1. 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
2. Why we need model in general? <ul><li>Data analysis </li></ul><ul><li>Epidemiological studies </li></ul><ul><li>- Decision making </li></ul>
5. Fig. 2: Schematic of the plant-virus insect-vector model.
6. Fig. 3: Area Under the Progressive Disease Curve (AUPDC). Time Disease index
7. <ul><li>Statistical models </li></ul><ul><li>Univariant and multivariant analysis </li></ul><ul><li>correlation analysis of disease intensity. </li></ul><ul><li>Nonparametric analysis; treatments and environmental factors. </li></ul><ul><li>Linear and Polynomial distributed regression analysis. </li></ul>
8. 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.
9. Needs of a new of Biomathematical models to overpass the disadvantages previously cited
10. <ul><li>Characteristics of the new biomathematical models </li></ul><ul><li>Give detailed description of the disease not only based on the analysis of the severity index meaning. (Growth model) </li></ul><ul><li>Generated equations by these model can be resolved without mathematical troubles. (LDE) </li></ul><ul><li>Good interpellation parameters of the disease to calculate more precisely the area under the fitted curve. (AUDPC) </li></ul><ul><li>Give a proper evaluation for the tested sample and not a relative theoretical compared methods. (Statistical models) </li></ul>
11. Objectives <ul><li>General </li></ul><ul><li>Reducing the effect of the biotic factors. </li></ul><ul><li>Elimination of the errors related to the results interpretation. </li></ul><ul><li>Validation of the Mathematical model as a new tool for the host-parasites relationships analysis. </li></ul><ul><li>Specific </li></ul><ul><li>Resolving the controversial results problem between the laboratories tests. </li></ul><ul><li>Determination of the inner resistance level of Citrus limon cultivars. </li></ul><ul><li>Screening of resistance genotypes to control the disease. </li></ul>
12. 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.
13. Disease evaluation tools Visual evaluation: scale of 6 degree (each degree is determined a class)
14. Class: 0 Chlorosis
15. Class:1
16. Class: 2
17. Class: 3
18. Class: 4
19. Class: 5 Extended brownishment
20. 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
21. 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
22. Area under the curve : AUC We use total integration from point 0 to 5 to calculate the area under the curve
23. Results
24. 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
25. 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
26. Fig. 1: Cumulative frequency classes Group: I A
27. Fig. 2: Cumulative frequency regression curve of group I A
28. Fig. 3: Cumulative frequency classes
29. Fig. 4: Cumulative frequency regression curve of isolates of Group I B
30. Fig. 5: Cumulative frequency classes
31. Fig. 6: Cumulative frequency regression curve of isolates of group II
32. What does it means? <ul><li>A positive corelation was found between statistical analysis of the virulence and the graphical representation of the polynom of the studied inoculated plants </li></ul><ul><li> The shape of the fitted curve (the polynom) indicates the level of the virulence of the isolates </li></ul><ul><li> 3 Types of curves indicate 3 level of virulence </li></ul>
33. 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
34. 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
35. Computing the plant resistance level <ul><li>- The parametric analysis of each polynomial curve by calculating its derivative near the convection points allows distinguishing between the three types. </li></ul><ul><li> </li></ul><ul><li>f’(x i ) = 5a x i 4 +4b x i 3 +3c x i 2 +2d x i +e </li></ul><ul><li> </li></ul><ul><li>The derivative calculation for each polynomial regression curve is performed from the point x i =1 to the point x i =4. </li></ul>
36. <ul><li>Analytical analysis </li></ul><ul><li>- Linear regression curve : </li></ul><ul><li>y i ’= ax i +b </li></ul><ul><li> </li></ul><ul><li>a= </li></ul><ul><li> </li></ul><ul><li>b= - a </li></ul><ul><li>with : </li></ul><ul><li>= means of xi </li></ul><ul><li>= means of y’ i </li></ul><ul><li> </li></ul><ul><li>Coefficient of determination: </li></ul><ul><li>The adjustment of the fitted linear derivative linear regression curve is appreciated using its R’ 2 value. </li></ul>
37. <ul><ul><li>Class II with a lower concave curve: </li></ul></ul><ul><ul><li>Mathematically this type of curve can be described anylatically as: </li></ul></ul><ul><ul><li>is decreasing </li></ul></ul><ul><ul><li>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. </li></ul></ul>
38. Derivative fitted curve with a<0 Fig. 7: Derivative linear regression curve of weak virulent isolate
39. <ul><ul><li>Class III with a mixed concave curve: </li></ul></ul><ul><ul><li>Mathematically this type of curve can be described anylatically as: </li></ul></ul><ul><ul><li>is increasing and decreasing </li></ul></ul><ul><ul><li>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. </li></ul></ul>
40. <ul><ul><li>Type IV uper concave curve: </li></ul></ul><ul><ul><li>Mathematically this type of curve can be described anylatically as: </li></ul></ul><ul><ul><li>is incresing </li></ul></ul><ul><ul><li>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. </li></ul></ul>
41. Derivative fitted curve with a>0 Fig. 8: Derivative linear regression curve of a virulent isoalte
42. 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
43. <ul><li>The Biomathematical model offer two decision tools for the virulence screening of the studied isolates: </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul>Conlusion
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