Considering behaviour to ensure the success of a disease control strategy

The success or failure of a disease control strategy can be significantly affected by the behaviour of individual agents involved, influencing the effectiveness of disease control, its cost and sustainability. This behaviour has rarely been considered in agricultural systems, where there is significant opportunity for impact. Efforts to increase the adoption of control while decreasing oscillations in adoption and yield, particularly through the administration of subsidies, could increase the effectiveness of interventions. We study individual behaviour for the deployment of clean seed systems to control cassava brown streak disease in East Africa, noting that high disease pressure is important to stimulate grower demand of the control strategy. We show that it is not necessary to invest heavily in formal promotional or educational campaigns, as word-of-mouth is often sufficient to endorse the system. At the same time, for improved planting material to have an impact on increasing yields, it needs to be of a sufficient standard to restrict epidemic spread significantly. Finally, even a simple subsidy of clean planting material may be effective in disease control, as well as reducing oscillations in adoption, as long as it reaches a range of different users every season.


Background
The adoption of a costly control strategy for disease can be viewed as a public goods problem, and has been studied in many systems, particularly for vaccination (e.g. [1]; for a review, see [2]). Oscillations in adoption of the control strategy are often observed; waves of infection and subsequent adoption are followed by a decrease in adoption as infection declines and control is deemed too costly in the absence, followed by a resurgence, of infection (e.g. [3,4]). Cost thresholds, above which adoption decreases dramatically as very few individuals different components of the system: the farming system and disease presence in the area, the clean seed material and how it is introduced, and the grower behaviour in response to this introduction. In this way we can compare the outcomes we might expect in different areas with different seed systems, evaluating the influence that grower behaviour has on our results

Population dynamics 2.1.1. Within season
For each field i we consider uninfected (S i ), latently infected (L i ) and infectious and symptomatic (I i ) plants and infectious vectors (V i , with a total whitefly vector population given by W). For fields i = 1 . . . N, we consider Plants are harvested at rate h and additional plants may be replanted at rate µ. Replanting occurs with planting material obtained at the beginning of a season, where we consider commercial (seasonal harvesting and replanting events), subsistence (continuous harvesting throughout the season with annual replanting events) and casual (continuous harvesting and replanting with original material throughout the season) growers. Plants may additionally be infected through contact with viruliferous vectors, which infect uninfected plants at density-dependent rate β p . Resistant plants are infected at a lower infection rate β p . Once infected, latently infected plants progress to a fully infectious state at rate γ p , when they may be rogued at rate g.
In terms of the vector, uninfected vectors (W − V i ) become infected at density-dependent rate β v through contact with infectious plants in the same field or a reservoir host (at density R). The vector loses this infectivity at rate λ, and dies at rate ω. Vector between-field dispersal, where we assume emigration and immigration rates are the same, occurs at rate m. We assume this dispersal is constant for emigration, while for immigration into field i we sum dispersal of infectious vectors from each field j to field i. This is given for Euclidean distance d ij between the centres of the pair of fields by the probability density dispersal function δ ij = (Aα 2 /2π )e −αd ij , for attractive area A (where the dispersal probability at every point in the field is equal) and mean distance of dispersal 1/α.

Between seasons
Replanting occurs with planting material obtained either from the grower's own field (q i = 1, with probability b c if not using a clean seed system) or through trade with the owner of another field (q i = 1, with probability 1 − b c if not using a clean seed system), or from a clean seed system (q i = 0). If the former, the proportion of cuttings in field i obtained from every field j is given by τ ij , so that N j=1 τ ij = 1. For trade with neighbouring fields, a grower at field i trades with a number of neighbours up to a given maximum b s , where grower j is chosen with probability ρ ij = re −d ij /d max for d max the maximum distance between any two fields and random, uniformly selected variable r ∈ [0, 1]. This latter variable r adds stochasticity to the trade process, ensuring that growers do not simply trade with their nearest neighbours. Growers may remain loyal to the same suppliers each season, or may alter suppliers, where a given number of growers b l across the system are loyal. We note that for each pair of fields i and j, the probability of trade ρ ij is unrelated to the quantity of trade τ ij , although τ ij > 0 if and only if ρ ij > 0. The level of infection in the planting material obtained from field j is determined by P j = 300 0 h(S j + ηL j ) dt + (S j + ηL j )| 300 and C j = the end of the season where all remaining plants are harvested. These are constant within a season, based on the dynamics of the previous season, but vary between seasons. Latently infected harvested plants may also have sufficiently low viral load as to undergo reversion at rate η, while infectious plants may be successfully identified as such and discarded with likelihood given by cutting selection parameter ξ . In this way, the proportion of infected cuttings replanted depends on whether or not the grower is using planting material directly obtained through a clean seed system, and the incidence of infection in planting material that the grower has collected both from her own and her neighbour's fields.

Grower behaviour 2.2.1. Between seasons
The epidemiological model is continuous within a fixed duration season, with an annual discrete complete harvesting and replanting event. The behavioural model and the epidemiological model are then linked through an annual decision by the growers on whether or not to alter their choice of replanting strategy (affecting q i above), based on the reward that each strategy is perceived to produce. This follows the methods of Milne et al. [4]. A detailed analysis of the effect of parameter q i is included in McQuaid et al. [16]; in essence, as would be expected when clean seed is used there is less disease in a field. The incidence across the region then quickly equilibrates, where the choice of growers that use clean seed (i.e. for whom q i = 1) affects the equilibrium disease level. In particular, the spatial clustering of users affects results, as does the consistency of those users between seasons.
Decision-making for a grower takes place at the beginning of every season, and is therefore not concurrent to the disease spread model. A grower changes strategy with probability Here θ measures the responsiveness of the grower to loss, while φ F is the reward that the grower obtained in the previous season and φ A is the reward that the grower perceived among those of her farmer's group, trade partners or the general population who used the alternative strategy. Parameter κ measures the likelihood of a grower to conform to the strategy of their neighbours (κ > 0) or to stubbornly persist with their own strategy (κ < 0), in either case regardless of the actual reward both strategies offer (see (2.1)). We also note that there is a probability of switching strategy even when the alternative results in lower levels of reward, given by contrariness ψ, simulating irrational choices by growers as well as accounting for ignorance of the cause of the disease. Reward is measured, for strategy s (either using a clean seed system or not), as where Y s is the proportion of yield that a grower using that strategy obtained in the previous season, p is the selling price of the crop that could have been obtained from an uninfected field, and c s is the cost that using the strategy incurs. Yield, given by the harvest taken over the season including the final harvest, where the yield of latently infected plants is unaffected by disease, while only a proportion (ι) of infectious plants are usable. Tolerant plants have a higher useable portion ι. All fields are assumed to be identical in size and are therefore comparable, where growers receive information on relative yields from their farmers group of neighbouring growers with probability r l , their trade suppliers with probability r t or across the district as a whole through extension workers with probability r d .
We initially presume that 70% of fields are infected with approximately 100% incidence (T. Alicai, 11 May 2015, personal communication) and 10% of growers use the clean seed system. See

How to increase use of a clean seed system
Complete adoption of clean planting material by the entire grower population is never reached with our default parameter set (figure 1), as approaching ubiquity of adoption reduces disease to such an extent as to ensure clean planting material is unnecessary. However, if external disease pressure and whitefly Table 1. Model parameters and default values. For simplicity, we assume that there is one growing season of 300 days per year, ignoring the initial two months of the season when there is no foliage and whitefly cannot transmit the pathogen, although we note that in reality there may be some transmission between 1 and 2 months. We assume that the majority of growers are subsistence farmers, as is often the case. This implies that fields are harvested continuously at a constant rate over a period of months during the season, commencing after the first two months, with one replanting event at the start of each season (μ = 0, h > 0). The size of a cassava field is taken to be 1.5 hectares (e.g. [22]), and we presume an increase in affinity for whitefly of cassava fields over other areas. We presume that whitefly are a third again as likely to land on a cassava field as on a bare patch of land. We base the cassava field density of our model on the area of cassava harvested in Nakasongola district, Uganda (10 000 hectares, http://kids.fao.org/agromaps/), and hence consider 6000 cassava fields. Whitefly migration is calculated from Riis & Nachman [23], using the total population to find the immigration rate at equilibrium, which we presume to be identical to emigration. To simulate the dispersal of the vector we use data from Isaacs & Byrne [24] and Byrne et al. [25] for the dispersal of the sweet potato whitefly (on average, 50-700 m). We do not include the long-distance dispersal of whitefly that Byrne et al. [25] observe in a second peak of migration, as this is not consistent with the exponential dispersal kernel that we have assumed. More importantly, however, whitefly may not remain infectious, or even survive, for the duration of these journeys [15,26,27]. See McQuaid et al. [16] for further details. For harvesting and replanting, one example of the maximum potential yield in Uganda is UGX 1 200 000, while the cost to growers using certified clean planting material is UGX 300 000 compared to UGX 0 for those that obtain planting material through the recycling of cuttings. It is the ratio of the maximum potential yield to the cost of technology that is important, which from the above is taken to be 1 : 0.25 : 0, although we vary this to consider other systems. We estimate the responsiveness of growers to loss from the known adoption of certified seed and seed systems for other crops by growers in East Africa (roughly 5-15%, see [28][29][30][31]) and the maximum potential benefit of using certified clean planting material (0.01-0.28). The latter is calculated from equation (2.3) and the equilibrium yield (approx. 100% for users of the clean seed system, 47-74% for nonusers) from equation (2.1) when 5-15% of growers dispersed across the landscape use the clean seed system, where the users vary each season. Solving equation (2.2) using these values gives the range 0.18 ≤ θ ≤ 16.25, where we pick θ from this range. We choose stubbornness κ to be of a similar order of magnitude, with a default value of zero. Contrariness ψ was chosen to be two orders of magnitude less than σ , to represent its rarity, although we investigated values around this range. The unit 'plants' refers to the number of plants in a field. parameter description value a source crop agronomy    numbers are particularly high, and the cost of clean planting material low, adoption can increase to very high levels as clean planting material becomes both necessary and readily available. In the absence of these factors, lower adoption, and oscillations in the adoption over time, are likely to occur. We examine a range of results, focusing on non-monotonic relationships here (figure 2). A summary of the effect of individual parameter values on adoption, yield and oscillations in both is outlined in table 2.
Agronomic system: the degree of coordination and synchronization in planting and harvesting affects adoption and disease levels. Commercial systems (involving seasonal planting and harvesting) are likely to see higher levels of adoption than casual or subsistence systems ( figure 3a,b), as seasonal harvesting ensures that more plants are in the field for longer, increasing their chances of infection. If there are high levels of trade in a region (a reasonable likelihood of trade with many, varying suppliers) we are likely to see an increase in adoption and a decrease in yields as the pathogen is able to spread more rapidly resulting in increased disease (figures 3c-e and 4c-e).
Disease pressure: areas with increased overall disease pressure are likely to incur lower yields and higher adoption. However, high initial disease pressure only affects the yield and adoption in the short term (figures 3f and 4f ), after which disease pressure and therefore adoption equilibrate to similar levels regardless of the initial disease pressure. Additionally, the adoption varies to compensate for whitefly numbers, resulting in little change to yield (figures 3g and 4g). The presence of an alternative, reservoir host also only increases adoption (due to increased disease) up to a given density, above which adoption    Parameter values are grouped into those describing the agronomic system, the disease pressure, the clean planting material, the actions of growers and their behaviour. Varying these parameters affects the adoption of clean planting material and the percentage yield obtained, averaged over a period of 25 seasons, as well as the amplitude and frequency of oscillations and their dampening in adoption and yield. Increasing parameter values may increase or decrease these results monotonically ( or respectively), have no effect at all (→) or have a non-monotonic effect ( , or ). Note that in all cases oscillations in adoption and yield are similarly affected by changes in parameter values. Clean planting material: costly clean planting material is less likely to be used as it begins to outweigh the benefit of its use, so that disease increases and yields decline (figures 3i and 4i). On the other hand, improved planting material, which is either tolerant or resistant and has low levels of infection, may increase yield but with a subsequent decrease in adoption (figures 3j-l and 4j-l) as growers' assessment of disease risk declines. However, the planting material must be highly improved, or it may lead to very little change in yield as the adoption varies to compensate. In such cases, an extreme reduction in disease means that growers no longer see the need for clean planting material, allowing for disease persistence and subsequent re-emergence.
Grower activity: practices that remove infected plants (roguing and selection) reduce adoption as they provide an alternative method of disease reduction, but have no effect on yields unless the rate is high enough to near-eradicate the disease, when yields rapidly increase (figures 3m-o and 4m-o). If growers are able to access information from a number of sources (i.e. have an increased likelihood of obtaining regional-level information) both adoption and yield increase (figures 3o and 4o) as growers across the region are able to assess the threat of disease. However, the increase in both adoption and yield is primarily noticeable for very low levels of regional information, above which there is less effect (figure 2b); there needs to be some access to global information, so that assessment of disease risk and clean planting material use is not confined to affected areas only, but this need not be ubiquitous as local dynamics are still important in determining risk of infection.
Grower behaviour: a high degree of responsiveness means that growers react quickly to the presence of disease, slowing disease dispersal and therefore reducing the need for adoption of clean planting material and increasing the yield (figures 3p and 4p). In terms of adoption, there is an optimum level   of stubbornness of growers; above and below this optimum the adoption decreases, although with an additional increase in adoption again as growers begin to conform (figure 2c). This represents the balance in rapidity of response; if growers are too stubborn the majority will persist with their current strategy of non-engagement with clean planting material, while if they conform too readily the majority will quickly take up clean planting material, but then just as rapidly abandon it until an equilibrium number of users is reached. Yield, however, is highest for conforming growers, with a lower peak at a medium level of stubbornness as growers are quicker to follow the example of their neighbours and use clean planting material. Contrariness of growers increases both adoption and yield, as growers continue to use the system when it is unnecessary (figures 3r and 4r).

How to reduce oscillations
We have chosen to present only a 25-year span here, as this is both more relatable for practitioners and more realistic in terms of a period of interest. Increasing the simulation time does not change our qualitative conclusions about what promotes oscillations (although quantitative differences exit).
Agronomic system: an abundance of commercial growers is likely to see increased oscillations compared with a population comprised of casual or subsistence growers (figures 3a,b and 4a,b), as the effect of clean planting material through replanting becomes seasonal rather than continuous. Higher levels of trade result in more frequent, higher amplitude oscillations occurring as both the pathogen and clean planting material are disseminated quickly through the region (figures 3c-e and 4c-e).
Disease pressure: the initial pressure has little effect on oscillations except to increase the amplitude for very high initial incidence (figures 3f and 4f ), as the dynamics quickly account for the initial state. However, consistent high disease pressure through high numbers of the vector or an alternative reservoir host leads to low amplitude, dampening oscillations as there is always disease present, so always a need for clean planting material (figures 3g,h and 4g,h).
Clean planting material: increasing the cost of clean planting material leads to reduced amplitude and dampening of oscillations as very few growers use the system (figures 3i and 4i). In comparison, tolerant or resistant planting material may dampen oscillations, decreasing the frequency and amplitude as plants are less affected by the disease (figures 3j,k and 4j,k). In contrast, infection in planting material leads to frequent, low amplitude oscillations with significant dampening, due (as with the presence of a reservoir host discussed above) to the constant presence of disease (figures 3l and 4l).
Grower activity: practices removing infected plants also dampen oscillations, lowering the amplitude and frequency (figures 3m,n and 4m,n) as disease is removed from the region through other means than clean planting material. However, if the disease is not completely eradicated it may return rapidly as control is no longer practiced. At the same time, very high or very low average numbers of information sources (due to high or low likelihood of obtaining regional level information) lead to dampening of oscillations (figures 3o and 4o), where growers make decisions based on either regional dynamics (dampening oscillations due to the interactions of different groups of growers) or very local dynamics (dampening oscillations as local trends have no opportunity to spread).
Grower behaviour: high or low responsiveness of growers leads to dampening of oscillations in adoption and yield; in the case of high responsiveness, growers react quickly to change before large population-wide trends develop, while for low responsiveness growers respond much slower to change than the disease dynamics, allowing for equilibration in the infection levels (figures 3p and 4p). Similarly, a high frequency of highly stubborn or conforming growers dampens oscillations (figures 3q and 4q). Contrariness also dampens oscillations in adoption and yield (figures 3r and 4r) as growers fail to respond to oscillations in disease dynamics.

How to implement an intervention
A subsidy of free clean planting material to a given group of growers increases adoption (figure 5) as more growers are exposed to the benefits of the material, and has the most effect if distributed to varying recipients as suggested previously (see [16]). This is likely because the majority of growers in the region are repeat users, using the system for more than one season, so any overlap in past and present recipients of a subsidy is wasteful. Indeed, varying the recipients of the subsidy has the additional incentive of dampening oscillations in both yield and adoption with a small but consistent effect. However, any clustering of these recipients makes very little difference to adoption (not shown), contrary to previous findings (see [16]).

Discussion
Most epidemic models do not account for grower behaviour explicitly: for some exploratory exceptions see [4][5][6]40]. In this paper, we have built on a well-established framework for a parsimonious epidemic model with dual sources of infection (via vectors and replanting) and introduced simple elaborations that allowed us to gain insight into some of the consequences of grower choices on epidemic dynamics. Future work will refine the socio-economic aspects of the model as more empirical data become available to inform the choice of framework. We note firstly that grower behaviour is, as expected, highly likely to impact disease spread through the adoption of clean planting material [5] and we are unlikely ever to see complete disease prevention (see [41]). Indeed, if external disease pressure or a reservoir host (although in reality this appears unlikely to be important) is present at sufficiently high levels, the release of disease resistant clean planting material may be necessary to control the disease at all. However, growers in areas with high disease pressure are more likely to show interest in clean planting material, as we might expect (see [8]), which allows for disease persistence in low pressure areas. Additionally, oscillations in adoption and yield are highly likely, as witnessed in other agronomic systems ( [4], and hearsay from varieties resistant to cassava mosaic disease in Uganda at the turn of the century). In particular, areas with many commercial plantings (rather than casual or subsistence growers) are highly likely to succumb to large oscillations in adoption, even when differences in behaviour are not considered. One aspect not covered here is the relative importance of the crop to grower, compared with their reliance on other crops, which will also affect whether they are likely to consider disruptive management practices.
In terms of the planting material, competitively low pricing can significantly increase use; indeed, there is a threshold (where the cost of planting material is equal to the minimum yield that can be obtained from a fully infected field) above which growers are unlikely to use the planting material at all. We assume that the cost of the planting material also implicitly includes a cost in terms of a reduction in quality or desirability of the variety for growers, who may prefer local landraces. We note on the other hand that seed companies in other agronomic systems often bundle a number of desirable traits together (similar here to increasing the reward that using the clean seed system is perceived to give), as well as varying the cost of planting material, in order to avoid oscillations in adoption, an aspect that is not explicitly considered in the current analyses [4]. Planting material that is partially resistant will increase yields, but in reality may require replenishment as these yields decline with repeated recycling, an aspect we do not consider here. On the other hand, planting material that is tolerant will maintain high yields, but will also result in a build-up in inoculum pressure that may affect other varieties. However, infection in planting material will, economically at least, not dissuade growers from its use, although work on opinion dynamics by A. Milne (personal communication, 12 May 2016) suggests that trust in a control strategy is vital for its success.
At the same time, over-investment in the spread of information on success of the clean seed system appears to be unnecessary (see also [7], but see [21]); word of mouth between growers may often be sufficient, with little additional reward in terms of yield for any increase in information distribution. Reducing the dissemination of information may also slow grower reactions to change, decreasing oscillations (see also [4]). This implies that an optimal amount of information disseminated will lead to high adoption and yield, but with a low chance of oscillations occurring.
Education, influencing grower behaviour, is also important (e.g. [21]). Education affects responsiveness and conformity, as well as potentially improving agronomic practices and affecting the management of additional pests and diseases not considered here. Responsiveness among growers reduces adoption while increasing yield, and also dampens oscillations (see also [4]). However, so does very low responsiveness. Ideally, growers should assess their particular situation individually (i.e. not be too conformist, although if conformism is particularly high that can also increase adoption) but also be reasonable (not be too stubborn), where it may be key to stress the folly of hasty decisions and the wisdom of a certain degree of persistence (ensuring some level of stubbornness, see also [4]). However, the behaviour described above may also lead to oscillations. This may reflect the 'prisoner's dilemma' faced by growers undertaking potentially costly action to benefit the community. Surprisingly, contrariness of growers leads to high adoption with dampened oscillations, as growers persist with an unnecessary strategy.

Conclusion
Our model identifies five key aspects to consider when implementing a clean seed control strategy. Importantly, it has been necessary to consider grower behaviour here; in the absence of grower choice the model tends towards stable steady states (see also [16]). However, when we allow for the reaction of growers to this, cycles in adoption and incidence can result. It is vital to bear these cycles in mind when planning disease control.
Firstly, high disease pressure and aspects that lead to this are likely to encourage individuals to engage with a control strategy. However, if disease pressure is too high or persistent, individuals may be discouraged from engagement as control has little effect. Aspects that lead to rapid dispersal of the pathogen may also lead to oscillations in engagement, although consistent disease pressure dampens these oscillations as there is a constant demand for control.
Secondly, improvements in clean planting material and field management can be effective in reducing disease, although also decreasing engagement as control is no longer necessary (which may in turn be useful if only a limited supply of clean planting material is available). However, these improvements must be of a sufficient standard or they risk leading to near but not complete control of disease, at which point individuals cease to engage with the control strategy allowing for re-emergence of the disease.
Thirdly, high levels of information need not be disseminated in order to have an effect on engagement, although there must be some chance of individuals gaining a broad view of disease risk. If this chance is high, but not guaranteed, it may lead to oscillations in engagement.
Fourthly, education of individuals is effective in promoting rapid response to changes in disease pressure, although if this response is not sufficiently rapid it may also promote oscillations in engagement and disease. However, we should not be discouraged by a certain degree of stubbornness or contrariness in the population, which may increase engagement over time as well as dampening oscillations in this.
Finally, the introduction of some level of 'free' control can have a marked positive effect on both engagement and disease reduction. This control need not be associated with a complex, coordinated approach to achieve high impact. However, in order to have maximum effect, as well as to dampen oscillations in engagement, it is important that the control reach different individuals over time, rather than the same select group consistently.
Data accessibility. The Matlab code for simulations described above can be found at http://doi.org/10.5281/zenodo. 556986.