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 Dynamics Day at Wilfrid Laurier University-April 7, 2004University-April 7, 2004

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*)

Stochastic Stability (chart)

A sym pto tica lM ea n-Sq ua re

S tab i li ty.

A sym pto tica lS tab i li ty

inM ean

A sym pto tica lM eanS tab i li ty

.

E xpo ne ntia lm ea n-sq ua re

s tab ili ty.

Stochastic Stability (definitions)

Exponential Mean-Square Stability

Asymptotical Mean-Square Stability

Asymptotical Stability in Mean

Asymptotical Mean Stability

Connection

Stochastic Epidemics Model of Bacteriophages in the Marine Bacteria Populations

(non-linear system of stochastic differential equations)

Equilibrium Points of Deterministic Model

Deterministic Model

Equilibrium Points

Equilibrium Point of Stochastic Model

Problems with the Non-linear Stochastic Model

(we need a new approach)

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

Asymptotical Mean Stability for Vector NLSDE near Equilibrium

Point

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

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

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

Asymptotical Mean-Square StabilityNon-linear Vector Stochastic Differential Equation

Main Results on Asymptotical Mean-Square Stability for Vector SDE

Asymptotical Mean-Square Stability for Epidemic Model

Equilibrium Point E_0=(0,0,0)

Equilibrium Point E_f=(1,0,0)

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

Exponential Mean-Square Stability for Vector NSDE

Vector NSDE

First-Order Approximated Vector SDE

Extended Vector NSDE

Main Results on Exponential Mean-Square Stability of NSDE

Continuation of Main Results on Exponential Mean-Square Stability I

Continuation of Main Results on Mean-Square Stability II

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 ModelsWhich include more than 3 equations (8 or 10, for example)

Deterministic SARS Model I

Stochastic SARS Model I(transmission coefficients are

stochastic)

Stochastic SARS Model II(additive or multiplicative noise)

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).