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.