## Abstract

The deployment of crop varieties that are partially resistant to plant pathogens is an important method of disease control. However, a trade-off may occur between the benefits of planting the resistant variety and a yield penalty, whereby the standard susceptible variety outyields the resistant one in the absence of disease. This presents a dilemma: deploying the resistant variety is advisable only if the disease occurs and is sufficient for the resistant variety to outyield the infected standard variety. Additionally, planting the resistant variety carries with it a further advantage in that the resistant variety reduces the probability of disease invading. Therefore, viewed from the perspective of a grower community, there is likely to be an optimal trade-off and thus an optimal cropping density for the resistant variety. We introduce a simple stochastic, epidemiological model to investigate the trade-off and the consequences for crop yield. Focusing on susceptible–infected–removed epidemic dynamics, we use the final size equation to calculate the surviving host population in order to analyse the yield, an approach suitable for rapid epidemics in agricultural crops. We identify a single compound parameter, which we call the efficacy of resistance and which incorporates the changes in susceptibility, infectivity and durability of the resistant variety. We use the compound parameter to inform policy plots that can be used to identify the optimal strategy for given parameter values when an outbreak is certain. When the outbreak is uncertain, we show that for some parameter values planting the resistant variety is optimal even when it would not be during the outbreak. This is because the resistant variety reduces the probability of an outbreak occurring.

## 1. Introduction

Plant diseases impose significant crop losses in agriculture, horticulture and forestry. Estimates of annual crop losses in agriculture range from 14% of crop yield worldwide [1–3], extending to 20–40% when weeds and pests are included [4], with total elimination of crops in some severe epidemics [5]. The deployment of crop varieties that are genetically resistant to plant pathogens provides an efficient means of disease control. The simplest case involves complete resistance to a pathogen that remains durable for long periods of time without being overcome by new virulent strains of the pathogen [6,7]. Newly released resistant varieties should ideally also match or surpass the agronomic properties of the susceptible varieties that they replace. Historically, many resistant varieties exhibited a qualitative resistance that is a form of complete resistance in which the pathogen is unable to infect the host [8]. There is increasing awareness, however, that qualitative resistance under single gene control in the host imposes such strong pressures on the pathogen population to overcome resistance so that failure becomes almost inevitable [9,10]. Accordingly, increasing attention is being paid to the release of partially resistant varieties that slow or otherwise reduce, but do not prevent, infection and multiplication of the pathogen on the host crop [10,11]. Such resistance is frequently under the control of few to many genes in the host crop [12,13]. Trade-offs may occur, however, between partial resistance and the agronomic properties of the host crop such that partial resistance may be associated with the lower yield for reduced quality in the crop variety compared with the standard [14–17]. This presents individual growers with a dilemma: growing the resistant crop imposes a yield penalty that may be compensated for only if disease occurs and is sufficient for the infected, partially resistant crop to outyield an equivalently infected standard crop. The decision is further complicated: as more growers elect to plant the partially resistant crop the probability of a severe epidemic occurring decreases. Viewed from the perspective of a community of growers, there is likely to be an optimal trade-off, and hence a cropping density for the proportion of sites that are planted to the partially resistant variety. Moreover, even if an outbreak is certain, it is not necessary for all growers to have planted the partially resistant variety for the benefit to be realized by the community of growers as a whole [18,19].

Previous epidemiological analyses on the introduction of partially resistant varieties in the landscape have focused on the effects of disease dynamics [18–21]. Here, we focus on the trade-off between disease and yield at the landscape scale. The primary aim of this paper is, therefore, to provide insight into when it is worth deploying a partially resistant variety and in what proportions relative to a standard, higher yielding susceptible variety. We do this using simple stochastic epidemiological models that characterize the spread of the pathogen, and hence disease, and the consequences for crop yield. We identify policy plots [22] that can be used to infer optimal strategies given some prior knowledge of the resistance–yield trade-offs. We make certain simplifying assumptions that we subsequently relax. Our intention in this paper is to address the generic problem of how to deploy partial resistance when there is a yield penalty and uncertainty. We intentionally propose a flexible modelling framework that encompasses a variety of epidemiological mechanisms that could be associated with partial resistance, with broad applicability to a range of host–pathogen systems. Hence, we consider *SIR* epidemics, in which susceptible hosts (*S*) become infected (*I*), remain infectious for a period of time and then are removed (*R*). Removal may occur naturally by disease-induced death or by deliberate removal, for example, by roguing of infected plants. *SIR* epidemics are typified by an increase in infected hosts followed by a decrease as the epidemic ‘burns itself out’ or is controlled. We assume that yield is a function of the amount of healthy, i.e. uninfected, host, allowing for a yield penalty for healthy hosts of the partially resistant compared with the susceptible variety. We initially assume that yield is accumulated over a long period of time relative to the period of crop growth and the time course of the epidemic. This assumption enables us to gain analytical insight to inform the deployment of partially resistant crops subject to yield penalties. We subsequently relax the assumptions to test the robustness of the conclusions. Our initial results apply to annual crops, in which epidemics happen fast, typified by potato late blight and rusts of small grain cereals. Here, *S* and *I* are consequently expressed as units of plant host tissue [23]. Our results also apply to cassava virus diseases, in which yield is accumulated over a long period and roguing is practised to remove infected hosts. In this case, *S* and *I* refer to whole plants. The model may also be applied, more generally, to perennial crops in which roguing of infected plants occurs, yield accumulates over long periods and there is continuing harvesting. We first consider the case where epidemics are inevitable and address two broad questions: under what circumstances is investing in the partially resistant variety likely to be profitable, and how much of the resistant variety should be deployed to maximize yield? We subsequently consider the robustness of the inferences about the deployment of the partially resistant variety when the occurrence of the epidemic is uncertain.

## 2. Methods

### 2.1. Epidemic model

We consider disease spreading through a metapopulation, comprising two types of host crop, a partially resistant and a fully susceptible ‘standard’ variety. Plants of each variety can be in one of three classes: susceptible (*S*, i.e. healthy), infected (*I*) or removed (*R*, i.e. post-infectious). The principal parameters used in the models for disease spread and yield are summarized in table 1, for ease of reference. We make the following assumptions:

— The epidemic follows

*SIR*compartmental dynamics, with density-dependent mixing [24]. The transition is realized either by disease-induced mortality, by roguing of the infected hosts, or by a combination of both.— Hosts in the infected and removed classes do not contribute to the yield.

— The resistant hosts are less likely to become infected upon contact with the pathogen (by a factor ), produce less inoculum (by a factor ), have a different infectious period (by a factor ) and contribute less to the final yield (by a factor ). It is natural to consider both shorter and longer infectious periods for the resistant variety, because the resistant hosts might take longer to die or they might take longer to develop visible symptoms upon infection and thus avoid detection and subsequent removal (by, for example, roguing of symptomatic hosts in certain crops) for longer than the standard hosts.

We assume a fraction of all the hosts to be resistant, where and are the numbers of standard and resistant hosts, respectively. The model for disease spread is given by (where subscripts S and R denote standard and resistant varieties) 2.1 2.2 2.3 2.4 2.5where is the transmission rate and () is the removal rate of the standard (resistant) hosts, respectively. It is convenient to introduce the basic reproductive number of the standard hosts in a monoculture, . This is defined in the standard way as the average number of secondary infections caused by a single infected host over the course of its infectious period.

### 2.2. Yield without uncertainty

When an outbreak is certain, we represent the yield *per host* and *per unit time*, accumulated by healthy plants over a fixed period *T*, by a straightforward adaptation of the integral over the susceptible hosts, a simple measure used in plant epidemiology [25,26]
2.6

To simplify the analysis, we consider the yield accumulated over a very long period of time relative to the duration of the longest epidemic ( in the above equation), so that it is effectively given by the proportions of susceptible hosts that survive the epidemic. We include an explanatory sketch in the electronic supplementary material, S1. Therefore, the yield *Y* is given by
2.7where is the proportion of the susceptible hosts in the population after the outbreak has ended. Note that this assumption is made purely for mathematical convenience to make analytical progress, from which initial insights into the optimal strategy can be inferred. We subsequently relax this assumption. We introduce an arbitrary, finite-time horizon to equation (2.6) and show that the inferences derived for an infinite-time horizon hold when yield is accumulated over a finite period of time (§3.4). The choice of , while arbitrary, is motivated by keeping the analysis generic.

### 2.3. Yield with uncertainty

When there is uncertainty about whether or not an outbreak will occur, the yield function is given by the expected yield 2.8 2.9where is the probability of an epidemic. To model this probability, we consider a situation in which the pathogen has a constant small rate of introduction into the host population over some period of time (for example, because the climatic conditions are favourable during this period). The number of introductions then follows a Poisson distribution with mean . Each time the pathogen is introduced into the system, it infects an initial host with probability , where is the probability of a standard host getting infected upon contact with the pathogen. For convenience, we define . Once the initial host has been infected, a large-scale outbreak will occur with probability which can be calculated using standard arguments; see the electronic supplementary material, S2. Putting this together and using the thinning property of Poisson processes leads to the expression for the overall probability of an epidemic as 2.10

### 2.4. Summary of the assumptions

In this section, for convenience and reference, we summarize the principal assumptions of the model and the subsequent analyses. We approach the problem generically, using biologically plausible parameter values and ranges to reflect classes of host–pathogen system rather than restricting the analysis to a single system. Instead, our analysis is designed to identify which parameters are important and what are their critical ranges. We introduce flexibility by allowing for the epidemiological mechanisms accounting for partial resistance to be expressed through changes in one or more of the following: the infectivity of infected hosts, the susceptibility of healthy hosts or the length of the infectious period. These effects can be tuned independently in our model through different parameters. In our model, the unit of host can be either a whole plant or healthy tissue and we assume the infected hosts eventually die or are removed through roguing. For simplicity, we assume that neither infected nor removed hosts contribute to the yield. This means there is no replanting or that it takes a long time for a replanted host to reach maturity, such as in the case of tree crops. The yield is modelled as an integral over the healthy hosts. Initially, in order to make the model analytically tractable, we assume the yield is accumulated over a time scale much longer than the epidemic duration so that it can be approximated by an infinite time horizon. In the second part of the analysis, we relax this assumption and consider yield accumulated only over the duration of the epidemic. The main purpose of this is to verify that our qualitative results are not simply an artefact of the infinite time horizon. Finally, when we allow for uncertainty in the occurrence of an epidemic, we assume that the pathogen has a constant rate of import into the host population over a certain time period preceding the potential epidemic.

## 3. Results

### 3.1. Model without uncertainty

Using the final size equations for an epidemic [24], it is possible to derive an analytic expression for the yield function (see the electronic supplementary material S3 for details)
3.1where is the final size of the susceptible class of the non-resistant hosts and depends on . It is not possible to obtain an analytic expression for the yield as a function of directly. However, the analytic solution (i.e. (3.1)) shows three possible control scenarios to optimize yield. These are illustrated for three different yield penalties associated with the partially resistant variety in figure 1. There are two extreme scenarios: no control, i.e. grow only the standard variety when the yield penalty is high (figure 1*a*); and ‘full’ control, i.e. grow sufficient resistant variety to bring the basic reproductive number below 1 (see below) when the yield penalty is low (figure 1*c*). As the yield penalty increases, so the cropping ratio of standard to resistant variety required to achieve an optimal yield increases leading to an intermediate control scenario (figure 1*b*). Note that the ‘full’ control (figure 1*c*) does not necessarily mean but corresponds to the density being given by
3.2where is a critical density such that the effective basic reproductive number of the system (2.1)–(2.4) in the presence of the resistant hosts falls below 1. This is because once the basic reproductive number falls below one, the epidemic is prevented and further deployment of the resistant variety has no effect. The effective basic reproductive number can be calculated using the next generation method [27]: it is given by
3.3where we introduce the parameter which we refer to as the *resistance efficacy*.

From equation (3.1), we conclude that the optimal proportion only depends on and through their product . This allows us to make two-dimensional policy plots showing where different types of control are optimal, for various values of yield penalty, *f* (e.g. figure 2). Note that when (the dark region in figure 2*a*), the control supports the spread of the pathogen and we therefore do not consider this region any further. Examination of figure 2*a* and corresponding figures for a range of values of *f* shows that the region of the parameter space where intermediate control is optimal is small. Accordingly, we investigate the potential for loss in yield if the option of intermediate control is ignored, using as a metric for yield loss in choosing no or ‘full’ control in place of intermediate control (cf*.* figure 1). The metric is a function of parameters *f*, , and . In figure 2*b*, we plot the worst yield loss as a function of *f*, that is, . The losses are small and decrease very quickly with *f*. Furthermore, in the () plane, they are only appreciable along the curve and close to , where the relative susceptibilities of resistant and standard varieties are identical. The explanation of this can be found in the electronic supplementary material, S4. This means that for the most part, we can focus on the extreme controls, because they are mostly optimal and when they are not, they provide a good approximation to the optimum.

### 3.2. Extreme control optimization

From equation (3.1), we can derive the conditions for deployment of a resistant variety under which ‘full’ control is better than no control. There are two cases to consider. If the resistance is effective enough so that the outbreak can be prevented altogether, that is if , full control is better than no control when 3.4where is the proportion of surviving hosts when no control is deployed. If the resistance is not effective enough to prevent the outbreak, that is if , full control is better than no control when 3.5

These conditions are derived in the electronic supplementary material, S4. Note that these conditions only depend on the parameters , and via the resistance efficacy and therefore there are effectively only three controlling parameters: , and *f*. This allows us to plot a policy diagram, showing when the resistant variety is worth deploying. The plot for can be found in figure 3; other basic reproductive numbers are not qualitatively different. Figure 3*a* shows the region where full control is optimal and the boundary, which corresponds to the critical value of .

Figure 3*b* shows whether the optimal value of the yield, *Y*, increases more quickly as the yield penalty *f* increases or when the resistance efficacy, , increases, using the quantity as a metric. When the metric is positive, increasing *f* is more important and when it is negative increasing is more important to increase yield. The effect of changing is most dramatic near the boundary and therefore when is close to this boundary from below, it is much more profitable to increase it above the boundary than to attempt to increase the yield factor.

### 3.3. Model with uncertainty

The probability of invasion, , derived from equation (2.10) decreases as the cropping ratio of the resistant hosts increases (figure 4). The decline in is steeper when the susceptibility factor (, table 1) of the resistant fraction of the host population becomes smaller, that is, when this fraction of hosts presents an increasing level of resistance. To investigate the impact on the yield of the deployment of the resistant variety in the presence of uncertainty, we repeat our analysis from the previous section. Similarly to the case of a deterministic, certain outbreak, numerical analysis reveals that a good approximation is provided by only considering extreme controls, that is none or ‘full’. Note that because of the introduction of the probability there are now four independent controlling parameters, , , *f* and . In figure 5, we show the policy plots corresponding to three different values of and for both the deterministic and the stochastic case. We can see that when the resistance significantly reduces the susceptibility to the pathogen, but its efficacy is low overall because , it can happen that the control is optimal only in the presence of uncertainty. This means that while the control is not desirable during the outbreak, the benefits of the possibility of preventing the outbreak altogether are significant. We can formalize this by defining , that is the yield gained by controlling correctly as opposed to not controlling at all. In figure 6, we plot the difference between the yield gained by controlling with uncertainty and without uncertainty. We can see that without uncertainty the control provides greater yield gains for high resistance efficacy but, in agreement with figure 5, when the resistance efficacy is low, the susceptibility factor is low and the yield parameter *f* is high, the control provides significantly greater yield gains when the uncertainty is present. This demonstrates the importance of reducing the probability of an outbreak occurring even when during the outbreak the control is not desirable.

### 3.4. Yield model with finite time horizon

So far, we have considered long-term yield. We have shown that contrary to intuition, the benefit of deploying a resistant variety can be greater when the disease outbreak is uncertain. We now verify this result when the assumption of the infinite time horizon is relaxed and the yield to be maximized is evaluated over a finite period of time. We selected the time of the duration of the epidemic as a representative of this finite time horizon. Therefore, the yield is now given by 3.6

Whereas our initial model focuses on the final state of the epidemic, the beginning of the epidemic and how quickly the pathogen invades now become important factors. Note that because increasing the cropping ratio of the resistant variety leads to a decrease in of the system, the duration of the epidemic depends on . It is no longer possible to use the final size of the epidemic to characterize the yield in this case and no analytic expression for the yield can be derived. To proceed, we ran numerical simulations of the model for randomly generated parameter values (see the electronic supplementary material, S5, for the technical details). The results (figure 7) are in good agreement with the idealized ‘long-time’ yield model (cf. figure 6). As expected, the benefits of full control when an outbreak is uncertain correlate with lower values of and higher values of . The average value of in the simulations where the uncertainty leads to greater benefits of control was 0.48, while in those where the benefits of control were greater without uncertainty it was 0.63. For , the values were 4.47 and 1.65, respectively.

## 4. Discussion

We have analysed the impacts on disease dynamics and crop yield at landscape scales of the resistance–yield trade-off when attempting to control a disease by deploying a resistant crop variety in two different situations. In both cases, we assume that the resistant variety carries a yield penalty, i.e. it yields less than the standard variety in the absence of disease. First, we assumed that an epidemic is inevitable in order to analyse how the deployment of the resistant variety impacts on disease dynamics and how these in turn affect the final yield averaged over multiple fields in the landscape. Subsequently, we considered the situation where only the probability of an epidemic is known at the time of planting. We have not considered detailed models for demographic or environmental stochasticity [24] other than by allowing for uncertainty as to whether an outbreak will occur, coupled with deterministic dynamics. We selected this approach for two reasons. Firstly, it concentrates on the main component of variability that we wanted to investigate, that is whether or not an epidemic occurs. Secondly, it allows us to carry over most of the methodology from the analysis of the problem without uncertainty and it simplifies the numerical analysis when a different yield model with a finite time horizon is considered. We used a simple approach of integrating over the susceptible hosts or host tissue as appropriate [25,26,28], which assumes that healthy tissue contributes to yield. Our intention throughout is to introduce a generic modelling framework. The framework is motivated by fast epidemics on agricultural crops in which the epidemic naturally burns itself out or by removal of infected hosts by roguing as may occur, for example, in cassava crops. The framework can, in principle, be extended to perennial crops. To simplify the mathematics and to minimize the number of free parameters, we initially assume that the yield is accumulated over a time period which is much longer than the time scale of the epidemic such as in the case of potato late blight and some rust diseases of small grain cereals. This allows us to approximate the yield by the amount of susceptible host that survives the epidemic.

When an outbreak is certain, we showed that the susceptibility, infectivity and the removal factors , and introduced in the model can be combined into a single parameter which we call the *efficacy of the resistance*. This aggregate parameter is a convenient means of integrating the components that characterize the differences between resistant and susceptible varieties. The approach is analogous to, yet different from, the approach advocated by Parlevliet [29] directed at quantifying the components of resistance when comparing different varieties. Thus, Parlevliet [29] first showed how to quantify epidemiological components of resistance such as infection frequency, latent period and spore production per unit time as well as the infectious period. From these analyses, Parlevliet [29] was able to quantify and ascribe the components of resistance that accounted for differences between susceptible and resistant hosts. Savary *et al.* [30] subsequently showed for peanut rust how to combine individual components into a product as a relative measure of resistance that reflected differences between epidemiological components, which is analogous to . The difference lies in that is constructed so that the components relate directly to parameters that define rates in an epidemiological dynamical model. Thus, is a measure of the reduction in the rate of transmission of infection; is a measure of the change in the infectious period; and is the measure of reduction in the transmission rate. We found the conditions that has to satisfy in order for deployment of the lower yielding, resistant variety to be optimal. Major gains can be achieved particularly when increases above , which is also the condition for being able to prevent the outbreak altogether.

A further trade-off arises when uncertainty about an outbreak is added. Deployment of the resistant variety is wasteful in the absence of an epidemic. However, given uncertainty about an epidemic outbreak, deploying a resistant variety renders the overall host population less susceptible to the pathogen and therefore decreases the probability that an outbreak will occur in the first place [31]. Figures 5 and 6 provide the resolution to this trade-off.

Our results indicate that when the resistance is sufficient to prevent an epidemic altogether ( large), the benefits of the control are greater when an outbreak is certain. However, when the resistance is not strong enough to prevent the outbreak ( small) but offers significant reduction in susceptibility to the pathogen ( is small), the benefits of control are overall greater when uncertainty is accounted for (cf. figure 6 and figure 5). In such a case, reducing the invasion probability and thus possibly preventing the outbreak altogether outweighs the risk of wasting resources by deploying resistant cultivars. Note that this is independent of the assumption that the invasions follow a Poisson process. Rather it is a consequence of the fact that when the resistance is not very effective during the outbreak, it can still significantly reduce the probability of the pathogen invading. Biologically, this can happen when the resistant hosts have a longer infectious period (that is ). This is possible, for example, when the disease eventually kills the hosts, but takes longer to kill the resistant hosts. Alternatively, in the analogous case when the disease does not kill the hosts but rather they are removed from the population via roguing, the resistant hosts might take longer to show symptoms and therefore avoid detection.

We have made a number of important simplifying assumptions about the epidemiological model. These have allowed some insights to be gained about the trade-offs to be considered in deploying resistant varieties with lower yield potential compared with a susceptible variety. We have used a simple *SIR* epidemiological model (equations (2.1)–(2.5)) that is parsimonious while allowing flexibility in attributing the effects of partial resistance to different epidemiological processes. Accordingly, there are just two parameters for the underlying epidemiological model (, the transmission rate; , a measure of the infectious period). To these we added three parameters (table 1) to allow for resistance. One or more of these could be set to one and effectively eliminated as a separate parameter. We also introduced a similarly tunable parameter for the ratio of resistant hosts in the population, while is a measure of uncertainty about whether or not disease is likely to occur. The other parameters listed in table 1, and , are compound parameters derived from the others.

We have also used a strongly simplifying assumption that crop yield can be assessed from the final level or the integral over time of susceptible hosts, which assumes that infected hosts do not contribute to yield. Our intention here, however, was to focus on principles relating to decision-making in relation to the deployment of resistance. For this, we have preferred to keep the model for the epidemic and for yield simple. It is possible that infected hosts might well contribute to yield and this could readily be included in the model as could additional feedback loops for the effects of different levels of infection on growth dynamics of the host [26,32,33].

Our results have shown the importance of accounting for uncertainty to inform policy on behalf of an agricultural planner. We restricted our analyses to considering the probability of an outbreak occurring. Future analyses could address the robustness of the conclusions to short-term fluctuations associated with demographic and environmental stochasticity. Of more probable importance, however, are longer term fluctuations that result in periodic epidemics, arising from repeated introductions of the pathogen from outside the system, and periods of long-term environmental suitability. The intricacy of this approach lies in the fact that while the environmental suitability forces outbreaks with a certain period, introducing the resistant variety would change the intrinsic time scale of the epidemic [19]. Combining these two effects could produce complex dynamics [19,24]. Finally, it would be interesting to include the effects of heterogeneity in growing behaviour or a more formal treatment of risk-aversion [34,35]. In this paper, we model the yield in the presence of uncertainty as the average of the yield when the outbreak epidemic does or does not occur. However, farmers and policy-makers tend to prefer control strategies that minimize the probability of losses rather than those that maximize the probability of gains [36,37]. Our model could be combined with game theoretic approaches to analyze these outcomes.

## Authors' contributions

M.V. conceived and designed the study. M.V. and N.C. implemented the study. M.V. and C.G. wrote the initial version of the paper. All three authors discussed the results and their implications throughout the study, and edited the final version of the paper.

## Competing interests

We declare we have no competing interests.

## Funding

M.V. was funded by Bakala Foundation and Trinity College, Cambridge.

## Acknowledgements

We thank the four anonymous referees for their helpful comments and Matt Castle and Andrew Craig for many helpful suggestions, which we feel have greatly improved the paper.

## Footnotes

Electronic supplementary material is available online at https://dx.doi.org/10.6084/m9.figshare.c.3491703.

- Received June 8, 2016.
- Accepted September 6, 2016.

- © 2016 The Author(s)

Published by the Royal Society. All rights reserved.