Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment

Li, Jinhui; Teng, Zhidong; Wang, Guangqing; Zhang, Long; Hu, Cheng

doi:10.1016/j.chaos.2017.03.047

Cited by 62 publications

(30 citation statements)

References 17 publications

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“…Li et al studied in [4] an SIR model with logistic growth rate, bilinear incidence rate and a saturated treatment function of the form /(1 + ). They studied the local stability of the disease-free and endemic equilibria and showed that the system exhibits backward bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation of codimension 2.…”

Section: Complexitymentioning

confidence: 99%

“…The study of nonlinear equations in the modeling of biological processes has gained attention in recent years due to the fact that many systems in nature present inherently nonlinear dynamics. Recent research has shown that the use of nonlinear, saturated functions in both epidemic models [2][3][4][5] and ecological (predator-prey) models [6,7] can lead to the appearance of very complex dynamics from the mathematical viewpoint, such as several types of bifurcation, bistability, heteroclinic and homoclinic orbits, and periodic oscillations. Although these functions make the models more difficult to analyze, they have been proved to represent more accurately the processes that we want to describe, so nonlinear models are worthy of being studied in greater detail.…”

Section: Complexitymentioning

confidence: 99%

“…Theorem 14 can be viewed as a generalization of Theorem 3.1 of [4] that takes into account the nonlinearity of the incidence rate. Notice that the critical value 0 at which the Hopf bifurcation takes place is given implicitly as a root of the function .…”

Section: Complexitymentioning

confidence: 99%

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Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment

2019

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There is a wide range of works that have proposed mathematical models to describe the spread of infectious diseases within human populations. Based on such models, researchers can evaluate the effect of applying different strategies for the treatment of diseases. In this article, we generalize previous models by studying an SIR epidemic model with a nonlinear incidence rate, saturated Holling type II treatment rate, and logistic growth. We compute the basic reproduction number and determine conditions for the local stability of equilibria and the existence of backward bifurcation and Hopf bifurcation. We also show that, when the disease transmission rate and treatment parameter are varied, our model undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. Simulations of the solutions and numerical continuation of equilibria are carried out to generate 2D and 3D bifurcation diagrams, as well as several related phase portraits that illustrate our results. Our work shows that incorporating these factors into epidemic models can lead to very complex dynamics.

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Section: Complexitymentioning

confidence: 99%

Section: Complexitymentioning

confidence: 99%

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Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment

2019

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“…where I 0 is the capacity of treatment. Recently, Li introduced the following saturated treatment function [21]:…”

Section: Introductionmentioning

confidence: 99%

“…where a represents the maximal medical resources supplied per united time and is half-saturation constant, which measures the effect of being delayed for treatment. Other works have investigated the effects of the treatment on an epidemic (see [19][20][21][22][23][24][25]) and also its optimal control (see [26]). The motivation of this work comes from [10,11], where the authors studied an SIR epidemic model with nonlinear incidence function, and from [19][20][21], where the authors considered a special type of treatment function.…”

Section: Introductionmentioning

confidence: 99%

Global stability analysis for a generalized delayed SIR model with vaccination and treatment

et al. 2019

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In this work, we investigate the stability of an SIR epidemic model with a generalized nonlinear incidence rate and distributed delay. The model also includes vaccination term and general treatment function, which are the two principal control measurements to reduce the disease burden. Using the Lyapunov functions, we show that the disease-free equilibrium state is globally asymptotically stable if R 0 ≤ 1, where R 0 is the basic reproduction number. On the other hand, the disease-endemic equilibrium is globally asymptotically stable when R 0 > 1. For a specific type of treatment and incidence functions, our analysis shows the success of the vaccination strategy, as well as the treatment depends on the initial size of the susceptible population. Moreover, we discuss, numerically, the behavior of the basic reproduction number with respect to vaccination and treatment parameters. MSC: 34D03; 92D30

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SIRS epidemiological model with ratio‐dependent incidence: Influence of preventive vaccination and treatment control strategies on disease dynamics

Kumar

Mandal

Tripathi

et al. 2021

Math Methods in App Sciences

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In this paper, we study an SIRS epidemic model with ratio‐dependent incidence rate function describing the mechanisms of infectious disease transmission. Impacts of vaccination and treatment on the transmission dynamics of the disease have been explored. The treatment rate is constant when the number of infected individuals is greater than the maximal capacity of treatment and proportional to the number of infected individuals when the number of infected individuals is less than the maximal capacity of treatment. Analysis shows that (1) the sufficiently large value of the preventive vaccination rate can control the spread of disease, and (2) a threshold level of the psychological (or inhibitory) effects in the incidence rate function is enough to decrease the infective population. It is also obtained that model undergoes transcritical and saddle‐node bifurcations with respect to disease contact rate. Moreover, in the presence of treatment strategy, the model has multiple endemic equilibria and undergoes a backward bifurcation. The maximal capacity of treatment plays important roles on the disease dynamics of the model. The number of infected individuals decreases with respect to the maximal capacity of treatment, and the disease completely dies out from the system for the large capacity of the treatment when constant treatment strategy is applied. Further, it is also found that the spread of disease can be suppressed by increasing treatment rate. From sensitivity analysis, we have observed that the transmission and treatment rates are most sensitive parameters. The effects of different parameters on the disease dynamics have also been investigated via numerical simulation.

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Stability and bifurcation analysis of an SIR epidemic model with logistic growth and saturated treatment

Cited by 62 publications

References 17 publications

Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment

Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment

Global stability analysis for a generalized delayed SIR model with vaccination and treatment

SIRS epidemiological model with ratio‐dependent incidence: Influence of preventive vaccination and treatment control strategies on disease dynamics

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