scholarly journals Bifurcation analysis of an SIR epidemic model with saturated treatment function

2021 ◽  
Author(s):  
Phuc Ngo
Keyword(s):  
Epidemic Models ◽  
Stable Node ◽  
Sir Epidemic Model ◽  
Treatment Rate ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
Treatment Function ◽  
Sir Epidemic Models ◽  
Saturated Treatment ◽  
Disease Free

In this thesis we investigate the dynamics and bifurcation of SIR epidemic models with horizontal and vertical transmissions and saturated treatment rate. It is proved that such SIR epidemic models always have positive disease free equilibria and also have three positive epidemic equilibria. The ranges of the parameters related in the model were found under which the equilibria of the models are positive. By applying the qualitative theory of planar systems, it is shown the disease free equilibria is a saddle, stable node and globally asymptotically stable. Furthermore, it is also shown that the interior equilibria are saddle, saddle node or saddle point.

scholarly journals Bifurcation analysis of an SIR epidemic model with saturated treatment function

2021 ◽  
Author(s):  
Phuc Ngo
Keyword(s):  
Epidemic Models ◽  
Stable Node ◽  
Sir Epidemic Model ◽  
Treatment Rate ◽  
Globally Asymptotically Stable ◽  
Sir Epidemic ◽  
Treatment Function ◽  
Sir Epidemic Models ◽  
Saturated Treatment ◽  
Disease Free

In this thesis we investigate the dynamics and bifurcation of SIR epidemic models with horizontal and vertical transmissions and saturated treatment rate. It is proved that such SIR epidemic models always have positive disease free equilibria and also have three positive epidemic equilibria. The ranges of the parameters related in the model were found under which the equilibria of the models are positive. By applying the qualitative theory of planar systems, it is shown the disease free equilibria is a saddle, stable node and globally asymptotically stable. Furthermore, it is also shown that the interior equilibria are saddle, saddle node or saddle point.

scholarly journals Qualitative and Bifurcation Analysis of an SIR Epidemic Model with Saturated Treatment Function and Nonlinear Pulse Vaccination

2016 ◽  
Vol 2016 ◽  
pp. 1-18
Author(s):  
Xiangsen Liu ◽  
Binxiang Dai
Keyword(s):  
Epidemic Model ◽  
Sufficient Conditions ◽  
Sir Epidemic Model ◽  
Pulse Vaccination ◽  
Sir Epidemic ◽  
Treatment Function ◽  
Existence And Stability ◽  
The Stability ◽  
Saturated Treatment ◽  
Disease Free

An SIR epidemic model with saturated treatment function and nonlinear pulse vaccination is studied. The existence and stability of the disease-free periodic solution are investigated. The sufficient conditions for the persistence of the disease are obtained. The existence of the transcritical and flip bifurcations is considered by means of the bifurcation theory. The stability of epidemic periodic solutions is discussed. Furthermore, some numerical simulations are given to illustrate our results.

Dynamic Behavior of an SIR Epidemic Model along with Time Delay; Crowley–Martin Type Incidence Rate and Holling Type II Treatment Rate

2019 ◽  
Vol 20 (7-8) ◽  
pp. 757-771 ◽  
Author(s):  
Abhishek Kumar ◽  
Nilam
Keyword(s):  
Incidence Rate ◽  
Time Lag ◽  
Type Ii ◽  
Functional Type ◽  
Sir Epidemic Model ◽  
Treatment Rate ◽  
Nonlinear Incidence Rate ◽  
Sir Epidemic ◽  
The Stability ◽  
Disease Free

Abstract In this article, we propose and analyze a time-delayed susceptible–infected–recovered (SIR) mathematical model with nonlinear incidence rate and nonlinear treatment rate for the control of infectious diseases and epidemics. The incidence rate of infection is considered as Crowley–Martin functional type and the treatment rate is considered as Holling functional type II. The stability of the model is investigated for the disease-free equilibrium (DFE) and endemic equilibrium (EE) points. From the mathematical analysis of the model, we prove that the model is locally asymptotically stable for DFE when the basic reproduction number {R_0} is less than unity ({R_0} \lt 1) and unstable when {R_0} is greater than unity ({R_0} \gt 1) for time lag \tau \ge 0. The stability behavior of the model for DFE at {R_0} = 1 is investigated using Castillo-Chavez and Song theorem, which shows that the model exhibits forward bifurcation at {R_0} = 1. We investigate the stability of the EE for time lag \tau \ge 0. We also discussed the Hopf bifurcation of EE numerically. Global stability of the model equilibria is also discussed. Furthermore, the model has been simulated numerically to exemplify analytical studies.

scholarly journals Global stability for a discrete SIR epidemic model with delay in the general incidence function

2019 ◽  
Vol 8 (2) ◽  
pp. 32
Author(s):  
Guiro Aboudramane ◽  
Dramane Ouedraogo ◽  
Harouna Ouedraogo
Keyword(s):  
Epidemic Model ◽  
Reproduction Number ◽  
Asymptotically Stable ◽  
Sir Epidemic Model ◽  
Step Size ◽  
Globally Asymptotically Stable ◽  
Incidence Function ◽  
Sir Epidemic ◽  
Disease Free ◽  
Boundedness Of Solution

In this paper, we construct a backward difference scheme for a class of general SIR epidemic model with general incidence function f. We use the step size h > 0, for the discretization. The dynamical properties are investigated (positivity and the boundedness of solution). By constructing the Lyapunov function, under the conditions that function f satisfies some assumptions. The global stabilities of equilibria are obtained. If the basic reproduction number R0<1, the disease-free equilibrium is globally asymptotically stable. If R0>1, the endemic equilibrium is globally asymptotically stable.

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

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.

DYNAMICS OF SIS EPIDEMIC MODEL WITH THE STANDARD INCIDENCE RATE AND SATURATED TREATMENT FUNCTION

2012 ◽  
Vol 05 (03) ◽  
pp. 1260003 ◽  
Author(s):  
JINGJING WEI ◽  
JING-AN CUI
Keyword(s):  
Incidence Rate ◽  
Epidemic Model ◽  
Recruitment Rate ◽  
Globally Asymptotically Stable ◽  
Sis Epidemic Model ◽  
Treatment Function ◽  
Two Factors ◽  
Saturated Treatment ◽  
Disease Free ◽  
Sis Epidemic

An SIS epidemic model with the standard incidence rate and saturated treatment function is proposed. The dynamics of the system are discussed, and the effect of the capacity for treatment and the recruitment of the population are also studied. We find that the effect of the maximum recovery per unit of time and the recruitment rate of the population over some level are two factors which lead to the backward bifurcation, and in some cases, the model may undergo the saddle-node bifurcation or Bogdanov–Takens bifurcation. It is shown that the disease-free equilibrium is globally asymptotically stable under some conditions. Numerical simulations are consistent with our obtained results in theorems, which show that improving the efficiency and capacity of treatment is important for control of disease.

TWO DIFFERENT VACCINATION STRATEGIES IN AN SIR EPIDEMIC MODEL WITH SATURATED INFECTIOUS FORCE

2008 ◽  
Vol 01 (02) ◽  
pp. 147-160 ◽  
Author(s):  
YONGZHEN PEI ◽  
SHAOYING LIU ◽  
LANSUN CHEN ◽  
CHUNHUA WANG
Keyword(s):  
Epidemic Model ◽  
Floquet Theory ◽  
Treatment Strategies ◽  
Analytic Method ◽  
Sir Epidemic Model ◽  
Globally Asymptotically Stable ◽  
Pulse Vaccination ◽  
Vaccination Strategies ◽  
Sir Epidemic ◽  
Disease Free

Two different vaccination and treatment strategies in the SIR epidemic model with saturation infectious force are analyzed. With the continuous vaccination and treatment, it is obtained that the disease free equilibrium and endemic equilibrium are globally asymptotically stable by using Lassall theorem and Pioncare–Bendixon trichotomy. Moreover, with pulse vaccination and treatment at different time, the dynamics of the epidemic model is globally investigated by using Floquet theory and comparison theorem of impulsive differential equation and analytic method. We obtain the conditions of global asymptotical stability of the infection-free periodic solution and permanence of the model. Finally, we compare the two different vaccination and treatment strategies, and obtain that the elimination of disease is independent of treatment in the case of the pulse vaccination.

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