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### Open Access Journals

MBE

In this paper, we study an age-structured SIS epidemic model with periodicity and vertical transmission. We show that the spectral radius of the Fréchet derivative of a nonlinear integral operator plays the role of a threshold value for the global behavior of the model, that is, if the value is less than unity, then the disease-free steady state of the model is globally asymptotically stable, while if the value is greater than unity, then the model has a unique globally asymptotically stable endemic (nontrivial) periodic solution. We also show that the value can coincide with the well-know epidemiological threshold value, the basic reproduction number $\mathcal{R}_0$.

DCDS-B

In this paper, we investigate the global asymptotic stability of multi-group SIR and SEIR age-structured models. These models allow the infectiousness and the death rate of susceptible individuals to vary and depend on the susceptibility, with which we can consider the heterogeneity of population. We establish global dynamics and demonstrate that the heterogeneity does not alter the dynamical structure of the basic SIR and SEIR with age-dependent susceptibility. Our results also demonstrate that, for age structured multi-group models considered, the graph-theoretic approach can be successfully applied by choosing an appropriate weighted matrix as well.

keywords:
Age-structured model
,
susceptibility
,
asymptotic smoothness
,
global attractor
,
global stability

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-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 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.

MBE

A recent paper [F. Brauer, Z. Shuai and P. van den Driessche,

*Dynamics of an age-of-infection cholera model*, Math. Biosci. Eng., 10, 2013, 1335--1349.] presented a model for the dynamics of cholera transmission. The model is incorporated with both the infection age of infectious individuals and biological age of pathogen in the environment. The basic reproduction number is proved to be a sharp threshold determining whether or not cholera dies out. The global stability for disease-free equilibrium and endemic equilibrium is proved by constructing suitable Lyapunov functionals. However, for the proof of the global stability of endemic equilibrium, we have to show first the relative compactness of the orbit generated by model in order to make use of the invariance principle. Furthermore, uniform persistence of system must be shown since the Lyapunov functional is possible to be infinite if $i(a, t)/i^* (a) =0$ on some age interval. In this note, we give a supplement to above paper with necessary mathematical arguments.
keywords:
global stability
,
Cholera model
,
Lyapunov functional
,
uniform persistence.
,
age-of-infection

MBE

A recent paper [Y. Xiao and X. Zou,

*On latencies in malaria infections and their impact on the disease dynamics*, Math. Biosci. Eng., 10(2) 2013, 463-481.] presented a mathematical model to investigate the spread of malaria. The model is obtained by modifying the classic Ross-Macdonald model by incorporating latencies both for human beings and female mosquitoes. It is realistic to consider the new model with latencies differing from individuals to individuals. However, the analysis in that paper did not resolve the global malaria disease dynamics when $\Re_0>1$. The authors just showed global stability of endemic equilibrium for two specific probability functions: exponential functions and step functions. Here, we show that if there is no recovery, the endemic equilibrium is globally stable for $\Re_0>1$ without other additional conditions. The approach used here, is to use a direct Lyapunov functional and Lyapunov- LaSalle invariance principle.
DCDS-B

This paper provides the first detailed analysis of a multi-group SIR epidemic model with age structure, which is given by a nonlinear system of $3n$ partial differential equations.
The basic reproduction number $\mathcal{R}_0$ is obtained as the spectral radius of the next generation operator, and it is shown that if $\mathcal{R}_0 < 1$, then the disease-free equilibrium is globally asymptotically stable, while if $\mathcal{R}_0 >1$, then an endemic equilibrium exists. The global asymptotic stability of the endemic equilibrium is also shown under additional assumptions such that the transmission coefficient is independent from the age of infective individuals and the mortality and removal rates are constant.
To our knowledge, this is the first paper which applies the method of Lyapunov functional and graph theory to a multi-dimensional PDE system.

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