Equilibriums of an SIS Epidemic Model

Epidemic Model ◽

Death Rate ◽

Birth Rate ◽

Reproduction Number ◽

Nonlinear Incidence Rate ◽

Sis Epidemic Model ◽

Disease Free ◽

Sis Epidemic ◽

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.

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):

Epidemic Model ◽

Reproduction Number ◽

Global Solutions ◽

Uhlenbeck Process ◽

Sis Epidemic Model ◽

Ornstein Uhlenbeck Process ◽

Brown Motion ◽

Sis Epidemic ◽

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.

Dynamics of Stochastically Perturbed SIS Epidemic Model with Vaccination

Abstract and Applied Analysis ◽

10.1155/2013/517439 ◽

2013 ◽

Vol 2013 ◽

pp. 1-12 ◽

Cited By ~ 5

Author(s):

Yanan Zhao ◽

Daqing Jiang

Keyword(s):

Epidemic Model ◽

Death Rate ◽

Deterministic Model ◽

Reproduction Number ◽

Dynamical Behavior ◽

Standard Technique ◽

Deterministic Models ◽

Sis Epidemic Model ◽

Disease Free ◽

Sis Epidemic

We introduce stochasticity into an SIS epidemic model with vaccination. The stochasticity in the model is a standard technique in stochastic population modeling. In the deterministic models, the basic reproduction numberR0is a threshold which determines the persistence or extinction of the disease. When the perturbation and the disease-related death rate are small, we carry out a detailed analysis on the dynamical behavior of the stochastic model, also regarding of the value ofR0. IfR0≤1, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model, whereas, ifR0>1, there is a stationary distribution, which means that the disease will prevail. The results are illustrated by computer simulations.

Simple MSEIR Model for Measles Transmission

Asian Research Journal of Mathematics ◽

10.9734/arjom/2019/v12i330090 ◽

2019 ◽

pp. 1-11

Author(s):

Mojeeb Al-Rahman EL-Nor Osman ◽

Appiagyei Ebenezer ◽

Isaac Kwasi Adu

Keyword(s):

Mathematical Model ◽

Recovery Rate ◽

Birth Rate ◽

Reproduction Number ◽

Progression Rate ◽

Asymptotically Stable ◽

The Stability ◽

Disease Free ◽

In this paper, an Immunity-Susceptible-Exposed-Infectious-Recovery (MSEIR) mathematical model was used to study the dynamics of measles transmission. We discussed that there exist a disease-free and an endemic equilibria. We also discussed the stability of both disease-free and endemic equilibria. The basic reproduction number is obtained. If , then the measles will spread and persist in the population. If , then the disease will die out. The disease was locally asymptotically stable if and unstable if . ALSO, WE PROVED THE GLOBAL STABILITY FOR THE DISEASE-FREE EQUILIBRIUM USING LASSALLE'S INVARIANCE PRINCIPLE OF Lyaponuv function. Furthermore, the endemic equilibrium was locally asymptotically stable if , under certain conditions. Numerical simulations were conducted to confirm our analytic results. Our findings were that, increasing the birth rate of humans, decreasing the progression rate, increasing the recovery rate and reducing the infectious rate can be useful in controlling and combating the measles.

On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation

Applied Sciences ◽

10.3390/app10228296 ◽

2020 ◽

Vol 10 (22) ◽

pp. 8296 ◽

Cited By ~ 1

Author(s):

Malen Etxeberria-Etxaniz ◽

Santiago Alonso-Quesada ◽

Manuel De la Sen

Keyword(s):

Equilibrium Point ◽

Epidemic Model ◽

Disease Transmission ◽

Reproduction Number ◽

Equilibrium Points ◽

Endemic Equilibrium ◽

Impulsive Vaccination ◽

Disease Free ◽

This paper investigates a susceptible-exposed-infectious-recovered (SEIR) epidemic model with demography under two vaccination effort strategies. Firstly, the model is investigated under vaccination of newborns, which is fact in a direct action on the recruitment level of the model. Secondly, it is investigated under a periodic impulsive vaccination on the susceptible in the sense that the vaccination impulses are concentrated in practice in very short time intervals around a set of impulsive time instants subject to constant inter-vaccination periods. Both strategies can be adapted, if desired, to the time-varying levels of susceptible in the sense that the control efforts be increased as those susceptible levels increase. The model is discussed in terms of suitable properties like the positivity of the solutions, the existence and allocation of equilibrium points, and stability concerns related to the values of the basic reproduction number. It is proven that the basic reproduction number lies below unity, so that the disease-free equilibrium point is asymptotically stable for larger values of the disease transmission rates under vaccination controls compared to the case of absence of vaccination. It is also proven that the endemic equilibrium point is not reachable if the disease-free one is stable and that the disease-free equilibrium point is unstable if the reproduction number exceeds unity while the endemic equilibrium point is stable. Several numerical results are investigated for both vaccination rules with the option of adapting through ime the corresponding efforts to the levels of susceptibility. Such simulation examples are performed under parameterizations related to the current SARS-COVID 19 pandemic.

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

Mathematics ◽

10.3390/math6120328 ◽

2018 ◽

Vol 6 (12) ◽

pp. 328 ◽

Cited By ~ 3

Author(s):

Yanli Ma ◽

Jia-Bao Liu ◽

Haixia Li

Keyword(s):

Lyapunov Function ◽

Epidemic Model ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Global Dynamics ◽

Asymptotically Stable ◽

Globally Asymptotically Stable ◽

Disease Free ◽

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.

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

Discrete Dynamics in Nature and Society ◽

10.1155/2020/5362716 ◽

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.

Analysis of COVID-19 based on SEIR epidemic models in a multi-patch environment

10.21203/rs.3.rs-306960/v1 ◽

2021 ◽

Author(s):

Lan Meng ◽

Wei Zhu

Keyword(s):

Numerical Simulations ◽

Dynamic Behavior ◽

Epidemic Model ◽

Migration Rate ◽

Reproduction Number ◽

Population Migration ◽

Theoretical Results ◽

Disease Free ◽

Abstract In this paper, an n-patch SEIR epidemic model for the coronavirus disease 2019 (COVID-19) is presented. It is shown that there is unique disease-free equilibrium for this model. Then, the dynamic behavior is studied by the basic reproduction number. Some numerical simulations with three patches are given to validate the effectiveness of the theoretical results. The influence of quarantined rate and population migration rate on the basic reproduction number is also discussed by simulation.

Bifurcation in Disease Dynamics with Latent Period of Infection and Media Awareness

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127416500978 ◽

2016 ◽

Vol 26 (06) ◽

pp. 1650097 ◽

Cited By ~ 4

Author(s):

Harkaran Singh ◽

Joydip Dhar ◽

Harbax Singh Bhatti

Keyword(s):

Latent Period ◽

Reproduction Number ◽

Endemic Equilibrium ◽

State Variables ◽

Sis Epidemic Model ◽

Number Formula ◽

Sensitive Parameters ◽

Disease Free ◽

In the present study, an SIS epidemic model with a latent period of infection and media awareness as control strategy is proposed. The asymptotic stability of the model is studied for both disease-free equilibrium and endemic equilibrium states with respect to the basic reproduction number [Formula: see text]. It is observed that the coefficient of media awareness [Formula: see text] does not affect [Formula: see text], but significantly affects the level of endemic equilibrium. Further, the specific conditions for the existence of Hopf bifurcation have been obtained for the endemic equilibrium state. We also performed the sensitivity analysis of the basic reproduction number and state variables at endemic steady state with respect to the model parameter and identified the respective sensitive parameters. Numerical simulations have been presented in support of our analytic findings.

THRESHOLD DYNAMICS IN A PERIODIC SVEIR EPIDEMIC MODEL

International Journal of Biomathematics ◽

10.1142/s1793524511001490 ◽

2011 ◽

Vol 04 (04) ◽

pp. 493-509 ◽

Cited By ~ 7

Author(s):

JINLIANG WANG ◽

SHENGQIANG LIU ◽

YASUHIRO TAKEUCHI

Keyword(s):

Epidemic Model ◽

Reproduction Number ◽

Dynamical Behavior ◽

Threshold Dynamics ◽

Globally Asymptotically Stable ◽

Infectious Risk ◽

Persistence Theory ◽

Disease Free ◽

In this paper, we investigate the dynamical behavior of a class of periodic SVEIR epidemic model. Since the nonautonomous phenomenon often occurs as cyclic pattern, our model is then a periodic time-dependent system. It follows from persistence theory that the basic reproduction number is the threshold parameter above which the disease is uniformly persistent and below which disease-free periodic solution is globally asymptotically stable. The threshold dynamics extends the classic results for the corresponding autonomous model. Furthermore, we show that eradication policy on the basis of the basic reproduction number of the autonomous system may overestimate the infectious risk when the disease follows periodic behavior. The according simulation results are also given.

Global stability and vaccination of an SEIVR epidemic model with saturated incidence rate

International Journal of Biomathematics ◽

10.1142/s1793524516500686 ◽

2016 ◽

Vol 09 (05) ◽

pp. 1650068 ◽

Cited By ~ 6

Author(s):

Muhammad Altaf Khan ◽

Yasir Khan ◽

Sehra Khan ◽

Saeed Islam

Keyword(s):

Global Stability ◽

Epidemic Model ◽

Reproduction Number ◽

Endemic Equilibrium ◽

Asymptotically Stable ◽

Saturated Incidence Rate ◽

Saturated Incidence ◽

Disease Free ◽

This study considers SEIVR epidemic model with generalized nonlinear saturated incidence rate in the host population horizontally to estimate local and global equilibriums. By using the Routh–Hurwitz criteria, it is shown that if the basic reproduction number [Formula: see text], the disease-free equilibrium is locally asymptotically stable. When the basic reproduction number exceeds the unity, then the endemic equilibrium exists and is stable locally asymptotically. The system is globally asymptotically stable about the disease-free equilibrium if [Formula: see text]. The geometric approach is used to present the global stability of the endemic equilibrium. For [Formula: see text], the endemic equilibrium is stable globally asymptotically. Finally, the numerical results are presented to justify the mathematical results.