Yoshiaki Muroya

Global stability for a class of discrete SIR epidemic models.

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.

Persistence and global stability in Lotka-Volterra delay differential systems

Abstract Consider the persistence and the global asymptotic stability of models governed by the following Lotka-Volterra delay differential system: d x i ( t ) d t = x i ( t ) { c i − a i x i ( t ) − ∑ j = 1 n a i j x j ( t − τ i j ) } , t ≥ t 0 , 1 ≤ i ≤ n , x i ( t ) = φ i ( t ) ≥ 0 , t ≤ t 0 , and φ i ( t 0 ) > 0 , 1 ≤ i ≤ n , where each i(t) is a continuous function for t ≤ t0, each ci, ai, and aij are finite and a i > 0 , a i + a i i > 0 , 1 ≤ i ≤ n , and τ i j ≥ 0 , 1 ≤ i , j ≤ n . In this paper, applying the former results [1], we obtain conditions for the persistence of the system, and extending a technique offered by Saito, Hara and Ma [2] for n = 2 to the above system for n ≥ 2, we establish new conditions for global asymptotic stability of the positive equilibrium which improve the well-known result of Gopalsamy for some special cases.

Global dynamics of difference equations for SIR epidemic models with a class of nonlinear incidence rates

In this paper, by applying a variation of the backward Euler method, we propose a discrete-time SIR epidemic model whose discretization scheme preserves the global asymptotic stability of equilibria for a class of corresponding continuous-time SIR epidemic models. Using discrete-time analogue of Lyapunov functionals, the global asymptotic stability of the equilibria is fully determined by the basic reproduction number , when the infection incidence rate has a suitable monotone property.

Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays

Abstract. 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 suitable Lyapunov functionals. 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 R0 and the distributed delays do not influence the global dynamics of the model.

A General Discrete Time Model of Population Dynamics in the Presence of an Infection

We present a set of difference equations which generalizes that proposed in the work of G. Izzo and A. Vecchio (2007) and represents the discrete counterpart of a larger class of continuous model concerning the dynamics of an infection in an organism or in a host population. The limiting behavior of this new discrete model is studied and a threshold parameter playing the role of the basic reproduction number is derived.

Global stability of a delayed SIRS computer virus propagation model

We propose a delayed SIRS computer virus propagation model. Applying monotone iterative techniques and Lyapunov functional techniques, we establish sufficient conditions for the global asymptotic stability of both virus-free and virus equilibria of the model.

Stability analysis of delayed SIR epidemic models with a class of nonlinear incidence rates

We analyze stability of equilibria for a delayed SIR epidemic model, in which population growth is subject to logistic growth in absence of disease, with a nonlinear incidence rate satisfying suitable monotonicity conditions. The model admits a unique endemic equilibrium if and only if the basic reproduction number R0 exceeds one, while the trivial equilibrium and the disease-free equilibrium always exist. First we show that the disease-free equilibrium is globally asymptotically stable if and only if R0 1. Second we show that the model is permanent if and only if R0 > 1. Moreover, using a threshold parameter R0 characterized by the nonlinear incidence function, we establish that the endemic equilibrium is locally asymptotically stable for 1 < R0 R0 and it loses stability as the length of the delay increases past a critical value for 1 < R0 < R0. Our result is an extension of the stability results in (J-J. Wang, J-Z. Zhang, Z. Jin, Analysis of an SIR model with bilinear incidence rate, Nonl. Anal. RWA. 11 (2009) 2390-2402).

GLOBAL STABILITY OF EXTENDED MULTI-GROUP SIR EPIDEMIC MODELS WITH PATCHES THROUGH MIGRATION AND CROSS PATCH INFECTION

Abstract In this article, we establish the global stability of an endemic equilibrium of multi-group SIR epidemic models, which have not only an exchange of individuals between patches through migration but also cross patch infection between different groups. As a result, we partially generalize the recent result in the article [16].

Persistence and global stability for discrete models of nonautonomous Lotka–Volterra type

Abstract Consider the following discrete models of nonautonomous Lotka–Volterra type: N i (p+1)=N i (p) exp c i (p)−a i (p)N i (p) − ∑ j=1 n ∑ l=0 m a ij l (p)N j (p−k l ) , 1⩽i⩽n, p=0,1,2,…, N i (p)=N ip ⩾0, p⩽0, and N i0 >0, 1⩽i⩽n, where each c i (p) , a i (p) and a ij l (p) are bounded for p⩾0 and inf p⩾0 a i (p)>0, a ii 0 (p)≡0, 1⩽i⩽n, a ij l (p)⩾0, 1⩽i⩽j⩽n, 0⩽l⩽m, k 0 =0, integers k l ⩾0, 1⩽l⩽m. In this paper, to the above discrete system, we apply the techniques offered by Ahmad and Lazer (Nonlinear Anal. 40 (2000) 37–49), and establish similar conditions of the persistence and global asymptotic stability of the system.

On the Attainable Order of Collocation Methods for Delay Differential Equations with Proportional Delay

AbstractTo analyze the attainable order of m-stage implicit (collocation-based) Runge-Kutta methods for the delay differential equation (DDE) y′(t) = by(qt), 0 < q ≤ 1 with y(0) = 1, and the delay Volterra integral equation (DVIE) y(t) = 1 +