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

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.

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Stability Analysis of a Delayed SIR Epidemic Model with Stage Structure and Nonlinear Incidence

Discrete Dynamics in Nature and Society ◽

10.1155/2009/979217 ◽

2009 ◽

Vol 2009 ◽

pp. 1-17 ◽

Cited By ~ 1

Author(s):

Xiaohong Tian ◽

Rui Xu

Keyword(s):

Basic Reproduction Number ◽

Epidemic Model ◽

Reproduction Number ◽

Sufficient Conditions ◽

Stage Structure ◽

Sir Epidemic Model ◽

Globally Asymptotically Stable ◽

Sir Epidemic ◽

The Stability ◽

The Basic Reproduction Number

We investigate the stability of an SIR epidemic model with stage structure and time delay. By analyzing the eigenvalues of the corresponding characteristic equation, the local stability of each feasible equilibrium of the model is established. By using comparison arguments, it is proved when the basic reproduction number is less than unity, the disease free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, sufficient conditions are derived for the global stability of an endemic equilibrium of the model. Numerical simulations are carried out to illustrate the theoretical results.

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A Stochastic SIS Epidemic Model with Ornstein-Uhlenbeck Process and Brown Motion

10.21203/rs.3.rs-679213/v1 ◽

2021 ◽

Author(s):

Xingzhi Chen ◽

Baodan Tian ◽

Xin Xu ◽

Ruoxi Yang ◽

Shouming Zhong

Keyword(s):

Basic Reproduction Number ◽

Epidemic Model ◽

Reproduction Number ◽

Global Solutions ◽

Uhlenbeck Process ◽

Sis Epidemic Model ◽

Ornstein Uhlenbeck Process ◽

Brown Motion ◽

Sis Epidemic ◽

The Basic Reproduction Number

Abstract This paper studies a stochastic differential equation SIS epidemic model, disturbed randomly by the mean-reverting Ornstein-Uhlenbeck process and Brownian motion. We prove the existence and uniqueness of the positive global solutions of the model and obtain the controlling conditions for the extinction and persistence of the disease. The results show that when the basic reproduction number Rs0 < 1, the disease will extinct, on the contrary, when the basic reproduction number Rs0 > 1, the disease will persist. Furthermore, we can inhibit the outbreak of the disease by increasing the intensity of volatility or decreasing the speed of reversion ϑ, respectively. Finally, we give some numerical examples to verify these results.

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Analisis dinamik model SVEIR pada penyebaran penyakit campak

Jambura Journal of Biomathematics (JJBM) ◽

10.34312/jjbm.v1i2.8482 ◽

2020 ◽

Vol 1 (2) ◽

pp. 57-64

Author(s):

Sitty Oriza Sativa Putri Ahaya ◽

Emli Rahmi ◽

Nurwan Nurwan

Keyword(s):

Equilibrium Point ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Total Population ◽

Reproduction Number ◽

Equilibrium Points ◽

Transmission Model ◽

Asymptotically Stable ◽

Disease Free ◽

The Basic Reproduction Number

In this article, we analyze the dynamics of measles transmission model with vaccination via an SVEIR epidemic model. The total population is divided into five compartments, namely the Susceptible, Vaccinated, Exposed, Infected, and Recovered populations. Firstly, we determine the equilibrium points and their local asymptotically stability properties presented by the basic reproduction number R0. It is found that the disease free equilibrium point is locally asymptotically stable if satisfies R01 and the endemic equilibrium point is locally asymptotically stable when R01. We also show the existence of forward bifurcation driven by some parameters that influence the basic reproduction number R0 i.e., the infection rate α or proportion of vaccinated individuals θ. Lastly, some numerical simulations are performed to support our analytical results.

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

BMC Bioinformatics ◽

10.1186/s12859-020-03839-1 ◽

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.

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

10.21203/rs.3.rs-23409/v1 ◽

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.

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Equilibriums of an SIS Epidemic Model

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.678.103 ◽

2014 ◽

Vol 678 ◽

pp. 103-106

Author(s):

Jing Hai Wang

Keyword(s):

Basic Reproduction Number ◽

Epidemic Model ◽

Death Rate ◽

Birth Rate ◽

Reproduction Number ◽

Nonlinear Incidence Rate ◽

Sis Epidemic Model ◽

Disease Free ◽

Sis Epidemic ◽

The Basic Reproduction Number

An SIS epidemic model with nonlinear incidence rate is considered. At the same time we decide that the birth rate is equal to death rate. We get the basic reproduction number . We analyze the existences of the equilibriums of the system. There are some endemic equilibriums and one disease-free equilibrium of the system.

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Discrete-Time Fractional Order SIR Epidemic Model with Saturated Treatment Function

International Journal of Nonlinear Sciences and Numerical Simulation ◽

10.1515/ijnsns-2019-0068 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Mahmoud A. M. Abdelaziz ◽

Ahmad Izani Ismail ◽

Farah A. Abdullah ◽

Mohd Hafiz Mohd

Keyword(s):

Numerical Simulations ◽

Discrete Time ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Reproduction Number ◽

Sir Epidemic Model ◽

Delayed Treatment ◽

Sir Epidemic ◽

Treatment Function ◽

Saturated Treatment

AbstractIn this paper, a discrete-time fractional-order SIR epidemic model with saturated treatment function is investigated. The local asymptotic stability of the equilibrium points is analyzed and the threshold condition basic reproduction number is derived. Backward bifurcation is shown when the model possesses a stable disease-free equilibrium point and a stable endemic point coexisting together when the basic reproduction number is less than unity. It is also shown that when the treatment is partially effective, a transcritical bifurcation occurs at $\Re_{0}=1$ and reappears again when the effect of delayed treatment is getting stronger at $\Re_{0}<1$. The analysis of backward and forward bifurcations associated with the transcritical, saddle-node, period-doubling and Neimark–Sacker bifurcations are discussed. Numerical simulations are carried out to illustrate the complex dynamical behaviors of the model. By carrying out bifurcation analysis, it is shown that the delayed treatment parameter ε should be less than two critical values ε1 and ε2 so as to avoid $\Re_{0}$ belonging to the dangerous range $\left[ \Re_{0},1\right]$. The results of the numerical simulations support the theoretical analysis.

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DYNAMICS OF COMPETING STRAINS WITH SATURATED INFECTIVITY AND MUTATION ON NETWORKS

Journal of Biological System ◽

10.1142/s0218339016500133 ◽

2016 ◽

Vol 24 (02n03) ◽

pp. 257-273 ◽

Cited By ~ 3

Author(s):

QINGCHU WU ◽

XINCHU FU

Keyword(s):

Dynamical Systems ◽

Complex Network ◽

Basic Reproduction Number ◽

Epidemic Model ◽

Reproduction Number ◽

Dynamical Behavior ◽

Critical Values ◽

Epidemic Threshold ◽

Persistence Theory ◽

The Stability

This paper deals with the dynamical behavior of a two-strain epidemic model in a complex network with saturated infectivity and mutation. The model is shown to exhibit the phenomena of strain coexistence where two epidemic strains co-exist and strain replacement where the strain with smaller basic reproduction number can become predominant. By using the stability and persistence theory of dynamical systems, we calculate the epidemic threshold, and show that above which at least one strain persists in a population, and also obtain the critical values to discriminate the strain coexistence and replacement. The results suggest that the newly mutated strain can spread in the population, even if it has a very small transmissivity.

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