort α (in man-hours) as a function of the efficacies q and γ (top left), search effort β (in man-hours) as a function of the efficacies q and γ (top right) total cost (in €) as a function of the efficacies q and γ (bottom left) and comparison between α and β (bottom right) for values of the parameters in Table VI.
times the cost obtained for PE = 0.1. The mean time of detection is the same obtained for the values in Table VI because the value of β does not depend on PE . 4. DISCUSSION This article presents a simple model for an optimal resources allocation to manage the initial phase of the invasion of a plant pest vectoring plant virus. The model is applied to the invasion of B. tabacitransmitted viruses in Europe. Resources can be allocated to both interception at the point of entry and detection and eradication of locally established populations. For a more realistic description, the probability of arrival of a B. tabaci viruliferous for a new virus is described by a probabilistic model that considers the trade volume and the outcome of the different processes involved in the arrival (association with the commodity at place of origin and survival of the propagules during transport). The establishment phase is described by means of a population dynamics model describing the population growth of the propagule population and the local dispersion eventually resulting in the generation of many locally establishing populations. In
the proposed approach, simple exponential growth has been considered for the vector insect (Equation (1)), but the model can be extended to account for the more complex population dynamics of B. tabaci, as presented in Gilioli et al.(27) The importance of describing more complex patterns of population growth and spread(24) will be considered in a future work. The optimization problem formulated in Section 2 allows to determine the best partition between search effort for interception and for detection/eradication of the vector/insect. The search effort has been studied as a function of the efficacy of search in the interception phase and its efficacy in the detection phase. When the efficacy in both interception and detection takes large values the search efforts required in interception and detection are small. In this case, it is convenient to allocate more effort in interception than in the detection at field level. For low search efficacy of detection and high search efficacy of interception, it is more convenient to allocate more time to the detection than to the interception. This can be explained by the high search efficacy in the interception that requires a small effort. With increasing vector population growth rate a, more time has to be allocated to both interception
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β (man−hours)
α (man−hours)
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20 0 1
0.1
0.5 q
40 20 0 1 0.5 q
0.05 0 0
γ
0.1 0.05 0 0
γ
5
x 10
α β
40
TC
man−hours
5
0 1
0.1
0.5 q
20 0 1 0.5 q
0.05 0 0
γ
0.1 0.05 0 0
γ
38.5
12.83
38.4
12.82 12.81
38.3 mean time of detection
mean time of detection
Fig. 2. Search effort α (in man-hours) as a function of the efficacies q and γ (top left), search effort β (in man-hours) as a function of the efficacies q and γ (top right), total cost (in €) as a function of the efficacies q and γ (bottom left), and comparison between α and β (bottom right) for PE = 0.1, a = 0.04/day.
38.2 38.1 38
12.8 12.79 12.78 12.77 12.76
37.9 12.75
37.8 37.7 0
12.74
0.02
0.04
γ
0.06
0.08
0.1
12.73 0
0.02
0.04
γ
0.06
0.08
0.1
Fig. 3. Mean time of detection as a function of the efficacy γ , for PE = 0.1, a = 0.014/day (left) and for PE = 0.1, a = 0.04/day (right).
and detection/eradication (Fig. 2). This is due to the fact that when vector populations grow rapidly after establishment, an early effective detection is required to be able to eradicate the vector. The increase of the probability of virus establishment results in an increase in the time allocated to the interception phase (Fig. 4). In this case, it is important to improve the interception phase, hence to
prevent the establishment of the insect (which is most likely) and, consequently, that of the virus. The model also allows a calculation of the mean time of detection (Fig. 3). For a growth rate of 0.014/day, the mean time of detection would approximately be 38 days. When the growth rate is increased by approximately three times (0.04/day), the mean time of detection decreases to one-third
40
β (man−hours)
α (man−hours)
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity
20 0 1 0.5 q
0.1
40 20 0 1 0.5 q
0.05 0 0
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γ
0.1 0.05 0 0
γ
5
x 10
α β
man−hours
40 TC
5
0 1 0.5 q
0.1 0.05 0 0
γ
20 0 1 0.5 q
0.1 0.05 0 0
γ
Fig. 4. Search effort α (in man-hours) as a function of the efficacies q and γ (top left), search effort β (in man-hours) as a function of the efficacies q and γ (top right), total cost (in €) as a function of the efficacies q and γ (bottom left), and comparison between α and β (bottom right) for PE = 0.5, a = 0.014/day.
(approximately 12–13 days) because an abundant population is relatively easier to find. The modeling approach here presented allows to determine the optimal allocation of resources to minimize the expected total costs due to the introduction of an invasive species. In the application an insect vectoring plant diseases is considered; this gives greater importance to the initial phase of invasion, since for vectored diseases introduced into new areas eradication efforts have to be concentrated at the very early stage of the invasion, when the vector population is limited in terms of numbers and restricted to a small area. The approach offers the advantages of combining data on trade volume with expert judgment on complex information usually not available (e.g., probabilities of arrival of a disease agent vectored by an insect for different commodities and different area of origin). A further advantage is offered by the possibility to separate search effort spent in interception and search effort spent in detection. This allows comparatively evaluating the efficacy of strategies defined in terms of different allocation of resources to interception and detection. No constraints have been considered on the total amount of resources (i.e., man-hours) available for
searching operations. To make the approach more realistic, an extension of the present research could consider constraints on the search effort or on the budget for interception, detection, and eradication. A further possible extension concerns the population dynamics. In this article, a simple model is presented based on the exponential growth of the vector insect, but more complex population dynamics aspects can be considered, for instance, a stage-structured population dynamics(28) or the temperature-dependent responses of development, mortality, and fecundity.(29) A further level of complexity can be introduced modeling the transmission of the virus to the host plant and the way in which the virus per se is able to establish (e.g., considering the competition with the already established virus strains). The approach here proposed could be useful to risk managers to plan control activities and to provide support for decision making in plant biosecurity. ACKNOWLEDGMENTS The authors would like to thank the members of the EFSA Panel on Plant Health and specifically the working group on B. tabaci for helpful suggestions and critical discussions. The authors are also
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grateful to two anonymous reviewers for their constructive comments that allowed to improve the article. APPENDIX: DETERMINATION OF THE OPTIMAL SEARCH EFFORT To find the minimum in Equation (4) it is necessary to calculate the first derivatives of the functional to be minimized with respect to α and β, and the Hessian matrix. Setting the derivative with respect to α equal to zero gives: qα
2αθ SI − q PE η AI e− TV N0
θβ 2 SD + γβ SE = 0, δ + γβ − a
and setting the derivative with respect to β equal to zero: qα
PE η AI TVe− TV N0 ·
θ γβ 2 SD + 2θ (δ − a)β SD + γ (δ − a)SE = 0. (δ + γβ − a)2
The system of these two equations allows to obtain β as in Equation (6) and α as solution to Equation (7). Values of α and β obtained represent the minimum in Equation (4) if the first element of the Hessian matrix and its determinant are positive. The second derivative of TC in Equation (2) with respect to α, which is the first element of the Hessian matrix, is given by: 2θ SI +
qα θβ 2 SD + γβ SE q2 PE η AI e− TV N0 TV δ + γβ − a
and it is positive under the assumption Equation (5). The second derivative with respect to β is: 2(δ − a) θ (δ − a)SD − γ 2 SE qα − TV PE η AI TVe N0 (δ + γβ − a)3 while the second derivative with respect to α and β is: qα
−PE η AI qe− TV N0 · θγβ
2
SD+2θ(δ−a)β SD+γ (δ−a)SE . (δ+γβ−a)2
It follows that the determinant of the Hessian matrix is positive and α and β satisfying Equation (6) and (7) represent the minimum in Equation (4). REFERENCES 1. Gennadius P. Disease of tobacco plantations in the Trikonia. The aleurodid of tobacco. Ellenike Georgia, 1889; 5:1–3. 2. Stansly PA, Naranjo S. Bemisia: Bionomics and Management of a Global Pest. Dordrecht: Springer, 2010.
3. Dinsdale A, Cook L, Riginos C, Buckley Y, De Barro P. Refined global analysis of Bemisia tabaci (Hemiptera: Sternorrhyncha: Aleyrodoidea: Aleyrodidae) mitochondrial cytochrome oxidase 1 to identify species level genetic boundaries. Annals of the Entomological Society of America, 2010; 103:196–208. 4. Brown JK. Phylogenetic biology of the Bemisia tabaci sibling species group. Pp. 31–67 in Stansly PA, Naranjo SE (eds). Bemisia: Bionomics and Management of a Global Pest. Dordrecht: Springer, 2010. ´ 5. Navas-Castillo J, Fiallo-Olive´ E, Sanchez-Campos S. Emerging virus diseases transmitted by whiteflies. Annual Review of Phytopathology, 2011; 49:219–248. 6. Moriones E, Navas-Castillo J. Tomato yellow leaf curl virus, an emerging virus complex causing epidemics worldwide. Virus Research, 2000, 71:123–134. 7. Celix A, Lopez-Sese A, Almarza N, Gomez-Guillamon ML, Rodriguez-Cerezo E. Characterization of cucurbit yellow stunting disorder virus, a Bemisia tabaci transmitted closterovirus. Phytopathology, 1996; 86:1370–1376. 8. Andersen MC, Adams H, Hope B, Powell M. Risk assessment for invasive species. Risk Analysis, 2004; 24(3): 787–793. 9. Epanchin-Niell RS, Hastings A. Controlling established invaders: Integrating economics and spread dynamics to determine optimal management. Ecology Letters, 2010; 13:528–541. 10. Carrasco LR, Baker R, MacLeod A, Knight JD, Mumford JD. Optimal and robust control of invasive alien species spreading in homogeneous landscape. Journal of the Royal Society Interface, 2010; 7:529–540. 11. Moore JL, Rout TM, Hauser CE, Moro D, Jones M, Wilcox C, Possingham HP. Protecting islands from pest invasion: Optimal allocation of biosecurity resources between quarantine and surveillance. Biological Conservation, 2010; 143:1068– 1078. 12. Rout TM, Moore JL, Possingham HP, McCarthy MA. Allocating biosecurity resources between preventing, detecting, and eradicating island invasions. Ecological Economics, 2011; 71:54–62. 13. Mehta SV, Haight RG, Homans FR, Polasky S, Venette RC. Optimal detection and control strategies for invasive species management. Ecological Economics, 2007; 61:237– 245. 14. Yokomizo H, Possingham HP, Thomas MB, Buckley YM. Managing the impact of invasive species: The value of knowing the density-impact curve. Ecological Applications, 2009; 19(2): 376–386. 15. Epanchin-Niell RS, Haight RG, Berec L, Kean JM, Liebhold AM. Optimal surveillance and eradication of invasive species in heterogeneous landscapes. Ecology Letters, 2012; 15:803– 812. 16. Cacho OJ, Hester SM. Deriving efficient frontiers for effort allocation in the management of invasive species. Australian Journal of Agricultural and Resource Economics, 2011; 55:72– 89. 17. Homans F, Horie T. Optimal detection strategies for an established invasive pest. Ecological Economics, 2011; 70:1129– 1138. 18. Sanchirico JN, Albers HJ, Fischer C, Coleman C. Spatial management of invasive species: Pathways and policy options. Environmental and Resource Economics, 2010; 45:517–535. 19. Hauser CE, McCarthy MA. Streamlining “search and destroy”: Cost-effective surveillance for invasive species management. Ecology Letters, 2009; 12:683–692. 20. Bartell SM, Nair SK. Establishment risks for invasive species. Risk Analysis, 2003; 24(4): 833–845. 21. Hulme PE. Trade, transport and trouble: Managing invasive species pathways in an era of globalization. Journal of Applied Ecology, 2009; 46:10–18.
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity 22. Meyerson LA, Mooney HA. Invasive alien species in an era of globalization. Frontiers in Ecology and the Environment, 2007; 5:199–208. 23. Perrings C, Dehnen-Schmutz K, Touza J, Williamson M. How to manage biological invasions under globalization. Trends in Ecology and Evolution, 2005; 20(5): 212–215. 24. Gilioli G, Pasquali S, Tramontini S, Riolo F. Modelling local and long-distance dispersal of invasive chestnut gall wasp in Europe. Ecological Modelling, 2013; 263:281–290. 25. Shigesada N, Kawasaki K, Takeda Y. Modeling stratified diffusion in biological invasions. American Naturalist, 1995; 146(2): 229–251. 26. EFSA Panel on Plant Health (PLH). Scientific opinion on the risks to plant health posed by Bemisia tabaci species complex
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and viruses it transmits for the EU territory. EFSA Journal, 2013; 11(4):3162, 302pp. doi:10.2903/j.efsa.2013. 3162. Available at: http://www.efsa.europa.eu/efsajournal. 27. Gilioli G, Pasquali S, Parisi S, Winter S. Modelling the potential distribution of Bemisia tabaci in Europe in light of the climate change scenario. Pest Management Science, 2014. doi:10.1002/ps.3734. 28. Buffoni G, Pasquali S. Structured population dynamics: Continuous size and discontinuous stage structures. Journal of Mathematical Biology, 2007; 54:555– 595. 29. Gilioli G, Pasquali S. Use of individual-based models for population parameters estimation. Ecological Modelling, 2007; 200:109–118.
DOI: 10.1111/risa.12278
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity Sara Pasquali,1,∗ Gianni Gilioli,2 Dirk Janssen,3 and Stephan Winter4
The introduction of invasive species causes damages from the economic and ecological point of view. Interception of plant pests and eradication of the established populations are two management options to prevent or limit the risk posed by an invasive species. Management options generate costs related to the interception at the point of entry, and the detection and eradication of established field populations. Risk managers have to decide how to allocate resources between interception, field detection, containment, and eradication minimizing the expected total costs. In this work is considered an optimization problem aiming at determining the optimal allocation of resources to minimize the expected total costs of the introduction of Bemisia tabaci-transmitted viruses in Europe. The optimization problem takes into account a probabilistic model for the estimation of the percentage of viruliferous insect populations arriving through the trade of commodities, and a population dynamics model describing the process of the vector populations’ establishment and spread. The time of field detection of viruliferous insect populations is considered as a random variable. The solution of the optimization problem allows to determine the optimal allocation of the search effort between interception and detection/eradication. The behavior of the search effort as a function of efficacy or search in interception and in detection is then analyzed. The importance of the vector population growth rate and the probability of virus establishment are also considered in the analysis of the optimization problem. KEY WORDS: Bemisia tabaci; invasive species; optimization; population dynamics; probabilistic model
1. INTRODUCTION
morphocryptic species(3) some of these, Med and MEAM1, occur in Europe and are highly invasive. B. tabaci is included in the 100 World’s Worst Invasive Species list (http://www.issg.org). These insects typically exhibit resistance to certain insecticides, have high fecundity and spread capacity, and a broader plant host range.(4) B. tabaci also transmits globally over 200 plant virus species, belonging to five different genera: Begomovirus, Crinivirus, Ipomovirus, Torradovirus, and Carlavirus.(5) Many of these viruses cause serious diseases in crops, and hence crop production requires extensive protection measures. The importance of newly emerging plant viruses is then directly related to the adaptability of invasive B. tabaci to new agricultural habitats, in open fields, and to greenhouses
The whitefly Bemisia tabaci(1) is a polyphagous sap-sucking insect commonly found in tropical and subtropical regions(2) and it is a serious pest of agricultural crops and ornamental plants. It causes direct damage by its feeding activity, leading to reduced host growth and yield, and indirectly by transmitting viruses. B. tabaci occurs worldwide and it constitutes a complex of at least 28 indistinguishable 1 CNR-IMATI, 2 DMMT,
Via Bassini 15, 20133 Milano, Italy. University of Brescia, Viale Europa 11, 25123 Brescia,
Italy. 3 IFAPA,
Centro La Mojonera, Almer´ıa, Spain. Institut DSMZ, Plant Virus Department, Germany. ∗ Address correspondence to Sara Pasquali, CNR-IMATI, Via Bassini 15, 20133 Milano, Italy; [email protected]. 4 Leibniz
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C 2014 Society for Risk Analysis 0272-4332/15/0100-1663$22.00/1
1664 of temperate climates where much of the vegetable production occurs in northern regions of Europe. To date, only 10 of these emerging B. tabacitransmitted viruses have been found in Europe, such as tomato yellow leaf curl begomovirus, in tomato,(6) and cucurbit yellow stunting disorder crinivirus in cucurbits(7) and those are mostly confined to the Mediterranean region. Nevertheless, the introduction of new viruses currently not present in the European Union presents a constant risk to crop production. Three major processes characterize biological invasions of plant pests like B. tabaci and the viruses it transmits. During the entry phase, a nonindigenous species arrives in an area, begins to reproduce, and builds up a local population, leading to the establishment. Locally established populations can grow numerically and spread by invading new areas suitable for establishment, causing irreversible ecological and economic damage.(8) Hence, preventing the introduction of new species and eradicating populations at establishment are the only strategies to prevent the risk posed by B. tabaci and the viruses it transmits. In recent years, the management of invasive species has been widely considered, and methods and approaches to decision making during different phases of invasion have been proposed. EpanchinNiell and Hastings(9) give a complete review of studies on economically optimal control of established invasive species. Three aspects of prevention through quarantine, surveillance, and removal have been taken into account in optimal management problems.(10–12) A model developed by Mehta et al.(13) captures also the stochasticity of the detection time and combines the population density with the economic impact. The “density-impact curve” is treated also by Yokomizo et al.(14) Also, spatial aspects are considered in management of invasive species.(15–19) In this article, the problem of optimal allocation of the search and control efforts aiming at preventing introduction and establishment of a new virus vectored by an insect is considered. The insect and the virus can enter into new areas via the international trade of plant products and planting materials. Interception is the opportunity to prevent entry of the vector and the virus in infected plants at the point of entry. If the insect escapes interception it may establish viable populations and, provided suitable conditions, populations grow and subsequently spread. The objective of this study is to determine the optimal allocation of resources between interception, detection
Pasquali et al. in the field, and eradication. For the sake of simplicity, containment is not taken into account, also because the wide host range and the high spread capacity of the vector make this option not feasible in the long term. The main issue is to minimize the expected total costs, which are composed of interception costs, detection costs up to the time of detection, and eradication costs depending on the population abundance at the time of detection. The time of detection is considered stochastic(13) and it is supposed that once the detection takes place, eradication is successfully carried out. This simplifying assumption is justified in an initial phase of establishment when dispersal is only limited around the area of entry and control measures can be highly effective. The damage costs, useful to encompass the range of costs associated with large population,(15) are not taken into account here because detection and eradication are supposed in the early stage of invasion. If this is not the case and the virus escapes interception and eradication in the early stage of establishment or spread, it causes an economic damage far more important that the costs for the management of invasion, and the control of the vector is performed by integrated pest management (IPM) tactics and strategies. The optimization problem, aimed at determining the optimal allocation of resources to minimize the expected total cost, is presented in Section 2 and it takes into account a population growth modeled through a differential equation. The importance of considering the population dynamics in management problems has been pointed out by other authors.(13,14,17,20) An application in plant biosecurity is considered in Section 3. In particular, as a case study the arrival and establishment of new B. tabacitransmitted viruses in Europe is considered, taking into account the trade volume of commodities carrying the insect and the percentage of viruliferous insect populations estimated by means of a probabilistic model. Other works remark on the importance of taking into account the trade in management invasive species.(21–23) Results of the optimization problem are presented in Section 3.3 considering the variation in the parameters of cost functions. Changes in the growth rate of the population and in the virus probability establishment have been considered and the results are discussed in Section 4. 2. OPTIMIZATION PROBLEM This section is devoted to the analysis of the resource allocation problem. The optimization
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity problem aims to minimize the costs deriving from different allocation of the search effort between the vector interception at the point of entry and the detection/eradication of established vector populations. Eradication costs depend on population abundance, which in turn depends on the time of detection. Before setting the optimal control problem it is necessary to make some definitions and assumptions.
r A single new virus is considered and is supposed r
r r r
to be vectored by an insect species. A propagule population of the vector is defined as the population of abundance A0 corresponding to the minimum number of individuals surviving the arrival process and able to establish a local population. The population abundance of the propagules is supposed to be constant until reaching the area of potential establishment. An area of potential establishment is an area with suitable environmental conditions (e.g., temperature) and availability of host plants. As soon as the propagule arrives in a suitable area it establishes a local population (establishment phase). The local population growths and the population abundance N(t) (with N(0) = A0 ) follows the dynamics: dN(t) = a N(t). dt
(1)
Equation (1) accounts for two processes, the local population growth and the possibility that local populations give rise to other local populations by a discontinuous spread process (branching process). The global pattern of population growth, which takes into account both short - and long-distance dispersal, is not limited by local constraints and it is well approximated, in the first phase of colonization, by exponential growth.(24,25)
r The probability of transmitting the virus to the
r
host plants in the area of establishment is related to the prevalence of the infection in the vector population and population abundance, and it also changes over time. The control operations may occur in three phases: at the point of entry (interception, I), searching for establishing local populations (detection, D), and eliminating local populations (eradication, E). Detection and eradication occur at the same time.
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r The r r
costs of eradication depend on the total population abundance. Economic losses to producers due, for example, to consignments rejected at port of entry are not considered here. The optimization problem does not take into account budget constraints for interception, detection, and eradication.
Denoting by α the search effort for the interception phase measured in man-hours at time t, by θ the productivity per man-hour, and by SI the interception cost for a man-hour, the total interception cost is supposed to be quadratic in the search effort: α 2 θ SI . The quadratic cost reflects, for example, overtime pay for increased surveying for more intensive interception effort.(13,17) Let TVt be the trade volume at time t, and q the search sensitivity of the interception process related to the type of goods, measured in ton/man-hour. Parameter q represents a measure of the efficacy of an operator in intercepting a pest considering a unit of search effort (man-hour) and a pest density one (A0 for unit of trade volume). Then, exp(−qα/TV) is the probability that a propagule population survives the interception phase. For the sake of simplicity, the trade volume is supposed constant over time. Denote by SP = AI TV exp(−qα/TV) the number of propagule populations that survive the interception phase, where AI is the number of propagule populations per unit of trade volume. The number of propagule populations that survive the interception phase is equal to the number of establishing local populations SP of dimension A0 at the end of the process of arrival. Let PE ηSP be the number of local foci of virus infection, depending on the number of propagule populations SP establishing in suitable areas, on the probability of virus establishment PE , and on the percentage of viruliferous insect populations η. Denote by β the search effort for the detection of established populations at time t (in man-hours), that is, the number of man-hours allocated to search an establishing insect vector population, by SD the cost unit per man-hour for the detection, and by SE the eradication cost (including labor and the use of control techniques) per individual. Let τ denote the time of detection, which depends on the search effort for the detection β and on the efficacy of the detection process γ , that is, the number of viruliferous individuals found per man-hour. The time of detection is distributed according to an exponential
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Pasquali et al.
distribution(13) and its probability density function is given by γβ exp(−γβτ ) for τ > 0. The expected discounted total cost composed of the costs of interception, detection, and eradication is given by: TC(α, β) = α 2 θ SI ∞ β 2 θ SD + 0
τ
qα
+SE PE η AI TVe
qα − TV
e−δτ N(τ ) γβe−γβτ dτ, (2)
where PE AI TVexp(−qα/TV)N(t) is the dimension of the establishing populations to be eradicated and δ is the discount rate. The objective is to find the optimal search strategy (α ∗ , β ∗ ) that minimize TC with constraints: α > 0,
β>0
Commodity C1 C2 C3
e−δt PE η AI TVe− TV N(t)dt
0
Table I. Notation for the Commodities and Macro-Areas from Which the Commodities Originate
A1 A2 A3 A4 A5 A6 A7 A8 A9
(3) Table II. Correspondence Between Values Assumed by the Variable “Arrival of Viruses with Bemisia” and Intervals Representing the Percentage of Infested Commodities
and Equation (1), that is: TC(α ∗ , β ∗ ) = min TC(α, β), α>0,β>0
(4)
subject to Equation (1). To solve the optimization problem (Equation (4)) subject to Equation (1), it is necessary to compute the first derivatives with respect to α and β and the determinant of the Hessian matrix (see the Appendix for details). Under the condition: δ < a < δ + γβ,
and α ∗ solution to: qα ∗
θ (β ∗ )2 SD +γβ ∗ SE = 0. δ+γβ ∗ − a
“Arrival of Viruses with Bemisia”
Percentage of Infested Commodities in the Interval
= very low = low = medium = high = very high
[0, 0.001] [0.001, 0.01] [0.01, 0.05] [0.05, 0.2] [0.2, 1]
H1 H2 H3 H4 H5
(5)
the first derivatives of TC with respect to α and β are null for: θ SD(a − δ) β∗ = θ γ SD 2 θ 2 (a − δ)2 SD + θ γ 2 (a − δ)SE SD + (6) θ γ SD 2α ∗ θ SI − q PE η AI e− TV N0
Plants for planting Cut flowers Fruits and leafy vegetables Macro-Area Africa North Africa SS Asia Australia Non - EU Europe Near East North America Oceania South America
(7)
Furthermore, it can be easily seen that the second derivative with respect to α is positive and the determinant of the Hessian matrix is positive (see the Appendix). Consequently, the minimum of TC(α, β) is reached in (α ∗ , β ∗ ) solution to Equtions (6)–(7). 3. APPLICATION TO RISK ASSESSMENT AND MANAGEMENT IN PLANT BIOSECURITY In this section, an application to the problem of preventing the invasion in Europe by B. tabaci-
transmitted viruses is considered. These viruses can enter into the risk assessment area with the vector insect or with the infected host plants. Because the focus is on a circulative virus, the vector insect plays a major role in the arrival of the virus; therefore, in this application only the contribution of B. tabaci to the virus arrival is considered. To estimate the value of η, the percentage of viruliferous individuals in the propagule population, a probabilistic model is applied. The model takes into account the trade volume and the estimation, based on expert judgment, of the outcome of the different processes involved in the arrival. This model is illustrated in Section 3.1. For further details on the probabilistic model of arrival, refer to the EFSA (European Food Safety Authority) Opinion on B. tabaci.(26) 3.1. Arrival Probability for a Virus by Infected B. tabaci The probability of arrival of a virus with B. tabaci is calculated using a formula combining expert judgment with data on the trade volumes of
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity
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Table III. Import Volumes in EU (in Tons) Relative to the Three Commodities, Summarized for Different Macro-Areas of Origin
C1 C2 C3 TOT.
A1
A2
A3
A4
A5
A6
A7
A8
A9
TOT.
3,671.97 1,387.73 1,641,167 1,646,227
6,932.57 1,558.27 1,969,635 1,978,126
30,621.73 1,476.03 1,207,587 1,239,685
812.07 254.5 77,175.73 78,242.3
5,370.43 1,967.43 1,655,379 1,662,717
3,656.8 8,216.63 692,715.3 704,588.7
18,221.97 38,532.73 1,633,474 1,690,229
92.47 0.7 400,276.8 400,370
28,475.03 19,440.13 7,020,722 7,068,637
97,855.03 72,834.17 16,298,132 16,468,821
different types of commodities. The model assumes that the probability of arrival depends on the type of commodity and on the geographical macro-area from which the commodities originate. Three different types of commodity (plants for planting, cut flowers, fruits, and leafy vegetables) have been considered and denoted by C j , j = 1, 2, 3. Nine different geographical macro-areas in the world are defined according to the homogeneity in the type and frequency of infestation of the commodities at the origin. The macro-areas are denoted by Ai , i = 1, 2, ..., 9. The arrival probability can be calculated for each different commodity separately or for all the commodities considered together. In the following, it is shown how to compute the probabilities for the latter case, the extension to the case of a single commodity is trivial. Moreover, it is possible to calculate the probabilities for the whole Europe and for the different EU countries. The notation reported in Table I will be used. The random variable “arrival of viruses with Bemisia” is supposed to assume five possible values: H1 = very low, H2 = low, H3 = medium, H4 = high, and H5 = very high. In Table II, a correspondence between these values and numerical intervals representing the percentage of infested commodities is given. The probability of arrival of viruses with B. tabaci can be calculated as:
=
P(arrival of viruses with Bemisia = Hk) 3 i=1 j=1 P(arrival of viruses with Bemisia = Hk|
9
arrival of C j from Ai )P(arrival of C j from Ai ), (8) where P(arrival of viruses with Bemisia = Hk|arrival of Cj from Ai ), that is, the probability that the arrival of viruses with Bemisia coming from the macro-area Ai with the commodity C j is Hk depends on expert evaluation. A panel of experts provided estimation of the probability of association of viruliferous vectors with the commodity at place of
origin and survival of the propagules during transport. The probability of arrival of C j coming from Ai , P(arrival of C j from Ai ), is given by “n. of plants C j coming from Ai /total n. of plants.” The values P(arrival of viruses with Bemisia = Hk) allows to obtain the probability distribution of the variable “arrival of viruses with Bemisia.” To estimate the value η representing the percentage of viruliferous individuals, the mean of the variable “arrival of viruses with Bemisia” is computed choosing as representative point of an interval its middle point. 3.2. Data To calculate the probabilities of arrival, it is necessary to know the trade matrix and the expert judgment on the probabilities of arrival with respect to a given commodity and macro-area of origin. The trade matrix reporting the trade volume in EU relative to different commodities and macro-areas is reported in Table III. Data on trade volume allow the computation of the second probability in Equation (8). An estimation of the arrival probabilities to be used in the computation of the first probability in Equation (8) is formulated by a panel of experts on B. tabaci and it is reported in Table IV.(26) These probabilities are used in combination with data on trade volume of commodities to estimate the probability of arrival of a virus with B. tabaci. The percentage η of viruliferous insect populations is calculated applying Equation (8). The probability distribution of the variable “arrival of viruses with Bemisia” is reported in Table V together with its mean used as an estimate of η. In the optimization problem, η is set equal to 7.2/1000. Other data necessary to solve the optimization problem are summarized in Table VI. It is supposed that the interception cost is smaller than the detection cost because the control in the interception phase is less heavy than the control
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Table IV. Probabilities of Arrival of a Virus with B. tabaci for Different Commodities and Macro-Areas A1
A2
A3
A4
A5
A6
A7
A8
A9
Very low
C1 C2 C3
0 0.7 0.8
0 0.5 0.9
0 0.5 0.8
0 1 1
0 1 1
0 0.6 0.8
0 1 1
0 1 1
0 0.7 0.9
Low
C1 C2 C3
0.2 0.3 0.2
0.2 0.5 0.1
0.2 0.5 0.2
0.2 0 0
0.2 0 0
0.2 0.4 0.2
0.2 0 0
0 0 0
0.2 0.3 0.1
Medium
C1 C2 C3
0.8 0 0
0.8 0 0
0.8 0 0
0.8 0 0
0.8 0 0
0.8 0 0
0.8 0 0
0.2 0 0
0.8 0 0
High
C1 C2 C3
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0.8 0 0
0 0 0
Very high
C1 C2 C3
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
0 0 0
Table V. Probability Distribution of the Variable “Arrival of Viruses with Bemisia” (Columns 1–5) for the Whole Europe and Mean of the Distribution (Column 6) Very Low 0.5336
Low
Medium
High
0.2822
0.01448
0.0393
Very High 0
Mean Value 7.2/1000
Table VI. Values of the Parameters Appearing in the Optimization Problem Parameter
Value
A0 = N0 a TV AI δ SI SD SE θ PE η
10 individuals 0.014/day 10 tons 2 individuals/ton 0.001/day 50€/man-hour 100€/man-hour 10/individual 1/man-hour 0.1 0.0072
relative to detection. The probability of virus establishment PE is supposed constant, but in the next section the model will be analyzed considering different values of PE and θ .
3.3. Results In this section, the solution of the optimization problem (Equation (4)) subject to Equation (1) is studied as a function of the parameters q and γ and for different values of the probability of establishment PE , and of the growth rate a. Different values of θ do not produce significant variations in the parameters α and β and in the total cost. For the sake of simplicity, here and in the following the superscript ∗ , for the optimal search efforts α and β, is omitted. In Fig. 1, the search efforts α and β and the total cost TC as a function of q and γ are represented using the parameter values indicated in Table VI. For small values of the efficacy of search and detection, the search efforts α and β are high and, consequently, also the total cost TC is high. On the contrary, high values of the efficacies q and γ require a smaller search effort and, consequently, a smaller cost. From the comparison between α and β it can be seen that for small values of the efficacy of detection γ (γ < 0.004) and large values of the efficacy of interception q (q > 0.45), it is more convenient to allocate more time to detection than to interception. In all the other cases, it is convenient to allocate more time to interception than to detection. The search effort for detection β does not depend on the efficacy of search in the interception process (Fig. 1). It follows that, for fixed γ , the time (in man-hours) spent in detection is the same for all values of the efficacy of interception q. Moreover, fixing q, it is possible to obtain a graphic representing the mean time of detection as a function of γ (Fig. 3). Increasing the growth rate a, it is necessary to spend more time for both interception and detection, and in particular for detection. For a = 0.04 (Fig. 2), the time spent in detection is more than two times that of the previous case, as the total cost. Moreover, for small values of the efficacy of detection γ , it is always convenient to spend more time in detection than in interception. For γ > 0.02, it is always convenient to spend more time in interception than in detection. In this case, the mean time of detection is less than in the previous case (Fig. 3), due to the fact that the population increases quickly and it is easier to find an infected insect than in the previous case. Increasing the probability of establishment PE there is again an increase in the time spent for control, in particular for interception (Fig. 4). For PE = 0.5, only in a few cases, for high q and small γ , it is more convenient to spend more time in detection than in interception. Moreover, the cost for γ small is approximately five
40
β (man−hours)
α (man−hours)
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity
20 0 1 0.5 q
0.1
40 20 0 1 0.5 q
0.05 0 0
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γ
0.1 0.05 0 0
γ
5
x 10
α β
40
TC
man−hours
5
0 1 0.5 q
0.1 0.05 0 0
γ
20 0 1 0.5 q
0.1 0.05 0 0
γ
Fig. 1. Search effort α (in man-hours) as a function of the efficacies q and γ (top left), search effort β (in man-hours) as a function of the efficacies q and γ (top right) total cost (in €) as a function of the efficacies q and γ (bottom left) and comparison between α and β (bottom right) for values of the parameters in Table VI.
times the cost obtained for PE = 0.1. The mean time of detection is the same obtained for the values in Table VI because the value of β does not depend on PE . 4. DISCUSSION This article presents a simple model for an optimal resources allocation to manage the initial phase of the invasion of a plant pest vectoring plant virus. The model is applied to the invasion of B. tabacitransmitted viruses in Europe. Resources can be allocated to both interception at the point of entry and detection and eradication of locally established populations. For a more realistic description, the probability of arrival of a B. tabaci viruliferous for a new virus is described by a probabilistic model that considers the trade volume and the outcome of the different processes involved in the arrival (association with the commodity at place of origin and survival of the propagules during transport). The establishment phase is described by means of a population dynamics model describing the population growth of the propagule population and the local dispersion eventually resulting in the generation of many locally establishing populations. In
the proposed approach, simple exponential growth has been considered for the vector insect (Equation (1)), but the model can be extended to account for the more complex population dynamics of B. tabaci, as presented in Gilioli et al.(27) The importance of describing more complex patterns of population growth and spread(24) will be considered in a future work. The optimization problem formulated in Section 2 allows to determine the best partition between search effort for interception and for detection/eradication of the vector/insect. The search effort has been studied as a function of the efficacy of search in the interception phase and its efficacy in the detection phase. When the efficacy in both interception and detection takes large values the search efforts required in interception and detection are small. In this case, it is convenient to allocate more effort in interception than in the detection at field level. For low search efficacy of detection and high search efficacy of interception, it is more convenient to allocate more time to the detection than to the interception. This can be explained by the high search efficacy in the interception that requires a small effort. With increasing vector population growth rate a, more time has to be allocated to both interception
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β (man−hours)
α (man−hours)
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0.1
0.5 q
40 20 0 1 0.5 q
0.05 0 0
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0.1 0.05 0 0
γ
5
x 10
α β
40
TC
man−hours
5
0 1
0.1
0.5 q
20 0 1 0.5 q
0.05 0 0
γ
0.1 0.05 0 0
γ
38.5
12.83
38.4
12.82 12.81
38.3 mean time of detection
mean time of detection
Fig. 2. Search effort α (in man-hours) as a function of the efficacies q and γ (top left), search effort β (in man-hours) as a function of the efficacies q and γ (top right), total cost (in €) as a function of the efficacies q and γ (bottom left), and comparison between α and β (bottom right) for PE = 0.1, a = 0.04/day.
38.2 38.1 38
12.8 12.79 12.78 12.77 12.76
37.9 12.75
37.8 37.7 0
12.74
0.02
0.04
γ
0.06
0.08
0.1
12.73 0
0.02
0.04
γ
0.06
0.08
0.1
Fig. 3. Mean time of detection as a function of the efficacy γ , for PE = 0.1, a = 0.014/day (left) and for PE = 0.1, a = 0.04/day (right).
and detection/eradication (Fig. 2). This is due to the fact that when vector populations grow rapidly after establishment, an early effective detection is required to be able to eradicate the vector. The increase of the probability of virus establishment results in an increase in the time allocated to the interception phase (Fig. 4). In this case, it is important to improve the interception phase, hence to
prevent the establishment of the insect (which is most likely) and, consequently, that of the virus. The model also allows a calculation of the mean time of detection (Fig. 3). For a growth rate of 0.014/day, the mean time of detection would approximately be 38 days. When the growth rate is increased by approximately three times (0.04/day), the mean time of detection decreases to one-third
40
β (man−hours)
α (man−hours)
Optimal Strategies for Interception, Detection, and Eradication in Plant Biosecurity
20 0 1 0.5 q
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40 20 0 1 0.5 q
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0.1 0.05 0 0
γ
5
x 10
α β
man−hours
40 TC
5
0 1 0.5 q
0.1 0.05 0 0
γ
20 0 1 0.5 q
0.1 0.05 0 0
γ
Fig. 4. Search effort α (in man-hours) as a function of the efficacies q and γ (top left), search effort β (in man-hours) as a function of the efficacies q and γ (top right), total cost (in €) as a function of the efficacies q and γ (bottom left), and comparison between α and β (bottom right) for PE = 0.5, a = 0.014/day.
(approximately 12–13 days) because an abundant population is relatively easier to find. The modeling approach here presented allows to determine the optimal allocation of resources to minimize the expected total costs due to the introduction of an invasive species. In the application an insect vectoring plant diseases is considered; this gives greater importance to the initial phase of invasion, since for vectored diseases introduced into new areas eradication efforts have to be concentrated at the very early stage of the invasion, when the vector population is limited in terms of numbers and restricted to a small area. The approach offers the advantages of combining data on trade volume with expert judgment on complex information usually not available (e.g., probabilities of arrival of a disease agent vectored by an insect for different commodities and different area of origin). A further advantage is offered by the possibility to separate search effort spent in interception and search effort spent in detection. This allows comparatively evaluating the efficacy of strategies defined in terms of different allocation of resources to interception and detection. No constraints have been considered on the total amount of resources (i.e., man-hours) available for
searching operations. To make the approach more realistic, an extension of the present research could consider constraints on the search effort or on the budget for interception, detection, and eradication. A further possible extension concerns the population dynamics. In this article, a simple model is presented based on the exponential growth of the vector insect, but more complex population dynamics aspects can be considered, for instance, a stage-structured population dynamics(28) or the temperature-dependent responses of development, mortality, and fecundity.(29) A further level of complexity can be introduced modeling the transmission of the virus to the host plant and the way in which the virus per se is able to establish (e.g., considering the competition with the already established virus strains). The approach here proposed could be useful to risk managers to plan control activities and to provide support for decision making in plant biosecurity. ACKNOWLEDGMENTS The authors would like to thank the members of the EFSA Panel on Plant Health and specifically the working group on B. tabaci for helpful suggestions and critical discussions. The authors are also
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grateful to two anonymous reviewers for their constructive comments that allowed to improve the article. APPENDIX: DETERMINATION OF THE OPTIMAL SEARCH EFFORT To find the minimum in Equation (4) it is necessary to calculate the first derivatives of the functional to be minimized with respect to α and β, and the Hessian matrix. Setting the derivative with respect to α equal to zero gives: qα
2αθ SI − q PE η AI e− TV N0
θβ 2 SD + γβ SE = 0, δ + γβ − a
and setting the derivative with respect to β equal to zero: qα
PE η AI TVe− TV N0 ·
θ γβ 2 SD + 2θ (δ − a)β SD + γ (δ − a)SE = 0. (δ + γβ − a)2
The system of these two equations allows to obtain β as in Equation (6) and α as solution to Equation (7). Values of α and β obtained represent the minimum in Equation (4) if the first element of the Hessian matrix and its determinant are positive. The second derivative of TC in Equation (2) with respect to α, which is the first element of the Hessian matrix, is given by: 2θ SI +
qα θβ 2 SD + γβ SE q2 PE η AI e− TV N0 TV δ + γβ − a
and it is positive under the assumption Equation (5). The second derivative with respect to β is: 2(δ − a) θ (δ − a)SD − γ 2 SE qα − TV PE η AI TVe N0 (δ + γβ − a)3 while the second derivative with respect to α and β is: qα
−PE η AI qe− TV N0 · θγβ
2
SD+2θ(δ−a)β SD+γ (δ−a)SE . (δ+γβ−a)2
It follows that the determinant of the Hessian matrix is positive and α and β satisfying Equation (6) and (7) represent the minimum in Equation (4). REFERENCES 1. Gennadius P. Disease of tobacco plantations in the Trikonia. The aleurodid of tobacco. Ellenike Georgia, 1889; 5:1–3. 2. Stansly PA, Naranjo S. Bemisia: Bionomics and Management of a Global Pest. Dordrecht: Springer, 2010.
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