Stability and bifurcation analysis of epidemic models with saturated incidence rates: An application to a nonmonotone incidence rate
Yoichi Enatsu Yukihiko Nakata
Mathematical Biosciences & Engineering 2014, 11(4): 785-805 doi: 10.3934/mbe.2014.11.785
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.
keywords: nonlinear incidence rate Hopf bifurcation linearized stability delay differential equation. Epidemic model
Dynamics of an ultra-discrete SIR epidemic model with time delay
Masaki Sekiguchi Emiko Ishiwata Yukihiko Nakata
Mathematical Biosciences & Engineering 2018, 15(3): 653-666 doi: 10.3934/mbe.2018029

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.

keywords: Epidemic model ultra-discretization time delay dynamics
On the global stability of an SIRS epidemic model with distributed delays
Yukihiko Nakata Yoichi Enatsu Yoshiaki Muroya
Conference Publications 2011, 2011(Special): 1119-1128 doi: 10.3934/proc.2011.2011.1119
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.
keywords: global asymptotic stability distributed delays SIRS epidemic model Lyapunov functional
Global stability for a class of discrete SIR epidemic models
Yoichi Enatsu Yukihiko Nakata Yoshiaki Muroya
Mathematical Biosciences & Engineering 2010, 7(2): 347-361 doi: 10.3934/mbe.2010.7.347
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.
keywords: globally asymptotic stability distributed delays Backward Euler method Lyapunov functional. discrete SIR epidemic model
Unbounded and blow-up solutions for a delay logistic equation with positive feedback
István Győri Yukihiko Nakata Gergely Röst
Communications on Pure & Applied Analysis 2018, 17(6): 2845-2854 doi: 10.3934/cpaa.2018134

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.

keywords: Delay logistic equation stability blow-up delay differential equation
Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays
Yoichi Enatsu Yukihiko Nakata Yoshiaki Muroya
Discrete & Continuous Dynamical Systems - B 2011, 15(1): 61-74 doi: 10.3934/dcdsb.2011.15.61
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.
keywords: Lyapunov functional. distributed delays permanence SIR epidemic models nonlinear incidence rate global asymptotic stability

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