scholarly journals Stability and Bifurcations in a Discrete-Time Epidemic Model with Vaccination and Vital Dynamics

2020 ◽  
Author(s):  
Mahmood Parsamanesh ◽  
Majid Erfanian ◽  
Saeed Mehrshad
Keyword(s):  
Discrete Time ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Euler Method ◽  
Continuous Version ◽  
Sis Epidemic Model ◽  
Intervention Measures ◽  
The Stability ◽  
Discretized Model

Abstract BackgroundThe spread of infectious diseases is such important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and investigate under which conditions it will be wiped out or continued.ResultsA discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number $\mathcal{R}_0$ . Then the stability of the equilibria is given in terms of $\mathcal{R}_0$ , and moreover, the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of model, a discretized model is obtained and analyzed. ConclusionIt is proved that the disease-free equilibrium and endemic equilibrium are stable if $\mathcal{R}_0<1$ and $\mathcal{R}_0>1$ , respectively. The system has a transcritical bifurcation when $\mathcal{R}_0=1$ and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

scholarly journals Stability and bifurcations in a discrete-time epidemic model with vaccination and vital dynamics

2020 ◽  
Vol 21 (1) ◽  
Author(s):  
Mahmood Parsamanesh ◽  
Majid Erfanian ◽  
Saeed Mehrshad
Keyword(s):  
Discrete Time ◽  
Epidemic Model ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
Euler Method ◽  
Continuous Version ◽  
Sis Epidemic Model ◽  
The Stability ◽  
Disease Free ◽  
Discretized Model

Abstract Background The spread of infectious diseases is so important that changes the demography of the population. Therefore, prevention and intervention measures are essential to control and eliminate the disease. Among the drug and non-drug interventions, vaccination is a powerful strategy to preserve the population from infection. Mathematical models are useful to study the behavior of an infection when it enters a population and to investigate under which conditions it will be wiped out or continued. Results A discrete-time SIS epidemic model is introduced that includes a vaccination program. Some basic properties of this model are obtained; such as the equilibria and the basic reproduction number $$\mathcal {R}_0$$ R 0 . Then the stability of the equilibria is given in terms of $$\mathcal {R}_0$$ R 0 , and the bifurcations of the model are studied. By applying the forward Euler method on the continuous version of the model, a discretized model is obtained and analyzed. Conclusion It is proven that the disease-free equilibrium and endemic equilibrium are stable if $$\mathcal {R}_0<1$$ R 0 < 1 and $$\mathcal {R}_0>1$$ R 0 > 1 , respectively. Also, the disease-free equilibrium is globally stable when $$\mathcal {R}_0\le 1$$ R 0 ≤ 1 . The system has a transcritical bifurcation when $$\mathcal {R}_0=1$$ R 0 = 1 and it might also have period-doubling bifurcation. The sufficient conditions for the stability of equilibria in the discretized model are established. The numerical discussions verify the theoretical results.

Disease extinction and persistence in a discrete-time sis epidemic model with vaccination and varying population size

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4735-4747 ◽  
Author(s):  
Rahman Farnoosh ◽  
Mahmood Parsamanesh
Keyword(s):  
Population Size ◽  
Discrete Time ◽  
Epidemic Model ◽  
Jacobian Matrix ◽  
Sufficient Conditions ◽  
Period Doubling ◽  
System Of Difference Equations ◽  
Sis Epidemic Model ◽  
Necessary And Sufficient ◽  
Sis Epidemic

A discrete-time SIS epidemic model with vaccination is introduced and formulated by a system of difference equations. Some necessary and sufficient conditions for asymptotic stability of the equilibria are obtained. Furthermore, a sufficient condition is also presented. Next, bifurcations of the model including transcritical bifurcation, period-doubling bifurcation, and the Neimark-Sacker bifurcation are considered. In addition, these issues will be studied for the corresponding model with constant population size. Dynamics of the model are also studied and compared in detail with those found theoretically by using bifurcation diagrams, analysis of eigenvalues of the Jacobian matrix, Lyapunov exponents and solutions of the models in some examples.

scholarly journals Stability of the equilibria in a discrete-time sivs epidemic model with standard incidence

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2393-2408 ◽  
Author(s):  
Mahmood Parsamanesh ◽  
Saeed Mehrshad
Keyword(s):  
Discrete Time ◽  
Basic Reproduction Number ◽  
Epidemic Model ◽  
Total Population ◽  
Reproduction Number ◽  
Flip Bifurcation ◽  
Sis Epidemic Model ◽  
Fold Bifurcation ◽  
The Stability ◽  
Conditions Of Stability

A discrete-time SIS epidemic model with vaccination is presented and studied. The model includes deaths due to disease and the total population size is variable. First, existence and positivity of the solutions are discussed and equilibria of the model and basic reproduction number are obtained. Next, the stability of the equilibria is studied and conditions of stability are obtained in terms of the basic reproduction number R0. Also, occurrence of the fold bifurcation, the flip bifurcation, and the Neimark-Sacker bifurcation is investigated at equilibria. In addition, obtained results are numerically discussed and some diagrams for bifurcations, Lyapunov exponents, and solutions of the model are presented.

scholarly journals On the Stability of the Endemic Equilibrium of A Discrete-Time Networked Epidemic Model

2020 ◽  
Vol 53 (2) ◽  
pp. 2576-2581
Author(s):  
Fangzhou Liu ◽  
Shaoxuan CUI ◽  
Xianwei Li ◽  
Martin Buss
Keyword(s):  
Discrete Time ◽  
Epidemic Model ◽  
Endemic Equilibrium ◽  
The Stability

Stability and Hopf Bifurcation in a Delayed SIS Epidemic Model with Double Epidemic Hypothesis

2018 ◽  
Vol 19 (6) ◽  
pp. 561-571
Author(s):  
Jiangang Zhang ◽  
Yandong Chu ◽  
Wenju Du ◽  
Yingxiang Chang ◽  
Xinlei An
Keyword(s):  
Hopf Bifurcation ◽  
Epidemic Model ◽  
Positive Equilibrium ◽  
Normal Form Theory ◽  
Center Manifold Theorem ◽  
Characteristic Roots ◽  
Sis Epidemic Model ◽  
Form Theory ◽  
The Stability ◽  
Sis Epidemic

AbstractThe stability and Hopf bifurcation of a delayed SIS epidemic model with double epidemic hypothesis are investigated in this paper. We first study the stability of the unique positive equilibrium of the model in four cases, and we obtain the stability conditions through analyzing the distribution of characteristic roots of the corresponding linearized system. Moreover, we choosing the delay as bifurcation parameter and the existence of Hopf bifurcation is investigated in detail. We can derive explicit formulas for determining the direction of the Hopf bifurcation and the stability of bifurcation periodic solution by center manifold theorem and normal form theory. Finally, we perform the numerical simulations for justifying the theoretical results.

scholarly journals Stability Analysis for Discrete-Time Stochastic Fuzzy Neural Networks with Mixed Delays

2019 ◽  
Vol 2019 ◽  
pp. 1-13
Author(s):  
YaJun Li ◽  
Quanxin Zhu
Keyword(s):  
Neural Networks ◽  
Discrete Time ◽  
Sufficient Conditions ◽  
Fuzzy Neural Networks ◽  
Linear Matrix ◽  
Mixed Delays ◽  
The Novel ◽  
Free Weight ◽  
Fuzzy Neural ◽  
The Stability

This paper is concerned with the stability problem of a class of discrete-time stochastic fuzzy neural networks with mixed delays. New Lyapunov-Krasovskii functions are proposed and free weight matrices are introduced. The novel sufficient conditions for the stability of discrete-time stochastic fuzzy neural networks with mixed delays are established in terms of linear matrix inequalities (LMIs). Finally, numerical examples are given to illustrate the effectiveness and benefits of the proposed method.

Stability analysis for discrete-time stochastic neural networks with mixed time delays and impulsive effects

2010 ◽  
Vol 88 (12) ◽  
pp. 885-898 ◽  
Author(s):  
R. Raja ◽  
R. Sakthivel ◽  
S. Marshal Anthoni
Keyword(s):  
Neural Networks ◽  
Equilibrium Point ◽  
Discrete Time ◽  
Time Delays ◽  
Sufficient Conditions ◽  
Impulsive Effects ◽  
Linear Matrix ◽  
Stochastic Neural Networks ◽  
Mixed Time Delays ◽  
The Stability

This paper investigates the stability issues for a class of discrete-time stochastic neural networks with mixed time delays and impulsive effects. By constructing a new Lyapunov–Krasovskii functional and combining with the linear matrix inequality (LMI) approach, a novel set of sufficient conditions are derived to ensure the global asymptotic stability of the equilibrium point for the addressed discrete-time neural networks. Then the result is extended to address the problem of robust stability of uncertain discrete-time stochastic neural networks with impulsive effects. One important feature in this paper is that the stability of the equilibrium point is proved under mild conditions on the activation functions, and it is not required to be differentiable or strictly monotonic. In addition, two numerical examples are provided to show the effectiveness of the proposed method, while being less conservative.

Switching Control for a Class of Discrete-Time Switched Fuzzy Systems Based on LMI

2014 ◽  
Vol 945-949 ◽  
pp. 2543-2546
Author(s):  
Hong Yang ◽  
Huan Huan Lü ◽  
Le Zhang
Keyword(s):  
Discrete Time ◽  
Fuzzy Systems ◽  
Sufficient Conditions ◽  
Switching Control ◽  
Linear Matrix ◽  
Common Lyapunov Function ◽  
Lyapunov Function Method ◽  
The Common ◽  
The Stability ◽  
Stability Issues

Switching control and stability issues for discrete-time switched systems whose subsystems are all discrete-time fuzzy systems are studied and new results derived. Innovated representation models for switched fuzzy systems are proposed. The common Lyapunov function method has been adopted to study the stability of this class of switched fuzzy systems. Sufficient conditions for asymptotic stability are presented. The main conditions are given in form of linear matrix inequalities (LMIs), which are easily solvable. The elaborated illustrative examples and the respective simulation experiments demonstrate the effectiveness of the proposed method.

scholarly journals Analysis of a Deterministic and a Stochastic SIS Epidemic Model with Double Epidemic Hypothesis and Specific Functional Response

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Mouhcine Naim ◽  
Fouad Lahmidi
Keyword(s):  
Epidemic Model ◽  
Deterministic Model ◽  
Sufficient Condition ◽  
Nonlinear Incidence Rate ◽  
Local Asymptotic Stability ◽  
Sis Epidemic Model ◽  
The Stability ◽  
Stochastic Sis Epidemic Model ◽  
Disease Free ◽  
Sis Epidemic

The purpose of this paper is to investigate the stability of a deterministic and stochastic SIS epidemic model with double epidemic hypothesis and specific nonlinear incidence rate. We prove the local asymptotic stability of the equilibria of the deterministic model. Moreover, by constructing a suitable Lyapunov function, we obtain a sufficient condition for the global stability of the disease-free equilibrium. For the stochastic model, we establish global existence and positivity of the solution. Thereafter, stochastic stability of the disease-free equilibrium in almost sure exponential and pth moment exponential is investigated. Finally, numerical examples are presented.

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