Bifurcations in an epidemic model with constant removal rate of the infectives

Wang, Wendi; Ruan, Shigui

doi:10.1016/j.jmaa.2003.11.043

Cited by 237 publications

(169 citation statements)

References 18 publications

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“…It is interesting to notice that not only epidemic models with nonlinear incidence rates but also simple epidemic models with bilinear mass-action incidence rates can have complex dynamics such as the occurrence of Bogdanov-Takens bifurcations. For instance, Wang and Ruan [27] considered an epidemic model with a bilinear mass-action incidence rate and a constant removal rate of infectious individuals and showed that the model undergoes a sequence of bifurcations, including saddle-node bifurcation, subcritical Hopf bifurcation, and homoclinic bifurcation.…”

Section: Degenerate Bogdanov-takens Bifurcation By Lemma 23 Whenmentioning

confidence: 99%

Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate

Tang¹,

Huang²,

Ruan³

et al. 2008

SIAM J. Appl. Math.

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Abstract. Recently, Ruan and Wang [J. Differential Equations, 188 (2003), pp. 135-163] studied the global dynamics of a SIRS epidemic model with vital dynamics and a nonlinear saturated incidence rate. Under certain conditions they showed that the model undergoes a Bogdanov-Takens bifurcation; i.e., it exhibits saddle-node, Hopf, and homoclinic bifurcations. They also considered the existence of none, one, or two limit cycles. In this paper, we investigate the coexistence of a limit cycle and a homoclinic loop in this model. One of the difficulties is to determine the multiplicity of the weak focus. We first prove that the maximal multiplicity of the weak focus is 2. Then feasible conditions are given for the uniqueness of limit cycles. The coexistence of a limit cycle and a homoclinic loop is obtained by reducing the model to a universal unfolding for a cusp of codimension 3 and studying degenerate Hopf bifurcations and degenerate Bogdanov-Takens bifurcations of limit cycles and homoclinic loops of order 2. In most epidemic models (see Anderson and May [3]), the incidence rate (the number of new cases per unit time) takes the mass-action form with bilinear interactions, namely, κS(t)I(t), where S(t) and I(t) are the numbers of susceptible and infectious individuals at time t, respectively, and the constant κ is the probability of transmission per contact. Epidemic models with such bilinear incidence rates usually have at most one endemic equilibrium and do not exhibit periodicity; the disease will be eradicated if the basic reproduction number is less than one and will persist otherwise (Anderson and May [3], Hethcote [11]). There are many reasons for using nonlinear incidence rates, and various forms of nonlinear incidence rates have

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Section: Degenerate Bogdanov-takens Bifurcation By Lemma 23 Whenmentioning

confidence: 99%

Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate

Tang¹,

Huang²,

Ruan³

et al. 2008

SIAM J. Appl. Math.

Self Cite

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“…To prevent the spread of the infectious disease, many researchers [33][34][35][36] began to investigate a treatment function in the epidemic models. In 2004, Wang et al [37] introduced a constant treatment function…”

Section: = Dn(t) -Ds(t) -λS(t)i(t) + νR(t) Di(t) Dt = λS(t)i(t) -(D mentioning

confidence: 99%

Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function

2018

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In this paper, we propose a new nonlinear stochastic SIRS epidemic model with standard incidence rate and saturated treatment function. The main purpose of this paper is to investigate the threshold dynamics of the nonlinear stochastic SIRS epidemic model by making use of stochastic inequality techniques. By using Lyapunov methods and Itô's formula, we first prove the existence and uniqueness of a global positive solution for the corresponding limiting system. Furthermore, we obtain sufficient conditions for the extinction and persistence in mean of the nonlinear stochastic SIRS epidemic model by using the techniques of a series of stochastic inequalities. Finally, we provide some numerical simulations to illustrate the performance of our theoretical findings.

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“…The experience with SARS epidemic sparked a renewed interest among modellers to investigate the implication of the capacity of the healthcare system. Authors in [29] considered an SIR epidemic model and assumed a Heaviside treatment function while Wang [30] restudied the same SIR model but assumed a piecewise linear treatment function. Here we will assume that the heroin users 1 ( ) receive treatment based on the following more general saturated treatment function:…”

Section: Model Formulationmentioning

confidence: 99%

Analysis of a Heroin Epidemic Model with Saturated Treatment Function

Wangari¹,

Stone

2017

Journal of Applied Mathematics

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A mathematical model is developed that examines how heroin addiction spreads in society. The model is formulated to take into account the treatment of heroin users by incorporating a realistic functional form that "saturates" representing the limited availability of treatment. Bifurcation analysis reveals that the model has an intrinsic backward bifurcation whenever the saturation parameter is larger than a fixed threshold. We are particularly interested in studying the model's global stability. In the absence of backward bifurcations, Lyapunov functions can often be found and used to prove global stability. However, in the presence of backward bifurcations, such Lyapunov functions may not exist or may be difficult to construct. We make use of the geometric approach to global stability to derive a condition that ensures that the system is globally asymptotically stable. Numerical simulations are also presented to give a more complete representation of the model dynamics. Sensitivity analysis performed by Latin hypercube sampling (LHS) suggests that the effective contact rate in the population, the relapse rate of heroin users undergoing treatment, and the extent of saturation of heroin users are mechanisms fuelling heroin epidemic proliferation.

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Bifurcations in an epidemic model with constant removal rate of the infectives

Cited by 237 publications

References 18 publications

Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate

Coexistence of Limit Cycles and Homoclinic Loops in a SIRS Model with a Nonlinear Incidence Rate

Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function

Analysis of a Heroin Epidemic Model with Saturated Treatment Function

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