Numerical Analysis of Bacterial Meningitis Stochastic Delayed Epidemic Model through Computational Methods

Based on theWorld Health Organization(WHO),Meningitis is a severe infection of the meninges,the membranes covering the brain and spinal cord.It is a devastating disease and remains a significant public health challenge.This study investigates a bacterial meningitis model through deterministic and stochastic versions.Four-compartment population dynamics explain the concept,particularly the susceptible population,carrier,infected,and recovered.The model predicts the nonnegative equilibrium points and reproduction number,i.e.,the Meningitis-Free Equilibrium(MFE),and Meningitis-Existing Equilibrium(MEE).For the stochastic version of the existing deterministicmodel,the twomethodologies studied are transition probabilities and non-parametric perturbations.Also,positivity,boundedness,extinction,and disease persistence are studiedrigorouslywiththe helpofwell-known theorems.Standard and nonstandard techniques such as EulerMaruyama,stochastic Euler,stochastic Runge Kutta,and stochastic nonstandard finite difference in the sense of delay have been presented for computational analysis of the stochastic model.Unfortunately,standard methods fail to restore the biological properties of the model,so the stochastic nonstandard finite difference approximation is offered as an efficient,low-cost,and independent of time step size.In addition,the convergence,local,and global stability around the equilibria of the nonstandard computational method is studied by assuming the perturbation effect is zero.The simulations and comparison of the methods are presented to support the theoretical results and for the best visualization of results.

Stochastic Maximum Principle for Optimal Advertising Models with Delay and Non-Convex Control Spaces

In this paper we study optimal advertising problems that model the introduction of a new product into the market in the presence of carryover effects of the advertisement and with memory effects in the level of goodwill. In particular, we let the dynamics of the product goodwill to depend on the past, and also on past advertising efforts. We treat the problem by means of the stochastic Pontryagin maximum principle, that here is considered for a class of problems where in the state equation either the state or the control depend on the past. Moreover the control acts on the martingale term and the space of controls U can be chosen to be non-convex but now the space of controls U can be chosen to be non-convex. The maximum principle is thus formulated using a first-order adjoint Backward Stochastic Differential Equations (BSDEs), which can be explicitly computed due to the specific characteristics of the model, and a second-order adjoint relation.

Asymptotic Analysis of a Stochastic Model of Mosquito-Borne Disease with the Use of Insecticides and Bet Nets

Ross’ epidemic model describing the transmission of malaria uses two classes of infection, one for humans and one for mosquitoes. This paper presents a stochastic extension of a deterministic vector-borne epidemic model based only on the class of human infectious. The consistency of the model is established by proving that the stochastic delay differential equation describing the model has a unique positive global solution. The extinction of the disease is studied through the analysis of the stability of the disease-free equilibrium state and the persistence of the model. Finally, we introduce some numerical simulations to illustrate the obtained results.

The Stochastic Asymptotic Stability Analysis in Two Species Lotka-Volterra Model

The asymptotic stability of two species stochastic Lotka-Volterra model is explored in this paper. Firstly, the Lotka-Volterra model with random parameter is built and reduced into the equivalent deterministic system by orthogonal polynomial approximation. Then, the linear stability theory and Routh-Hurwitz criterion for nonlinear deterministic systems are applied to the equivalent one. At last, at the aid of Lyapunov second method, we obtain that as the random intensity or statistical parameter of random variable is changed, the stability about stochastic Lotka-Volterra model is different from the deterministic system.

Virtualized Wireless SDNs:Modelling Delay Through the Use of Stochastic Network Calculus

Software-defined networks （SDN） have attracted much attention recently because of their flexibility in terms of network management. Increasingly, SDN is being introduced into wireless networks to form wireless SDN. One enabling technology for wireless SDN is network virtualization, which logically divides one wireless network element, such as a base station, into multiple slices, and each slice serving as a standalone virtual BS. In this way, one physical mobile wireless network can be partitioned into multiple virtual networks in a software-defined manner. Wireless virtual networks comprising virtual base stations also need to provide QoS to mobile end-user services in the same context as their physical hosting networks. One key QoS parameter is delay. This paper presents a delay model for software-defined wireless virtual networks. Network calculus is used in the modelling. In particular, stochastic network calculus, which describes more realistic models than deterministic network calculus, is used. The model enables theoretical investigation of wireless SDN, which is largely dominated by either algorithms or prototype implementations.

Stochastic Analysis of Interconnect Delay in the Presence of Process Variations

Process variations can reduce the accuracy in estimation of interconnect performance. This work presents a process variation based stochastic model and proposes an effective analytical method to estimate interconnect delay. The technique decouples the stochastic interconnect segments by an improved decoupling method. Combined with a polynomial chaos expression （PCE）, this paper applies the stochastic Galerkin method （SGM） to analyze the system response. A finite representation of interconnect delay is then obtained with the complex approximation method and the bisection method. Results from the analysis match well with those from SPICE. Moreover, the method shows good computational efficiency, as the running time is much less than the SPICE simulation＇s.

Dynamics of stochastic predator-prey models with distributed delay and stage structure for prey

In this paper,we analyze two stochastic predator-prey models with distributed delay and stage structure for prey.For the nonautonomous periodic case of the model,by using Khasminskii’s theory of periodic solution,we show that the system has at least one positive T-periodic solution.For the model which is disturbed by both white and telegraph noises,we obtain sufficient criteria for positive recurrence of the solutions to the model by constructing a suitable stochastic Lyapunov function with regime switching.The positive recurrence implies that both prey and predator populations will be persistent in the long term.

A stochastic threshold of a delayed epidemic model incorporating Levy processes with harmonic mean and vaccination

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 77s 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 TZS.Furthermore,using different scenarios of Tuberculosis disease in Morocco,we perform some numerical simulations to support the analytical results.

Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance

In this paper,stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied,and we use distribution delay to simulate the delay in nutrient conversion.By the linear chain technique,we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations.First,we state that this model has a unique global positive solution for any initial value,which is helpful to explore its stochastic properties.Furthermore,we prove the stochastic ultimate boundness of the solution of system.Then sufficient conditions for solution of the system tending toward the boundary equilibrium point at exponential rate are established,which means the microorganism will be extinct.Moreover,we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions.Finally,we provide some numerical examples to illustrate theoretical results,and some conclusions and analysis are given.

Dynamics of a stochastic periodic SIRS model with time delay

Dynamical behaviors of a siocluustic periodic SIRS epidemic model with time delay are investigated.By constructing suitable Lyapunov functions and applying Ito's formula,the existence of the global positive solution and the property of stochastically ultimate boundedness of model(1.1)are proved.Moreover,the extinction and the persistence of the disease are established.The results are verified by numerical simulations.

Dynamics of a stochastic delayed avian influenza model with mutation and temporary immunity

In this paper,the dynamic behaviors are studied for a stochastic delayed avian influenza model with mutation and temporary immunity.First,we prove the existence and uniqueness of the global positive solution for the stochastic model.Second,we give two different thresholds R_(01)^(s) and,R_(02)^(s) and further establish the sufficient conditions of extinction and persistence in the mean for the avian-only subsystem and avian-human system,respectively.Compared with the corresponding deterministic model,the thresholds affected by the white noises are smaller than the ones of the deterministic system.Finally,numerical simulations are carried out to support our theoretical results.It is concluded that the vaccination immunity period can suppress the spread of avian influenza during poultry and human populations,while prompt the spread of mutant avian influenza in human population.

Stability of High-Order Linear ItôEquations with Delays

A novel general stability analysis scheme based on a non-Lyapunov framework is explored. Several easy-to-check sufficient conditions for exponential p-stability are formulated in terms of M-matrices. Stability analysis of applied second-order It? equations with delay is provided as well. The linearization technique, in combination with the tests obtained in this paper, can be used for local stability analysis of a wide class of nonlinear stochastic differential equations.

Dynamics of delayed stochastic predator-prey models with feedback controls based on discrete observations

In this paper, we concern a class of the generalized delayed stochastic predator-prey models with feedback coutrols based on discrete observations. The existence of global positive solution is given first. Then we discuss the deterministic model briefly, and establish the necessary conditions and the sufficient conditions for almost-sure extinction and persistence in mean for the stochastic system, where we show that the feedback controls can change the properties of the population systems significantly. Finally, numerical simulations are introduced to support the main results.

Study on a susceptible—infec ted—vaccina ted model with delay and proportional vaccination

A susceptible-infected-vaccinated epidemic model with proportional vaccination and generalized nonlinear rate is formulated and investigated in the paper. We show that the stochastic epidemic model admits a unique and global positive solution with probability one when constructing a proper C2-function therewith. Then a sufficient condition that guarantees the disappearances of diseases is derived when the indicator R0< 1. Further, if R0> 1, then we obtain that the solution is weakly permanent with probability one. We also derived the sufficient conditions of the persistence in the mean for the susceptible and infected when another indicator R0> 1.

Optimal Reinsurance and Investment Strategy with Delay in Heston’s SV Model

In this paper,we consider an optimal investment and proportional reinsurance problem with delay,in which the insurer’s surplus process is described by a jump-diffusion model.The insurer can buy proportional reinsurance to transfer part of the insurance claims risk.In addition to reinsurance,she also can invests her surplus in a financial market,which is consisted of a risk-free asset and a risky asset described by Heston’s stochastic volatility(SV)model.Considering the performance-related capital flow,the insurer’s wealth process is modeled by a stochastic differential delay equation.The insurer’s target is to find the optimal investment and proportional reinsurance strategy to maximize the expected exponential utility of combined terminal wealth.We explicitly derive the optimal strategy and the value function.Finally,we provide some numerical examples to illustrate our results.

Accuracy Improvement of Zenith Tropospheric Delay Estimation Based on GPS Precise Point Positioning Algorithm

In the precise point positioning(PPP),some impossible accurately simulated systematic errors still remained in the GPS observations and will inevitably degrade the precision of zenith tropospheric delay(ZTD) estimation.The stochastic models used in the GPS PPP mode are compared.In this paper,the research results show that the precision of PPP-derived ZTD can be obviously improved through selecting a suitable stochastic model for GPS measurements.Low-elevation observations can cover more troposphere information that can improve the estimation of ZTD.A new stochastic model based on satellite low elevation cosine square is presented.The results show that the stochastic model using satellite elevation-based cosine square function is better than previous stochastic models.