We analyze local asymptotic stability of an SIRS epidemic model with a distributed delay. The incidence rate is given by a general saturated function of the number of infective individuals.
Our first aim is to find a class of nonmonotone incidence rates such that a unique endemic equilibrium is always asymptotically stable.
We establish a characterization for the incidence rate, which shows that nonmonotonicity with delay in the incidence rate is necessary for destabilization of the endemic equilibrium. We further elaborate the stability analysis for a specific incidence rate. Here we improve a stability condition obtained in [Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Disc. Cont. Dynam. Sys. B 13 (2010) 195-211], which is illustrated in a suitable parameter plane. Two-parameter plane analysis together with an application of the implicit function theorem facilitates us to obtain an exact stability condition. It is proven that as increasing a parameter, measuring saturation effect, the number of infective individuals at the endemic steady state decreases, while the equilibrium can be unstable via Hopf bifurcation. This can be interpreted as that reducing a contact rate may cause periodic oscillation of the number of infective individuals, thus disease can not be eradicated completely from the host population, though the level of the endemic equilibrium for the infective population decreases. Numerical simulations are performed to illustrate our theoretical results.
In this paper, we propose a class of discrete SIR epidemic models which are derived from SIR epidemic models with distributed delays by using a variation of the backward Euler method.
Applying a Lyapunov functional technique, it is shown that the global dynamics of each discrete SIR epidemic model are fully determined by a single threshold parameter and the effect of discrete time delays are harmless for the global stability of the endemic equilibrium of the model.
In this paper, we establish the global asymptotic stability of equilibria for an SIR model of infectious diseases with distributed time delays governed by a wide class of nonlinear incidence rates. We obtain the global properties of the model by proving the permanence and constructing a suitable Lyapunov functional. Under some suitable assumptions on the nonlinear term in the incidence rate, the global dynamics of the model is completely determined by the basic reproduction number $R_0$ and the distributed delays do not influence the global dynamics of the model.
In this paper, we establish the global asymptotic stability of an
endemic equilibrium for an SIRS epidemic model with distributed time delays.
It is shown that the global stability holds for any rate of immunity loss, if the
basic reproduction number is greater than 1 and less than or equals to a critical
value. Otherwise, there is a maximal rate of immunity loss which guarantees
the global stability. By using an extension of a Lyapunov functional established
by [C.C. McCluskey, Complete global stability for an SIR epidemic model with
delay-Distributed or discrete, Nonlinear Anal. RWA. 11 (2010) 55-59], we
provide a partial answer to an open problem whether the endemic equilibrium
is globally stable, whenever it exists, or not.
We propose an ultra-discretization for an SIR epidemic model with time delay. It is proven that the ultra-discrete model has a threshold property concerning global attractivity of equilibria as shown in differential and difference equation models. We also study an interesting convergence pattern of the solution, which is illustrated in a two-dimensional lattice.
We study bounded, unbounded and blow-up solutions of a delay logistic equation without assuming the dominance of the instantaneous feedback. It is shown that there can exist an exponential (thus unbounded) solution for the nonlinear problem, and in this case the positive equilibrium is always unstable. We obtain a necessary and sufficient condition for the existence of blow-up solutions, and characterize a wide class of such solutions. There is a parameter set such that the non-trivial equilibrium is locally stable but not globally stable due to the co-existence with blow-up solutions.