AACIMP 2011 Summer School. Neuroscience Stream. Lecture by Khaled Khanchouch.

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

4. Fig.1: Growth model.

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

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

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

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

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

38. Derivative fitted curve with a<0 Fig. 7: Derivative linear regression curve of weak virulent isolate

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