DCDS-B

In this paper, we consider global stability for a heroin model with
two distributed delays. The basic reproduction number of the heroin
spread is obtained, which completely determines the stability of the
equilibria. Using the direct Lyapunov method with Volterra type
Lyapunov function, we show that the drug use-free equilibrium is
globally asymptotically stable if the basic reproduction number is
less than one, and the unique drug spread equilibrium is globally
asymptotically stable if the basic reproduction number is greater
than one.

DCDS-B

This paper focuses on the study of an age-structured SIRS epidemic
model with a vaccination program. We first give the explicit
expression of the
reproductive number $ \mathcal{R}(\psi) $ in the presence of vaccine, and
show that the infection-free steady state is locally asymptotically stable
if $ \mathcal{R}(\psi)<1 $ and unstable if $ \mathcal{R}(\psi)>1 $.
Second, we prove that the infection-free state is globally stable if
the basic reproductive number $ \mathcal{R}_0 <1 $, and that an endemic
equilibrium exists when the reproductive number $ \mathcal{R}(\psi)>1 $.

MBE

In this paper, a partial differential equation (PDE) model is proposed to explore the transmission dynamics of vector-borne diseases. The model includes both incubation age of the exposed hosts and infection age of the infectious hosts which describe incubation-age dependent removal rates in the latent period and the variable infectiousness in the infectious period, respectively. The reproductive number $\mathcal R_0$ is derived. By using the method of Lyapunov function, the global dynamics of the PDE model is further established, and the results show that the basic reproduction number $\mathcal R_0$ determines the transmission dynamics of vector-borne diseases: the disease-free equilibrium is globally asymptotically stable if $\mathcal R_0≤ 1$, and the endemic equilibrium is globally asymptotically stable if $\mathcal{R}_0>1$. The results suggest that an effective strategy to contain vector-borne diseases is decreasing the basic reproduction number $\mathcal{R}_0$ below one.

MBE

In this paper, an age-structured epidemic
model is formulated to describe the transmission dynamics of
cholera. The PDE model incorporates direct and indirect transmission
pathways, infection-age-dependent infectivity and variable periods
of infectiousness. Under some suitable assumptions, the PDE model
can be reduced to the multi-stage models investigated in the
literature. By using the method of Lyapunov function, we established
the dynamical properties of the PDE model, and the results show that
the global dynamics of the model is completely determined by the
basic reproduction number $\mathcal R_0$: if $\mathcal R_0 < 1$
the cholera dies out, and if $\mathcal R_0
>1$ the disease will persist at the endemic
equilibrium. Then the global results obtained for multi-stage models
are extended to the general continuous age model.

MBE

This article
focuses on the study of an age-structured
two-strain model with super-infection. The explicit expression of
basic reproduction numbers and the invasion reproduction numbers
corresponding to strain one and strain two are obtained. It is
shown that the infection-free steady state is globally stable if
the basic reproductive number $ R_0 $ is below one. Existence
of strain one and strain two exclusive equilibria is established.
Conditions for local stability or instability
of the exclusive equilibria of the
strain one and strain two are established. Existence of
coexistence equilibrium is also obtained under the condition that both
invasion reproduction numbers are larger than one.

DCDS-B

An infection-age-structured epidemic model
with environmental bacterial infection is investigated in this
paper. It is assumed that the infective population is
structured according to age of infection, and the infectivity of the treated individuals is reduced but
varies with the infection-age.
An explicit formula for the reproductive number $ \Re_0$ of the model
is obtained. By constructing a suitable Lyapunov function, the
global stability of the infection-free equilibrium in the system
is obtained for $\Re_0<1$. It is also shown that if the
reproduction number $\Re_0>1$, then the system has a unique endemic
equilibrium which is locally asymptotically stable. Furthermore, if
the reproduction number $\Re_0>1$, the system is permanent. When
the treatment rate and the transmission rate are both independent
of infection age, the system of partial differential equations
(PDEs) reduces to a system of ordinary differential equations
(ODEs). In this special case, it is shown that the global dynamics
of the system can be determined by the basic reproductive number.

MBE

We consider a model for a disease with
two competing strains and vaccination.
The vaccine provides complete protection against one of the strains
(strain 2) but only
partial protection
against the other (strain 1). The partial protection leads to
existence of subthreshold equilibria of strain 1. If the first strain
mutates into the second, there are subthreshold coexistence equilibria
when both vaccine-dependent reproduction numbers are below one.
Thus, a vaccine that is specific toward the second strain
and that, in absence of other strains, should be able to eliminate
the second strain
by reducing its reproduction number below one,
cannot do so because it provides only
partial protection to another strain that mutates into the second strain.

keywords:
latent stage
,
coexistence
,
strongly
subthreshold coexistence
,
vaccine enhanced pathogen polymorphism.
,
multiple coexistence equilibria
,
multiple endemic equilibria
,
mutation
,
backward bifurcation
,
latent-stage progression age structure
,
alternating stability
,
vaccination