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### Open Access Journals

DCDS-B

In this paper, using an approach of Lyapunov functional, we establish the complete global stability of a multi-group SIS epidemic model in which the effect of population migration among different regions is considered.
We prove the global asymptotic stability of the disease-free equilibrium of the model for $R_0\leq 1$, and that of an endemic equilibrium for $R_0>1$. Here $R_0$ denotes the well-known basic reproduction number defined by the spectral radius of an irreducible nonnegative matrix called the next generation matrix. We emphasize that the graph-theoretic approach, which is typically used for multi-group epidemic models, is not needed in our proof.

DCDS-S

In this paper, for a Lotka-Volterra system with infinite delays and patch structure related to a multi-group SI epidemic model, applying Lyapunov functional techniques without using the form of diagonal dominance of the instantaneous negative terms over the infinite delay terms, we establish the complete global dynamics by a threshold parameter $s(M(0))$, that is, the trivial equilibrium is globally asymptotically stable if $s(M(0)) \leq 0$ and the positive equilibrium is globally asymptotically stable if $s(M(0))>0$, respectively. This offer new type condition of global stability for Lotka-Volterra systems with patch structure.

DCDS-B

In this paper, we investigate the global stability of a delayed multi-group SIRS epidemic model which includes not only nonlinear incidence rates but also rates of immunity loss and relapse of infection. The model analysis can be regarded as an extension to a multi-group epidemic analysis in [Muroya, Li and Kuniya, Complete global analysis of an SIRS epidemic model with graded cure rate and incomplete recovery rate,

*J. Math. Anal. Appl.***410**(2014) 719-732] is studied. Applying a Lyapunov functional approach, we prove that a disease-free equilibrium of the model, is globally asymptotically stable, if a threshold parameter $R_0 \leq 1$. For the global stability of an endemic equilibrium of the model, we establish a sufficient condition for small recovery rates $\delta_k \geq 0$, $k=1,2,\ldots,n$, if $R_0>1$. Further, by a monotone iterative approach, we obtain another sufficient condition for large $\delta_k$, $k=1,2,\ldots,n$. Both results generalize several known results obtained for, e.g., SIS, SIR and SIRS models in the recent literature. We also offer a new proof on permanence which is applicable to other multi-group epidemic models.
MBE

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.

DCDS-B

In this note, under the condition for the permanence used by [Beretta and Breda, An SEIR epidemic model with constant latency time and infectious period,

*Math. Biosci. Eng.***8**(2011) 931-952], applying modified monotone sequences, we establish the global asymptotic stability of the endemic equilibrium of this SEIR epidemic model, without any other additional conditions on the global stability.
MBE

In this paper, we formulate an SIR epidemic model with hybrid of multigroup and patch structures, which can be regarded as a model for the geographical spread of infectious diseases or a multi-group model with perturbation. We show that if a threshold value, which corresponds to the well-known basic reproduction number $R_0$, is less than or equal to unity, then the disease-free equilibrium of the model is globally asymptotically stable. We also show that if the threshold value is greater than unity, then the model is uniformly persistent and has an endemic equilibrium. Moreover, using a Lyapunov functional technique, we obtain a sufficient condition under which the endemic equilibrium is globally asymptotically stable. The sufficient condition is satisfied if the transmission coefficients in the same groups are large or the per capita recovery rates are small.

keywords:
multigroup
,
SIR epidemic model
,
patch
,
global asymptotic stability
,
Lyapunov functional.

DCDS-B

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.

PROC

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.
MBE

In this paper, we study the global stability of a multistrain SIS model with superinfection. We present an iterative procedure to calculate a sequence of reproduction numbers, and we prove that it completely determines the global dynamics of the system. We show that for any number of strains with different infectivities, the stable coexistence of any subset of the strains is possible, and we completely characterize all scenarios. As an example, we apply our method to a three-strain model.

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