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1
Content available The impact of two independent gaussian white noises on the behavior of a stochastic epidemic model
Yavuz Mehmet, Boulaasair Lahcen, Bouzahir Hassane, Diop Mamadou Abdoul, Benaid Brahim
Journal of Applied Mathematics and Computational Mechanics
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2024
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Vol. 23, nr 1
121-134
EN
The aim of this paper is to investigate a stochastic SIS (Susceptible, Infected, Susceptible) epidemic model in which the disease transmission coefficient and the death rate are subject to random disturbances. Using the convergence theorem for local martingales and solving the Fokker-Planck equation associated with the one-dimensional stochastic differential equation, we demonstrate that the disease will almost surely persist in the mean. In the case of global asymptotic stability of the endemic equilibrium for a SIS deterministic epidemic model, we formulate suitable conditions guaranteeing that the stochastic SIS model has a unique ergodic stationary distribution. Furthermore, we deal with the exponential extinction of the disease. Finally, some numerical simulations are provided to illustrate the obtained analytical results.
2
Content available Stochastic models of the slow/fast type of atrioventricular nodal reentrant tachycardia and tachycardia with conduction aberration
Jackowska-Zduniak Beata
International Journal of Applied Mathematics and Computer Science
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2022
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Vol. 32, no. 3
429--440
EN
Models are proposed to describe the heart’s action potential. A system of stochastic differential equations is used to recreate pathological behaviour in the heart such as atrioventricular nodal reentrant tachycardia (AVNRT) and also AVNRT with conduction aberration. Part of the population has abnormal accessory pathways: fast and slow. An additional pathway is not always induced, since the deterministic model is not proper due to a stochasticity in this process. Introduction of a stochastic term allows modelling a pre-excitation perturbation (such as unexpected excitation by premature contractions in atrium (PAC)) which triggers the mechanism of AVNRT. Also, a system of AVNRT with additional conduction aberration, which is a rare type of arrhythmia, is considered. The aim of this work is to propose a mathematical model superior to the deterministic one that recreates this disease better and allows understanding its mechanism and physical dependencies, which may help to propose a new therapy of AVNRT. Results are illustrated with numerical solutions.
3
Content available Optimization of periodic maintenance for wind turbines based on stochastic degradation model
Su Hongsheng, Duan Xuping, Wang Dantong
Archives of Electrical Engineering
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2021
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Vol. 70, nr 3
585--599
EN
The degradation process of wind turbines is greatly affected by external factors. Wind turbine maintenance costs are high. The regular maintenance of wind turbines can easily lead to over and insufficient maintenance. To solve the above problems, a stochastic degradation model (SDE, stochastic differential equation) is proposed to simulate the change of the state of the wind turbine. First, the average degradation trend is obtained by analyzing the properties of the stochastic degradation model. Then the average degradation model is used to describe the predictive degradation model. Then analyze the change trend between the actual degradation state and the predicted state of the wind turbine. Secondly, according to the update process theory, the effect of maintenance on the state of wind turbines is comprehensively analyzed to obtain the availability. Then based on the average degradation process, the optimal maintenance period of the wind turbine is obtained. The optimal maintenance time of wind turbines is obtained by optimizing the maintenance cycle through availability constraints. Finally, an onshore wind turbine is used as an example to verification. Based on the historical fault data of wind turbines, the optimized maintenance decision is obtained by analyzing the reliability and maintenance cost of wind turbines under periodic and non-equal cycle conditions. The research results show that maintenance based on this model can effectively improve the performance of wind turbines and reduce maintenance costs.
4
Content available A dynamic bi-orthogonal field equation approach to efficient Bayesian inversion
Tagade P. M., Choi H. L.
International Journal of Applied Mathematics and Computer Science
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2017
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Vol. 27, no. 2
229--243
EN
This paper proposes a novel computationally efficient stochastic spectral projection based approach to Bayesian inversion of a computer simulator with high dimensional parametric and model structure uncertainty. The proposed method is based on the decomposition of the solution into its mean and a random field using a generic Karhunen–Loève expansion. The random field is represented as a convolution of separable Hilbert spaces in stochastic and spatial dimensions that are spectrally represented using respective orthogonal bases. In particular, the present paper investigates generalized polynomial chaos bases for the stochastic dimension and eigenfunction bases for the spatial dimension. Dynamic orthogonality is used to derive closed-form equations for the time evolution of mean, spatial and the stochastic fields. The resultant system of equations consists of a partial differential equation (PDE) that defines the dynamic evolution of the mean, a set of PDEs to define the time evolution of eigenfunction bases, while a set of ordinary differential equations (ODEs) define dynamics of the stochastic field. This system of dynamic evolution equations efficiently propagates the prior parametric uncertainty to the system response. The resulting bi-orthogonal expansion of the system response is used to reformulate the Bayesian inference for efficient exploration of the posterior distribution. The efficacy of the proposed method is investigated for calibration of a 2D transient diffusion simulator with an uncertain source location and diffusivity. The computational efficiency of the method is demonstrated against a Monte Carlo method and a generalized polynomial chaos approach.
5
Content available remote Nonlinear filtering for Markov systems with delayed observations
Calzolari A., Florchinger P., Nappo G.
International Journal of Applied Mathematics and Computer Science
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2009
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Vol. 19, no 1
49-57
EN
This paper deals with nonlinear filtering problems with delays, i.e., we consider a system (X,Y ), which can be represented by means of a system [...], in the sense that [...], where a(t) is a delayed time transformation. We start with X being a Markov process, and then study Markovian systems, not necessarily diffusive, with correlated noises. The interest is focused on the existence of explicit representations of the corresponding filters as functionals depending on the observed trajectory. Various assumptions on the function a(t) are considered.
6
Content available remote Euler's approximations of weak solutions of reflecting SDEs with discontinuous coefficients
Semrau A.
Bulletin of the Polish Academy of Sciences. Mathematics
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2007
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Vol. 55, no 2
171-182
EN
We study convergence in law for the Euler and Euler-Peano schemes for stochastic differential equations reflecting on the boundary of a general convex domain. We assume that the coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. The proofs are based on new estimates of Krylov's type for the approximations considered.
7
Content available remote The probabilistic solution of the Dirichlet problem for degenerate elliptic equations
Liu C., Morimoto H.
Bulletin of the Polish Academy of Sciences. Mathematics
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2006
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Vol. 54, no 2
163-174
EN
We study the Dirichlet problem for degenerate elliptic equations, and show that the probabilistic solution is a unique viscosity solution.
8
Content available remote Existence of strong solutions to stochastic differential equations in conuclear spaces
Jakubowski J.
Bulletin of the Polish Academy of Sciences. Mathematics
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2000
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Vol. 48, no 3
247-252
EN
The existence of an unique strong solutions to stochastic differential equations with respect to a generalized non-homogeneous Wiener process in the dual of a nuclear space is proved under monotonicity condition and conditions which guarantee e~stence of weak solutions.
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