Modeling and Analysis of Stochastic Model for a Marine Bacteria Populations

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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

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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)

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Stochastic Stability (chart)
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Stochastic Stability (definitions)
Exponential Mean-Square Stability
Asymptotical Mean-Square Stability
Asymptotical Stability in Mean
Asymptotical Mean Stability
Connection
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Stochastic Epidemics Model of Bacteriophages
in the Marine Bacteria Populations(non-linea
r system of stochastic differential equations)
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Equilibrium Points of Deterministic Model
Deterministic Model
Equilibrium Points
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Equilibrium Point of Stochastic Model
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Problems with the Non-linear Stochastic Model(we
need a new approach)
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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

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Asymptotical Mean Stability for Vector NLSDE near
Equilibrium Point
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Asymptotical Mean Stability of Epidemic Model
Near E_0(0,0,0).
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Asymptotical Mean Stability of Epidemic Model
Near E_f(1,0,0)
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Asymptotical Mean Stability of Epidemic Model
near E_(s, i,v)
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Asymptotical Mean-Square Stability
Non-linear Vector Stochastic Differential Equation
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Main Results on Asymptotical Mean-Square
Stability for Vector SDE
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Asymptotical Mean-Square Stability for Epidemic
Model
Equilibrium Point E_0(0,0,0)
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Equilibrium Point E_f(1,0,0)
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Equilibrium Point E_(s,iv)
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Exponential Mean-Square Stability for Vector NSDE
Vector NSDE
First-Order Approximated Vector SDE
Extended Vector NSDE
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Main Results on Exponential Mean-Square Stability
of NSDE
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Continuation of Main Results on Exponential
Mean-Square Stability I
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Continuation of Main Results on Mean-Square
Stability II
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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)
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Deterministic SARS Model I
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Stochastic SARS Model I(transmission
coefficients are stochastic)
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Stochastic SARS Model II(additive or
multiplicative noise)
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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).