1. Modeling and Analysis of Stochastic Model for a Marine Bacteria Populations Anatoliy Swishchuk Laboratory for Industrial & Applied Mathematics, Department of Mathematics & Statistics, York University (So-joint work with D. Liang, J.Wu and F. Zang ) Dynamics Day at Wilfrid Laurier University-April 7, 2004

2. Outline General Results on Stability for NSDE near Equilibrium Points Asymptotical Mean Stability Asymptotical Mean- Square Stability Exponential Mean-Square Stability Applications for Epidemic Model Equilibrium Points E_0=(0,0,0) E_f=(1,0,0) E_+=(s*,I*,v*)

3. Stochastic Stability (chart)

4. Stochastic Stability (definitions) Exponential Mean-Square Stability Asymptotical Mean-Square Stability Asymptotical Stability in Mean Asymptotical Mean Stability Connection

5. Stochastic Epidemics Model of Bacteriophages in the Marine Bacteria Populations (non-linear system of stochastic differential equations)

6. Equilibrium Points of Deterministic Model Deterministic Model Equilibrium Points

7. Equilibrium Point of Stochastic Model

8. Problems with the Non-linear Stochastic Model (we need a new approach)

9. First Order Approximated and Extended Vector Non-linear Stochastic Differential Equation (Mean Value) Vector Non-linear Stochastic Differential Equation First Order Approximated NLSDE Extended NLSDE

10. Asymptotical Mean Stability for Vector NLSDE near Equilibrium Point

11. Asymptotical Mean Stability of Epidemic Model Near E_0=(0,0,0).

12. Asymptotical Mean Stability of Epidemic Model Near E_f=(1,0,0)

13. Asymptotical Mean Stability of Epidemic Model near E_+=(s*, i*,v*)

14. Asymptotical Mean-Square Stability Non-linear Vector Stochastic Differential Equation

15. Main Results on Asymptotical Mean- Square Stability for Vector SDE

16. Asymptotical Mean-Square Stability for Epidemic Model Equilibrium Point E_0=(0,0,0)

17. Equilibrium Point E_f=(1,0,0)

18. Equilibrium Point E_+=(s*,i*v*)

19. Exponential Mean-Square Stability for Vector NSDE Vector NSDE First-Order Approximated Vector SDE Extended Vector NSDE

20. Main Results on Exponential Mean- Square Stability of NSDE

21. Continuation of Main Results on Exponential Mean- Square Stability I

22. Continuation of Main Results on Mean-Square Stability II

23. Conclusions: -We have exponential, asymptotical mean-square and mean stability for Vector NSDE -We applied it to study of epidemic model of Marine Bacteria Population (system of 3 NSDE) -We can apply our theory to the study of stochastic SARS Models Which include more than 3 equations (8 or 10, for example)

24. Deterministic SARS Model I

25. Stochastic SARS Model I (transmission coefficients are stochastic)

26. Stochastic SARS Model II (additive or multiplicative noise)

27. Future Work on Stochastic SARS Models I and II Stochastic SARS Model I Averaging, merging, diffusion approximation, normal approximation, stochastic stability, using the results from the recent book by J. Wu and A. Swishchuk “Evolution of Biological Systems in Random Media: Limit Theorems and Stability”, Kluwer AP, 2003. Stochastic SARS Model II Stochastic stability (mean, mean square, exponential, etc.) using the results from this talk and the working paper by D.Liang, J. Wu, F. Zang and A. Swishchuk “Modeling and Analysis of a Marine Bacteria Population” (2004).