scholarly journals Deterministic and Stochastic Study for an Infected Computer Network Model Powered by a System of Antivirus Programs

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Youness El Ansari ◽  
Ali El Myr ◽  
Lahcen Omari
Keyword(s):  
Equilibrium State ◽  
Stationary Distribution ◽  
Computer Network ◽  
Reproduction Number ◽  
Sufficient Condition ◽  
Stochastic Perturbations ◽  
Stochastic Fluctuations ◽  
Spread Model ◽  
The Stability ◽  
Theoretical Results

We investigate the various conditions that control the extinction and stability of a nonlinear mathematical spread model with stochastic perturbations. This model describes the spread of viruses into an infected computer network which is powered by a system of antivirus software. The system is analyzed by using the stability theory of stochastic differential equations and the computer simulations. First, we study the global stability of the virus-free equilibrium state and the virus-epidemic equilibrium state. Furthermore, we use the Itô formula and some other theoretical theorems of stochastic differential equation to discuss the extinction and the stationary distribution of our system. The analysis gives a sufficient condition for the infection to be extinct (i.e., the number of viruses tends exponentially to zero). The ergodicity of the solution and the stationary distribution can be obtained if the basic reproduction number Rp is bigger than 1, and the intensities of stochastic fluctuations are small enough. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals A Stochastic SIR Epidemic System with a Nonlinear Relapse

2018 ◽  
Vol 2018 ◽  
pp. 1-10 ◽  
Author(s):  
Ali El Myr ◽  
Abdelaziz Assadouq ◽  
Lahcen Omari ◽  
Adel Settati ◽  
Aadil Lahrouz
Keyword(s):  
Numerical Simulations ◽  
Stationary Distribution ◽  
Stochastic Perturbations ◽  
Sir Epidemic ◽  
Spread Model ◽  
Stochastic Sir ◽  
Theoretical Results

We investigate the conditions that control the extinction and the existence of a unique stationary distribution of a nonlinear mathematical spread model with stochastic perturbations in a population of varying size with relapse. Numerical simulations are carried out to illustrate the theoretical results.

scholarly journals Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuai Yang ◽  
Haijun Jiang ◽  
Cheng Hu ◽  
Juan Yu ◽  
Jiarong Li
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Model Parameters ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Reductio Ad Absurdum ◽  
The Stability ◽  
The Impact ◽  
Theoretical Results ◽  
The Basic Reproduction Number

Abstract In this paper, a novel rumor-spreading model is proposed under bilingual environment and heterogenous networks, which considers that exposures may be converted to spreaders or stiflers at a set rate. Firstly, the nonnegativity and boundedness of the solution for rumor-spreading model are proved by reductio ad absurdum. Secondly, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. Whereafter, the global stability of rumor-prevailing equilibrium is explored by utilizing Lyapunov method and LaSalle’s invariance principle. Finally, the sensitivity analysis and the numerical simulation are respectively presented to analyze the impact of model parameters and illustrate the validity of theoretical results.

The asymptotic behavior of a stochastic multigroup SIS model

2018 ◽  
Vol 11 (03) ◽  
pp. 1850037 ◽  
Author(s):  
Chunyan Ji ◽  
Daqing Jiang
Keyword(s):  
Asymptotic Behavior ◽  
Numerical Simulations ◽  
Stationary Distribution ◽  
Endemic Equilibrium ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Sis Model ◽  
Stochastic Perturbations ◽  
Long Time ◽  
Theoretical Results

In this paper, we explore the long time behavior of a multigroup Susceptible–Infected–Susceptible (SIS) model with stochastic perturbations. The conditions for the disease to die out are obtained. Besides, we also show that the disease is fluctuating around the endemic equilibrium under some conditions. Moreover, there is a stationary distribution under stronger conditions. At last, some numerical simulations are applied to support our theoretical results.

scholarly journals Stationary Distribution and Extinction of a Stochastic SIQR Model with Saturated Incidence Rate

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Qiuhua Zhang ◽  
Kai Zhou
Keyword(s):  
Lyapunov Function ◽  
Incidence Rate ◽  
Stationary Distribution ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Sufficient Condition ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Dynamical Properties ◽  
Theoretical Results

In this paper, we consider a stochastic SIQR epidemic model with saturated incidence rate. By constructing a proper Lyapunov function, we obtain the existence and uniqueness of positive solution for this SIQR model. Furthermore, we study the dynamical properties of this stochastic SIQR model; that is, (i) we establish the sufficient condition for the existence of ergodic stationary distribution of the model; (ii) we obtain the extinction of the disease under some conditions. At last, numerical simulations are introduced to illustrate our theoretical results.

scholarly journals Robust Quadratic Stabilizability and H∞ Control of Uncertain Linear Discrete-Time Stochastic Systems with State Delay

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Xiushan Jiang ◽  
Xuemin Tian ◽  
Weihai Zhang
Keyword(s):  
Discrete Time ◽  
Stochastic Systems ◽  
Linear Matrix ◽  
Sufficient Condition ◽  
Feedback Controllers ◽  
State Delay ◽  
Quadratic Stability ◽  
Stability And Stabilization ◽  
The Stability ◽  
Theoretical Results

This paper mainly discusses the robust quadratic stability and stabilization of linear discrete-time stochastic systems with state delay and uncertain parameters. By means of the linear matrix inequality (LMI) method, a sufficient condition is, respectively, obtained for the stability and stabilizability of the considered system. Moreover, we design the robust H∞ state feedback controllers such that the system with admissible uncertainties is not only quadratically internally stable but also robust H∞ controllable. A sufficient condition for the existence of the desired robust H∞ controller is obtained. Finally, an example with simulations is given to verify the effectiveness of our theoretical results.

scholarly journals Fractional SEIRP model for COVID-19 dynamics incorporatingsocial distancing and environment

2021 ◽  
Author(s):  
Saheed Ojo Akindeinde ◽  
E. Okyere ◽  
A. O. Adewumi ◽  
R. S. Lebelo ◽  
O. O. Fabelurin ◽  
...  
Keyword(s):  
Compartmental Model ◽  
Reproduction Number ◽  
Banach Fixed Point Theorem ◽  
The Novel ◽  
Disease Dynamics ◽  
Stability Criteria ◽  
Proposed Model ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Theoretical Results

Abstract We propose a Caputo-based fractional compartmental model for the dynamics of the novel COVID-19 epidemic taking into consideration social distancing and the influence of the environment. Using basic concepts such as continuity and Banach fixed-point theorem, the existence and uniqueness of the solution to the proposed model were shown. Furthermore, we analyze the stability of the model in the context of Ulam-Hyers and generalized Ulam-Hyers stability criteria. The concept of next-generation matrices was used to compute the basic reproduction number $R_0,$ a number that determines the spread or otherwise of the disease into the general population. Numerical simulation of the disease dynamics was carried out using the fractional Adam-Bashforth-Moulton method to validate the obtained theoretical results.

scholarly journals An extended SEIARD model for COVID-19 vaccination in Mexico: analysis and forecast

2021 ◽  
Author(s):  
Ángel G. C. Pérez ◽  
David Adeyemi Oluyori
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Reproduction Number ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Stability Of Equilibria ◽  
The Stability ◽  
Reported Data ◽  
The Impact ◽  
Disease Free

In this study, we propose and analyze an extended SEIARD model with vaccination. We compute the control reproduction number Rc of our model and study the stability of equilibria. We show that the set of disease-free equilibria is locally asymptotically stable when Rc<1 and unstable when Rc>1, and we provide a sufficient condition for its global stability. Furthermore, we perform numerical simulations using the reported data of COVID-19 infections and vaccination in Mexico to study the impact of different vaccination, transmission and efficacy rates on the dynamics of the disease.

scholarly journals An extended SEIARD model for COVID-19 vaccination in Mexico: analysis and forecast

2021 ◽  
Author(s):  
Ángel G. C. Pérez ◽  
David A. Oluyori
Keyword(s):  
Global Stability ◽  
Numerical Simulations ◽  
Reproduction Number ◽  
Sufficient Condition ◽  
Asymptotically Stable ◽  
Stability Of Equilibria ◽  
The Stability ◽  
Reported Data ◽  
The Impact ◽  
Disease Free

In this study, we propose and analyse an extended SEIARD model with vaccination. We compute the control reproduction number $\mathcal{R}_c$ of our model and study the stability of equilibria. We show that the set of disease-free equilibria is locally asymptotically stable when $\mathcal{R}_c<1$ and unstable when $\mathcal{R}_c>1$, and we provide a sufficient condition for its global stability. Furthermore, we perform numerical simulations using the reported data of COVID-19 infections and vaccination in Mexico to study the impact of different vaccination, transmission and efficacy rates on the dynamics of the disease.

scholarly journals Global Dynamics for a Novel Differential Infectivity Epidemic Model with Stage Structure

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Yunguo Jin
Keyword(s):  
Epidemic Model ◽  
Lyapunov Functions ◽  
Reproduction Number ◽  
Stage Structure ◽  
Threshold Dynamics ◽  
Uniform Persistence ◽  
Global Dynamics ◽  
Differential Infectivity ◽  
The Stability ◽  
Theoretical Results

A novel differential infectivity epidemic model with stage structure is formulated and studied. Under biological motivation, the stability of equilibria is investigated by the global Lyapunov functions. Some novel techniques are applied to the global dynamics analysis for the differential infectivity epidemic model. Uniform persistence and the sharp threshold dynamics are established; that is, the reproduction number determines the global dynamics of the system. Finally, numerical simulations are given to illustrate the main theoretical results.

Behavioral Modeling of Malicious Objects in a Highly Infected Network Under Quarantine Defence

2019 ◽  
Vol 13 (1) ◽  
pp. 17-29 ◽  
Author(s):  
Yerra Shankar Rao ◽  
Prasant Kumar Nayak ◽  
Hemraj Saini ◽  
Tarini Charana Panda
Keyword(s):  
Control Strategy ◽  
Computer Network ◽  
Reproduction Number ◽  
Malicious Code ◽  
Behavioral Modeling ◽  
Asymptotically Stable ◽  
Matlab Simulation ◽  
Safety Measures ◽  
Malicious Codes ◽  
The Stability

This article describes a highly infected e-epidemic model in a computer network. This article establishes the Basic reproduction number R0, which explicitly brings out the stability conditions. Further, the article shows that if R0&lt; 1 then the infected nodes ceases the spreading of malicious code in computer network as it dies down and consequently establishes the asymptotically stable, when R0&gt; 1, the alternative aspect is that infected nodes stretch out into the network and becomes asymptotically unstable. The pivotal, impact of quarantine node on e-epidemic models has been verified along with its control strategy for a high infected computer network. In the MATLAB simulation, the quarantine class shows its explicit relationship with respect to high as well as low infected class, exposed class, and finally, with recovery class in order to yield increasing safety measures on transmission of malicious codes.

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