DCDS-B

In this paper, we consider a class of epidemic models
described by five nonlinear ordinary differential equations. The population
is divided into susceptible, vaccinated, exposed, infectious, and recovered subclasses.
One main feature of this kind of models is that treatment and vaccination
are introduced to control and prevent infectious diseases. The existence and local stability
of the endemic equilibria are studied.
The
occurrence of backward bifurcation is established by using center manifold theory.
Moveover, global dynamics are studied by applying the geometric approach.
We would like to mention that in the case of bistability, global results are difficult
to obtain since there is no compact absorbing set. It is the first time
that higher (greater than or equal to four) dimensional systems are discussed.
We give sufficient conditions in terms of the system parameters by extending
the method in Arino et al. [2]. Numerical simulations are also provided to
support our theoretical results. By carrying out sensitivity analysis of the basic
reproduction number in terms of some parameters, some effective measures to control
infectious diseases are analyzed.

MBE

As an important ecosystem, alpine meadow in China has been degraded severely over the past few decades. In order to restore degraded alpine meadows efficiently, the underlying causes of alpine meadow degradation should be identified and the efficiency of restoration strategies should be evaluated. For this purpose, a mathematical modeling exercise is carried out in this paper. Our mathematical analysis shows that the increasing of raptor mortality and the decreasing of livestock mortality (or the increasing of the rate at which livestock increases by consuming forage grass) are the major causes of alpine meadow degradation. We find that controlling the amount of livestock according to the grass yield or ecological migration, together with protecting raptor, is an effective strategy to restore degraded alpine meadows; while meliorating vegetation and controlling rodent population with rodenticide are conducive to restoring degraded alpine meadows. Our analysis also suggests that providing supplementary food to livestock and building greenhouse shelters to protect livestock in winter may contribute to alpine meadow degradation.

MBE

Infection age is an important factor affecting the transmission of
infectious diseases. In this paper, we consider an SIRS model
with infection age, which is described by a mixed system of
ordinary differential equations and partial differential
equations. The expression of the basic reproduction number
$\mathscr {R}_0$ is obtained. If $\mathscr{R}_0\le 1$ then the
model only has the disease-free equilibrium, while if
$\mathscr{R}_0>1$ then besides the disease-free equilibrium the
model also has an endemic equilibrium. Moreover, if
$\mathscr{R}_0<1$ then the disease-free equilibrium is globally
asymptotically stable otherwise it is unstable; if
$\mathscr{R}_0>1$ then the endemic
equilibrium is globally asymptotically stable under additional conditions. The local stability
is established through linearization. The global stability of the
disease-free equilibrium is shown by applying the fluctuation
lemma
and that of the endemic equilibrium is proved by employing Lyapunov functionals.
The theoretical results are illustrated with numerical simulations.

MBE

A multi-group model is proposed to describe a general relapse phenomenon of infectious diseases
in heterogeneous populations.
In each group, the population is divided into
susceptible, exposed, infectious, and recovered subclasses. A general
nonlinear incidence rate is used in the model. The results show that the global dynamics are completely
determined by the basic reproduction number $R_0.$ In particular, a matrix-theoretic method is used to prove
the global stability of the disease-free equilibrium when $R_0\leq1,$
while a new combinatorial identity (Theorem 3.3 in Shuai and van
den Driessche [29]) in graph theory is applied to prove
the global stability of the endemic equilibrium when $R_0>1.$
We would like to mention that by applying the new combinatorial identity, a graph of 3n (or 2n+m) vertices can be converted into
a graph of n vertices in order to deal with the global stability of the endemic equilibrium in this paper.