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

Abstract

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

Bifurcations of an SIRS model with generalized non-monotone incidence rate

“…Recently, in [19,20], Ruan et al studied the bifurcation of an SIRS epidemic model of incidence rate H(S, I) with p = q = 2, i.e., g(I) = κI 2 1 + αI 2 .…”

“…In particular, they referred to the nonlinear incidence H(S, I) in [20] and classified it into three classes. (i) Unbounded incidence function: p > q; (ii) Saturated incidence function: p = q; (iii) non-monotone incidence function: p < q.…”

Complex Dynamics of an SIR Epidemic Model with Nonlinear Saturate Incidence and Recovery Rate

“…where βI l S measures the infection force of the disease, 1/(1 + αI h ) describes the inhibition effect from the behavioral change of the susceptible individuals when the number of infectious individuals increases, l, h, β are positive constants and α is non-negative constant [6,7]. The nonlinear function g(I) given in (1) has three types, for details one can see [7].…”

“…Capasso and Serio [8] used the case when l = h = 1, i.e., g(I) = investigating the cholera epidemic in Bari in 1973. Due to the nonlinearity and saturation property of these incidence rates, SIR epidemic models usually possess multiple endemic equilibria and rich nonlinear dynamics [5][6][7][8][9][10][11][12]. Furthermore, a compartmental model with nonlinear incidence rate is usually used to explore the impact of intervention strategies on the transmission dynamics of infectious diseases.…”

“…The nonlinear function g(I) given in (1) has three types, for details one can see [7]. Capasso and Serio [8] used the case when l = h = 1, i.e., g(I) = investigating the cholera epidemic in Bari in 1973.…”