Backward bifurcation in a cholera model with a general treatment function

AbstractWe consider a general cholera model with a nonlinear treatment function. The treatment function describes the saturated treatment scenario due to the limited availability of resources. The sufficient conditions for the existence of backward bifurcation have been obtained using the central manifold theory. At last, we illustrate the results by considering some special types of treatment functions.

Download Full-text

Stability and Bogdanov-Takens Bifurcation of an SIS Epidemic Model with Saturated Treatment Function

Mathematical Problems in Engineering ◽

10.1155/2015/745732 ◽

2015 ◽

Vol 2015 ◽

pp. 1-14 ◽

Cited By ~ 1

Author(s):

Yanju Xiao ◽

Weipeng Zhang ◽

Guifeng Deng ◽

Zhehua Liu

Keyword(s):

Sufficient Conditions ◽

Global Dynamics ◽

Sis Model ◽

Delayed Treatment ◽

Sis Epidemic Model ◽

Treatment Function ◽

Lyapunov Coefficient ◽

Saturated Treatment ◽

Theoretical Results ◽

Disease Free

This paper introduces the global dynamics of an SIS model with bilinear incidence rate and saturated treatment function. The treatment function is a continuous and differential function which shows the effect of delayed treatment when the rate of treatment is lower and the number of infected individuals is getting larger. Sufficient conditions for the existence and global asymptotic stability of the disease-free and endemic equilibria are given in this paper. The first Lyapunov coefficient is computed to determine various types of Hopf bifurcation, such as subcritical or supercritical. By some complex algebra, the Bogdanov-Takens normal form and the three types of bifurcation curves are derived. Finally, mathematical analysis and numerical simulations are given to support our theoretical results.

Download Full-text

Backward bifurcation and stability analysis of a network-based SIS epidemic model with saturated treatment function

Physica A Statistical Mechanics and its Applications ◽

10.1016/j.physa.2019.121407 ◽

2019 ◽

Vol 527 ◽

pp. 121407 ◽

Cited By ~ 2

Author(s):

Yi-Jie Huang ◽

Chun-Hsien Li

Keyword(s):

Stability Analysis ◽

Epidemic Model ◽

Backward Bifurcation ◽

Sis Epidemic Model ◽

Treatment Function ◽

Bifurcation And Stability ◽

Saturated Treatment ◽

Sis Epidemic

Download Full-text

Analysis of an SEIR Epidemic Model with Saturated Incidence and Saturated Treatment Function

The Scientific World JOURNAL ◽

10.1155/2014/910421 ◽

2014 ◽

Vol 2014 ◽

pp. 1-11 ◽

Cited By ~ 14

Author(s):

Jinhong Zhang ◽

Jianwen Jia ◽

Xinyu Song

Keyword(s):

Epidemic Model ◽

Reproduction Number ◽

Equilibrium Points ◽

Sufficient Conditions ◽

Asymptotical Stability ◽

Treatment Function ◽

Threshold Conditions ◽

Saturated Incidence Rate ◽

Saturated Incidence ◽

Saturated Treatment

The dynamics of SEIR epidemic model with saturated incidence rate and saturated treatment function are explored in this paper. The basic reproduction number that determines disease extinction and disease survival is given. The existing threshold conditions of all kinds of the equilibrium points are obtained. Sufficient conditions are established for the existence of backward bifurcation. The local asymptotical stability of equilibrium is verified by analyzing the eigenvalues and using the Routh-Hurwitz criterion. We also discuss the global asymptotical stability of the endemic equilibrium by autonomous convergence theorem. The study indicates that we should improve the efficiency and enlarge the capacity of the treatment to control the spread of disease. Numerical simulations are presented to support and complement the theoretical findings.

Download Full-text

Backward bifurcation of an epidemic model with saturated treatment function

Journal of Mathematical Analysis and Applications ◽

10.1016/j.jmaa.2008.07.042 ◽

2008 ◽

Vol 348 (1) ◽

pp. 433-443 ◽

Cited By ~ 112

Author(s):

Xu Zhang ◽

Xianning Liu

Keyword(s):

Epidemic Model ◽

Backward Bifurcation ◽

Treatment Function ◽

Saturated Treatment

Download Full-text

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

Discrete Dynamics in Nature and Society ◽

10.1155/2016/9146481 ◽

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.

Download Full-text

A Theoretical Model for the Transmission Dynamics of the Buruli Ulcer with Saturated Treatment

Computational and Mathematical Methods in Medicine ◽

10.1155/2014/576039 ◽

2014 ◽

Vol 2014 ◽

pp. 1-14 ◽

Cited By ~ 8

Author(s):

Ebenezer Bonyah ◽

Isaac Dontwi ◽

Farai Nyabadza

Keyword(s):

Human Population ◽

Reproduction Number ◽

Buruli Ulcer ◽

Coupled System ◽

Model Parameters ◽

The Public ◽

Treatment Function ◽

Medical Facilities ◽

The Stability ◽

Saturated Treatment

The management of the Buruli ulcer (BU) in Africa is often accompanied by limited resources, delays in treatment, and macilent capacity in medical facilities. These challenges limit the number of infected individuals that access medical facilities. While most of the mathematical models with treatment assume a treatment function proportional to the number of infected individuals, in settings with such limitations, this assumption may not be valid. To capture these challenges, a mathematical model of the Buruli ulcer with a saturated treatment function is developed and studied. The model is a coupled system of two submodels for the human population and the environment. We examine the stability of the submodels and carry out numerical simulations. The model analysis is carried out in terms of the reproduction number of the submodel of environmental dynamics. The dynamics of the human population submodel, are found to occur at the steady states of the submodel of environmental dynamics. Sensitivity analysis is carried out on the model parameters and it is observed that the BU epidemic is driven by the dynamics of the environment. The model suggests that more effort should be focused on environmental management. The paper is concluded by discussing the public implications of the results.

Download Full-text