Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment (original) (raw)

Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment

Applied Mathematics and Computation, 2009

The dynamical behaviors of an SIR epidemic model with nonlinear incidence and treatment is investigated. It is assumed that treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low. Theoretical and numerical results suggest that decreasing the basic reproduction number below one is insufficient for disease eradication.

Stability and bifurcation analysis of a SIR model with saturated incidence rate and saturated treatment

We study the dynamics of a SIR epidemic model with nonlinear incidence rate, vertical transmission vaccination for the newborns and the capacity of treatment, that takes into account the limitedness of the medical resources and the efficiency of the supply of available medical resources. Under some conditions we prove the existence of backward bifurcation, the stability and the direction of Hopf bifurcation. We also explore how the mechanism of backward bifurcation affects the control of the infectious disease. Numerical simulations are presented to illustrate the theoretical findings.

Dynamics of an SIR Model with Nonlinear Incidence and Treatment Rate

2015

In this paper, global dynamics of an SIR model are investigated in which the incidence rate is being considered as Beddington-DeAngelis type and the treatment rate as Holling type II (saturated). Analytical study of the model shows that the model has two equilibrium points (diseasefree equilibrium (DFE) and endemic equilibrium (EE)). The disease-free equilibrium (DFE) is locally asymptotically stable when reproduction number is less than one. Some conditions on the model parameters are obtained to show the existence as well as nonexistence of limit cycle. Some sufficient conditions for global stability of the endemic equilibrium using Lyapunov function are obtained. The existence of Hopf bifurcation of model is investigated by using Andronov-Hopf bifurcation theorem. Further, numerical simulations are done to exemplify the analytical studies.

Bifurcation and Stability Analysis of a Discrete Time Sir Epidemic Model with Vaccination

International Journal of Analysis and Applications, 2019

In this paper, we study the qualitative behavior of a discrete-time epidemic model with vaccination. Analysis of the model shows forth that the Disease Free Equilibrium (DFE) point is asymptotically stable if the basic reproduction number R 0 is less than one, while the Endemic Equilibrium (EE) point is asymptotically stable if the basic reproduction number R 0 is greater than one. The results are reinforced with numerical simulations and enhanced with graphical representations like time trajectories, phase portraits and bifurcation diagrams for different sets of parameter values.

Qualitative and bifurcation analysis using an SIR model with a saturated treatment function

Mathematical and Computer Modelling, 2012

In this paper, we introduce a saturated treatment function into the SIR epidemic model with a bilinear incidence rate and density-dependent demographics, where the treatment function is limited for increasing number of infected individuals. By carrying out global qualitative and bifurcation analysis, it is shown that the system exhibits some new and complicated behaviors: if the basic reproduction number is larger than unity, the number of infected individuals will show persistent behavior, either converging to some positive constant or oscillating; and if the basic reproduction number is below unity, the model may exhibit complicated behaviors including: (i) backward bifurcation; (ii) almost sure disease eradication where the number of infective individuals tends to zero for all initial positions except the interior equilibria; (iii) ''oscillating'' backward bifurcation where either the number of infective individuals oscillates persistently, if the initial position lies in a region covering the stable endemic equilibrium, or disease eradication, if the initial position lies outside this region; (iv) disease eradication for all initial positions if the basic reproduction number is less than a turning point value. Numerical simulations are presented to illustrate the conclusions.

Stability Analysis of an SIR Epidemic Model with Non-Linear Incidence Rate and Treatment

We consider a SIR epidemic model with saturated incidence rate and treatment. We show that if the basic reproduction number, R0 is less than unity and the disease free equilibrium is locally asymptotically stable. Moreover, we show that if R0 > 1, the endemic equilibrium is locally asymptotically stable. In the end, we give some numerical results to compare our model with existing model and to show the effect of the treatment term on the model.

On the dynamics of an SIR epidemic model with a saturated incidence rate

Journal of Advanced Studies in Topology, 2017

In this paper, discrete-time epidemic model with a saturated incidence rate is considered. Firstly, we introduce the local stability analysis of the system by details. Next, we study the bifurcation phenomena and the sufficient condition to verify flip bifurcation and Neimark-sacker bifurcation by using bifurcation theory and the center manifold theorem. Finally, numerical simulation including bifurcation diagrams, phase portraits and Chaotic attractors is carried out by using matlab to verify theoretical results obtained.

Qualitative analysis and optimal control of an SIR model with logistic growth, non-monotonic incidence and saturated treatment

2021

This paper describes an SIR model with logistic growth rate of susceptible population, non-monotonic incidence rate and saturated treatment rate. The existence and stability analysis of equilibria have been investigated. It has been shown that the disease free equilibrium point (DFE) is globally asymptotically stable if the basic reproduction number is less than unity and the transmission rate of infection less than some threshold. The system exhibits the transcritical bifurcation at DFE with respect to the cure rate. We have also found the condition for occurring the backward bifurcation, which implies the value of basic reproduction number less than unity is not enough to eradicate the disease. Stability or instability of different endemic equilibria has been shown analytically. The system also experiences the saddle-node and Hopf bifurcation. The existence of Bogdanov-Takens bifurcation (BT) of co-dimension 2 has been investigated which has also been shown through numerical simul...

SIR Model with Vaccination: Bifurcation Analysis

Qualitative Theory of Dynamical Systems

There are few adapted SIR models in the literature that combine vaccination and logistic growth. In this article, we study bifurcations of a SIR model where the class of Susceptible individuals grows logistically and has been subject to constant vaccination. We explicitly prove that the endemic equilibrium is a codimension two singularity in the parameter space (\mathcal {R}_0, p)(R0,p),where( R 0 , p ) , where(R0,p),where\mathcal {R}_0R0isthebasicreproductionnumberandpistheproportionofSusceptibleindividualssuccessfullyvaccinatedatbirth.WeexhibitexplicitlytheHopf,transcritical,Belyakov,heteroclinicandsaddle−nodebifurcationcurvesunfoldingthesingularity.ThetwoparametersR 0 is the basic reproduction number and p is the proportion of Susceptible individuals successfully vaccinated at birth. We exhibit explicitly the Hopf, transcritical, Belyakov, heteroclinic and saddle-node bifurcation curves unfolding the singularity. The two parametersR0isthebasicreproductionnumberandpistheproportionofSusceptibleindividualssuccessfullyvaccinatedatbirth.WeexhibitexplicitlytheHopf,transcritical,Belyakov,heteroclinicandsaddle−nodebifurcationcurvesunfoldingthesingularity.Thetwoparameters(\mathcal {R}_0, p)$$ ( R 0 , p ) are written in a useful way to evaluate the proportion of vaccinated individuals necessary to eliminate the disease and to conclude how the vaccination may affect the outcome of the epidemic. We also exhibit the region in the parameter space where the disease p...

Stability Analysis of a Complex Dynamics of a SIR Epidemic Model with Bilinear Incidence Rate and Treatment

Journal of scientific research

In this article, A SIR epidemic model with bilinear incidence rate has been proposed and the existing threshold requirements of all classifications of equilibrium points are obtained. Further, we study the global and local stability of the disease-free and endemic equilibriums of the model. An optimal control problem is formed and solved. Some numerical simulations works are carried out to demonstrate our results. In this process, our results generalized and improved any results in existing literature.