Modelling induced resistance to plant diseases

Nurul S. Abdul Latif a,b, Graeme C. Wake b,n, Tony Reglinski c, Philip A.G. Elmer c

a Faculty of Agro Based Industry, Universiti Malaysia Kelantan, Jeli, Kelantan, Malaysiab Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealandc The New Zealand Institute for Plant and Food Research Limited, Hamilton, New Zealand

H I G H L I G H T S

� Determining a plausible dynamical system that describes the SIR compartments for disease.� Investigating control strategies that will minimise the onset of disease.� Using a case study of experimental data enables parameter values to be found.

a r t i c l e i n f o

Article history:Received 1 June 2013Received in revised form25 November 2013Accepted 20 December 2013Available online 5 January 2014

Keywords:ElicitorInduced resistanceDiplodia pineaMethyl jasmonateDynamical system

a b s t r a c t

Plant disease control has traditionally relied heavily on the use of agrochemicals despite their potentiallynegative impact on the environment. An alternative strategy is that of induced resistance (IR). However,while IR has proven effective in controlled environments, it has shown variable field efficacy, thus raisingquestions about its potential for disease management in a given crop. Mathematical modelling of IRassists researchers with understanding the dynamics of the phenomenon in a given plant cohort againsta selected disease-causing pathogen. Here, a prototype mathematical model of IR promoted by achemical elicitor is proposed and analysed. Standard epidemiological models describe that, underappropriate environmental conditions, Susceptible plants (S) may become Diseased (D) upon exposure toa compatible pathogen or are able to Resist the infection (R) via basal host defence mechanisms. Theapplication of an elicitor enhances the basal defence response thereby affecting the relative proportion ofplants in each of the S, R and D compartments. IR is a transient response and is modelled using reversibleprocesses to describe the temporal evolution of the compartments. Over time, plants can move betweenthese compartments. For example, a plant in the R-compartment can move into the S-compartment andcan then become diseased. Once in the D-compartment, however, it is assumed that there is no recovery.The terms in the equations are identified using established principles governing disease transmission andthis introduces parameters which are determined by matching data to the model using computer-basedalgorithms. These then give the best match of the model with experimental data. The model predicts therelative proportion of plants in each compartment and quantitatively estimates elicitor effectiveness.An illustrative case study will be given; however, the model is generic and will be applicable for a rangeof plant–pathogen–elicitor scenarios.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Conventional crop production systems rely heavily on the useof synthetic agro-chemicals to manage plant disease. However,concerns about potentially negative effects of agro-chemicals onthe environment have increased the demand for the developmentof biologically based disease control methods. One strategy thathas gained more popularity in the last decade involves theapplication of inducing agents known as elicitors to stimulatethe natural plant resistance response to pathogen attack; this

approach is referred to as induced resistance (IR). A range ofnaturally occurring elicitors have been identified and these includepolysaccharides such as glucan and chitosan and also planthormones such as salicylates and jasmonates. Elicitors are notantimicrobial per se but operate primarily by stimulating plantdefences to enable a more rapid and intense resistance response tofuture pathogen attacks. The elicited responses are orchestratedvia complex signalling cascades and typically there is a lag ofseveral days between the elicitor application and the onset of IR.

Some studies suggest that elicitors can induce a broad spec-trum and long lasting resistance; however, more commonly it hasbeen shown that IR is transient and lasts for a few weeks only. Theefficacy of IR has been reported to range from 20 to 85% (Walterset al., 2005) and this variability is a function of the dynamic

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Journal of Theoretical Biology

0022-5193/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jtbi.2013.12.023

n Corresponding author.E-mail address: G.C.Wake@massey.ac.nz (G.C. Wake).

Journal of Theoretical Biology 347 (2014) 144–150

relationships between the plant, the pathogen and the environ-ment. As we gain a greater understanding of these relationships sowill we be better placed to develop smarter elicitor applicationstrategies and so reduce the variability of IR in the field. This isessential for the practical implementation and acceptance of IR asa crop protection strategy in the future.

In this paper, we describe a generic mathematical model thatincorporates the elicitor effect to combat disease infection thatwas initially introduced in Abdul Latif et al., 2013 Numericalevaluation of this IR model provides a baseline to determine theoptimum timing of the elicitor application. We extended thisproposed model to include the effects of the following factors ondisease development: (i) the timing of elicitor application, (ii) themultiple applications of elicitor, and (iii) post-disease elicitortreatment.

2. Model development

2.1. Background and definitions

Standard epidemiological models for plant disease develop-ment are based upon interactions between three critical compo-nents: the plant, the pathogen and the environment. Spread ofdisease is the result of migration of the pathogen. Dispersal is themovement of the pathogens0 dispersal units (e.g spores) from theplace where they are formed to the plant tissue they infect. Somehave referred this as the “conquest of space” (Zadoks and Schein,1979). Only a few fungi can move actively from host to host andthey belong to the wood rotting fungi where they form rhizo-morphs (underground stems, up to 10 m to progress from one rootto another). Active movement of fungi is the exception and passivemovement is the rule. There are many mechanisms of dispersalsuch as leaf rubbing, rain splash, turbulent transfer (wind borne),water-borne, animal or vertebrate-borne, insect (invertebrate-borne) and human-borne.

Under disease conducive conditions susceptible (S) plants becomeinfected and develop disease (D) when inoculated with a compatiblepathogen. However, this is a dynamic relationship and a subtlechange to any one parameter may result in a proportion of the plantpopulation being able to exhibit resistance (R) to infection. In thisstudy a plant defence elicitor was applied to susceptible plants inorder to elevate their basal resistance and so enable a proportion offormerly susceptible plants to express IR i.e. a shift in the populationfrom S to R. This prototype IR model is based on the model by Jegeret al. (2009) and Xu et al. (2010). Their models were developed for ageneric biocontrol system. We extended these models to include IRas a specific biocontrol system and to determine the effectiveness ofthe elicitor used as the agent for the IR mechanism. There are nomodels in the literature which specifically describe the interactionsamong the plant, the pathogen, and the elicitor. Typically, inducibledefences are triggered in response to pathogen attacks. Therefore, themodel is formulated with the elicitor treated plants divided into tworegimes: (i) pre-inoculation (before pathogen arrives) and (ii) post-inoculation (after pathogen arrives). The assumptions for the model0sformulation can be summarised as follows:

1. The plant population is divided into three compartmentsaccording to the above definitions where SþRþD¼ 1.

2. At the time when plants are treated with an elicitor ðt ¼ 0Þ, aproportion of the plant population will exhibit natural resis-tance ðRiÞ.

3. The induction period ðtpÞ describes the time interval betweenelicitor treatment and pathogen inoculation. Upon inoculation,a proportion of plants ðDiÞ will become infected immediately.This is shown schematically in Fig 1.

The model0s equations for the treated plants are as follows:Pre-inoculation: For 0ototp

dRdt

¼ ðeðtÞ�γRÞð1�RÞ; Rð0Þ ¼ Ri ð1Þ

Post-inoculation: For tpotoT

dRdt

¼ ðeðtÞ�γRÞð1�R�DÞ; RðtpÞ ¼ Rp ð2Þ

dDdt

¼ βDð1�R�DÞ; DðtpÞ ¼Di ð3Þ

and we take the elicitor effectiveness in the plants as

eðtÞ ¼ kt

t2þL2ð4Þ

where k=2Lðdays�1Þ is the maximum elicitor effect and L (days) isthe time where this is at its peak, γðdays�1Þ is the rate thatresistant tissue becomes susceptible and βðdays�1Þ is the rate atwhich the disease spreads. The elicitor effectiveness e(t) isassumed to be a time dependent function (see comments in theintroduction). Other functional forms for e(t) were tried, and thisone gave a best fit and matched the biological insight fromexperiments.

Rp is the degree of the resistance at the time of the pathogeninoculation obtained from Eq. (1). Also, T is a finite (sufficientlylarge) time after the pathogen inoculation. Note that, Eqs. (1)–(3)are based on the assumption that the rate of change of R and D isdirectly proportional to the amount of S available at a particulartime (that is, S¼ 1�R�D) using the usual “mass-action-kinetics”(see Keener and Sneyd, 1998). This obviously applies in a standardway for Eqs. (3) and (6), however, it is less clear for the Rcompartment. The only mechanism that could cause this wouldbe genetic in origin and this is beyond the scope of this study. Thedata sets (see Fig. 3) in our case study show that the steady stateattained for D (o1) is not unique and is dependent on the initialpoint ðRi;DiÞ which can be modelled by the inclusion ofS¼ 1�R�D (or 1�R) in the kinetics of the R compartment. ForEq. (1), D is not in the equation because the pathogen is absentduring this period. Pathogen is inoculated at time tp, therefore theD term occurs only in Eqs. (2) and (3).

The model0s equation for the untreated plants is as follows:

dRdt

¼ �αRð1�R�DÞ; Rð0Þ ¼ Ri ð5Þ

dDdt

¼ βDð1�R�DÞ; Dð0Þ ¼Di ð6Þ

where α (days�1) is the rate at which the untreated plants lose theresistance due to the pathogen attack. The untreated model sharedthe same parameters β, Ri, Di as the treated model as theseuntreated plants are characterised as the control group. Thismeans there is no elicitor treatment for the control group.

Fig. 1. Here S, D, R are the proportion of plants in the respective compartments andthe parameters γ, β and e(t) are the specific rates of movement between the linkedcompartments.

N.S. Abdul Latif et al. / Journal of Theoretical Biology 347 (2014) 144–150 145

Therefore, the untreated model will not have the e(t) term in themodel0s equations and so are autonomous, that is, independent oftime except implicitly. Note that, the rate of R and D for theuntreated plants is also directly proportional to the amount of Savailable this is shown schematically in Fig 2.

2.2. Model simulation

The seven unknown parameters (α, β, γ, k, L, Ri, Di) aredetermined by matching data to the model using MATLAB tool-box ‘fminsearch’. For illustration, this proposed model was fitted tothe experimental data based on the effect of the elicitor, methyljasmonate (MeJA), on the resistance of Pinus radiata seedlings toDiplodia pinea the causal agent of pine stem canker and tipdieback. The pine seedlings were divided into groups and sprayedwith 0.1% MeJA at 35, 27, 21, 14, 13, 7, 6 or 3 days before inoculationwith D. pinea using methods previously described in Gould et al.(2008). There was also an untreated control group for thisexperiment. Disease assessments for each treatment commencedat 1 week after inoculation and continued at 3–4 day intervalsthereafter for 5 weeks.

Here, the induction times are tp¼[3 6 7 13 14 21 27 35]. Ourassumption is that for each induction time tp there will be differentdynamics outcomes for the plant population, but share the sameparameter values that apply for each tp. The ordinary least squaresmethod was used to estimate the unknown parameter values. Themodel has three compartments (S, R, D), but the data sets have

only two (½SþR�;D). The susceptible cohort is given by S¼ 1�R�Dand can be separated out by this model.

The solution curve is plotted as in Fig. 3 using the optimalparameter values as in Table 1, and shows the dynamics of eachcompartment corresponding to its induction time tp¼3 days. Inthe figure the comparison between the treated and untreatedmodel is shown. The calculated coefficient of determination for thewhole system i.e. for all induction case tp is R2 ¼ 0:6598. Theexperimental data is available for D (and hence SþR¼ 1�D) onlyand is shown in the marked points. Similar graphs are available forall tp and are similar to those chosen.

Fig. 4 illustrates the characteristics of the two compartments Rand D for Eqs. (1)–(6) with phase plane. The flows in the figureindicate the time-evolution of the system of the differentialequations described above based on the different values of tpand the untreated case. It shows that when tp is between 3 and7 days, the subsequent development of disease is less severecompared with the other induction times and with the untreatedcontrol plants. That is, the resistance induced by the elicitorapplication is at its peak. The figure shows that the trajectoriesfor each induction case will eventually approach that straight lineRþD¼ 1. The model has a line of equilibrium states in the R, Dplane which are attracting, and shows that disease developmentdepends on the induction time tp.

Fig. 5 shows the proportion of diseased seedlings which weretreated at the final time against the induction time tp. The IRmodel can determine the optimal induction time tp that gives thebest outcome for disease control for the treated pine seedlings.This can be done by taking the point at the final time tf as a fixedvariable e.g. let tf ¼ tpþ35, and minimise the trajectories ofDFinal ¼Dðt ¼ tf Þ. For tp¼0 days, it means the elicitor treatmentand the pathogen inoculation were introduced at the same time.Experimental data for diseased seedlings measured at tpþ35 daysfor each tp case is taken into account to compare with the modeloutcome. This numerical experiment shows that induced resis-tance was greatest when MeJA was applied to pine seedlings6 days before pathogen inoculation. Although the error gapsbetween the model outcomes and the experimental data arenoticeable, especially for the case tp¼14 days, it can be attributedto the variability of the data sets or the experimental error. Also,note that we assumed for each induction time tp, the dynamics ofeach tp share the same parameter values (see Table 1).

Fig. 2. Here S, D, R are the proportion of plants in the respective compartmentsand the parameters are the specific rates of movement between the compartments.

0 10 20 300

0.5

1

Dis

ease

d (D

)

Time (days)0 10 20 30

0

0.5

1

Hea

lhty

(S +

R)

Time (days)

0 10 20 300

0.5

1

Res

ista

nt (R

)

Time (days)0 10 20 30

0

0.5

1

Sus

cept

ible

(S)

Time (days)

Fig. 3. The vertical dashed line (here at 3 days) is shown at the induction time t ¼ tp at which time the pathogen arrives and causes the disease to develop further. The bluelines (—-) are the treated model and the red lines (– –) are the untreated model. The data is shown in the marked points. (For interpretation of the references to colour in thisfigure caption, the reader is referred to the web version of this paper.)

N.S. Abdul Latif et al. / Journal of Theoretical Biology 347 (2014) 144–150146

3. Extensions to the IR model

3.1. Varying the elicitor concentration

Next, this IR Model was extended to include the effect ofdifferent concentrations of the elicitor on its capability to induceresistance since some experimental studies have reported elicitordose responses (Godard et al., 1999; Boughton et al., 2006; Gouldet al., 2009; Vivas et al., 2012; Hindumathy, 2012). For model0sillustrations, liquid suspensions of MeJA at concentrations of 0.0%,0.023%, 0.025%, 0.1% or 0.4% were applied to P. radiata seedlings asa foliar spray. Each concentration was applied once and theseedlings were inoculated with D. pinea7 days after MeJA applica-tion. We assumed that the elicitor concentration is a sub-lineareffect to the e(t) term. We choose this term as f ðaÞ ¼ areð1�aÞ wherea is the scale for concentration and r¼0.6650 is the sub-lineareffect which determined by the least squares method. Note thatthe function f(a) is maximised at a¼r. Therefore the effect ofelicitor concentration to IR can be defined as

concentration� eðtÞ ¼ areð1�aÞ � kt

t2þL2:

The values of a as a concentration were scaled as a¼0, 0.23,0.25, 1, 4. If a¼1, then f ð1Þ ¼ 1. The parameter values, including thephysiological ones for the case study of pine seedlings, are this inline with those in Table 1 which we therefore use. Of course we areassuming the dynamics of the physiological response remainsthe same, expect for the elicitor effect when the dose is changed.For 0% concentration, the seedlings are assumed to behave asuntreated seedlings. We say that the seedlings had a “water”treatment. From the model equations given above, this is true.If a¼0, then the term above is equal to zero which implies theseedlings should be evaluated as the untreated model. Therefore,the model formulation will be:

For 0% elicitor concentration assume as untreated caseFor pre-inoculation: 0ototp

dRdt

¼ �αRð1�RÞ; Rð0Þ ¼ Ri ð7Þ

For post-inoculation: tpotoT

dRdt

¼ �αRð1�R�DÞ; RðtpÞ ¼ Rpunt ð8Þ

dDdt

¼ βDð1�R�DÞ; DðtpÞ ¼Di ð9Þ

Rpunt is obtained from (7) and is the value of RðtpÞ for the untreatedplants.

For other elicitor concentrations

Table 1Description of variables and parameters used in the IR model for pine seedlings with elicitor MeJA with values obtained from matching model simulating with data.

Variable / Parameter Description Value [Units]

R Proportion of plant population being able to express resistance to infection (dimensionless)D Proportion of plant population being infected and become diseased (dimensionless)S Proportion of plant population which is susceptible (dimensionless)α The rate at which untreated plants lose the resistance due to the pathogen attack 0.0908 (days�1)β The rate at which the disease spreads 0.7379 (days�1)γ The rate of the resistant plant become susceptible 0.2801 (days�1)k Determines the effectiveness of the elicitor 5.3749 (dimensionless)L The time where the elicitor effectiveness is at the peak 10.7607 (days)tp The induction time of the pathogen i.e. the time interval between the elicitor application and the pathogen challenge (days)Ri A proportion of the plant population exhibit natural resistance at the initial time t¼0 0.6118 (dimensionless)Di A proportion of the plant population which becomes infected immediately after the pathogen challenge 0.0168 (dimensionless)te Time of the elicitor application (see later) (days)r The sublinear effect of the elicitor % concentration (see later) 0.6650 (dimensionless)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Resistant

Dis

ease

d

Untreatedtp = 3 days

tp = 6 daystp = 7 days

tp = 13 days

tp = 14 days

tp = 21 days

tp = 27 days

tp = 35 days

Pathogen is introduced here

Fig. 4. Phase plane for different induction time tp. These trajectories of the R, Dcompartments are plotted in the feasible region ðR;D40;RþDo1Þ. The linesschematically represent the values of the two compartments R and D as time passesfor Eqs. (1)–(5) based on each induction case tp and the untreated case. When D¼0,the straight lines illustrate the dynamics of the R compartment before the pathogeninoculation. These lines will have a discontinuity when the pathogen is introducedat time t ¼ tp , and show the state of the R compartment at that particular time.As can be seen in the figure, there is a jump in the D values of Di at t ¼ tp and thetrajectories of the R, D compartments will continue to approach the straight lineRþD¼ 1. For the untreated case, the dynamics of the R, D compartments willdepend on the initial condition of the systems i.e. Ri and Di.

0 5 10 15 20 25 30 35 400.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Induction time tp days

The

prop

ortio

n of

dis

ease

d se

edlin

gs

at a

fixe

d fin

al ti

me

( DFi

nal )

Treated ModelUntreated ModelExperimental data

Fig. 5. Long-term outcome ðtpþ35Þ for diseased plants for different inductiontimes tp.

N.S. Abdul Latif et al. / Journal of Theoretical Biology 347 (2014) 144–150 147

For pre-inoculation: 0ototp

dRdt

¼ areð1�aÞ kt

t2þL2�γR

� �ð1�RÞ; Rð0Þ ¼ Ri ð10Þ

For post-inoculation: tpotoT

dRdt

¼ areð1�aÞ kt

t2þL2�γR

� �ð1�R�DÞ; RðtpÞ ¼ Rp ð11Þ

dDdt

¼ βDð1�R�DÞ; DðtpÞ ¼Di ð12Þ

Figs. 6 and 7 show the model simulation for some differentelicitor concentrations and match reasonably well the experimen-tal behaviour.

The results in Fig. 8 demonstrate that the effect of elicitor MeJAon seedling health is dose dependent. For example, it shows that ifthe MeJA concentration used is between 0.02% and 0.1% it willreduce disease incidence by approximately 70% compared with the0.0% “water” treatment. On the other hand, if lower concentrationðe:g:o0:02%Þ is used, the outcome will be the same as theuntreated seedlings (i.e. no disease efficacy). The numerical com-putation demonstrated that induced resistance was greatest whenMeJA was applied at a concentration of at � 0:067% the effect ofthe elicitor treatment to induced resistance is at the highest value.Note that the optimal value obtained is close to the value whichmaximises the function f(a), in the elicitor effectiveness factor. Thisfinding has important cost benefit implications as this “optimal”dose of MeJA would increase its cost effectiveness making it moreattractive as a plant protection product. In a study by Gould et al.(2009), it was indicated that if higher elicitor concentration(40:1%) was used, there was evidence of phytotoxicity in thetreated seedlings where the disease incidence was higher.

3.2. Effect of multiple elicitor applications

Next, we introduced multiple elicitor applications to theproposed IR Model. The elicitors were applied before the pathogenchallenge. The scenarios are described in Fig. 9.

Here, we assumed that the term f ðaÞ ¼ areð1�aÞ is true. There-fore, the equations for a multiple application will be:

First application: for 0otote2

dRdt

¼ ðf ðaÞ kt

t2þL2�γRÞð1�RÞ ð13Þ

Second application: for te2 ototp

dRdt

¼ f ðaÞ kt

t2þL2þ f ðaÞ kðt�te2 Þ

ðt�te2 Þ2þL2

" #�γR

!ð1�RÞ; Rðte2 Þ ¼ Re2

ð14Þ

Pathogen inoculation: tpotoT

dRdt

¼ f ðaÞ kt

t2þL2þ f ðaÞ kðt�te2 Þ

ðt�te2 Þ2þL2

" #�γR

!

� ð1�R�DÞ; RðtpÞ ¼ Rp ð15Þ

dDdt

¼ βDð1�R�DÞ; DðtpÞ ¼Di ð16Þ

Here we only consider two elicitor applications. The equationsabove are formulated the same way as the single application.The same concentration of elicitor is applied at each applicationtime. Eq. (13) describes the first application and Eq. (14) thesecond application at time te2 . Eqs. (15) and (16) demonstrate thepathogen inoculation to the seedlings. Fig. 10 shows the dynamicsof these multiple elicitor applications for each S,R and D compart-ments. The marked points are the experimental data. For thisillustration, with 0.1% MeJA was applied 14 and 7 days before thepathogen inoculation. That is te1 ¼ 0, te2 ¼ 7 and tp¼14.

It must be noted that no parameterisation was done for thismodel extension. The same parameter values as in Table 1 and ther value stated in Section 3.1 were used to plot Fig. 10. Theillustrations demonstrate the model that have the same patternsobserved in the experiment.

0 10 20 30 40

0

0.2

0.4

0.6

0.8

10% MeJA concentration

Time (days)

Dis

ease

d

0 10 20 30 400

0.2

0.4

0.6

0.8

1

Time (days)

Hea

lthy

0 10 20 30 400

0.2

0.4

0.6

0.8

1

Res

ista

nt

Time (days)0 10 20 30 40

0

0.2

0.4

0.6

0.8

1

Sus

cept

ible

Time (days)

Fig. 6. Disease progression when tp¼7 days for the 0% elicitor concentration (untreated plants) with marked points (⋄; �) giving the separate experimental values. Thecalculated SSE with these points are 0.3811.

N.S. Abdul Latif et al. / Journal of Theoretical Biology 347 (2014) 144–150148

3.3. Elicitor treatment after the pathogen challenge

The model is flexible enough to investigate different pathogenscenarios and in this next example, we investigated the effect ofelicitor treatment applied after pathogen inoculation. Here isthe model:

Treated modelFor pre-treatment: 0otote

dRdt

¼ �αRð1�R�DÞ; Rð0Þ ¼ Ri ð17Þ

dDdt

¼ βDð1�R�DÞ; Dð0Þ ¼Di ð18Þ

For post-treatment: teotoT

dRdt

¼ kðt�teÞðt�teÞ2þL2

�γR

!ð1�R�DÞ; RðteÞ ¼ Re ð19Þ

dDdt

¼ βDð1�R�DÞ; DðteÞ ¼De ð20Þ

The pre-treatment scenario (Eqs. (17) and (18)) describes theseedlings after they have been challenged by the pathogen at t¼0.We have assumed that they behave like the untreated seedlings asin Eqs. (1) and (2). Then, after some time te, the seedlings aretreated with the elicitor. Therefore, the post-treatment formula-tions are given by (19) and (20) with eðtÞ ¼ kðt�teÞ=ðt�teÞ2þL2.Values for Re and De are obtained from (17) and (18) respectively attime te. This model extension0s dynamics are plotted in Fig. 11. Thesame parameter values in Table 1 are used to plot the graph. Themodel demonstrates that if the elicitor treatment is done after thepathogen challenge, these treated seedlings will be less severelyaffected compared to the untreated seedlings. This shows that thepost-inoculation elicitor treatment assists seedlings recovery frombeing infected and becoming severely diseased. The model0sequations (Eqs. (17)–(20)) show that the proportion of diseasedplants is an increasing function of te. Thus, delaying the treatmentwill result in no significant difference in the treatment0s outcomewhen compared to the untreated seedlings.

4. Conclusion

The potential for using a plant-defence elicitor to induceresistance against plant disease is evaluated by use of a compart-ment model, which is in turn based on a SIR-type model, whichhere has the three compartments: Susceptible–Diseased–Resis-tant. This leads to a potentially powerful tool for quantifying thedevelopment of more potent elicitors and strategies for theirapplication, both before and/or after pathogen attack and subse-quent disease development. This enables the better managementof disease progression which is critical for commercial operations

0 10 20 30 40

0

0.2

0.4

0.6

0.8

10.1% MeJA concentration

Dis

ease

d

Time (days)0 10 20 30 40

0

0.2

0.4

0.6

0.8

1

Hea

lhty

Time (days)

0 10 20 30 400

0.2

0.4

0.6

0.8

1

Res

ista

nt

Time (days)0 10 20 30 40

0

0.2

0.4

0.6

0.8

1

Sus

cept

ible

Time (days)

Fig. 7. Disease progression when tp¼7 days and for the treated plants with higher elicitor concentration (0.1%). The marked points (⋄, �, n) are the separate experimentaloutcomes and the calculated SSE with these points are 0.1080.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.40.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

a (concentration)

Dis

ease

d at

the

final

tim

e ( D

Fina

l= D

(tp+

35) )

Varying elicitor concentration when inoculation time tp = 7 days

ModelExperimental data

Fig. 8. Final disease as a function of elicitor concentration. The marked points ðnÞindicate the final experimental outcome. The fact that there are two points ata¼0.4 is because there are two different experiment used.

N.S. Abdul Latif et al. / Journal of Theoretical Biology 347 (2014) 144–150 149

(Gent et al., 2011). Further the model enables the likely long-termeffect of the disease. The attracting steady states are in this modeldependent on the initial conditions and not, as is usually the case,asymptotically stable. The case study used here was pine seedling(Pinus radiata) inoculated by the causal agent of terminal crook(Diplodia pinea) and was used to develop and validate the model0spredictive capability. The model is generic and can (with different

parameters of course) be used for any combination of plant,elicitor, and pathogen.

Of particular interest is the optimal induction time tp whichleads to a minimisation of the long-term effect for elicitorapplication, either before or after (or both) the inoculation of theplants by the pathogen. This was determined for the pine seedlingexperiments described in this paper. Secondly, the effect ofdifferent concentration of elicitor and the frequency of elicitorapplication can be evaluated without further expensive experi-ments by simulated studies using this model.

Accordingly we put forward this model as a significant tool forunderpinning the understanding of disease control and the devel-opment of more effective elicitor agents for plant protection in thefuture.

Acknowledgements

The authors wish to thank J.T. Taylor for the experimental datacollection and Ministry of Higher Education of Malaysia (KPTBS-851110105416) for funding to support this study.

References

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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ease

d (D

)

te = 2 days : the seedlings were treated with the elicitor

te = 6 days : the seedlings were treated with the elicitor

Untreated Model

Treated Model

Pathogen is inoculatedat t = 0

Fig. 11. For this illustration, the seedlings were challenged by the pathogen at timet¼0 day, and then 2 days later or 6 days later (i.e. te¼2 or te¼6) the seedlings weretreated with the elicitor. The blue curve illustrating the treated seedlings and thered curve corresponds to the untreated seedlings. (For interpretation of thereferences to colour in this figure caption, the reader is referred to the web versionof this paper.)

Fig. 9. Time-line for multiple elicitor applications (illustrative only).

0 5 10 15 20 25 30 35 40 450

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Multiple application at 14 & 7 days before inoculation at 0.4% concentration

0 5 10 15 20 25 30 35 40 450

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lhty

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ista

nt

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cept

ible

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Fig. 10. Multiple application at 14 and 7 days before inoculation at 0.4% elicitorconcentration. The marked points (n) are the experimental outcomes and thecalculated SSE¼0.0178.

N.S. Abdul Latif et al. / Journal of Theoretical Biology 347 (2014) 144–150150