Multiparametric bifurcations for a model in epidemiology

Lizana, Marcos; Rivero, Jesus

doi:10.1007/s002850050040

Cited by 45 publications

(31 citation statements)

References 5 publications

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“…Early work on studying the dynamics of epidemic models focused on Hopf bifurcation, homoclinic bifurcation, or saddle-node bifurcation separately by using only one bifurcation parameter (Derrick and van den Driessche [6], Hethcote and van den Driessche [14], Liu et al [17,18]). Recent studies indicate that some epidemic models undergo codimension 2 bifurcations near degenerate equilibria; i.e., a Bogdanov-Takens bifurcation, which includes a Hopf bifurcation, a homocline bifurcation and a saddle-node bifurcation, can occur when two parameters vary near their critical values (Lizana and Rivero [19], Ruan and Wang [25], Alexander and Moghadas [1,2], Moghadas [21], Wang [26]). 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.…”

Section: Degenerate Bogdanov-takens Bifurcation By Lemma 23 Whenmentioning

confidence: 99%

“…It is very important to understand such epidemic patterns in order to introduce public health interventions and control the spread of diseases. Recent studies have demonstrated that the incidence rate plays a crucial role in producing periodic oscillations in epidemic models (Alexander and Moghadas [1,2], Derrick and van den Driessche [6], Hethcote and van den Driessche [14], Liu et al [17,18], Lizana and Rivero [19], Moghadas [21], Moghadas and Alexander [22], Ruan and Wang [25], Wang [26]). …”

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confidence: 99%

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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%

mentioning

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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“…The horizontal transmission is assumed to take the form of direct contact between infectious and susceptible hosts. The incidence rate term H(I, S) is assumed to be differentiable, ∂H/∂I and ∂H/∂S are nonnegative and finite for all I and S. For special forms of the incidence rate H(I, S), see [3,10,11,13,14,16]. Here, b is the natural birth rate of the host population which is assumed to have a constant density 1.…”

Section: (T) E(t) I(t) and R(t)mentioning

confidence: 99%

On stability and bifurcation of solutions of an SEIR epidemic model with vertical transmission

El-Sheikh

El-Marouf

2004

International Journal of Mathematics and Mathematical Sciences

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“…In some SIR type models, the incidence rate is bilinear in the infective fraction I and the susceptible fraction S, i.e., the form of κ I S, e.g., [4][5][6]. Recently, many researchers, e.g., [7][8][9][10], pay attention to a nonlinear incidence rate of the form κ I p S q for two reasons: the corresponding models have a much wider range of dynamical behaviors; and the underlying assumption of homogeneous mixing in bilinear rate models may be invalid. More related results are also surveyed, e.g., in [7] or [11].…”

Section: Introductionmentioning

confidence: 99%

Global dynamics of an epidemic model with an unspecified degree

Tang

2007

Computers & Mathematics with Applications

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In this paper, we give the global dynamical behaviors of a reduced SIRS epidemic model with a nonlinear incidence rate κ I p S q . We first discuss the qualitative properties of the equilibria in the interior of the first quadrant, and study the bifurcations including saddle-node bifurcation, transcritical bifurcation, Hopf bifurcation and Bogdanov-Takens bifurcation. Then we consider equilibria at infinity, determining the number of orbits in exceptional directions for the global tendency. In this discussion, the unspecified degree p, q of polynomials and their high degeneracy prevent us from using the methods of blowing-up or normal sectors in some cases. We lastly discuss the existence and uniqueness of limit cycles.

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Multiparametric bifurcations for a model in epidemiology

Abstract: In the present paper we make a bifurcation analysis of an SIRS epidemiological model depending on all parameters. In particular we are interested in codimension-2 bifurcations.

Cited by 45 publications

References 5 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

On stability and bifurcation of solutions of an SEIR epidemic model with vertical transmission

Global dynamics of an epidemic model with an unspecified degree

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