Purpose: The main motivation of this paper is to present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established.
Design/methodology/approach: The authors establish a result concerning the L2-convergence rate of the solution of backward stochastic differential equation with jumps with respect to the Yosida approximation.
Findings: The authors carry out a convergence rate of Yosida approximation to the semi-linear backward stochastic differential equation in infinite dimension.
Originality/value: In this paper, the authors present the Yosida approximation of a semi-linear backward stochastic differential equation in infinite dimension. Under suitable assumption and condition, an L2-convergence rate is established.
In this paper, we present the stability result of a spatial semi-discrete scheme to backward stochastic differential equations taking values in a Hilbert space. Under suitable assumptions of the final value and the drift, a convergence rate is established.
In this paper, we present convergence results of a spatial semi-discrete approximation of a Hilbert space-valued backward stochastic differential equations with noise driven by a cylindrical Q-Wiener process. Both the solution and its space discretization are formulated in mild forms. Under suitable assumptions of the final value and the drift, a convergence rate is established.
In epidemic modeling, interpretation of compartment quantities, such as s, i, and r in relevant equations, is not always straightforward. Ambiguities regarding whether these quantities represent numbers or fractions of individuals in each compartment rise questions about significance of the involved parameters. In this paper, we address these challenges by considering a density-dependent epidemic modelling by a birth-death process approach inspired by Kurtz from 1970s'. In contrast to existing literature, which employs population size scaling under constant population condition, we scale with respect to the area. Namely, under the assumption of spatial homogeneity of the population, we consider the quantities of susceptible, infective and recovered per unit area. This spatial scaling allows diffusion approximation for birth-death type epidemic models with varying population size. By adopting this approach, we anticipate to contribute to a clear and transparent description of compartment quantities and parameters in epidemic modeling.
Some diseases such as herpes, bovine and human tuberculosis exhibit relapse in which the recovered individuals do not acquit permanent immunity but return to infectious class. Such diseases are modeled by SIRI models. In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model with relapse and jumps. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work.
In this paper, we study a generalization of reflected backward doubly stochastic differential equations (RBDSDEs) and present a link to a general mean field game. In our case, the RBDSDEs are associated with a lower optional not right continuous barrier. First, we establish the existence and uniqueness of a solution of such RBDSDEs. We then study a mean field game with a new type of common noise related to an electricity grid with storage allowing jumps and prove the existence of a mean field Nash equilibrium.
In this paper, we investigate reflected backward doubly stochastic differential equations (RBDSDEs) with a lower not necessarily right-continuous obstacle. First, we establish the existence and uniqueness of a solution to RBDSDEs with Lipschitz drivers. In the second part, we present a comparison theorem and we prove the existence of a minimal solution to the RBDSDE with the continuous driver.
In this paper, we are interested in the study of a stochastic viral infection model with immune impairment driven by Levy noise. First we prove the existence of a unique global solution to the model. By means of the Lyapunov method we study the stability of the equilibria. We present sufficient conditions for the extinction and persistence in mean. Furthermore, we present some numerical results to support the theoretical work.
In this paper, we establish the existence of a unique global positive solution for a stochastic epidemic model, incorporating media coverage and driven by Levy noise. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we present some numerical results to support the theoretical work. (C) 2017 Elsevier Ltd. All rights reserved.
In this paper, we study the dynamic properties of an SIRI epidemic model incorporating media coverage, and stochastically perturbed by a Lévy noise. We establish the existence of a unique global positive solution. We investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model depending on the basic reproduction number under some noise excitation. Furthermore, we present some numerical simulations to support the theoretical results. © 2019 World Scientific Publishing Company.
The COVID-19 pandemic has triggered a groundbreaking reliance on mathematical modelling as an important tool for studying and managing the spread of the virus since its emergence. Public health preventive measures such as vaccination and therapeutics can effectively reduce or eradicate an infectious disease. This work investigates these two strategies for controlling the COVID-19 epidemic through a stochastic epidemiological modelling approach. The existence and uniqueness of a positive solution of the stochastic system is studied. A priori estimates of the vaccination and treatment controls are established. Sufficient and necessary conditions are obtained for the near-optimal control problem of the stochastic model using the maximum condition of the Hamiltonian function and the Ekeland principle. Finally, to support our theoretical results, numerical simulations for a combination of optimized vaccination and treatment strategies were presented to understand the challenges posed by COVID-19 in Brazil.
In this work, we investigate a stochastic epidemic model with relapse and distributed delay. First, we prove that our model possesses and unique global positive solution. Next, by means of the Lyapunov method, we determine some sufficient criteria for the extinction of the disease and its persistence. In addition, we establish the existence of a unique stationary distribution to our model. Finally, we provide some numerical simulations for the stochastic model to assist and show the applicability and efficiency of our results. (C) 2020 Elsevier Ltd. All rights reserved.
This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results.
A stochastic SIRS epidemic model with generalized nonlinear incidence and Levy noise is investigated. First, we show the existence and uniqueness of a global positive solution. Then, we establish sufficient conditions for the extinction and persistence of the disease. The main results are proved under weak assumptions regarding the incidence function, the obtained results are proved under a Levy-type perturbation without requiring the finiteness of its activity. Finally, numerical simulations are realized to illustrate the main results.(c) 2023 Elsevier B.V. All rights reserved.
In a Covid-19 susceptible-infected-recovered-dead model with time-varying rates of transmission, recovery, and death, the parameters are constant in small time intervals. A posteriori parameters result from the Euler-Maruyama approximation for stochastic differential equations and from Bayes' theorem. Parameter estimates and 10-day predictions are performed based on Moroccan and Italian Covid-19 data. Mean absolute errors and mean square errors indicate that predictions are of good quality.
In this paper, we consider a stochastic epidemic model with relapse, reinfection, and a general incidence function. Using stochastic tools, we establish a stochastic thresholdRs0and prove the extinction of the disease when its value is equal or less than unity. We also show the persistence in mean of the disease whenRs0>1.Moreover, we prove the existence and uniqueness of a stationary distribution. Finally, numerical simulations are presented to show the effectiveness of theoretical results.
This work is devoted to study the existence and uniqueness of global positive solution for a stochastic epidemic model with media coverage driven by Levy noise. We also investigate the dynamic properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Numerical simulations are presented to confirm the theoretical results.
In this paper, we present a stochastic epidemic model with relapse. First, we prove global positivity of solutions. Then we discuss stability of the disease-free equilibrium state and we show extinction of epidemics using Lyapunov functions. Furthermore we show persistence of the disease under some conditions on parameters of the model. Our numerical simulations confirm the analytical results. (C) 2017 Elsevier Inc. All rights reserved.
A birth-death process is considered as an epidemic model with recovery and transmittance from outside. The fraction of infected individuals is for huge population sizes approximated by a solution of an ordinary differential equation taking values in [0, 1]. For intermediate size or semilarge populations, the fraction of infected individuals is approximated by a diffusion formulated as a stochastic differential equation. That diffusion approximation however needs to be killed at the boundary {0}boolean OR{1}. An alternative stochastic differential equation model is investigated which instead allows a more natural reflection at the boundary.
In this paper, a delayed SIQR epidemic model with vaccination and elimination hybrid strategies is analysed under a white noise perturbation. We prove the existence and the uniqueness of a positive solution. Afterwards, we establish a stochastic threshold R-s in order to study the extinction and persistence in mean of the stochastic epidemic system. Then we investigate the existence of a stationary distribution for the delayed stochastic model. Finally, some numerical simulations are presented to support our theoretical results.
The aim of this paper is to investigate a stochastic threshold for a delayed epidemic model driven by Levy noise with a nonlinear incidence and vaccination. Mainly, we derive a stochastic threshold R-s which depends on model parameters and stochastic coefficients for a better understanding of the dynamical spreading of the disease. First, we prove the well posedness of the model. Then, we study the extinction and the persistence of the disease according to the values of R-s. Furthermore, using different scenarios of Tuberculosis disease in Morocco, we perform some numerical simulations to support the analytical results.
This paper is devoted to a continuous-time stochastic differential system which is derived by incorporating white noise to a deterministic SIRI epidemic model with mass action incidence, cure and relapse. We focus on the impact of a relapse on the asymptotic properties of the stochastic system. We show that the relapse encourages the persistence of the disease in the population and we determine the threshold of the relapse rate, above the threshold the disease prevails in the population. Furthermore, we show that there exists a unique density function of solutions which converges in L-1, under certain conditions of the parameters to an invariant density.
Bioenergy from logging residues is an important contributor to Swedish energysupplies. Thus, accurate measurements of delivered logging residues’ energycontents are very important for both sellers and buyers. Deliveries’ energycontents are highly correlated with their moisture contents, and thus aredetermined in southern Sweden (and elsewhere) by measuring their masses andmoisture contents. There is insufficient knowledge, however, about the variation inmoisture content within and between deliveries, and hence the minimum numberof samples needed to obtain the required precision. Thus, these variations wereexamined in detail in the presented study. Nested analysis of the variance of theacquired data shows that at least nine samples are required to obtain estimates ofa delivery’s moisture content with a 3% margin of error. For high volume trade,such as that between forest companies and the energy-conversion industry,current measurement practices are sufficiently accurate. For private forest ownersmaking single deliveries, however, higher precision is required as inaccuratemeasurements can strongly affect prices.
A short introduction to option pricing under exponential Lévy process stock price models is presented. Emphasis is on appropriate change of probability measures, in particular the Esscher transform.
The note may serve as an inspiration for readers that are curious about option pricing outside the Black-Scholes framework.
A jump process with finite variation and infinite activity is approximated by replacing the small jumps by their mean and a suitably scaled Brownian motion. Integration by parts for the approximated process with respect to the Brownian motion is investigated by numerical experiments. In particular, a Monte Carlo method, involving integration by parts for computation of a sensitivity measure Delta of a European put option in models with Normal Inverse Gaussian log returns, is applied.
A time multipoint nonlocal problem for a Schrödinger equation driven by cylindrical Q-Wiener process is presented. The initial value depends on a finite number of future values. Existence and uniqueness of a solution formulated as a mild solution is obtained. A single-step implicit Euler-Maruyama difference scheme, a Rothe-Maryuama scheme, is suggested as a numerical solution. Convergence rate for the solution of the difference scheme is established. The theoretical statements for the solution of this difference scheme is supported by a numerical example.