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A biomathematical model for Phoma tracheiphila Citrus resistance screening K. Khanchouch 1,4 , E. Ustimovich 2 , H. Kutucu 3 and M.R. Hajlaoui 4 1 Department of techniques, ISAJC University of Tunis, Tunis, Tunisia 2 Informatization center, Kiev, Ukraine 3 Department of Mathematics, Izmir institute of technology, Ural-Izmir, Turkey 4 Laboratory of plant protection, National research agronomic institute, Tunis, Tunisia
Nonparametric analysis; treatments and environmental factors.
Linear and Polynomial distributed regression analysis.
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Disadvantage Growth model: Gives an indispensable description of the disease. However Some 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.
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Needs of a new of Biomathematical models to overpass the disadvantages previously cited
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.
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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.
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Disease evaluation tools Visual evaluation: scale of 6 degree (each degree is determined a class)
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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 interpellation f(x i ) = a x i 5 + b x i 4 + c x i 3 + d x i 2 + e x i 1 + f
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Y 0 = a x 0 5 +b x 0 4 +c x 0 3 +d x 0 2 +e x 0 1 +f Y 1 = a x 1 5 +b x 1 4 +c x 1 3 +d x 1 2 +e x 1 1 +f Y 2 = a x 2 5 +b x 2 4 +c x 2 3 +d x 2 2 +e x 2 1 +f Y 3 = a x 3 5 +b x 3 4 +c x 3 3 +d x 3 2 +e x 3 1 +f Y 4 = a x 4 5 +b x 4 4 +c x 4 3 +d x 4 2 +e x 4 1 +f Y 5 = a x 5 5 +b x 5 4 +c x 5 3 +d x 5 2 +e x 5 1 +f The linear regression system - To calculate the coefficients a, b, c, d, e and f we use Gaussian elimination method
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Area under the curve : AUC We use total integration from point 0 to 5 to calculate the area under the curve
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Table 1: Infection severity rating of Lemon cultivars infected by the tested isolates of the pathogen. Phytopathological test 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
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Table 2: Classification of tested isolates 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
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Fig. 1: Cumulative frequency classes Group: I A
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Fig. 2: Cumulative frequency regression curve of group I A
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
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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
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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
- The parametric analysis of each polynomial curve by calculating its derivative near the convection points allows distinguishing between the three types.
f’(x i ) = 5a x i 4 +4b x i 3 +3c x i 2 +2d x i +e
The derivative calculation for each polynomial regression curve is performed from the point x i =1 to the point x i =4.
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.
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Derivative fitted curve with a<0 Fig. 7: Derivative linear regression curve of weak virulent isolate
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.
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.
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Derivative fitted curve with a>0 Fig. 8: Derivative linear regression curve of a virulent isoalte
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Table 3: Classification of tested isolats 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
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.
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