scholarly journals Mathematical Analysis and Stochastic Stability of Nonlinear Epidemic Model with Incidence Rate

2020 ◽  
pp. 8-19
Author(s):  
Laid Chahrazed
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability ◽  
Disease Free ◽  
Temporary Immunity

In this work, we consider a nonlinear epidemic model with temporary immunity and saturated incidence rate. Size N(t) at time t, is divided into three sub classes, with N(t)=S(t)+I(t)+Q(t); where S(t), I(t) and Q(t) denote the sizes of the population susceptible to disease, infectious and quarantine members with the possibility of infection through temporary immunity, respectively. We have made the following contributions: The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determine by the ratio called the basic reproductive number, This paper study the reduce model with replace S with N, which does not have non-trivial periodic orbits with conditions. The endemic -disease point is globally asymptotically stable if R0 ˃1; and study some proprieties of equilibrium with theorems under some conditions. Finally the stochastic stabilities with the proof of some theorems. In this work, we have used the different references cited in different studies and especially the writing of the non-linear epidemic mathematical model with [1-7]. We have used the other references for the study the different stability and other sections with [8-26]; and sometimes the previous references.

SVIR epidemic model with age structure in susceptibility, vaccination effects and relapse

2017 ◽  
Vol 82 (5) ◽  
pp. 945-970 ◽  
Author(s):  
Jinliang Wang ◽  
Min Guo ◽  
Shengqiang Liu
Keyword(s):  
Age Structure ◽  
Epidemic Model ◽  
Lyapunov Functional ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Combined Effects ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
The Stability ◽  
Disease Free

Abstract An SVIR epidemic model with continuous age structure in the susceptibility, vaccination effects and relapse is proposed. The asymptotic smoothness, existence of a global attractor, the stability of equilibria and persistence are addressed. It is shown that if the basic reproductive number $\Re_0<1$, then the disease-free equilibrium is globally asymptotically stable. If $\Re_0>1$, the disease is uniformly persistent, and a Lyapunov functional is used to show that the unique endemic equilibrium is globally asymptotically stable. Combined effects of susceptibility age, vaccination age and relapse age on the basic reproductive number are discussed.

scholarly journals Stochastic Stability and Analytical Solution with Homotopy Perturbation Method of Multicompartment Non-Linear Epidemic Model with Saturated Rate

2021 ◽  
pp. 149-457
Author(s):  
Laid Chahrazed
Keyword(s):  
Analytical Solution ◽  
Perturbation Method ◽  
Epidemic Model ◽  
Homotopy Perturbation Method ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Homotopy Perturbation ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
The Stability

In this work, we consider a nonlinear epidemic model with a saturated incidence rate. we consider a population of size N(t) at time t, this population is divided into six subclasses, with N(t)=S(t)+I(t)+I₁(t)+I₂(t)+I₃(t)+Q(t). Where S(t), I(t), I₁(t), I₂(t), I₃(t), and Q(t) denote the sizes of the population susceptible to disease, infectious members, and quarantine members, respectively. We have made the following contributions: 1. The local stabilities of the infection-free equilibrium and endemic equilibrium are; analyzed, respectively. The stability of a disease-free equilibrium and the existence of other nontrivial equilibria can be determined by the ratio called the basic reproductive number. 2. We find the analytical solution of the nonlinear epidemic model by Homotopy perturbation method. 3. Finally the stochastic stabilities. The study of its sections are justified with theorems and demonstrations under certain conditions. In this work, we have used the different references cited in different studies in the three sections already mentioned.

scholarly journals Global Stability of an SEIS Epidemic Model with General Saturation Incidence

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Hui Zhang ◽  
Li Yingqi ◽  
Wenxiong Xu
Keyword(s):  
Latent Period ◽  
Epidemic Model ◽  
Convergence Theorem ◽  
Incidence Rates ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Global Dynamics ◽  
Asymptotically Stable ◽  
Globally Asymptotically Stable ◽  
Disease Free

We present an SEIS epidemic model with infective force in both latent period and infected period, which has different general saturation incidence rates. It is shown that the global dynamics are completely determined by the basic reproductive number R0. If R0≤1, the disease-free equilibrium is globally asymptotically stable in T by LaSalle’s Invariance Principle, and the disease dies out. Moreover, using the method of autonomous convergence theorem, we obtain that the unique epidemic equilibrium is globally asymptotically stable in T0, and the disease spreads to be endemic.

Deterministic and stochastic stability of an SIRS epidemic model with a saturated incidence rate

2017 ◽  
Vol 25 (1) ◽  
Author(s):  
Modeste N’zi ◽  
Jacques Tano
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Deterministic Model ◽  
Random Noise ◽  
Stochastic Version ◽  
Noise Perturbation ◽  
Sirs Epidemic Model ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Disease Free

AbstractIn this paper, we formulate an epidemic model for the spread of an infectious disease in a population of varying size. The total population is divided into three distinct epidemiological subclass of individuals (susceptible, infectious and recovered) and we study a deterministic and stochastic models with saturated incidence rate. The stochastic model is obtained by incorporating a random noise into the deterministic model. In the deterministic case, we briefly discuss the global asymptotic stability of the disease free equilibrium by using a Lyapunov function. For the stochastic version, we study the global existence and positivity of the solution. Under suitable conditions on the intensity of the white noise perturbation, we prove that there are a

scholarly journals Global Stability for a Viral Infection Model with Saturated Incidence Rate

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Huaqin Peng ◽  
Zhiming Guo
Keyword(s):  
Viral Infection ◽  
Global Stability ◽  
Incidence Rate ◽  
Sufficient Conditions ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Infection Model ◽  
Saturated Incidence Rate ◽  
Saturated Incidence ◽  
Viral Infection Model

A viral infection model with saturated incidence rate and viral infection with delay is derived and analyzed; the incidence rate is assumed to be a specific nonlinear formβxv/(1+αv). The existence and uniqueness of equilibrium are proved. The basic reproductive numberR0is given. The model is divided into two cases: with or without delay. In each case, by constructing Lyapunov functionals, necessary and sufficient conditions are given to ensure the global stability of the models.

scholarly journals An SEIRS Epidemic Model with Immigration and Vertical Transmission

2020 ◽  
pp. 48-53
Author(s):  
Ruksana Shaikh ◽  
Pradeep Porwal ◽  
V. K. Gupta
Keyword(s):  
Vertical Transmission ◽  
Epidemic Model ◽  
Equilibrium Points ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Asymptotically Stable ◽  
Seirs Epidemic Model ◽  
Model Equations ◽  
The Stability ◽  
Disease Free

The study indicates that we should improve the model by introducing the immigration rate in the model to control the spread of disease. An SEIRS epidemic model with Immigration and Vertical Transmission and analyzed the steady state and stability of the equilibrium points. The model equations were solved analytically. The stability of the both equilibrium are proved by Routh-Hurwitz criteria. We see that if the basic reproductive number R0<1 then the disease free equilibrium is locally asymptotically stable and if R0<1 the endemic equilibrium will be locally asymptotically stable.

scholarly journals Dynamical Analysis of a Delayed HIV Virus Dynamic Model with Cell-to-Cell Transmission and Apoptosis of Bystander Cells

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Tongqian Zhang ◽  
Junling Wang ◽  
Yi Song ◽  
Zhichao Jiang
Keyword(s):  
Time Delay ◽  
Basic Reproductive Number ◽  
Dynamical Analysis ◽  
Reproductive Number ◽  
Globally Asymptotically Stable ◽  
Local Hopf Bifurcation ◽  
Hiv Virus ◽  
Bystander Cells ◽  
The Stability ◽  
Disease Free

In this paper, a delayed viral dynamical model that considers two different transmission methods of the virus and apoptosis of bystander cells is proposed and investigated. The basic reproductive number R0 of the model is derived. Based on the basic reproductive number, we prove that the disease-free equilibrium E0 is globally asymptotically stable for R0<1 by constructing suitable Lyapunov functional. For R0>1, by regarding the time delay as bifurcation parameter, the existence of local Hopf bifurcation is investigated. The results show that time delay can change the stability of endemic equilibrium and cause periodic oscillations. Finally, we give some numerical simulations to illustrate the theoretical findings.

Sign in / Sign up

Export Citation Format

Share Document