modeling and analysis of stochastic model for a marine bacteria populations anatoliy swishchuk...
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![Page 1: Modeling and Analysis of Stochastic Model for a Marine Bacteria Populations Anatoliy Swishchuk Laboratory for Industrial & Applied Mathematics, Department](https://reader036.vdocument.in/reader036/viewer/2022062515/56649d245503460f949fad01/html5/thumbnails/1.jpg)
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
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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*)
![Page 3: Modeling and Analysis of Stochastic Model for a Marine Bacteria Populations Anatoliy Swishchuk Laboratory for Industrial & Applied Mathematics, Department](https://reader036.vdocument.in/reader036/viewer/2022062515/56649d245503460f949fad01/html5/thumbnails/3.jpg)
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.
![Page 4: Modeling and Analysis of Stochastic Model for a Marine Bacteria Populations Anatoliy Swishchuk Laboratory for Industrial & Applied Mathematics, Department](https://reader036.vdocument.in/reader036/viewer/2022062515/56649d245503460f949fad01/html5/thumbnails/4.jpg)
Stochastic Stability (definitions)
Exponential Mean-Square Stability
Asymptotical Mean-Square Stability
Asymptotical Stability in Mean
Asymptotical Mean Stability
Connection
![Page 5: Modeling and Analysis of Stochastic Model for a Marine Bacteria Populations Anatoliy Swishchuk Laboratory for Industrial & Applied Mathematics, Department](https://reader036.vdocument.in/reader036/viewer/2022062515/56649d245503460f949fad01/html5/thumbnails/5.jpg)
Stochastic Epidemics Model of Bacteriophages in the Marine Bacteria Populations
(non-linear 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)
![Page 13: Modeling and Analysis of Stochastic Model for a Marine Bacteria Populations Anatoliy Swishchuk Laboratory for Industrial & Applied Mathematics, Department](https://reader036.vdocument.in/reader036/viewer/2022062515/56649d245503460f949fad01/html5/thumbnails/13.jpg)
Asymptotical Mean Stability of Epidemic Model near E_+=(s*, i*,v*)
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Asymptotical Mean-Square StabilityNon-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*,i*v*)
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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 ModelsWhich 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).