We formulate differential susceptibility and differential infectivity
models for disease transmission in this paper. The susceptibles are divided into
n groups based on their susceptibilities, and the infectives are divided into m
groups according to their infectivities. Both the standard incidence and the
bilinear incidence are considered for different diseases. We obtain explicit
formulas for the reproductive number. We define the reproductive number
for each subgroup. Then the reproductive number for the entire population
is a weighted average of those reproductive numbers for the subgroups. The
formulas for the reproductive number are derived from the local stability of
the infection-free equilibrium. We show that the infection-free equilibrium is
globally stable as the reproductive number is less than one for the models with
the bilinear incidence or with the standard incidence but no disease-induced
death. We then show that if the reproductive number is greater than one,
there exists a unique endemic equilibrium for these models. For the general
cases of the models with the standard incidence and death, conditions are
derived to ensure the uniqueness of the endemic equilibrium. We also provide
numerical examples to demonstrate that the unique endemic equilibrium is
asymptotically stable if it exists.
To study the impact of media coverage on spread and control of infectious diseases, we use a susceptible-exposed-infective (SEI) model, including individuals' behavior changes in their contacts due to the influences of media coverage, and fully investigate the model dynamics. We define the basic reproductive number $\Re_0$ for the model, and show that the modeled disease dies out regardless of initial infections when $\Re_0 < 1$, and becomes uniformly persistently endemic if $\Re_0>1$. When the disease is endemic and the influence of the media coverage is less than or equal to a critical number, there exists a unique endemic equilibrium which is asymptotical stable provided $\Re_0 $ is greater than and near one. However, if $\Re_0 $ is larger than a critical number, the model can undergo Hopf bifurcation such that multiple endemic equilibria are bifurcated from the unique endemic equilibrium as the influence of the media coverage is increased to a threshold value. Using numerical simulations we obtain results on the effects of media coverage on the endemic that the media coverage may decrease the peak value of the infectives or the average number of the infectives in different cases. We show, however, that given larger $\Re_0$, the influence of the media coverage may as well result in increasing the average number of the infectives, which brings challenges to the control and prevention of infectious diseases.
In this paper, we formulate a mathematical model for malaria transmission that
includes incubation periods for both infected human hosts and mosquitoes. We
assume humans gain partial immunity after infection and divide the infected human
population into subgroups based on their infection history. We derive an explicit
formula for the reproductive number of infection, $R_0$, to determine threshold
conditions whether the disease spreads or dies out. We show that there exists an
endemic equilibrium if $R_0>1$. Using an numerical example, we demonstrate that
models having the same reproductive number but different numbers of progression
stages can exhibit different transient transmission dynamics.
To study the impact of releasing transgenic mosquitoes on malaria transmission, we
formulate discrete-time models for interacting wild and transgenic mosquitoes
populations, based on systems of difference equations. We start with models
including all homozygous and heterozygous mosquitoes. We then consider either
dominant or recessive transgenes to reduce the 3-dimensional model systems to
2-dimensional systems. We include density-dependent vital rates and incorporate
Allee effects in the functional mating rates. Dynamics of these models are explored
by investigating the existence and stability of boundary and positive fixed points.
Numerical simulations are provided and brief discussions are given.
In this paper, we include two time delays in a mathematical model for the CD8$^+$ cytotoxic
T lymphocytes (CTLs) response to the Human T-cell leukaemia virus type I (HTLV-I) infection,
where one is the intracellular infection delay and the other is the immune delay to account for a
series of immunological events leading to the CTL response. We show that the global dynamics
of the model system are determined by two threshold values $R_0$, the corresponding
reproductive number of a viral infection, and $R_1$, the corresponding reproductive number
of a CTL response, respectively. If $R_0<1$, the infection-free equilibrium is globally
asymptotically stable, and the HTLV-I viruses are cleared. If $R_1 < 1 < R_0$, the immune-free
equilibrium is globally asymptotically stable, and the HTLV-I infection is chronic but with no
persistent CTL response. If $1 < R_1$, a unique HAM/TSP equilibrium exists, and the HTLV-I
infection becomes chronic with a persistent CTL response. Moreover, we show that the immune
delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations. Our numerical
simulations suggest that if $1 < R_1$, an increase of the intracellular delay may stabilize the
HAM/TSP equilibrium while the immune delay can destabilize it. If both delays increase, the
stability of the HAM/TSP equilibrium may generate rich dynamics combining the ``stabilizing"
effects from the intracellular delay with those ``destabilizing" influences from immune delay.
A simple SEIR model for malaria transmission dynamics is formulated as
our baseline model. The metamorphic stages in the mosquito population are
then included and a simple stage-structured mosquito population model is
introduced, where the mosquito population is divided into two classes, with
all three aquatic stages in one class and all adults in the other class, to keep
the model tractable in mathematical analysis. After a brief investigation of
this simple stage-structured mosquito model, it is incorporated into the
baseline model to formulate a stage-structured malaria model. A basic
analysis for the stage-structured malaria model is provided and it is shown
that a theoretical framework can be built up for further studies on the impact
of environmental or climate change on the malaria transmission. It is also
shown that both the baseline and the stage-structured malaria models
undergo backward bifurcations.
We formulate and study epidemic models with differential susceptibilities and
staged-progressions, based on systems of ordinary differential equations, for
disease transmission where the susceptibility of susceptible individuals vary
and the infective individuals progress the disease gradually through stages
with different infectiousness in each stage. We consider the contact rates to
be proportional to the total population or constant such that the infection
rates have a bilinear or standard form, respectively. We derive explicit
formulas for the reproductive number $R_0$, and show that the infection-free
equilibrium is globally asymptotically stable if $R_0<1$ when the infection
rate has a bilinear form. We investigate existence of the endemic equilibrium
for the two cases and show that there exists a unique endemic equilibrium for
the bilinear incidence, and at least one endemic equilibrium for the standard
incidence when $R_0>1$.
Thomas Guy Hallam began his career as a faculty member in
the Department of Mathematics at Florida State University, working
in the area of comparison theorems for ordinary differential
equations. While at Florida State he organized a mathematical
modeling course and thus became interested in mathematical biology.
He began to wonder how he, as a mathematician, might address the
mounting environmental problems. He took courses in oceanography and
ecology and delved deeply into the literature. During the summer of
1974, he gave a full series of lectures on mathematical biology at
the University of São Carlos in São Paulo, Brazil. In 1976,
he took a year's leave at the University of Georgia, Athens, in the
Departments of Mathematics, Zoology, and the Institute of Ecology,
where he met Tom Gard and Ray Lassiter, with whom he has had
career-long interactions. In Athens, he became interested in
ecotoxicology and partial differential equation models of
physiologically structured populations.
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Color image demosaicing consists in recovering full resolution color information from color-filter-array (CFA) samples with 66.7% amount of missing data. Most of the existing color demosaicing methods [14, 25, 16, 2, 26] are based on interpolation from inter-channel correlation and local geometry, which are not robust to highly saturated color images with small geometric features. In this paper, we introduce wavelet frame based methods by using a sparse wavelet [8, 22, 9, 23] approximation of individual color channels and color differences that recovers both geometric features and color information. The proposed models can be efficiently solved by Bregmanized operator splitting algorithm . Numerical simulations of two datasets: McM and Kodak PhotoCD, show that our method outperforms other existing methods in terms of PSNR and visual quality.
Zhien Ma's love for mathematics has strongly shaped his educational pursuits. He received formal training from the strong Chinese School of Dynamical System over a period of two years at Peking University and during his later visit to Nanjing University where the internationally renowned professor Yanqian Ye mentored him. Yet, it is well known that Zhien's curiosity and love of challenges have made him his own best teacher. Hence, it is not surprising to see his shift from an outstanding contributor to the field of dynamical systems to a pioneer in the field of mathematical biology. Zhien's vision and courage became evident when he abandoned a promising career in pure mathematics and enthusiastically embraced a career in the field of mathematical biology soon after his first visit to the United States in 1985. His rapid rise to his current role as an international leader and premier mentor to 11 Ph.D. students has facilitated the placement of Chinese scientists and scholars at the forefront of research in the fields of mathematical, theoretical and computational biology.
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