Modelling the impact and control of an infectious disease in a plant nursery with infected plant material inputs

Highlights • The plant trade is an important pathway for the spread of plant pathogens.• We model a nursery that constantly buys, grows and sells potentially infected plants• Analytic results give optimal levels of two management tools, restriction and removal.• For not very infectious diseases, removal and restriction are substitutes.• For highly infectious diseases, removal and restriction can be complements.


Introduction
Increases in the movement of people and traded goods as a consequence of 2 globalisation have led to growing concerns about the threat posed by invasive species. 3 especially invasive pathogens of humans, plants and animals [e.g. [1][2][3][4][5]. Recent disease 4 outbreaks in plants, such as the Chalara fungus (Hymenoscyphus pseudoalbidus) 5 affecting ash trees across Europe [6] and the oomycete Phytophthora ramorum 6 affecting many plants including larch in Europe [7] and oaks in the US [8], have 7 focused attention on the policy options to reduce the risks of similar plant disase 8 outbreaks occurring in the future, and the management options to reduce damage 9 from newly established pathogen populations. These disease outbreaks have also 10 raised concerns about patterns of plant trade, which has been identified as a key 11 introduction pathway for invasive pathogens [9], and on the need for a more prominent 12 role of the private sector in biosecurity practices to mitigate existing risk [10]. 13 Understanding the economic impacts of damage and mitigation is essential for 14 determining optimal policy and management options for invasive pathogens [11]. 15 The body of the literature that combines invasion ecology with economic analysis 16 to deal with these issues has drastically increased in the last decade (for an overview 17 see [12,13]). Bioeconomic studies explore the management problem from a central 18 authority perspective, focusing on the potential social welfare benefits from policy 19 intervention to limit the risk of invasive species damages using instruments that 20 include port inspections, quarantine and import tariffs [14,15], import risk screening 21 programs [16,17], the use of public funds to detect, eradicate and/or control 22 established invaders, and habitat restoration [e.g. [18][19][20]. Other studies have 23 examined the trade-off between preventive measures before the arrival and control 24 measures after the invader is known to be in the country in order to determine the 25 optimal allocation of limited public resources between these two strategies [e.g. 26

21-26]
Here we add to this literature by adopting a private sector perspective, in order 27 to understand the biosecurity vulnerability and management incentives affecting 28 individual businesses. 29 One of the challenges for developing policy to reduce the risk of outbreaks of 30 pathogens is the fact that the potential routes of invasion are not only diverse, but this framework, [37] are concerned on the management problem characterized by 63 livestock-wildlife interactions in disease transmission; and [38] studied the role of 64 government policies as regular testing on encouraging farmers' biosecurity investments. 65 More recently, [39] focused on assessing whether trade always increase risk or whether 66 it can act as a disease management mechanism. 67 Our focus, however, is the threat associated with private trading decisions, as 68 infected goods can be bought in and sold on. We contribute to the above work by   To combat the spread of the infection within the nursery, the nursery owner has 89 two different control measures. The owner can invest (i ) in restriction to reduce the 90 1 Although there is no recovery, infected plants can leave the system via being sold on or being removed and be replaced by a susceptible plant. This means there is some kind of pseudo-recovery, meaning the system behaves more like a classic SIS system than SI. the nursery, avoiding additional secondary invasions, but provides no revenue. 94 Schematically, the plant-disease dynamics can be described as (see Fig 1):  For simplicity, we assume that the stock of plants at the nursery is fixed, N , which 98 may mean for example that the nursery is always full (this is a simplifying assumption 99 that is not necessarily realistic; we address this in the Discussion). To do this, we set replacement of any removed plant is assumed; when something is either sold or 104 removed by control, it is immediately replaced to keep the stock at nursery constant. 105 We also set removal as proportional to the infected plant stock, i.e. removal of 106 2 Another approach is to have assume that infected plants stay longer in the nursery due to slower growth. However, this approach would ultimately lead to the same reduction in revenue, since revenue is price×output. Consequently, the only real difference would be that different output rates would lead to a more complex replacement term. I = u rem I, where u rem is removal control effort (with units of removal effort per 107 infected plant per unit time). We will assume that u rem is bounded between 0 and 108 u remmax , the maximum possible effort spent on removal. Incorporating this, we have: This input is split between susceptible and infected plants; p(u ins ) is the proportion of 110 plant inputs that are infected (as a function of restriction effort per unit time u ins , 111 which is a control variable) and thus (1 − p(u ins )) is the proportion of plant inputs 112 that are susceptible.

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Incorporating the control measures into standard SI equations [40][41][42], and assuming density dependent transmission (βSI), we get: Given the assumption of constant total plant stock at the nursery (S + I = N ), we 114 can reduce the system down to one equation by substitution S = N − I. We can also 115 rescale the infected population by the total population and consider disease prevalence, 116 i = I N , the proportion of infected plants in the population (0 ≤ i ≤ 1).

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Then we get: Furthermore, we rescale time by δ −1 , the expected time a susceptible plant stays in 119 the nursery. Consequently, τ (= δt) is the number of generations. Thus: whereû rem = u rem δ −1 , the removal effort per plant generation (which is bounded 121 above byû remmax = u remmax δ −1 ), and R 0 = βN δ −1 , the basic reproductive number, stock. The basic reproductive number is fundamental to whether a disease will spread 125 and is discussed in the results section.

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As mentioned previously, the proportion of plants brought into the nursery being 127 infected (p(u ins )) is a function of restriction (u ins ). We assume that the proportion of 128 infected plant inputs has the following properties:

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• p(u ins ) is a continuously differentiable function of the restriction effort u ins .

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• With any finite restriction effort, some proportion of infected plant will enter the 133 nursery, i.e. p(u ins ) > 0 for all finite u ins . This means that it is not possible to 134 completely stop infected inputs from arriving no matter how high the level of 135 effort, be it from the difficulty to recognise asypmtomatic infected inputs, or 136 machine and human error.

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• For all restriction effort, increasing restriction effort reduces the proportion of 138 infected plant entering the nursery, i.e. p(u ins ) is a monotonically decreasing 139 function of u ins (equivalently, dp duins ≤ 0 everywhere).

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Any function that is (a) continuous, (b) bounded below (by zero in this case) and (c) 141 monotonically decreasing, must converge to some limit as u ins goes to infinity. We   and P I representing the unit net price of those outputs, respectively 3 . We assume that 153 P I < P S since the infection would likely decrease the plants value when mature and 154 could incur higher production costs 4 . The dynamics of the proportion of infected 155 plants within the nursery is given by equation (5). In addition, we assume that disease subject to Equation (5) whereû rem = u rem δ −1 ∈ [0,û remmax ] (as before), 193û remmax = u remmax δ −1 andû ins = c ins u ins (δN ) −1 ).

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Note that u ins has been rescaled toû ins , which now represents restriction control  the costs of management (removal and restriction). To simplify notation further, we 211 will henceforth remove all the hats (i.e. setû rem as u rem ,û ins as u ins ,p(û ins ) as 212 p(u ins ) andd as d).

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Consequently, the nursery management decision is to choose between the two 214 control strategies to minimise these costs of the infection, subject to where u ins ≥ 0 and u rem ∈ [0, u remmax ].  Putting this all together, we have six different cases, three of which are where the 274 disease is not particular infectious (which will collectively be known as Scenario 1) and 275 three of which consider a highly infectious disease (collectively known as Scenario 2).

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A summary of all six Scenarios, including results, is in Table 1. Here, '↓ p' is the reduction of infected inputs from an increase in costs of restriction in one unit (i.e.

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(1 − exp(−d)) × 100% rounded to the nearest percentage point). 'Do nothing' means zero removal and zero restriction. hence the disease will spread out from any single introduction. Hence, the only stable 287 steady state is the endemic steady state i * = 1 − 1 R0 and thus any introduction will 288 result in the disease being endemic (Fig 3(a)).  (Fig 3(a)). For R rem 0 < 1, the disease will not become 294 endemic from any single introduction (Fig 3(b)).

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Now, for u rem > 0, we have that R rem 0 < R 0 . Thus, the disease will find it harder  With imperfect restriction, the disease will always persist in the nursery plant stock 302 to some level ( Figure 4). There is always only one steady state that is non-negative, and it is always stable. The lack of a disease-free steady state is due to the constant 304 inflow of infected plants into the system. In particular, di dτ = p > 0 at i = 0 and thus 305 disease prevalence will always increase when starting with a disease-free nursery.

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Despite the disease always persisting in the nursery, we wish to distinguish between 307 two cases. If R p 0 = R0 1+urem(1−p) > 1 (Fig 4(a)), the disease spreads through the plant . This is because the removal control is For both figures have only one steady state that is stable; there is no disease-free steady state unlike the case with p = 0.
inputs (as shown in the previous subsection for perfect restriction). If  Table 2 summarises the results about when the disease is endemic in the nursery 325 for both the perfect and imperfect restriction.

Endemic
Disease-free Perfect Restriction, no removal

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Working with the prevalence steady state, we seek to find the optimal combination 328 of removal and restriction, u rem and u ins that minimises the costs of the plant disease 329 at the nursery: To find the combination of u rem and u ins that minimise Q, we need to consider the partial derivatives of Q to find internal and boundary minima. When optimal prevention and control policies are interior they satisfy the first order conditions: where As expected, Equation (13) (Equation (14)) requires a nursery owner to allocate  (14)), can be found in Appendices A and B, respectively.

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Looking at Equations (13) and (14) and incorporating the results found in  From this and by looking at Equations (13) and (14), we can establish some rules 365 of thumb. Firstly, by looking at Equation (14), we can see that increasing L and/or C, The effects of R 0 and the parameters in p(u ins ) on Equations (13) and (14) Table 1 provides a summary of the results for all the scenarios analysed.  In Scenario 1a (Fig 5(a)), we have that the marginal benefit of removal is always 389 greater than the marginal cost since ∂Q ∂urem < 0 at u rem = 0 . Consequently, the 390 optimal removal is maximum removal u rem = u remmax . This is to be expected, since 391 removing an infected plant prevents not only losses from that infected plants (which 392 are assumed to be equal to the removal cost, L = C) but also losses from secondary 393 infections. Given that R 0 > p(u ins ) this additional loss from secondary infections is 394 considerably greater than the potential loss that could result from the possibility of 395 buying infected inputs when replacing plants that were subject to removal.

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In Fig 5(a) and all other contour plots, the optimal level of restriction is 397 determined by the line MB ins =MC ins . For Scenario 1a (Fig 5(a)), with no removal 398 effort, the optimal level of restriction is around u ins = 1.2. As the nursery increases its 399 capacity to remove infected plants, it slowly reduces the optimal level of restriction. nursery, because the costs of removing and replacing an infected plant is too expensive 404 relative to the revenue loss associated to its lower net price. 405 Now, in contrast to Scenario 1a, Scenario 1c (Fig 5(b)) simulates a situation where 406 restriction is more costly. This is represented by decreasing d from 1 to 0. strategy in Scenario 1c is maximum removal with no restriction (Fig 5(c)).  However, restriction does have a mild effect on disease prevalence when prevalence in 420 the nursery is high as the 'cleaner' inputted plants that replace those leaving the 421 nursery will have a mild rinsing effect. Thus, without removal effort, restriction is 422 often not viable (i.e. no restriction is optimal) when the disease is highly infectious.

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This is particularly the case here when contrasting the viable restriction in Scenario 1a 424 (Fig 5(a) where R 0 = 0.5) and the inviable restriction in Scenario 2a (Fig 6(a)) when 425 there is no removal.

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In Scenario 2a (Fig 6(a)) there are up to two local minima. We know from the 427 analytical results that optimal removal is either u rem = 0 or u rem = u remmax .

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Consequently we can argue about the importance of u remmax by varying 429 u rem = u remmax in the contour plots, following the MB ins =MC ins line. If the nursery 430 capacity to remove is small, in particular such that u remmax is below the intersection 431 of the MB ins =MC ins and MB rem =MC rem curves, then there is only one local (and 432 thus global) minimum, which is to do nothing and let the disease take its course. If 433 u remmax is beyond the intersection, then there are two local minima, the 434 aforementioned 'do nothing' and u rem = u remmax with the corresponding restriction 435 level given by MB ins =MC ins . The global minimum is one of these two local minima 436 and which one depends on the value of u remmax ; if u remmax is small enough that the 437 contour is either blue or green (below u remmax ≈ 3.5) then 'do nothing' is optimal, 438 whereas beyond u remmax ≈ 3.5 where the contours are yellow to red, then maximum 439 removal (u rem = u remmax ) is the optimal strategy. Consequently, there is a great 440 range of values u remmax where the optimal solution is to 'do nothing', that it is futile 441 to try and control the disease without being able to really get on top of it.

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One particularly interesting result in Scenario 2a (Fig 6(a)) is the kink that occurs  substitutes when u rem is substantially larger than R 0 .

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Going from Scenario 2a to 2b (Fig 6(b)), there is a reduction in the loss in revenue 455 from selling an infected plant from L = 10 to L = 1 (note that this is a considerably 456 smaller revenue loss than in Scenario 1b). The effect of this small revenue loss in the 457 optimal effort of controlling the disease is relatively minor with respect to Scenario 2a;

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MB ins =MC ins has shifted a little to the left, and thus the optimal level of restriction 459 is reduced everywhere and MB rem =MC rem has shifted a bit to the right and a little 460 up. The consequence of the move in MB rem =MC rem is that removal is also less viable 461 everywhere. In particular, the intersection between these two lines that separates the removal is optimal. This is because the disease will still spread through the nursery 477 since R p 0 is still considerably larger than 1, making removal efforts futile.

478
Now, consider the case where restriction is less cost-effective as d is decreased to Fig 6(c)). This decrease has a relatively minor effect on the removal 480 line MB rem =MC rem in Fig 6(c), the line keeps the same intercept with the y-axis and 481 it is flatter than in Fig 6(a). This is predictable since decreasing cost-effectiveness 482 means that more needs to be spent in restriction in order to have the same effect in

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The numerical analysis of the Scenarios (summarised in Table 1)  Olson and Roy [48] examine the conditions under which the optimal policy relies solely 532 on either prevention or control. Kim et al. [49] examine the optimal combination of 533 pre-discovery prevention, post-discovery prevention and post-discovery control where 534 the discovery time is stochastic, and find that post-discovery prevention and control 535 are substitutes. Leung et al. [22] consider that if there is expensive control activities, 536 this reduces social welfare at the post-invasion state, and consequently higher social 537 welfare can be achieved from avoiding invasion, and substituting control by prevention 538 efforts. Similarly, Finnoff et al. [24] conclude that a risk averse agent would substitute 539 more prevention expenditures with control policies when compared to a risk neutral 540 agent. Here, we found that the optimal level of restriction is complementary with 541 removal efforts if the disease is beyond the nursery owner's ability to limit its spread.

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The underlying reason for this is that, restriction measures may not be very effective in the human health literature in [50,51]. Hennessy et al. [51] argue that for 554 'prevention' and 'cure' being complements is that increasing prevention reduces the 555 chance that cured individuals become sick again and thus improving the long term

629
Finally, note that this paper deals with one disease of concern for the nursery To find out what the optimal solutions with respect to u rem , we need to investigate: where M (u ins , u rem ) = R 0 − 1 − (1 − p(u ins ))u rem . First, we need to manipulate this into something more manageable.  and thus internal solutions are always local maxima with respect to u rem . As there is no internal minimum with respect to u rem , the global minimum must occur on the boundary, either at u rem = 0 or u rem = u remmax . If ∂Q ∂urem < 0 at u rem = 0 then u rem = 0 is a local (global) maximum and u rem = u remmax is the global minimum.
Conversely, if ∂Q ∂urem > 0 at u rem = u remmax then u rem = u remmax is a local (global) maximum and thus u rem = 0 is a global minimum. If ∂Q ∂urem > 0 at u rem = 0 and ∂Q ∂urem < 0 at u rem = u remmax , then you have must compare Q for u rem = 0 and u rem = u remmax since both are local minima.
Appendix B. Optimal control with respect to restriction u ins : 'do something or do nothing' We need to find out the global minimum with respect to restriction u ins by analysing: First, we will look at the second partial derivative to see if ∂Q ∂uins is an increasing or decreasing function of u ins : . Armed with this, we have: Firstly, we note that if L + Cu rem ≤ 0 (which could be true if L < 0), there are no internal solutions from possible for Equation (17) from the main text and we have ∂Q ∂uins is monotonically increasing to -1. Hence, ∂Q ∂uins < 0 always and thus zero restriction is always the best (a disease that is beneficial should not be restricted). For L + Cu rem > 0, we have that ∂Q ∂uins is monotonically increasing (to 1 as u ins → ∞). In other words, increasing restriction has even diminishing returns, reducing the marginal benefit, whereas the marginal cost remains the same. Given we have that ∂Q ∂uins is monotonically increasing to 1 (and is continuous), we know that there exists one and only one admissible solution with respect to u ins (for fixed u rem ) if ∂Q ∂uins < 0 at u ins = 0 and that this solution is a global minimum with respect to u ins , i.e. the optimal control involves some restriction. Otherwise, ∂Q ∂uins ≥ 0 at u ins = 0, there is no internal solution and the global minimum with respect to u ins is at u ins = 0, i.e. no restriction is optimal.
If such solutions do not exist within admissible controls (u rem ∈ [0, u remmax ] and u ins ≥ 0), we need to pick the minimising values on the boundary, i.e. if ∂Q ∂uins > 0 at u ins = 0, then either u ins = 0 and u ins = ∞ are the global maximum. However, since ∂Q ∂uins → 1 as u ins → ∞ (because p(u ins ) is converging to b and thus ∂p(uins) ∂uins → 0, u ins = ∞ is always a local maximum and thus u ins = 0 is the global minimum, i.e. the cost minimising strategy, when ∂Q ∂uins > 0 at u ins = 0.

Appendix C. Linking dynamic and stationary approaches
Taking Equation (6) and following the rescaling and rearrangement that occur between Equation (7) and (9)  whereT = T δ andr = r δ (henceforth, we will drop these hats for simplicity, being consistent with what was done in the main text). First, we establish and analyse the Hamiltonian of Equations (9) and (10). This Hamiltonian is: Consequently, the adjoint equation is: 2i)) .
To link the solutions in this paper to those of this Hamiltonian, we will assume an infinite time interval, and treat u rem , u ins as constants. On top of this, we will insert the steady state value of i * from Equation (11) given from the population dynamics. Notice that the right hand side is dQ duins = M C ins − M B ins from Equation (14). Thus for zero discounting (r = 0), dQ duins = 0 gives the optimal restriction, whereas for a positive discounting rate (r > 0), the optimal restriction satisfies dQ duins = − r √ M 2 +4R0p(uins) . However, since dQ duins is monotonically increasing function, we know that increasing the discount rate (r) would lower the optimal level of restriction. This effect is very dependent on how long the plant is expected to be in the nursery due to the time rescaling (i.e. sincer = r δ ). If the average plant stay is short (i.e. weeks to months) then this discounting effect is negligible, whereas for longer period (i.e. years), this term becomes larger, having more impact on the optimal restriction.
Moving on to optimal removal, (C.5) is generally never satisfied, and instead the optimal removal is a 'bang-bang' control (i.e. all or nothing) which is consistent with the static analysis. Consequently, the optimal solution is either u rem = 0 or u rem = u remmax , which depends on the sign of λ(p(u ins ) − 1) − Ce −rt .
To determine the sign, we will focus on the threshold λ(p(u ins ) − 1) − Ce −rt = 0.
Substituting Equation C.6 and rearranging gives: This of condition is analogous with the static problem, with the right hand side being dQ durem = M C rem − M B rem from Equation (13). This alone does not give the global optimal since there are two λ's to compare, one where u rem = 0, the other where u rem = u remmax . In cases where λ(u rem = 0)(p(u ins ) − 1) − Ce −rt < 0 but λ(u rem = u remmax )(p(u ins ) − 1) − Ce −rt > 0, a comparison in terms of profit must be made, which is analogous to the two local optima solutions found in the static solutions. Again, like with restriction, we have that no discounting gives the same result, and increasing the discount rate makes u rem = u remmax less likely to be globally optimal.