scholarly journals Global Stability of Nonlinear Stochastic SEI Epidemic Model with Fluctuations in Transmission Rate of Disease

2017 ◽  
Vol 2017 ◽  
pp. 1-7 ◽  
Author(s):  
Olusegun Michael Otunuga
Keyword(s):  
Epidemic Model ◽  
Transmission Rate ◽  
External Noise ◽  
Basic Reproductive Number ◽  
Disease Spread ◽  
Reproductive Number ◽  
Stochastic Version ◽  
Threshold Criterion ◽  
Diffusion Rates ◽  
The Stability

We derive and analyze the dynamic of a stochastic SEI epidemic model for disease spread. Fluctuations in the transmission rate of the disease bring about stochasticity in model. We discuss the asymptotic stability of the infection-free equilibrium by first deriving the closed form deterministic (R0) and stochastic (R0) basic reproductive number. Contrary to some author’s remark that different diffusion rates have no effect on the stability of the disease-free equilibrium, we showed that even if no epidemic invasion occurs with respect to the deterministic version of the SEI model (i.e., R0<1), epidemic can still grow initially (if R0>1) because of the presence of noise in the stochastic version of the model. That is, diffusion rates can have effect on the stability by causing a transient epidemic advance. A threshold criterion for epidemic invasion was derived in the presence of external noise.

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 $\Re_0&lt;1$, then the disease-free equilibrium is globally asymptotically stable. If $\Re_0&gt;1$, 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 Analisis Kestabilan Model Penyebaran Penyakit Toxoplasmosis pada Populasi Manusia dan Kucing

2019 ◽  
Vol 1 (3) ◽  
pp. 37-46
Author(s):  
Febrina Tedjo Utami ◽  
Retno Wahyu Dewanti ◽  
Subchan Subchan
Keyword(s):  
Stability Analysis ◽  
Epidemic Model ◽  
Human Population ◽  
Compartment Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Final Project ◽  
Spread Model ◽  
The Stability ◽  
Models Of Disease

This final project studies the contruction and analyzes the stability of Toxoplasmosis epidemic model. Using a compartment model, Toxoplasmosis epidemic model classified into human population and cat population. The human population is divided into three subpopulations such as susceptible human, infected human, and recovered human. The cat population is divided into two subpopulations such as susceptible cat and infected cat. Stability analysis performed on two equilibrium models of disease free equilibrium and endemic equilibrium point. Stability analysis result showed that cat infected spread, natural birthd rate of cat population, natural death rate of cat population, and the probability of cat natality in the form of basic reproductive number (R0) is value that affect human population and cat population changed spread. When R0<1 the population will be free of disease and when R0>1 the disease will be spread. Model of the simulation is numerically analyzed by Runge Kutta 4th Order methods.

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.

A reliable numerical analysis for stochastic gonorrhea epidemic model with treatment effect

2019 ◽  
Vol 12 (06) ◽  
pp. 1950072 ◽  
Author(s):  
Ali Raza ◽  
Muhammad Shoaib Arif ◽  
Muhammad Rafiq
Keyword(s):  
Treatment Effect ◽  
Epidemic Model ◽  
Numerical Scheme ◽  
Deterministic Model ◽  
Basic Reproductive Number ◽  
Disease Spread ◽  
Reproductive Number ◽  
Step Size ◽  
Realistic Approach ◽  
Stochastic Setting

The phenomena of disease spread are unpredictable in nature due to random mixing of individuals in a population. It is of more significance to include this randomness while modeling infectious diseases. Modeling epidemics including their stochastic behavior could be a more realistic approach in many situations. In this paper, a stochastic gonorrhea epidemic model with treatment effect has been analyzed numerically. Numerical solution of stochastic model is presented in comparison with its deterministic part. The dynamics of the gonorrhea disease is governed by a threshold quantity [Formula: see text] called basic reproductive number. If [Formula: see text], then disease eventually dies out while [Formula: see text] shows the persistence of disease in population. The standard numerical schemes like Euler Maruyama, stochastic Euler and stochastic Runge–Kutta are highly dependent on step size and do not behave well in certain scenarios. A competitive non-standard finite difference numerical scheme in stochastic setting is proposed, which is independent of step size and remains consistent with the corresponding deterministic model.

scholarly journals Stability and Hopf Bifurcation of a Vector-Borne Disease Model with Saturated Infection Rate and Reinfection

2019 ◽  
Vol 2019 ◽  
pp. 1-17 ◽  
Author(s):  
Zhixing Hu ◽  
Shanshan Yin ◽  
Hui Wang
Keyword(s):  
Hopf Bifurcation ◽  
Infection Rate ◽  
Disease Model ◽  
Endemic Equilibrium ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Cure Rate ◽  
Vector Borne Disease ◽  
Vector Borne ◽  
The Stability

This paper established a delayed vector-borne disease model with saturated infection rate and cure rate. First of all, according to the basic reproductive number R0, we determined the disease-free equilibrium E0 and the endemic equilibrium E1. Through the analysis of the characteristic equation, we consider the stability of two equilibriums. Furthermore, the effect on the stability of the endemic equilibrium E1 by delay was studied, the existence of Hopf bifurcations of this system in E1 was analyzed, and the length of delay to preserve stability was estimated. The direction and stability of the Hopf bifurcation were also been determined. Finally, we performed some numerical simulation to illustrate our main results.

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.

THE DYNAMICS OF THE CONSTANT AND PULSE BIRTH IN AN SIR EPIDEMIC MODEL WITH CONSTANT RECRUITMENT

2007 ◽  
Vol 15 (02) ◽  
pp. 203-218 ◽  
Author(s):  
WENJUN CAO ◽  
ZHEN JIN
Keyword(s):  
Epidemic Model ◽  
Disease Model ◽  
Basic Reproductive Number ◽  
Reproductive Number ◽  
Free Solution ◽  
Sir Epidemic Model ◽  
Sir Epidemic ◽  
Stable Period ◽  
Number Formula ◽  
Globally Stable

In this paper, an SIR epidemic model with constant recruitment is considered. The dynamic behavior of this disease model with constant and pulse birth are analyzed. With constant birth, the infection-free equilibrium is locally and globally stable when the basic reproductive number R0 < 1. However, with pulse birth the system converges to a stable period solution with the number of infectious individuals equal to zero. Furthermore, the local and global stability of the periodic infection-free solution is obtained if the basic reproductive number [Formula: see text]. Numerical simulation shows that the periodic infection-free solution is unstable and the disease will persist when [Formula: see text]. The effectiveness of the constant and pulse birth to eliminating the disease are compared.

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