Modelling effects of vector acquisition threshold on disease progression in a perennial crop following deployment of a partially resistant variety

Abstract

Deployment of resistant varieties is a key strategy to mitigating economic losses due to arthropod-transmitted plant pathogens of perennial crops. In many cases, the best available resistant traits for introgression confer only partial resistance. Plants displaying partial resistance have lower pathogen titres than susceptible counterparts, but remain hosts for the pathogen. As partially resistant varieties maintain yield after infection, infected plants are unlikely to be rogued (i.e. removed). Accordingly, there is a risk that partially resistant plants could serve as a source of inoculum for pathogen spread to susceptible plants. Here, a mathematical model that tracked spread of an arthropod-transmitted pathogen in a plant population consisting of susceptible and partially resistant plants was used to identify a threshold acquisition rate from partially resistant plants that resulted in limited spread of the pathogen from partially resistant plants to susceptible plants. The acquisition threshold from partially resistant plants varied with parameters influenced by disease management decisions such as number of vectors per plant, vector turnover, replacement of susceptible plants, and proportion of plants that were partially resistant. In model simulations, effects of deploying a partially resistant variety on disease incidence in a susceptible variety depended on the extent to which pathogen spread among susceptible plants was suppressed and acquisition rates from partially resistant plants. Collectively, the results indicate that risk of partially resistant plants serving as inoculum sources could be assessed prior to deployment, thereby enabling design of complementary disease management tactics to minimize economic losses in susceptible varieties following deployment.

Introduction

Several management options may be employed to slow spread of arthropod-transmitted plant pathogens in perennial crops. Common management tactics include using pathogen-free planting material, vector suppression, roguing (i.e. removal), and deployment of resistant varieties (Perring et al., 1999; Laimer et al., 2009; Sisterson & Stenger, 2013; Barba et al., 2015; Rimbaud et al., 2015). Of these strategies, deployment of resistant varieties is the favoured option for effective long-term management of a number of arthropod-transmitted pathogens of perennial crops (Laimer et al., 2009; Barba et al., 2015; Rimbaud et al., 2015).

Perennial crops bred for disease resistance are evaluated for many traits, including pathogen titre, horticultural characteristics and fruit quality (Egea et al., 1999; Krivanek & Walker, 2005; Ramming et al., 2009; Butac et al., 2013; Walker et al., 2014). To develop a resistant variety using conventional breeding methods, plant breeders often use sources of resistance from wild relatives or cultivars lacking in qualities unrelated to pathogen infection (Krivanek et al., 2005; Maule et al., 2007; Laimer et al., 2009; Marandel et al., 2009). The extent to which pathogen titre may be reduced in a resistant line is entirely dependent on the source of resistance that was identified. For example, the gene PdR1 has been identified as a key source of resistance to Pierce's disease of grapevine (Krivanek et al., 2006) and is inherited as a single dominant gene, simplifying introgression into commercial varieties (Krivanek et al., 2005; Walker et al., 2014). Titres of Xylella fastidiosa (the causal agent of Pierce's disease) in PdR1 source plants are lower than in susceptible varieties, but are still detectable (Krivanek & Walker, 2005; Fritschi et al., 2007). Although plant breeders may wish to develop varieties with undetectable pathogen levels, conventional breeding programmes are limited to resistance traits available in compatible germplasm (Scorza et al., 2013; Ilardi & Tavazza, 2015). In many cases, the only available sources of compatible resistance traits may confer only partial resistance or tolerance (e.g. Plum pox virus: Hartmann & Neumüller, 2006; huanglongbing disease: Albrecht & Bowman, 2011; Bowman & McCollum, 2015).

Costs associated with establishing plantings of perennial crops are substantial, with upfront costs of planting offset by long productive lifetimes. For example, a newly planted vineyard is anticipated to be productive for c. 27 years and citrus orchards for c. 40 years (Vasquez et al., 2007; O'Connell et al., 2010). If a partially resistant variety were available, the decision to remove a susceptible variety and replant with a partially resistant variety is likely to depend on the age of the planting, costs associated with replanting, and losses due to disease. Multiple varieties of a particular perennial crop may be cultivated in a region. A newly released partially resistant variety may be an acceptable substitute for a portion of available varieties, but often not all. As a result, it may take decades for a region to transition from susceptible varieties to partially resistant varieties, resulting in a complex landscape consisting of blocks of susceptible and partially resistant varieties. As partially resistant varieties remain hosts of the pathogen, there is a risk that partially resistant varieties could serve as inoculum sources for spread to neighbouring susceptible varieties. Risk of a partially resistant variety serving as an inoculum source is further compounded by the fact that growers are unlikely to rogue infected partially resistant plants, as these plants are anticipated to maintain yield after infection.

For an arthropod-transmitted plant pathogen to spread from an infected partially resistant plant to a susceptible plant, vectors must acquire the pathogen from an infected partially resistant plant, move to a healthy plant, and subsequently inoculate the healthy plant. As the probability of vector acquisition appears to be related to pathogen titre in the plant (Lapidot et al., 2001; Galetto et al., 2014), a key question that needs to be addressed is: how low do acquisition rates from partially resistant plants need to be for partially resistant plants to not significantly contribute to pathogen spread? Theoretical and empirical studies evaluating deployment of resistant plants are abundant (Jeger et al., 1981a,b; Mundt & Leonard, 1986; van den Bosch et al., 1990; Garrett & Mundt, 1999; Skelsey et al., 2005). However, the majority of such studies have focused on wind-dispersed pathogens affecting annual crops, with resistant plants often conferring immunity (Mundt, 2002). By comparison, strategies for deploying varieties of perennial crops partially resistant to arthropod-transmitted pathogens have received less attention.

Here, a coupled differential equation model was used to identify a threshold value for acquisition from partially resistant plants that resulted in limited pathogen spread from partially resistant plants to susceptible plants. Identification of a threshold acquisition rate from partially resistant plants serves two purposes. First, it provides the opportunity to identify parameters influenced by disease management tactics that could be augmented to decrease risk posed by deploying a partially resistant variety. Secondly, it provides plant breeders with a target for acceptable levels of acquisition from partially resistant plants.

Materials and methods

The model

The model consisted of a set of compartmentalized differential equations, similar to other plant pathogen models that included a vector (Jeger et al., 1998; Madden et al., 2000; Sisterson, 2009; Sisterson & Stenger, 2016). For simplicity, the term resistant will be used to refer to partially resistant plants, or plants that maintain lower pathogen titres than susceptible plants, which in turn is assumed to reduce probability of vector acquisition. Likewise, the term susceptible will be used to refer to plants that are susceptible to the pathogen. Spread of an arthropod-transmitted plant pathogen in a population consisting of susceptible and resistant varieties requires a set of equations that tracks changes in abundance of uninfected plants (U), latently infected plants (L; infected but do not serve as acquisition sources), and infectious plants (I; infected and serve as acquisition sources) of both varieties. The subscript S was used to denote the susceptible variety and the subscript R was used to denote the resistant variety. The model does not explicitly designate spatial relationships between resistant and susceptible plants. Rather, vector dispersal is assumed to be sufficiently high that a vector is equally likely to contact any plant. Conceptually, spatial structure should be thought of as a block (or blocks) of susceptible plants that are in direct proximity to a block (or blocks) of resistant plants, such that vectors can freely move between susceptible and resistant plants.

The total number of susceptible plants (P_S) is the sum of the number of uninfected (U_S), latently infected (L_S) and infectious plants (I_S) of the susceptible variety (P_S = U_S + L_S + I_S). Likewise, the total number of resistant plants (P_R) is the sum of the number of uninfected (U_R), latently infected (L_R) and infectious plants (I_R) of the resistant variety (P_R = U_R + L_R + I_R). The total number of all plants (P) is the sum of the total number of susceptible and resistant plants (P_S + P_R = P). The set of equations also tracks changes in abundance of non-inoculative (X) and inoculative (Z) vectors, with the total number of vectors represented by N (N = X + Z). Vectors that acquire the pathogen are assumed to immediately become inoculative, with no latent stage. The system of equations is written as follows:

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Equations 1-3 describe the changes in abundance of uninfected, latently infected and infectious susceptible plants. Specifically, the abundance of uninfected susceptible plants (U_S) increases as latently infected susceptible plants (L_S) and infectious susceptible plants (I_S) are replaced (r_S) with uninfected susceptible plants (U_S). The abundance of uninfected susceptible plants (U_S) decreases as uninfected susceptible plants become infected, which occurs as a function of the number of inoculative vectors per plant , inoculation rate for susceptible plants (θ_S), and number of plants visited by a vector each day (v). The abundance of latently infected susceptible plants (L_S) increases as susceptible plants become infected () and decreases as latently infected susceptible plants (L_S) are replaced (r_S), or as latently infected susceptible plants become infectious (τ). The abundance of infectious susceptible plants (I_S) increases as latently infected susceptible plants (L_S) become infectious (τ), and decreases as infectious susceptible plants are replaced (r_S).

The equations for change in abundance of uninfected, latently infected, and infectious resistant plants (Eqns 4-6) are similar to those for susceptible plants (Eqns 1-3), with some key differences. First, the inoculation rate has the subscript R, indicating that the probability of successful inoculation of resistant plants differs from that for susceptible plants. Secondly, as resistant plants are anticipated to maintain yield even when infected, the rate at which resistant plants are rogued is anticipated to be substantially lower than for susceptible plants. Accordingly, the roguing rates for susceptible and resistant plants are also differentiated based on the subscripts S and R.

Following Jeger et al. (1998) and Madden et al. (2000), vector population size is fixed. Consequences of this simplifying assumption are discussed in detail by Sisterson & Stenger (2016). To fix vector population size, vectors that die are replaced with non-inoculative vectors (X). Thus, abundance of non-inoculative vectors (X) increases as inoculative vectors (Z) die (m) and decreases as non-inoculative vectors (X) acquire the pathogen. With fixed vector population size, vector mortality has no effect on vector abundance. Rather, vector mortality determines the length of time an inoculative vector lives and thus determines the period over which vectors remain inoculative. Following Jeger et al. (1998), vector mortality will be referred to as vector turnover. While transmission of the pathogen is presumed to be persistent, as no term for loss of inoculativity is explicitly included, it is important to consider that vector turnover and loss of inoculativity are mathematically equivalent processes that result in moving vectors from the inoculative category (Z) to the non-inoculative category (X; Sisterson & Stenger, 2016). As a result, increasing the magnitude of vector turnover could be due to decreasing vector lifespan or decreasing the period a vector remains inoculative, or both.

Vectors are assumed to have no preference for resistant or susceptible plants. Accordingly, the number of vectors acquiring the pathogen from susceptible plants is a function of the relative abundance of infectious susceptible plants , the probability of acquisition from a susceptible plant (α_S), and the number of plants visited by a vector each day (v). Likewise, the number of vectors acquiring the pathogen from resistant plants is a function of the relative abundance of infectious resistant plants , the probability of acquisition from a resistant plant (α_R), and the number of plants visited by a vector each day (v).

Basic reproductive numbers

The basic reproductive number of the full model (Eqns 1-8) was determined using the next generation method (van den Driessche & Watmough, 2002; Text S1):

(9)

The basic reproductive number of the full model describes the conditions under which the pathogen may persist in the whole plant population. With values of the basic reproductive number ˃1 the pathogen persists; with values of the basic reproductive number ˂1 the pathogen goes extinct. The basic reproductive number of the full model (Eqn 9) is the weighted sum of the basic reproductive number assuming only susceptible plants and the basic reproductive number assuming only resistant plants, with the weights being the relative proportion of each plant type (F or [1 − F]). This may be demonstrated in two ways. First, by setting the percentage of resistant plants to 0 or 1 and simplifying Eqn 9, or secondly, by obtaining the basic reproductive number by the next generation method for the model assuming only resistant or susceptible plants are present (Text S2 & S3). If the proportion of resistant plants is zero (F = 0), the basic reproductive number simplifies to:

(10)

The subscript S is used to denote that the basic reproductive number above (Eqn 10) assumes that only susceptible plants are present. Similarly, if all plants are assumed to be resistant (F = 1), the basic reproductive number of the full model (Eqn 9) simplifies to:

(11)

The subscript R is used in the basic reproductive number above (Eqn 11) to denote that the basic reproductive number assumes that only resistant plants are present. The structure of the basic reproductive number assuming only resistant plants is similar to the basic reproductive number assuming only susceptible plants (compare Eqns 10 and 11) and is similar to R₀ values reported for other vector models (Wonham et al., 2004; Zeilinger & Daugherty, 2014). Using the expression for the basic reproductive number assuming only susceptible (Eqn 10) or resistant (Eqn 11) plants, values for plant replacement (r_S or r_R), vector turnover (m), and vector abundance that result in basic reproductive numbers with values <1 may be determined (Fig. 1). In both cases, pathogen spread is slowed by increasing plant replacement (Fig. 1a) and vector turnover (Fig. 1b), and decreasing vector abundance (Fig. 1c).

Figure 1

Effects of (a) plant replacement (r_S or r_R), (b) vector turnover (m), and (c) vectors per plant (N⁄P) on the basic reproductive number (R_0,S [Eqn 10] or R_0,R [Eqn 11]) derived assuming only susceptible (F = 0) or resistant (F = 1) plants. With R₀ > 1 pathogen spread occurs, whereas with R₀ < 1 pathogen spread does not occur. Parameters not varied in a panel were set to the following values: τ = 0.017, v = 1, m = 0.2, = 0.5, and r = 0.033.

The structure of the basic reproductive number of the full model (R_0,full, Eqn 9) suggests two important observations with regards to deploying a resistant variety. First, if management tactics to suppress pathogen spread among susceptible plants are insufficient (R_0,S > 1), the pathogen is likely to persist regardless of the extent to which resistant plants serve as inoculum sources. Secondly, persistence of the pathogen is dependent on plant replacement (Fig. 1a). Importantly, resistant plants are assumed to maintain yield after infection and to show mild symptoms compared to susceptible plants. Accordingly, resistant plants are unlikely to be replaced and therefore replacement rates of resistant plants (r_R) are likely to be low. As replacement rates of resistant plants (r_R) decrease to zero, the denominator of R_0,R (Eqn 11) decreases, making it more likely that the basic reproductive number for resistant plants will be >1 and, in turn, that the basic reproductive number of the full model (Eqn 9) will be >1.

Acquisition threshold

While basic reproductive numbers (Eqns 9-11) provide insight into the role of parameters in the model, basic reproductive numbers are largely used to assess potential for the pathogen to persist (R₀ > 1) or go extinct (R₀ < 1). For systems in which the economic incentive for breeding resistant plants exist, it seems reasonable to presume that losses due to disease or costs associated with disease control are sufficiently high to warrant investment in a breeding programme. In such systems, basic reproductive numbers are anticipated to be >1 or to be <1 as a result of aggressive management practices (i.e. low number of vectors per plant [], high vector turnover [m] and/or high plant replacement rates [r_S]). In the case of the latter (R₀ < 1 due to aggressive management), failure to maintain aggressive management is expected to result in basic reproductive numbers increasing to a value >1. As the pathogen is presumed to be endemic, establishing whether or not the pathogen will persist is not of primary interest. Rather, the key question that needs to be addressed is whether or not deployment of a resistant variety that may serve as an inoculum source is likely to improve or detract from disease management practices already in place for susceptible varieties that are established in the field.

The risk of resistant plants serving as inoculum sources is related to the probability of the vector acquiring the pathogen from resistant plants. Consider that pathogen titres of resistant plants are expected to be lower than pathogen titres in susceptible plants (e.g. Krivanek & Walker, 2005; Fritschi et al., 2007) and that the probability of vector acquisition appears to decline as pathogen titre declines (Lapidot et al., 2001; Galetto et al., 2014). Thus, the key question that needs to be addressed is how low do acquisition rates from resistant plants need to be for deployment of resistant plants to not disrupt disease management in susceptible varieties that are already established in the field? To address this question, an acquisition threshold from resistant plants that results in limited spread to susceptible plants was identified.

A number of assumptions are used to derive the acquisition threshold from resistant plants that results in limited spread to susceptible plants. First, resistant plants are assumed to maintain yield and display mild symptoms relative to susceptible plants. Accordingly, replacement rates of resistant plants are assumed to be low and approaching zero. In addition, acquisition (α_R) and inoculation (θ_R) rates of resistant plants are assumed to be >0. Under the aforementioned assumptions, all resistant plants are expected to become infected once the system reaches equilibrium. Given that perennial crops are maintained for long time periods (often >20 years), even modest rates of pathogen spread will result in high levels of infection within the lifetime of the crop. If the pathogen is present in the resistant plant population, some susceptible plants will become infected due to movement of vectors between resistant and susceptible plants. Thus, a parameter representing the proportion of susceptible plants that are infectious that growers are willing to accept due to spread from resistant plants to susceptible plants is defined and is referred to as acceptable loss (Φ). Finally, to isolate infections that arise solely due to resistant plants, susceptible plants are assumed to not contribute to pathogen spread (α_S = 0). Under the assumptions detailed above, an acquisition threshold from resistant plants that results in limited spread to susceptible plants is identified (Text S4):

(12)

If the observed acquisition rate from resistant plants is less than the estimated threshold (), the equilibrium proportion of susceptible plants that are infectious due to spread from resistant plants to susceptible plants is expected to be less than the acceptable loss (Φ). In contrast, if the observed acquisition rate from resistant plants is greater than the estimated threshold (), the equilibrium proportion of susceptible plants that are infectious due to spread from resistant plants to susceptible plants is expected to be greater than the acceptable loss.

Throughout, two values for acceptable loss (Φ) are used: a conservative value of 0.001 (or one infectious susceptible plant for every 1000 susceptible plants) and a non-conservative value of 0.05 (or five infectious susceptible plants for every 100 susceptible plants). There is leeway in choosing an appropriate value for acceptable loss with some caveats. First, acceptable loss quantifies the proportion of infectious susceptible plants . As latently infected plants (L_S) are excluded from the measure, the total proportion of infected plants will be greater than the acceptable loss . Secondly, increasing the value for acceptable loss will, by definition, increase the number of infectious susceptible plants present at any given time and also may increase the number of susceptible plants replaced to maintain that level of disease. Finally, as the acquisition threshold is developed assuming that only resistant plants contribute to pathogen spread, there is a risk that susceptible plants infected due to spread from resistant to susceptible plants will contribute to increased spread among susceptible plants. This risk is expected to be negligible with conservative values of acceptable loss, and to increase as acceptable loss takes larger values.

Threshold acquisition rates from resistant plants () are affected by parameters that depend on management decisions (Fig. 2). Specifically, threshold acquisition rates () from resistant plants decrease as vector abundance increases () and as the proportion of plants that are resistant (F) increases (Fig. 2). In contrast, threshold acquisition rates from resistant plants increase as replacement of infected susceptible plants (r_S) and vector turnover (m) increase (Fig. 2). Threshold acquisition rates assuming a conservative value for acceptable loss (Φ) of 0.001 are lower than threshold acquisition rates using a non-conservative value of acceptable loss of 0.05 (compare Fig. 2a,c to Fig. 2b,d).

Figure 2

Effects of vector turnover (m), number of vectors per plant (), and replacement of susceptible plants (r_S) on the threshold acquisition rate from resistant plants (α_R,threshold; Eqn 12) with (a) a conservative value for acceptable loss (Φ = 0.001) and low proportion of resistant plants (F = 0.05); (b) a non-conservative value for acceptable loss (Φ = 0.05) and low proportion of resistant plants (F = 0.05); (c) a conservative value for acceptable loss (Φ = 0.001) and a high proportion of resistant plants (F = 0.50); and (d) a non-conservative value for acceptable loss (Φ = 0.05) and a high proportion of resistant plants (F = 0.50). If acquisition rates from resistant plants are greater than the acquisition threshold (α_R > α_R,threshold), the proportion of infected susceptible plants due solely to resistant plants is expected to exceed acceptable loss (Φ). Parameters not varied in the figure were set to the following values: τ = 0.017, v = 1, θ_S = 0.40.

Using thresholds to categorize scenarios for deploying resistant plants

Using the basic reproductive number for susceptible plants (R_0,S, Eqn 10) and the acquisition threshold from resistant plants (, Eqn 12), four categories describing deployment of resistant plants may be defined (Table 1). First, resistant plants may be deployed into a region where pathogen spread among susceptible plants is not effectively controlled (R_0,S > 1), with the impact of deploying the resistant variety depending on the acquisition rate from resistant plants. Thus, Category 1 defines the situation where pathogen spread is uncontrolled among susceptible plants (R_0,S > 1) and acquisition rates from resistant plants are below the theoretical threshold , whereas Category 2 describes the situation where pathogen spread is uncontrolled among susceptible plants (R_0,S > 1) and acquisition rates from resistant plants are above the theoretical threshold . Resistant plants may also be deployed in a region where pathogen spread among susceptible plants is held at bay due to aggressive management (R_0,S < 1). Thus, Category 3 represents the situation where pathogen spread among susceptible plants is controlled (R_0,S < 1) and acquisition rates from resistant plants are below the theoretical threshold , whereas Category 4 represents the situation where pathogen spread among susceptible plants is controlled (R_0,S < 1), but acquisition from resistant plants is above the theoretical threshold .

Table 1. Simulations using random draws for parameter values were placed into categories based on the estimated value of (Eqn 10) and the relative value of the acquisition rate from resistant plants (α_R) to the estimated threshold acquisition rate from resistant plants (Eqn 12, )

Category	Mathematical definitionab	Verbal description
1		Spread from susceptible to susceptible occurs, with limited spread from resistant to susceptible
2		Spread from susceptible to susceptible occurs, with spread from resistant to susceptible common
3		Spread from susceptible to susceptible does not occur, with limited spread from resistant to susceptible
4		Spread from susceptible to susceptible does not occur, with spread from resistant to susceptible common

• ^a R_0,S (Eqn 10) was determined assuming that the proportion of resistant plants was zero (F = 0) and therefore assessed spread from susceptible plants to susceptible plants.
• ^b (Eqn 12) was determined assuming no acquisition from susceptible plants (α_S = 0) and therefore assessed spread from resistant plants to susceptible plants. Two values for acceptable loss (Φ) were used: 0.001 (or 1 infectious susceptible plant for every 1000 susceptible plants) and 0.05 (or 5 infectious susceptible plants for every 100 susceptible plants).

Simulations

Numerical simulations of the model (Eqns 1-8) were conducted for several reasons. First, to describe the range of possible outcomes for each of the four categories described above (Table 1). Secondly, to test the robustness of predictions made by the acquisition threshold (Eqn 12), as derivation of the acquisition threshold requires making several limiting assumptions (Text S4). Finally, numerical simulations also allows assessment of whether or not effects described by the thresholds are observed over a timescale representative of a perennial crop.

Numerical simulations were completed by converting Eqns 1-8 to the respective recursion equation analogues with time steps of 1 day (Otto & Day, 2007; Sisterson & Stenger, 2016). The recursion equations were simulated using C++ (Microsoft Visual C++). For each simulation, equations were iterated for 1800 days; a time period that equates to approximately 10 years, assuming a 6 month growing period and a 6 month dormant period each year . At the end of each simulation, the percentage of susceptible plants that were infected and percentage of resistant plants that were infected on day 1800 were determined.

To understand model behaviour across a wide range of conditions, simulations were conducted with random draws for each parameter (Table 2). Accordingly, results are presented as distributions of outcomes for simulations satisfying each of the four categories in Table 1. Acquisition and inoculation rates of resistant plants were assumed to take values equal to or lower than values for susceptible plants. This was accomplished by obtaining random draws for acquisition and inoculation rates of susceptible plants first. Subsequently, a random value between 0 and 1 was obtained to determine the extent to which acquisition and inoculation of resistant plants was reduced relative to susceptible plants (Table 2). All simulations were initiated with a single latently infected susceptible plant.

Table 2. Model parameters, initial values, and range of values used in simulations with random draws

Parameter	Description	Range of values used or initial value
U S	Uninfected plants (susceptible variety)	(1-F)P - 1
L S	Latently infected plants (susceptible variety)	1
I S	Infectious plants (susceptible variety)	0
U R	Uninfected plants (resistant variety)	FP
L R	Latently infected plants (resistant variety)	0
I R	Infectious plants (resistant variety)	0
P S	Total number of susceptible plants (PS = US + LS + IS)	5000–9500
P R	Total number of resistant plants (PR = UR + LR + IR)	500–5000
P	Total number of plants (P = PS + PR)	10 000
F	Proportion of plants that were resistant (F = PR⁄P)	0.05–0.50
X	Non-inoculative vectors	N
Z	Inoculative vectors	0
N	Total number of vectors (N = X + Z)	100–40 000
r S	Replacement rate for susceptible plants (plant per day)	0.001–0.12
r R	Replacement rate for resistant plants (plant per day)	0.0001–0.001
θ S	Inoculation rate of susceptible variety (vector per day)	0.05–0.99
θ R	Inoculation rate of resistant variety (vector per day)	(1 − dθ) θS
d θ	Proportional reduction in inoculation rate due to resistance	0–1
v	Plants visited (vector per day)	0.5–1.5
τ	Rate latently infected plants become infectious (plant per day)	0.01 to 0.07
α S	Acquisition rate from susceptible variety (vector per day)	0.05–0.99
α R	Acquisition rate from resistant variety (vector per day)	(1 − dα) αS
d α	Proportional reduction in acquisition rate due to resistance	0–1
m	Vector turnover (vector per day)	0.01–0.60
Φ	Acceptable loss (proportion of susceptible plants)	0.001 or 0.05

Two sets of simulations with random draws were completed. The first set of simulations was conducted to quantify percentage of infected susceptible and resistant plants. To accomplish this, 40 000 simulations with random draws for each parameter were completed that satisfied each of the four categories. For each set of 40 000 simulations, a random draw of parameters was obtained from the whole range of parameters described in Table 2. Threshold values (Eqns 10-12) for that set of parameters were then determined. If the threshold values satisfied the category under consideration, the simulation was run and the simulation count increased by one. If the threshold values did not satisfy the category under consideration, the parameter set was discarded and another random parameter set obtained. This process was repeated until 40 000 simulations were completed for the designated category.

The second set of simulations was conducted to compare percentage of infected susceptible plants in simulations with and without the resistant variety. To accomplish this, a random draw of parameters was obtained as described above. Then, two simulations were completed: a simulation assuming that all plants were susceptible (F = 0) and a simulation assuming that the proportion of plants that were resistant was a random draw (F > 0.05 and F < 0.50; Table 2). This process was repeated 40 000 times for parameter sets satisfying each of the four categories. At the end of each paired simulation, the difference in percentage of susceptible plants that were infected was determined. Positive differences indicated that deploying resistant plants increased the percentage of susceptible plants that were infected compared to simulations without resistant plants. Negative differences indicated that deploying resistant plants decreased the percentage of susceptible plants that were infected compared to simulations without resistant plants.

Results

Category 1: Spread among susceptible plants uncontrolled (R_0,S > 1) and acquisition rates from resistant plants below the threshold

For simulations in Category 1, the percentage of infected susceptible and resistant plants at the end of 1800 days was high (Fig. 3a,b). As pathogen spread among susceptible plants was uncontrolled (R_0,S > 1), values for acquisition from resistant plants had to be low to be below the theoretical threshold (Fig. 3c). Specifically, the median acquisition rate from resistant plants was 0.0004 and 0.027 with acceptable loss (Φ) equal to 0.001 or 0.05, respectively (Fig. 3c). As acquisition rates from resistant plants were low, deployment of resistant plants generally decreased disease incidence in susceptible plants in simulations with resistant plants compared to simulations without resistant plants (Fig. 3d). In this case, deployment of resistant plants slowed pathogen spread as susceptible plants associated with high acquisition rates were replaced with resistant plants associated with low acquisition rates.

Figure 3

Results of simulations satisfying Category 1 . (a) Distribution of percentage of infected susceptible plants ; (b) distribution of percentage of infected resistant plants ; (c) distribution of observed acquisition rates (α_R); and (d) distribution of difference in percentage of infected susceptible plants in paired simulations with (F > 0) and without (F = 0) the resistant variety. A positive difference indicates that planting the resistant variety increased disease incidence in susceptible plants (to the right of the dashed line) and a negative difference indicates that planting the resistant variety decreased disease incidence in susceptible plants (to the left of the dashed line). Distributions were generated by completing 40 000 simulations using random draws for parameters (Table 2) that satisfied Category 1 (Table 1) and evaluating the outcome at the end of 1800 days. Two values for acceptable loss are shown: Φ = 0.001 and Φ = 0.05.

Category 2: Spread among susceptible plants uncontrolled (R_0,S > 1) and acquisition rates from resistant plants above the threshold

For simulations in Category 2, the percentage of infected susceptible plants was generally >50% after 1800 days (Fig. 4a). Similarly, the percentage of infected resistant plants after 1800 days was nearly always >90% (Fig. 4b). As acquisition rates from resistant plants were above the theoretical threshold in Category 2, acquisition rates from resistant plants in simulations from Category 2 took a wide range of values (Fig. 4c). For simulations in Category 2, the percentage of susceptible plants that were infected at the end of a simulation was similar for simulations with and without resistant plants (Fig. 4d). Results for simulations with and without resistant plants were similar, as in both cases the majority of susceptible plants were infected.

Figure 4

Results of simulations satisfying Category 2 . (a) Distribution of percentage of infected susceptible plants ; (b) distribution of percentage of infected resistant plants ; (c) distribution of observed acquisition rates (α_R); and (d) distribution of difference in percentage of infected susceptible plants in paired simulations with (F > 0) and without (F = 0) the resistant variety. A positive difference indicates that planting the resistant variety increased disease incidence in susceptible plants (to the right of the dashed line) and a negative difference indicates that planting the resistant variety decreased disease incidence in susceptible plants (to the left of the dashed line). Distributions were generated by completing 40 000 simulations using random draws for parameters (Table 2) that satisfied Category 2 (Table 1) and evaluating the outcome at the end of 1800 days. Two values for acceptable loss are shown: Φ = 0.001 and Φ = 0.05.

Category 3: Spread among susceptible plants controlled (R_0,S < 1) and acquisition rates from resistant plants below the threshold

For simulations in Category 3, disease incidence in susceptible plants after 1800 days was always <1% with acceptable loss (Φ) equal to 0.001 (Fig. 5a). Likewise, disease incidence in resistant plants was <1% in 99% of simulations with acceptable loss equal to 0.001 (Fig. 5b). With acceptable loss equal to 0.001, acquisition rates from resistant plants were generally low (Fig. 5c, median = 0.01). Accordingly, the pathogen did not spread among resistant plants (i.e. R_0,R < 1) or did not spread quickly enough to infect a high number of resistant plants within the timeframe of a simulation. With acceptable loss equal to 0.05, acquisition rates from resistant plants were greater than with acceptable loss equal to 0.001 (Fig. 5c; median = 0.09), resulting in greater spread among resistant plants within the timeframe of a simulation (Fig. 5b).

Figure 5

Results of simulations satisfying Category 3 . (a) Distribution of percentage of infected susceptible plants ; (b) distribution of percentage of infected resistant plants ; (c) distribution of observed acquisition rates (α_R); and (d) distribution of difference in percentage of infected susceptible plants in paired simulations with (F > 0) and without (F = 0) the resistant variety. A positive difference indicates that planting the resistant variety increased disease incidence in susceptible plants (to the right of the dashed line) and a negative difference indicates that planting the resistant variety decreased disease incidence in susceptible plants (to the left of the dashed line). Distributions were generated by completing 40 000 simulations using random draws for parameters (Table 2) that satisfied Category 3 (Table 1) and evaluating the outcome at the end of 1800 days. Two values for acceptable loss are shown: Φ = 0.001 and Φ = 0.05.

In Category 3, the basic reproductive number for susceptible plants was <1. Thus, in the absence of resistant plants, disease incidence in susceptible plants was expected to be zero. With acceptable loss equal to 0.001, deployment of resistant plants resulted in a small increase in the percentage of infected susceptible plants compared to simulations without resistant plants, with the proportion of infectious susceptible plants remaining below acceptable loss (Fig. 5a,d). With acceptable loss equal to 0.05, 27% of simulations with resistant plants resulted in an increase in the proportion of infected susceptible plants that exceeded 0.05 (Fig. 5a,d). As acceptable loss quantified the proportion of infectious susceptible plants and did not include latently infected susceptible plants , evaluation of the performance of the acquisition threshold is best accomplished by comparing the proportion of infectious susceptible plants to the assigned value for acceptable loss. With acceptable loss equal to 0.05, the proportion ofinfectious susceptible plants exceeded acceptable loss in only 6% of simulations (Fig. S1). While the acquisition threshold generally performed well for simulations in Category 3, the percentage of infectious susceptible plants exceeded acceptable loss in some cases when a non-conservative value for acceptable loss was used. Increasing the value for acceptable loss increased the number of susceptible plants infected due to deployment of resistant plants, which in turn destabilized disease control in susceptible plants for some parameter combinations.

Category 4: Spread among susceptible plants controlled (R_0,S < 1) and acquisition rates from resistant plants above the threshold

For simulations in Category 4, 48% and 83% of simulations resulted in >1% of susceptible plants infected at the end of 1800 days with values for acceptable loss of 0.001 and 0.05, respectively (Fig. 6a). As parameter values in Category 4 required that the basic reproductive number for susceptible plants was <1, disease incidence in the absence of resistant plants was expected to be zero. Accordingly, all disease observed in susceptible plants in Category 4 was due to deployment of resistant plants (Fig. 6d). As acquisition from resistant plants was above the theoretical threshold, acquisition rates from resistant plants were high (Fig. 6c). Thus, deployment of resistant plants with acquisition rates above the theoretical threshold resulted in spread of the pathogen among resistant plants (Fig. 6b) and subsequent spread to susceptible plants (Fig. 6a,d).

Figure 6

Results of simulations satisfying Category 4 . (a) Distribution of percentage of infected susceptible plants ; (b) distribution of percentage of infected resistant plants , (c) distribution of observed acquisition rates (α_R); and (d) distribution of difference in percentage of infected susceptible plants in paired simulations with (F > 0) and without (F = 0) the resistant variety. A positive difference indicates that planting the resistant variety increased disease incidence in susceptible plants (to the right of the dashed line) and a negative difference indicates that planting the resistant variety decreased disease incidence in susceptible plants (to the left of the dashed line). Distributions were generated by completing 40 000 simulations using random draws for parameters (Table 2) that satisfied Category 4 (Table 1) and evaluating the outcome at the end of 1800 days. Two values for acceptable loss are shown: Φ = 0.001 and Φ = 0.05.

Discussion

Deployment of resistant varieties is a key strategy for mitigating losses due to arthropod-transmitted pathogens of perennial crops (Laimer et al., 2009; Butac et al., 2013; Ilardi & Tavazza, 2015). While development of plants that are immune to infection would be ideal (i.e. have no detectable pathogen), in some cases the best available resistance traits may only confer partial resistance, with partially resistant plants remaining hosts for the pathogen (Krivanek & Walker, 2005; Hartmann & Neumüller, 2006; Walker et al., 2014). As partially resistant varieties are expected to maintain yield after infection, they are less likely to be rogued than susceptible varieties. Thus, there is a risk that partially resistant varieties may serve as inoculum sources for pathogen spread to susceptible varieties. Here, a threshold acquisition rate from partially resistant plants that resulted in limited pathogen spread from partially resistant to susceptible plants was identified (Eqn 12). Accordingly, observed acquisition rates from partially resistant plants could be determined by conducting transmission assays and compared to an estimated threshold rate (Eqn 12) to assess risk prior to deployment.

Threshold acquisition rates from partially resistant plants (Eqn 12) were dependent on a number of parameters (Fig. 2). Parameters such as the rate latently infected plants become infectious (τ), inoculation rate of susceptible plants (θ_S), and plant visitation rate (v) are inherent to the system under consideration. In contrast, parameters such as vector turnover (m), vectors per plant (), replacement of susceptible plants (r_S), and proportion of plants that were partially resistant (F) may be influenced by human actions. Accordingly, disease management decisions within a region will dictate levels of acquisition from partially resistant plants that result in limited or rapid spread of a pathogen from partially resistant plants to susceptible plants. For example, threshold acquisition rates will be higher in a region with an area-wide vector control programme (low ) that also engages in aggressive roguing of susceptible plants (high r_S) than in a region without an area-wide vector control programme (high ) that engages in haphazard roguing of susceptible plants (low r_S; Fig. 2). Finally, values for acceptable loss (Φ) may be designated based on the degree of loss a group of growers are willing to incur. An economic model could be used to identify values for acceptable loss that maximize economic return.

For tolerant plants, which maintain pathogen titres similar to susceptible plants, it seems reasonable to presume that acquisition rates of vectors held on tolerant and susceptible varieties will be similar, potential differences in vector feeding behaviour or preference notwithstanding. In contrast, partially resistant plants have lower pathogen titres than susceptible plants, with acquisition from resistant plants presumed to be lower than acquisition from susceptible plants. However, it is unclear if there is a general relationship between pathogen titre and probability of acquisition. For example, acquisition of Tomato yellow leaf curl virus by whiteflies and acquisition of the phytoplasma that causes grapevine flavescence dorée by Scaphoideus titanus were both positively related to pathogen titre (Lapidot et al., 2001; Galetto et al., 2014). However, transmission efficiency of Xylella fastidiosa by the glassy-winged sharpshooter was not related to pathogen titre in a range of commonly planted susceptible grapevine cultivars that varied in X. fastidiosa titre (Rashed et al., 2011). As X. fastidiosa titres in the cultivars used by Rashed et al. (2011) were higher than those reported for resistant grapevines (Krivanek & Walker, 2005), empirical data are needed to evaluate the potential of X. fastidiosa resistant grapevines to serve as inoculum sources. Ultimately, it would be of value to plant breeders to identify the relationship between pathogen titre and probability of acquisition for a specified plant–vector–pathogen system, as it may provide the opportunity to directly estimate a pathogen titre below which acquisition is unlikely to occur.

Model results were placed into one of four categories based on the extent to which pathogen spread among susceptible plants was suppressed and on whether acquisition rates from partially resistant plants were above or below the theoretical threshold (Table 1). Identifying which of the four categories a system conforms to prior to release of a resistant variety would be of value. For example, release of a partially resistant variety into a situation conforming to Category 1 will improve a bad situation. In this category, pathogen spread among susceptible plants was uncontrolled and acquisition rates from partially resistant plants were low (Fig. 3c). As a result, replacement of susceptible plants associated with high acquisition rates with partially resistant plants associated with low acquisition rates slowed pathogen spread, thereby providing a benefit to farmers maintaining susceptible varieties (Fig. 3d). In Category 2, pathogen spread among susceptible plants was uncontrolled and acquisition rates from partially resistant plants were high (Fig. 4c). While growers maintaining susceptible plants in Category 2 will experience no benefit from deployment of partially resistant plants (Fig. 4d), this point is moot as pathogen spread was not effectively controlled to begin with. Given losses anticipated in Category 2, rapid widespread adoption of the partially resistant variety is likely to be the best solution. Category 3 represents the ideal situation, with pathogen spread among susceptible plants suppressed and partially resistant plants not contributing to pathogen spread (Fig. 5). Situations conforming to Category 4 will require the most planning prior to deployment of a partially resistant variety. Category 4 describes a situation where, in the absence of partially resistant plants, disease control in the susceptible variety is effective but deployment of a partially resistant variety is likely to increase disease levels in susceptible varieties (Fig. 6d). Perhaps the simplest solution to the problem confronted in Category 4 is to spatially separate susceptible and partially resistant varieties to reduce likelihood of vector dispersal between fields maintaining susceptible and partially resistant varieties. Alternatively, management tactics that increase the acquisition threshold from partially resistant plants (Fig. 2) could be implemented to shift the system from Category 4 to Category 3.

Use of coupled differential equation models is a common approach in plant pathology (Jeger et al., 1998; Madden et al., 2000; Sisterson, 2009; Zeilinger & Daugherty, 2014; Sisterson & Stenger, 2016). One of the benefits of using coupled differential equation models is the ability to identify equilibrium values and threshold conditions in terms of model parameters, thereby providing broadly applicable results (Jeger, 1986; Madden et al., 2000, 2007). However, the model format used here has a number of limitations that should be considered when interpreting results. For example, the model included no spatial structure and assumed that vector dispersal was sufficiently high that a vector was equally likely to visit any plant. Vector dispersal rates are variable and distance between blocks of resistant and susceptible plants are likely to have an important effect on pathogen spread. Further, use of insecticides or other vector suppression tactics may be different in blocks of susceptible and partially resistant varieties which could have regional effects on vector population dynamics that cannot be considered using the model presented here (Sisterson & Stenger, 2016). In addition, latent periods of susceptible and resistant plants were assumed to be equal (Table 2). Increasing the latent period of resistant plants relative to susceptible plants would tend to decrease rates of pathogen spread and warrant additional investigation. Finally, the model did not include seasonality or alternative non-crop hosts for the vector or pathogen. Ultimately, there would be value in expanding the model to include spatial structure, explicit vector population dynamics, and seasonality.

Emphasis of the analyses conducted here was on the effects of deploying a partially resistant variety on disease incidence in a susceptible variety. Such effects are of concern only to growers maintaining susceptible varieties and not to growers that have switched to the partially resistant variety. For growers that switch to a partially resistant variety, the economic benefits are likely to be tangible. For example, the Citrus tristeza virus (CTV) susceptible rootstock ‘sour orange’ is regarded as the most vigorous rootstock for a number of citrus scions and outperforms CTV tolerant rootstocks in the absence of CTV (Louzada et al., 2008; Barnier et al., 2010). However, in areas where CTV is endemic there is a clear economic benefit to using CTV tolerant rootstocks. Results of the analyses conducted here indicate that risk of partially resistant plants serving as sources of inocula could be assessed prior to deployment, thereby enabling design of strategies to minimize losses in susceptible varieties.

Acknowledgments

The authors thank Sean Uchima and Lindsey P. Burbank for comments on an earlier draft of the manuscript and Adam Porter for sharing code for the random number generator. Funding for this work was from the United States Department of Agriculture (USDA) Agricultural Research Service appropriated project 2034-22000-012-00D. Mention of trade names or commercial products in this publication is solely for the purpose of providing specific information and does not imply recommendation or endorsement by the USDA. USDA is an equal opportunity employer.