**15)
***The Canada-China Minisymposium Session on Mathematical
Epidemiology, I, II, and III ***(3
minisymposia)**

Principal organisers: Professor Jianhong Wu (York), Yican
Zhou (Xian Jiaotong) and

Xingfu Zou (Western Ontario)

The
special session is an important part of the Canada-China Thematic
Program on Disease Modeling. This session will be coordinated
by Professor Yicang Zhou (Xi-an Jiaotong University), Professor
Xingfu Zou (University of Western Ontario) and Jianhong Wu
(York University). It is hoped this session will bring together
distinguished speakers from both Canada and China in order
to facilitate the scientific exchange and to enhance the bilateral
collaboration between scientists in Canada and China.

The
Canada-China Thematic Program on Disease Modeling was officially
launched at Beijing University in May of 2007, and it has
been a major component of a collaborative program between
The Mathematical Centre of the Chinese Ministry of Education
and the center Mathematics for Information Technology and
Complex Systems (MITACS), one of the Network of Centers of
Excellence (NCE) funded by the federal government of Canada.
This collaborative program has been funded by the International
Development Research Center and the NCE.

** Minisymposium
***I*

Friday,
10:00-10:30am.

**1. ****Robert Smith?** Department of Mathematics
and Statisitcs, University of Ottawa

Evaluating human papillomavirus vaccination programs in Canada:
should provincial healthcare pay for voluntary adult vaccination?

Recently,
provincial health programs in Canada and elsewhere have begun
rolling out vaccination against human papillomavirus for girls
aged 9-13. While vaccination is voluntary, the cost of vaccination
is waived, to encourage parents to have their daughters vaccinated.
Adult women who are eligible for the vaccine may still receive
it, but at a cost of approximately CAN$400. Given the high
efficacy and immuno-genicity of the vaccine, the possibility
of eradicating targeted types of the virus may be feasible,
assuming the vaccination programs are undertaken strategically.
We develop a mathematical model to describe the epidemiology
of vaccination against human papillomavirus, accounting for
a widespread childhood vaccination program that may be supplemented
by voluntary adult vaccination. A stability analysis is performed
to determine the stability of the disease-free equilibrium.
The critical vaccine efficacy and immunogenicity thresholds
are derived, and the minimum level of adult vaccination required
for eradication of targeted types is determined. We demonstrate
that eradication of targeted types is indeed feasible, although
the burden of coverage for a childhood-only vaccination program
may be high. However, if a small, but non-negligible, proportion
of eligible adults can be vaccinated, then the possibility
of eradication of targeted types becomes much more favourable.
We provide a threshold for eradication in general communities
and illustrate the results with numerical simulations. We
also investigate the effects of suboptimal efficacy and immunogenicity
and show that there is a critical efficacy below which eradication
of targeted types is not possible. If eradication is possible,
then there is a critical immunogenicity such that even 100%
childhood vaccination will not eradicate the targeted types
of the virus and must be supplemented with voluntary adult
vaccination. However, the level of adult vaccination coverage
required is modest and may be achieved simply by removing
the cost burden to vaccination. Consequently, we recommend
that provincial healthcare programs should pay for voluntary
adult vaccination for women aged 14-26.

**2.**Friday,
10:30-11:00am.

**Jie
Lou** Department of Mathematics, Shanghai University Shanghai,
P.R. China.

HIV? what do we know? What can we do using mathematics?

In this report, we will introduce some mechanisms about HIV
and how to solve some clinical problems using dynamic models.
We will mainly introduce some models in vivo: The first one
is about the function of APC during the HIV infection; The
second one is about cancers on HIV infected individuals; The
third one is about drug therapy for HIV individual. Some interesting
results about HIV will be presented in this report.

3.
Friday, 11:00-11:30am.

**Junling Ma** Department of Mathematics and Statisitcs,
University of Victoria.

*Epidemics on an evolving network*

Network
epidemiological models have recently become popular. They
provide a more realistical description to the underlying contact
process than most ODE and PDE models. Most previous research
on network models focused on static networks. However, many
diseases spread on evolving networks, because of either the
evolving nature of the underlying social networks, or the
control measures such as quarantine/isolation and social distancing.
In this talk, we study a simple SIS model spreading on an
evolving network with one of the following two types of rewiring:
rewiring independent of the disease, and social distancing
(the susceptibles rewire away from the infectives). The effect
of the rewiring process on the basic reproduction number is
discussed.

4.
Friday, 11:30-12:00am

**Sanyi Tang** College of Mathematical and Information
Science, Shaanxi Normal University.

**Hybrid Host-Parasitoid/Pathogen Models for Integrated
Pest/Disease Management**

A hybrid biological system is a dynamic system that exhibits
both continuous and discrete dynamic behavior. This broad
system class includes continuous systems intervened by discrete
events, such as biological cell growth, genetic regulation,
integrated pest/disease management (IPM/IDM) and drug design.
In this talk, we firstly provided several examples (economic
threshold in IPM/IDM and disease treatment with therapeutic
window) to show that the hybrid biological systems are much
more realistic. Then we focus on the development, investigation
and biological implications of continuous and discrete host-parasitoid/pathogen
models concerning IPM/IDM with impulsive effects. The theoretical
results with regard to host-eradication, host-parasitoid/pathogen
persistence and host-outbreak solutions are obtained. Solutions
for all three categories can coexist, with switch-like transitions
among their attractors showing that varying dosages and frequencies
of insecticide applications and the numbers of parasitoids/pathogens
released are crucial. The dosages and frequencies of IPM interventions
for these solutions are much reduced in comparison with the
pest-eradication periodic solution. Our results, which are
robust to inclusion of stochastic effects and with a wide
range of parameter values, confirm that IPM is more effective
than any single control tactic.

**Minisymposium
II **

Friday, 3:00-3:30pm.

5.** Jane Heffernan** Department of Mathematics and Statistics,
York University.

*Correction factors for the basic reproduction ratio
R0*

The basic reproductive ratio, R0, is defined as the expected
number of secondary infections arising from a single infected
individual during his or her entire infectious period, in
a population of susceptibles. In both within-host and epidemiological
models of pathogen dynamics R0 is a powerful tool for gauging
the risk associated with an emerging pathogen, or for estimating
the magnitude of required control measures. One limitation,
however, is that estimates of R0 often rely on mean transition
times (i.e. infection, recovery, death) and do not include
information about the dispersal of these transition times
about the mean. We have derived a correction factor that can
be applied to improve estimates of R0 when both the mean and
the standard deviation of transitions times are known. This
tool can be used to determine the sensitivity of R0 to underlying
lifetime distributions, which has numerous applications both
in mathematical immunology and epidemiology.

6.Friday
3:30-4:00pm

**Han
Litao** School of Information, Renmin University of China
Beijing

Study on Epidemic Models of Two Interaction Species

Some epidemic models of two interaction species (competition
or predation) are studied. For the models of two competitive
species, the SIS and SIRS types are discussed for different
incidence forms. The influence of the inter-infection of disease
on transmission and its real significance are revealed. Namely,
when the two species coexist and disease persists in both
of them, if the inter-infection is removed, it is found that
three different cases may occur:

1) Disease still persists in both species;

2) Disease persists in one species, but dies out in the other
species;

3) Disease dies out in both species.

For the prey-predator models, the SIS and SIR types are discussed
for the different incidence forms. The influence of the predation-infection
of disease on transmission and its real significance are revealed.
Namely, when the two species coexist and disease persists
in the prey species, it is found that disease must persist
in the predator species because of the predation-infection.

7.Friday
4:00-4:30pm

**Kaifa Wang** Department of Computer Science, College
of Medicine Third Military Medical University.

**A Viral infection model with a nonlinear infection rate**

A viral infection model with a nonlinear infection rate is
constructed based on the empirical evidences and the dynamical
behaviours are investigated by using qualitative method, bifurcation
theory and numeric simulations. The results indicated that
our model can display Allee effect, static and dynamical bifurcations
including saddle node bifurcation, Hopf bifurcation , homoclinic
bifurcation and bifurcation of cusp-type with codimension
two (i.e., Bogdanov-Takens bifurcation), which are important
for making strategies for controlling the invasion of virus.

8.
Friday 4:30-5:00pm

**Chris Bauch** Department of Mathematics and Statistics,
University of Guelph

**Wealth as a source of density dependence in human population
growth: evidence from the demographic transition**

The phenomenon of density dependence, whereby higher population
density results in a reduced population growth rate due to
resource limitations, is a topic of intensive research in
natural populations. In modern industrialized nations, human
birth rates have been declining persistently for decades,
and have now fallen below the replacement threshold in many
countries. However, unlike in natural populations, lower birth
rates in modern industrialized countries appear to be positively
correlated with resource availability, e.g. gross domestic
product (GDP) per capita. Here, we show that declining birth
rates in human populations are actually a manifestation of
density-regulated population growth brought on by socioeconomic
development, as reflected by GDP per capita. This is demonstrated
by combining empirical power law relations between population
size, GDP per capita, and fertility in a simple theoretical
model describing population dynamics in developed countries.
For a closed population, the model exhibits growth to a globally-stable
equilibrium population size, for both national and city populations.
The growth dynamics are highly sensitive to demographic and
economic power law exponents. An extended model for a country
that is open to the flow of labor, technology and capital
form other countries exhibits a good fit to long-term time
series data on population size, GDP per capita, and births
rates for the United States, France, and Japan, and provides
future projections as well. An implication of this work for
sustainability is that the earth's human population may limit
itself before environmental constraints cause widespread premature
death, as predicted by Malthusean models.

**Minisymposium
III**

Saturday
10:00-10:30am

9.
**Yan Wang**, Department of Applied Mathematics, Xian Jiaotong.

**Oscillatory Viral Dynamics in a HIV Pathogenesis Incorporating
Antiretroviral Therapy and Time Delay**

We consider a HIV pathogenesis model incorporating antiretroviral
therapy and HIV replication time. We investigate the existence
and stability of equilibria, as well as hopf bifurcations
of sustained oscillation when drug efficacy is

less than 100\%. We derive sufficient conditions for the globally
asymptotic stability of the uninfected steady state. We show
that time delay has no effect on the local asymptotic stability
of the infected steady state, but can destabilize the infected
steady state, leading to a hopf bifurcation of periodic solutions
in the realistic parameter ranges.

10.
Saturday 10:30-11:00am

**Peixuan Weng**, Department of Mathematics, South China
Normal University

*Spreading Speed and Traveling Waves for Multi-type SIS
Epidemic Model*

The theory of asymptotic speeds of spread and monotone traveling
waves for monotone semiflows is applied to a multi-type SIS
epidemic model to obtain the spreading speed c_, and the nonexistence
of traveling waves with wave speed c < c_. Then the method
of upper and lower solutions is used to establish the existence
of monotone traveling waves connecting the disease-free and
endemic equilibria for c _ c_. This shows that the spreading
speed coincides with the minimum wave speed for monotone traveling
waves. We also give an affirmative answer to an open problem
presented by Rass and Radcliffe.