Summary of vector, pathogen and disease symptoms of plants used to fit the apple proliferation (AP) joint model.
Phytoplasma diseases cause major economic damage on crops worldwide. To draw inferences from such a system, joint estimation of dependencies and high flexibility in the model structure are required. Using Bayesian inference, the aim of this chapter was to infer the apple proliferation (AP) disease epidemiology in South Tyrol, Italy. The data consisted of (1) presence/absence of the AP vector Cacopsylla picta collected in 44 orchards in 2014; (2) prevalence of the AP pathogen “Candidatus Phytoplasma mali” in the vector population; and (3) AP symptomatic trees visually assessed in 2015. Generalized linear mixed models evaluated in a Bayesian framework were used to test species-environment relationships. The model results indicated that the occurrence of the AP vector and symptomatic plants are positively influenced by elevation and temperature and negatively by management. Vector and pathogen predictions in the disease symptoms model correlated negatively or not at all with the prevalence of AP symptoms occurrence. In conclusion, the model results suggest that the presence/absence of the AP vector alone may not be the only cause for disease occurrence. Considering factors such as phytoplasma transmission via root-bridges and specific management strategies, may help to improve inference and finally to optimize the existing pest management.
- apple proliferation
- Bayesian inference
- habitat modeling
- imperfect detection
- latent infections
- occupancy model
- phytoplasma disease
- pest insect
- psyllid vector
Phytoplasma-induced diseases occur in a range of economically important crops and are therefore major threats in agriculture worldwide . Phytoplasma are cell wall-less plant pathogenic bacteria vectored by insects belonging to the order Hemiptera .
From an ecological perspective, phytoplasma diseases are complex biological systems. Complexity is linked to many sources of uncertainty which in most cases are difficult to measure. Among others, these sources of uncertainty include vector-pathogen-plant interactions, but also the presence of unknown vectors (e.g.,
Besides complexity, the statistical treatment of the inherent dependencies of such biological systems represents another challenge in the modeling process. Traditional statistical methods [such as generalized linear models (GLM), generalized linear mixed models (GLMM)] could be used in a step-wise approach. In a first step, the vector-environment relationship is identified (vector model). Second, using the results of the vector model, the pathogen-environment relationship is established (pathogen model). Finally, the results of both previous models are used to fit the plant disease model. However, this approach does not consider the dependencies between the responses at the same time. In contrast, methods that allow for combined dependencies such as structural equation modeling lack the flexibility in model specification . One solution is using Bayesian inference which allows to jointly estimate the model parameters and at the same time offers high flexibility in defining the model structure.
1.1. Case study: apple proliferation
In this study, the phytoplasma disease apple proliferation (AP) was chosen as a modeling system. AP-specific disease symptoms on apple trees are the proliferation of auxiliary shots (formation of witches’ broom) and enlarged stipulae. AP nonspecific disease symptoms include early leaf reddening, small, taste- and colorless fruits, chlorosis and premature bud break. The causal AP agent is ‘
The aim of this chapter was to jointly infer the AP disease epidemiology in South Tyrol, Italy using Bayesian inference. Imperfect detection was accounted for in the vector and symptomatic plant models. The AP insect vector was modeled using an occupancy model. To account for detection bias during the vector sampling, information on sampling effort was used as a predictor in an additional Bernoulli process conditional on the AP vector’s true presence or absence. Based on molecular analyses of AP prevalences in apple trees, I estimated the proportion of latent infected trees to account for imperfect detection of truly AP phytoplasma-infected apple trees.
2. Materials and methods
The AP vector
A summary of the final data set including the AP vector, the AP phytoplasma and AP symptoms of trees is provided in Table 1. Metric environmental predictors included elevation (m a.s.l.) and annual mean temperature (°C). Orchards were classified into integrated/not integrated management to account for different pest management strategies.
2.2. Modeling approach
Bayesian inference was used to jointly estimate the dependencies of all responses (AP vector, AP phytoplasma prevalences of the vector, AP symptoms of apple trees) and the environment. To fit the model, all environmental predictors (except vector and phytoplasma predictions) were scaled and centered (i.e., mean subtracted and divided by the standard deviation) to allow a faster convergence of the model fitting algorithm. To decide whether to account for unimodal response-curves, in a pre-step, I fitted multivariate GLMs including quadratic terms of elevation and temperature . As the unimodal response curves were not found to be ecologically sensible, only linear relationships were considered in subsequent analysis. Generalized linear models were developed using a binomial error distribution and a logit link function (GLMM; [12, 13]). The GLMMs were then evaluated in a Bayesian framework. As prior distributions for the fixed effects, zero-centered normal distributions were used. Except for the intercept, priors were defined to be mildly informative which results in a shrinkage effect similar to a ridge-regression .
The vector data set, as is common for ecological data, contained many zero values due to the rarity and detectability of the species. To account for imperfect detection, I used a site-occupancy model [15, 16]. These models rely on the “closure assumption” stating that the occupancy state remains unchanged between survey times. The occupancy model combines (1) an ecological process and (2) an observation process. The ecological process of the true occupancy state z (which is a latent or unobserved variable) can be described using a Bernoulli distribution with the occupancy probability
In the observation process, real observations (detections/nondetections) for each survey time (indexed by j) follow a Bernoulli distribution conditional on the true occupancy state z:
where p is defined as the detection probability at site i and survey time j given the site was actually occupied. The detection probability was modeled using a logistic regression and sampling effort as explanatory variable. Sampling effort was defined as the number of sampled trees in proportion to the total number of surveyed trees for AP symptoms.
Field surveys on the prevalence of plant diseases caused by plant-pathogenic bacteria are often based on visual diagnosis of disease symptoms [17, 18, 19]. Given trained and experienced plant inspectors, the false-positive rate can be assumed to be close to zero. The false negative rate is also often considered very small because latent infections are mostly ignored. Based on molecular analyses latent infections for the AP disease were found to be 2.32 and 10.48% depending on age of the apple trees . To account for imperfect detection caused by latent infections, an informative beta prior was used for the detection probability p with parameters
where N is the total number of survey trees for each site.
MCMC sampling was carried out by the STAN software (RStan version 2.12.1), which uses the No-U-Turn sampler (NUTS) [21, 22]. Model specifications included three chains with 3000 iterations each and considered a chain to be converged when the potential scale reduction statistic, Ȓ < 1.05 . To access model, fit posterior predictive checks were applied on each model separately using the DHARMa package . The DHARMa package calculates scaled residuals (Bayesian p-values) by comparing observations simulated from the fitted model with observed values. All statistical analyses were carried out in the R statistical environment (version 3.2.2; ).
The RStan code for the joint model is available in Appendix A.
The marginal posterior distributions of the parameter of interest of the AP joint model are shown in Figure 2. For the AP vector
As the vector model, AP symptom occurrences on apple trees were likewise positively correlated with elevation and temperature and negatively with integrated pest management measures. Moreover, the model estimated a negative correlation between the AP vector and AP symptoms. No relationship between phytoplasma infection rates within the AP vector and AP symptoms was found.
Regarding the model performance, the potential scale reduction statistic, Ȓ, for each parameter was close to 1 (not shown). Hence, I found no indication of non-convergence of the three chains. Figure 3 shows the results of the residual diagnosis. The plots show no serious violations of distributional assumptions. To confirm the overall uniformity of the scaled residuals, I applied one-sample Kolmogorov–Smirnov tests, which were not significant for all three models.
The modeling case study presented in this chapter illustrated the use of Bayesian inference to jointly investigate the influence of environment on the occurrence of the AP vector
4.1. Influence of environment on apple proliferation epidemiology
Using the 80% credible interval, I found that AP vector and AP symptoms on apple trees were positively associated with elevation and temperature and negatively with integrated pest management. While having similar ecological requirements, the joint model indicated a negative relationship between vector and symptoms. Elevation, temperature and integrated pest management did not affect AP phytoplasma prevalences within the vector. No correlation was found between prevalences of AP phytoplasma and symptoms.
Even though the joint model did not identify a clear correlation between the predictor variable integrated pest management and pathogen occurrences, overall, it seems that integrated pest management is an important environmental driver, negatively influencing vector, and disease symptom occurrences. But it is also possible that the AP responses are influenced by different management measures. For example, the presence/absence of the vectors may be influenced by application time, quantity and type of insecticides, while new disease incidences in plants may also relate to different levels in the effort of uprooting AP symptomatic trees, thereby eliminating sources of new vector infections or root transmissions to adjacent trees . Hence, in a follow-up study, it would be worth to further investigate which specific management measure leads to a decrease in the responses to optimize insect pest management strategies.
4.2. Advantages of Bayesian inference
Besides jointly estimating the disease system, Bayesian inference allows high flexibility in the model specifications. Models can be easily extended to include detection probabilities, overdispersion or zero-inflation [29, 30, 31]. The present joint model could be further extended by including AP symptoms detection probabilities depending on the cultivar and observed symptoms. The high flexibility is also important when data is collected for purposes different than statistical inference and prediction. For example, if vector data was collected to determine the first appearance of the vector in the orchard (to timely optimize the application of insecticides), vector prediction probabilities need to be constraint by probabilities of the true flight period of the pest insect.
Some parameter estimates in this study were associated with large credible intervals, meaning high uncertainty. One solution would be to use a higher number of observations, which is not always feasible in ecological studies. Another possibility is to include informative priors derived from the literature or previous analysis as illustrated for the informative beta prior to account for imperfect AP symptom detection due to latent infections. Priors play an essential role in every Bayesian analysis. For the environmental parameter estimates included in this chapter, no prior information from previous analysis was available. However, the identified relationships could be used to define prior distributions in future studies.
Finally, the results of a Bayesian inference (posterior distributions) can be summarized using, for example, credible intervals which allow an intuitive interpretation of the parameter estimates associated with well-defined uncertainties. Given chain convergence and successful posterior predictive checks, Bayesian credible intervals are also appropriate for small data sets . This is especially true in observational studies on animal and plant populations where data collection is often time- and cost-consuming.
In summary, the results of the AP joint model suggested that the presence of the AP vector is not necessarily positively correlated with disease occurrence. Instead, other factors such as phytoplasma transmission via root-bridges or specific management strategies should be additionally considered in future studies. In case of the AP disease system, Bayesian inference allowed to jointly fit combined dependencies which are common to phytoplasma epidemiological diseases. Unlike maximum likelihood methods, posterior distributions for all quantities of interest are obtained which could be further summarized using credible intervals and allowed intuitive interpretation of the results. The provided example of a joint Bayesian modeling framework can be used as a basis to infer species-environment relationships of phytoplasma disease systems.
The work was performed as part of the project APPLClust and was funded by the Autonomous Province of Bozen/Bolzano (Italy) and the South Tyrolean Apple Consortium. The author would like to thank Stefanie Fischnaller, Martin Parth, Manuel Messner, Robert Stocker, Christine Kerschbamer and Katrin Janik for providing data on insect vectors, phytoplasma prevalences and occurrences of disease symptoms of apple trees.
Supplementary data associated with this chapter is available online: https://doi.org/10.6084/m9.figshare.5896789