Random perturbations of recursive sequences with an application to an epidemic model

1995 ◽  
Vol 32 (3) ◽  
pp. 559-578 ◽  
Author(s):  
Daniel Pierre Loti Viaud
Keyword(s):  
Dynamical System ◽  
Lyapunov Function ◽  
Finite Number ◽  
Epidemic Model ◽  
Sample Path ◽  
Random Perturbations ◽  
Main Condition ◽  
Limit Sets ◽  
Discrete Time Dynamical System ◽  
Recursive Sequences

We investigate the asymptotic sample path behaviour of a randomly perturbed discrete-time dynamical system. We consider the case where the trajectories of the non-perturbed dynamical system are attracted by a finite number of limit sets and characterize a case where this property remains valid for the perturbed dynamical system when the perturbation converges to zero. For this purpose, no further assumptions on the perturbation are needed and our main condition applies to the limit sets of the non-perturbed dynamical system. When the limit sets reduce to limit points we show that this main condition is more general than the usual assumption of the existence of a Lyapunov function for the non-perturbed dynamical system. An application to an epidemic model is given to illustrate our results.

Random perturbations of recursive sequences with an application to an epidemic model

1995 ◽  
Vol 32 (03) ◽  
pp. 559-578 ◽  
Author(s):  
Daniel Pierre Loti Viaud
Keyword(s):  
Dynamical System ◽  
Lyapunov Function ◽  
Finite Number ◽  
Epidemic Model ◽  
Sample Path ◽  
Random Perturbations ◽  
Main Condition ◽  
Limit Sets ◽  
Discrete Time Dynamical System ◽  
Recursive Sequences

We investigate the asymptotic sample path behaviour of a randomly perturbed discrete-time dynamical system. We consider the case where the trajectories of the non-perturbed dynamical system are attracted by a finite number of limit sets and characterize a case where this property remains valid for the perturbed dynamical system when the perturbation converges to zero. For this purpose, no further assumptions on the perturbation are needed and our main condition applies to the limit sets of the non-perturbed dynamical system. When the limit sets reduce to limit points we show that this main condition is more general than the usual assumption of the existence of a Lyapunov function for the non-perturbed dynamical system. An application to an epidemic model is given to illustrate our results.

scholarly journals A non-linear discrete-time dynamical system related to epidemic SISI model

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sobirjon K. Shoyimardonov
Keyword(s):  
Dynamical System ◽  
Discrete Time ◽  
Epidemic Model ◽  
Death Rate ◽  
Birth Rate ◽  
Evolution Operator ◽  
Crucial Point ◽  
Linear Evolution ◽  
Discrete Time Dynamical System ◽  
Non Linear

Abstract We consider SISI epidemic model with discrete-time. The crucial point of this model is that an individual can be infected twice. This non-linear evolution operator depends on seven parameters and we assume that the population size under consideration is constant, so death rate is the same with birth rate per unit time. Reducing to quadratic stochastic operator (QSO) we study the dynamical system of the SISI model.

scholarly journals STATIONARY DISTRIBUTION AND EXTINCTION OF STOCHASTIC CORONAVIRUS (COVID-19) EPIDEMIC MODEL

Fractals ◽  
2021 ◽  
Author(s):  
AMIR KHAN ◽  
HEDAYAT ULLAH ◽  
MOSTAFA ZAHRI ◽  
USA WANNASINGHA HUMPHRIES ◽  
TOURIA KARITE ◽  
...  
Keyword(s):  
Brownian Motion ◽  
Lyapunov Function ◽  
Unique Solution ◽  
Epidemic Model ◽  
Feasible Region ◽  
Random Perturbations ◽  
Suggested Model ◽  
Susceptible Population ◽  
Ergodic Distribution ◽  
Very High

The aim of this paper is to model corona-virus (COVID-19) taking into account random perturbations. The suggested model is composed of four different classes i.e. the susceptible population, the smart lockdown class, the infectious population, and the recovered population. We investigate the proposed problem for the derivation of at least one unique solution in the positive feasible region of nonlocal solution. For one stationary ergodic distribution, the necessary result of existence is developed by applying the Lyapunov function and the condition for the extinction of the disease is also established. The obtained results show that the effect of Brownian motion and noise terms on the transmission of the epidemic is very high. If the noise is large the infection may decrease or vanish. For validation of our obtained scheme, the results for all the classes of the problem have been simulated numerically.

scholarly journals Extinction and persistence of a stochastic SIRV epidemic model with nonlinear incidence rate

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ramziya Rifhat ◽  
Zhidong Teng ◽  
Chunxia Wang
Keyword(s):  
Epidemic Model ◽  
Control Strategies ◽  
Random Perturbations ◽  
Disease Dynamics ◽  
Nonlinear Incidence Rate ◽  
Nonlinear Incidence ◽  
Analysis Theory ◽  
Lyapunov Function Method ◽  
Random Fluctuations ◽  
Theoretical Results

AbstractIn this paper, a stochastic SIRV epidemic model with general nonlinear incidence and vaccination is investigated. The value of our study lies in two aspects. Mathematically, with the help of Lyapunov function method and stochastic analysis theory, we obtain a stochastic threshold of the model that completely determines the extinction and persistence of the epidemic. Epidemiologically, we find that random fluctuations can suppress disease outbreak, which can provide us some useful control strategies to regulate disease dynamics. In other words, neglecting random perturbations overestimates the ability of the disease to spread. The numerical simulations are given to illustrate the main theoretical results.

Heteroclinic Trajectory and Hopf Bifurcation in an Extended Lorenz System

2018 ◽  
Vol 28 (09) ◽  
pp. 1850111
Author(s):  
Xianyi Li ◽  
Haijun Wang
Keyword(s):  
Hopf Bifurcation ◽  
Lyapunov Function ◽  
Numerical Simulations ◽  
Lorenz System ◽  
The Other ◽  
Hopf Bifurcations ◽  
Limit Sets ◽  
Initial Values ◽  
Heteroclinic Trajectory ◽  
The One

This note revisits an extended Lorenz system, which was presented in the paper entitled “Hopf bifurcations in an extended Lorenz system” by Zhou et al. [2017]. On the one hand, one points out and corrects some wrong results in that paper on the Hopf bifurcation at the symmetric equilibria [Formula: see text] and [Formula: see text]. On the other hand, combining Lyapunov function and the concepts of [Formula: see text]- and [Formula: see text]-limit sets, it is rigorously proved that there exists two and only two heteroclinic trajectories but no homoclinic trajectories under some certain conditions of its parameters and initial values. In addition, numerical simulations illustrate the consistence with the theoretical conclusions. The results together not only improve and complement the known ones, but also provide support in some future applications.

Ocean ecosystem discrete time dynamics generated by ℓ-Volterra operators

2019 ◽  
Vol 12 (02) ◽  
pp. 1950015 ◽  
Author(s):  
U. A. Rozikov ◽  
S. K. Shoyimardonov
Keyword(s):  
Dynamical System ◽  
Fixed Points ◽  
Discrete Time ◽  
Volterra Operator ◽  
Saddle Points ◽  
Time Dynamics ◽  
Discrete Time Dynamical System ◽  
Starting Point ◽  
Ocean Ecosystem ◽  
Initial Starting

We consider a discrete-time dynamical system generated by a nonlinear operator (with four real parameters [Formula: see text]) of ocean ecosystem. We find conditions on the parameters under which the operator is reduced to a [Formula: see text]-Volterra quadratic stochastic operator mapping two-dimensional simplex to itself. We show that if [Formula: see text], then (under some conditions on [Formula: see text]) this [Formula: see text]-Volterra operator may have up to three or a countable set of fixed points; if [Formula: see text], then the operator has up to three fixed points. Depending on the parameters, the fixed points may be attracting, repelling or saddle points. The limit behaviors of trajectories of the dynamical system are studied. It is shown that independently on values of parameters and on initial (starting) point, all trajectories converge. Thus, the operator (dynamical system) is regular. We give some biological interpretations of our results.

Feedback Anticontrol of Discrete Chaos

1998 ◽  
Vol 08 (07) ◽  
pp. 1585-1590 ◽  
Author(s):  
Guanrong Chen ◽  
Dejian Lai
Keyword(s):  
Dynamical System ◽  
Discrete Time ◽  
State Feedback ◽  
Design Method ◽  
Control Design ◽  
Rigorous Approach ◽  
Discrete Time Dynamical System ◽  
Discrete Chaos ◽  
Simple Feedback ◽  
Discrete Time Dynamical Systems

In this paper, a simple feedback control design method earlier proposed by us for discrete-time dynamical systems is proved to be a mathematically rigorous approach for anticontrol of chaos, in the sense that any given discrete-time dynamical system can be made chaotic by the designed state-feedback controller along with the mod-operations.

scholarly journals Global Dynamics of an SIQR Model with Vaccination and Elimination Hybrid Strategies

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 328 ◽  
Author(s):  
Yanli Ma ◽  
Jia-Bao Liu ◽  
Haixia Li
Keyword(s):  
Lyapunov Function ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Reproduction Number ◽  
Endemic Equilibrium ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free ◽  
The Basic Reproduction Number

In this paper, an SIQR (Susceptible, Infected, Quarantined, Recovered) epidemic model with vaccination, elimination, and quarantine hybrid strategies is proposed, and the dynamics of this model are analyzed by both theoretical and numerical means. Firstly, the basic reproduction number R 0 , which determines whether the disease is extinct or not, is derived. Secondly, by LaSalles invariance principle, it is proved that the disease-free equilibrium is globally asymptotically stable when R 0 < 1 , and the disease dies out. By Routh-Hurwitz criterion theory, we also prove that the disease-free equilibrium is unstable and the unique endemic equilibrium is locally asymptotically stable when R 0 > 1 . Thirdly, by constructing a suitable Lyapunov function, we obtain that the unique endemic equilibrium is globally asymptotically stable and the disease persists at this endemic equilibrium if it initially exists when R 0 > 1 . Finally, some numerical simulations are presented to illustrate the analysis results.

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