2017 CMS Winter Meeting

Waterloo, December 8 - 11, 2017


Contributed Papers
Org: Robert Andre and Dan Wolczuk (University of Waterloo)

MO'TASSEM AL-ARYDAH, Khalifa University

We formulate an age-structured model based on a system of nonlinear partial differential equations to assist the early and catch up female vaccination programs for HPV types 6 and 11. Since these HPV types do not induce permanent immunity, the model, which stratifies the population based on age and gender, has a susceptible- infectious-susceptible (SIS) structure.We calculate the effective reproduction number $R_v$ for the model and describe the local-asymptotic stability of the disease-free equilibrium using $R_v$. We prove the existence of an endemic equilibrium for $R_v > 1$ for the no vac- cine case. However, analysis of the model for the vaccine case reveals that it undergoes the phenomenon of backward bifurcation. To support our theoretical results we estimate the age and time solution with given data for Toronto population, when an early and catch up female vaccine program is adopted, and when there is no vaccine. We show that early and catch up female vaccine program eliminates the infection in both male and female populations over a period of 30 years. Finally, we introduce optimal control to an age-dependent model based on ordinary differential equations and solve it numerically to obtain the most cost-effective method for introducing the catch up vaccine into the population.

LIN JIU, Dalhousie University
Probabilistic and combinatorial interpretations of Bernoulli symbol  [PDF]

The Bernoulli symbol comes from umbral calculus, with simple evaluation rule that identifying super index, i.e., power, and lower index. Recently, the probabilistic interpretation allows to view the evaluation as expectation of the Bernoulli random variable, having hyperbolic secant square as its density on the whole real line. Further study on the corresponding moment problem, cumulants, especially orthogonal polynomial sequence and continued fractions, links Bernoulli numbers to generalized Motzkin numbers, providing combinatorial interpretations.

APITA KAR, Queen's University, Kingston, Canada
On a Conjecture of Bateman about $r_5(n)$  [PDF]

Let $r_5(n)$ be the number of ways of writing $n$ as a sum of five integer squares. In his study of this function, Bateman was led to formulate a conjecture regarding the sum $$\sum_{|j| \leq \sqrt{n}}\sigma(n-j^2)$$ where $\sigma(n)$ is the sum of positive divisors of $n$. We give a proof of Bateman's conjecture in the case $n$ is square-free and congruent to $1$ (mod $4$). This is joint work with Prof. Ram Murty (Queen's University).


Centre de recherches mathématiques Pacific Institute for the Mathematical Sciences Fields Institute AARMS: Atlantic Association for Research in the Mathematical Sciences University of Waterloo

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