A biomathematical model for Phoma tracheiphila

Citrus resistance screening

K. Khanchouch1,4, E. Ustimovich2, H. Kutucu3 and M.R. Hajlaoui4

1Department of techniques, ISAJC University of Tunis, Tunis, Tunisia 2 Informatization center, Kiev, Ukraine

3 Department of Mathematics, Izmir institute of technology, Ural-Izmir, Turkey 4Laboratory of plant protection, National research agronomic institute, Tunis, Tunisia

Why we need modelin general?

-Data analysis

-Epidemiological studies

- Decision making

Mathematical models

- Disease progressive curve ( Growth model) MonomolecularExponentialLogisticGompertz

- Linked differential equations (LDE)

-Area under disease progress curve (AUDPC)

Fig.1: Growth model.

Fig. 2: Schematic of the plant-virus insect-vector model.

Time

Disease index

Fig. 3: Area Under the Progressive Disease Curve (AUPDC).

Statistical models

-Univariant and multivariant analysis

-correlation analysis of disease intensity.

-Nonparametric analysis; treatments and environmental factors.

-Linear and Polynomial distributed regression analysis.

DisadvantageGrowth model:

Gives an indispensable description of the disease. HoweverSome missing components and the non detailed description of the disease development can

led to imprecise interpretation of the obtained results.

Linked differential equations (LDE):Equation generated are extremely troublesome for mathematical analysis.

Area under disease progress curve (AUDPC):Give misleading results when AUDPC is summarized over the specific period of the disease.

Statistical models:Supposes that data are normally distributed, the proposed models are mainly based on

theory and allow relative comparing of the tested samples.

Needs of a new of Biomathematical models to overpass

the disadvantages previously cited

Characteristics of the new biomathematical models

-Give detailed description of the disease not only based on the analysis of the severity index meaning. (Growth model)

- Generated equations by these model can be resolved without mathematical troubles. (LDE)

- Good interpellation parameters of the disease to calculate more precisely the area under the fitted curve. (AUDPC)

- Give a proper evaluation for the tested sample and not a relative theoretical compared methods. (Statistical models)

Objectives

General- Reducing the effect of the biotic factors. - Elimination of the errors related to the results interpretation. -Validation of the Mathematical model as a new tool for the host-parasites relationships analysis.

Specific-Resolving the controversial results problem between the laboratories tests.- Determination of the inner resistance level of Citrus limon cultivars.- Screening of resistance genotypes to control the disease.

The experimental biological model

Hosts: Citrus limon cultivars.Parasite: highly virulent isolate of Phoma tracheiphila.

Artificial Inoculation: Green house foliar inoculation method. Inoculation points: 120 inoculation points.

Disease evaluation tools

Visual evaluation: scale of 6 degree(each degree is determined a class)

Class: 0Chlorosis

Class:1

Class: 2

Class: 3

Class: 4

Class: 5

Extended brownishment

Mathematical model

- The cumulative frequency is determined as described below: Yi= /120]*100

Yi= The cumulative frequency at the respective class, Xi. Xi= class ‘’i’’ varying from “0” to “5” 120, it’s the number of the inoculation points tested

- The polynomial interpellationf(xi) = a xi 5+ b xi 4+ c xi 3+ d xi 2+ e xi 1+ f

Y0 = a x0 5+b x0

4+c x03+d x0

2+e x01+f

Y1= a x1 5+b x1

4+c x13+d x1

2+e x11+f

Y2= a x2 5+b x2

4+c x23+d x2

2+e x21+f

Y3= a x3 5+b x3

4+c x33+d x3

2+e x31+f

Y4= a x45+b x4

4+c x43+d x4

2+e x41+f

Y5= a x55+b x5

4+c x53+d x5

2+e x51+f

The linear regression system

- To calculate the coefficients a, b, c, d, e and f we use Gaussian elimination method

Area under the curve: AUC

We use total integration from point 0 to 5 to calculate the area under the curve

Results

Isolate Disease total score Disease Severity means

L01 271 2,258

D02 276 2,3

K001 276 2, 3

A12 515 4,291

Z35 519 4,325

T46 301 2,508

Table 1: Infection severity rating of Lemon cultivars infected by the tested isolates of the pathogen.

Phytopathological test

Isolates Test Newman-Keuls

Test LSD Test Ducan

L01 I I I

D02 I I I

K001 I I I

A12 II II II

Z35 II II II

T46 I I I

Table 2: Classification of tested isolates

Fig. 1: Cumulative frequency classes

Group: I A

Fig. 2: Cumulative frequency regression curve of group I A

Fig. 3: Cumulative frequency classes

Fig. 4: Cumulative frequency regression curve of isolates of Group I B

Fig. 5: Cumulative frequency classes

Fig. 6: Cumulative frequency regression curve of isolates of group II

What does it means?

• A positive corelation was found between statistical analysis of the virulence and the graphical representation of the polynom of the studied inoculated plants

The shape of the fitted curve (the polynom) indicates the level of the virulence of the isolates

3 Types of curves indicate 3 level of virulence

Three types of polynomial curve can be described:

Type A: with an lower concave convection

Group I A

lower degree of virulence

Type B: with a mixed convection curve

Group I B

intermediate degree of virulence

Type C: with a upper concave convection

Group II virulent isolate

Three types of polynomial curve can be described:

Type A: with an lower concave convection

Group I A

lower degree of virulence

Type B: with a mixed convection curve

Group I B

intermediate degree of virulence

Type C: with a upper concave convection

Group II virulent isolate

Avirulent class : I

Weak virulent class : II

class : III

Virulent class : IV

Higly virulent class : V

Computing the plant resistance level

- The parametric analysis of each polynomial curve by calculating its derivative near the convection points allows distinguishing between the three types. f’(xi) = 5a xi

4+4b xi3+3c xi

2+2d xi+e -The derivative calculation for each polynomial regression curve is performed from the point xi =1 to the point xi =4.

Analytical analysis

- Linear regression curve :

yi’= axi+b

a=

b= - a

with :

= means of xi = means of y’i

-Coefficient of determination:The adjustment of the fitted linear derivative linear regression curve is appreciated using its R’2 value.

• Class II with a lower concave curve:

Mathematically this type of curve can be described anylatically as:

is decreasing

If the derivative of the polynomial is decreasing in its values from the points x= 1 to the point 4, then the tested isoalte is determined as a weak virulent and to its we attribute the degree ‘‘II ’’ of virulence.

Derivative fitted curve with a<0

Fig. 7: Derivative linear regression curve of weak virulent isolate

• Class III with a mixed concave curve:

Mathematically this type of curve can be described anylatically as:

is increasing and decreasing

If the derivative of the polynomial is not totally increasing or not totally decreasing in its values from the points x= 1 to the point 4 then the tested isolate is determined as Virulent and to its we attribute the degree ‘‘III ’’ of virulence.

• Type IV uper concave curve:

Mathematically this type of curve can be described anylatically as:

is incresing

If the derivative of the polynomial is increasing in its values from the points x= 1 to 4, then the tested plant is determined as Virulent and to its we attribute the degree ‘‘IV ’’ of resistance.

Derivative fitted curve with a>0

Fig. 8: Derivative linear regression curve of a virulent isoalte

Cultivars Test Newman-Keuls Bio-Math Model

L01 I III

D02 I II

K001 I III

A12 II IV

Z35 II IV

T46 I III

Table 3: Classification of tested isolats

The Biomathematical model offer two decision tools for the virulence screening of the studied isolates:

-The first one its based on comparing the area under the polynom curve. More the value of the area under the curve is reduced more the isolate is virulent.

- The second one consists to classify the isolates into their respective degree of virulence (II, III and IV) based on the analytical analysis of the polynomial functions.

Conlusion