Mathematics of Infectious Diseases
Org: Abba Gumel (Manitoba) [PDF]
 CHRISTOPHER BOWMAN, Inst. for Biodiagnostics

 FRED BRAUER, University of British Columbia
Epidemic models with heterogeneous mixing
[PDF] 
We formulate and analyze an epidemic model in a population with groups
having different contact rates and proportionate mixing between
groups. We obtain a general expression for the reproduction number
and the final size of the epidemic. We are able to carry out these
calculations also if treatment of infectives is incorporated into the
model.
 ELAMIN ELBASHA, Merck Research Laboratories
Global dynamics of an HPV vaccination model
[PDF] 
We analyze a simple fourdimensional model of the susceptible,
infective, and recovered (SIR) type. The model describes the
transmission dynamics of human papillomavirus (HPV) infection
following the introduction of a mass vaccination program. By
constructing suitable Lyapunov functions, we prove that the global
dynamics of this model are determined by the reproduction number
R_{v}. If R_{v} is less than unity, there is a unique infectionfree
equilibrium which is globally asymptotically stable. For R_{v}
greater than unity, the infectionfree equilibrium is unstable, and
there is a unique endemic equilibrium which is globally asymptotically
stable. Future extensions of the model will be discussed.
 HONGBIN GUO, University of Alberta, 632 CAB University of Alberta,
Edmonton, AB, T6G 2G1
Global Dynamics of a Staged Progression Model for Infectious
Diseases with Amelioration
[PDF] 
A mathematical model for infectious diseases that progress through
distinct stages within infected hosts is considered. An example of
such diseases is AIDS which results from HIV infection. For a general
nstage stageprogression (SP) model with amelioration, we prove
that the global dynamics are completely determined by the basic
reproduction number R_{0}. If R_{0} £ 1,then the diseasefree
equilibrium P_{0} is globally asymptotically stable and the disease
always dies out. If R_{0} > 1, P_{0} is unstable, and a unique endemic
equilibrium P^{*} is globally asymptotically stable, and the disease
persists at the endemic equilibrium.
This is joint work with Michael Y. Li, Dept. of Mathematical and
Statistical Sciences, University of Alberta.
 MUDASSAR IMRAN, North Carolina State University
The Pharmacodynamics of Antibiotic Treatment
[PDF] 
We derive models of the effects of periodic, discrete dosing or
constant dosing of antibiotics on a bacterial population whose growth
is checked by nutrientlimitation and possibly by host defenses.
Mathematically rigorous results providing sufficient conditions for
treatment success, i.e., the elimination of the bacteria, as well as
for treatment failure, are obtained. Our models can exhibit
bistability where the infectionfree state and an infectionstate are
locally stable when antibiotic dosing is marginal. In this case,
treatment success may occur only for subthreshold level infections.
 ZACK JACOBSON, Health Canada
How does vaccination affect transient population dynamics?
[PDF] 
The talk focuses on the following question: if disease spread in a
population is a dynamic process, how will that be affected by
vaccination? By exploring transient dynamics and nonlinearities in
disease spread within a population and actions of vaccination as an
external impulse or force that modifies population dynamics, it is
possible to evaluate the impact of the vaccination program and also to
compare different vaccination strategies and schedules. We
investigate all these using suitable vaccination models (stochastic
and/or deterministic) with an eye on optimizing the impact of
vaccination on disease spread. In particular, we seek to determine
ways the system can be made to converge to a stable equilibrium point
where the number of infected individuals is small or zero. Some
pertinent modeling questions associated with the use of pulse
vaccination are also enumerated.
This is joint work involving Dragos Calitoiu and Zachary Jacobson,
Health Canada.
 EDWARD LUNGU, University of Botswana, P. Bag 0022, Gaborone, Botswana
AntiTuberculosis resistance in patients coinfected with
HIV and TB
[PDF] 
Globally, levels of HIV and TB coinfection are high and continue to
increase rapidly. UNAIDS and WHO data indicate that one third of all
people with HIV have TB coinfection and similarly that up to 70% of
TB cases are HIV positive. In subSaharan Africa, lack of access to
affordable TB screening and treatment and poor referral and followup
systems are some of the reasons why individuals default on their TB
treatment, a situation which can lead to patients developing
resistance to TB drugs. Botswana implemented 100% coverage of the
DOTS (directly observed therapy short course) strategy in 1986 for all
newly infected HIV individuals in all health centers. Two surveys
undertaken in Botswana in the 1990's recorded low rates of
antituberculosis drug resistance despite a threefold rise in
tuberculosis since 1989. Sputum specimens obtained from patients
nationwide in 2002 who also underwent anonymous rapid HIV testing by
use of Oraquick showed that of the 2200 sputum smear positive patients
and 219 previously treated patients with suspected recurrent
tuberculosis, 1457 (60%) were infected with HIV.
Resistance to atleast one drug in new patients rose from 3.7% in 1995
to 10.4% in 2002. We construct a model to investigate
(i) whether the 100% coverage of DOTS is responsible for the
rise in TB drug resistance, and
(ii) whether screening newly infected HIV patients for TB and
administering DOTS to those without TB and treating those with TB
would reduce the possibility of developing resistance.
 TUFAIL MALIK, Arizona State UniversityMathematics, P.O. Box 871804,
Tempe, AZ 852871804
Microbial Quiescence, A Survival Strategy In Environmental
Stress
[PDF] 
Quiescence, or dormancy, is a strategy for microbial survival through
an environmental stress such as lack of a resource. To investigate
when quiescence is a beneficial strategy, we quantify and compare
fitness of a quiescencecapable population and that of a `sleepless'
quiescenceincapable population under various growth conditions.
Fitness is defined as the top Lyapunov exponent of certain
nonautonomous linear ordinary differential equations forced by
resource availability. Special attention is given to the case of
periodic and stochastic resource availability. Nonlinear models are
also considered where resource limitation is assumed to trigger
transition to and from the quiescent state.
 RONALD MICKENS, Clark Atlanta University
SIR Models with Ö{SI} Dynamics
[PDF] 
We investigate various forms of SIR models where the disease dynamics
is modeled by a Ö{SI} term in contrast to the standard SI
representation. It is shown, for the general case, that two
fixedpoints exist, one stable, one unstable. The unstable state
consists of only susceptible individuals, while the stable fixedpoint
has both susceptibles and infective individuals. Using nullclines, we
construct geometrically, in the 2dim SI phase space, the general
behavior of the associated trajectories. To obtain numerical
solutions, we show the construction of a nonstandard finite difference
(NSFD) scheme for this set of SIR differential equations.
The work reported here is supported by a grant from DOE and funds from
the MBRSSCORE Program at Clark Atlanta University.
 JEFF MUSGRAVE, UNB  Fredericton
An evaluation of control strategies for the HIV epidemic
[PDF] 
Since the discovery of HIV/AIDS there have been numerous mathematical
models proposed to explain the epidemic of the disease and to evaluate
possible control measures. In particular, several recent studies have
looked at the potential impact of condom usage on the epidemic
(Greenhalgh 2001, Gumel 2005, Hethcote 2000, Hyman 1999). We propose
a model of the effect of condom use and withdrawl on the spread of the
virus, similar to that of Gumel et al. (2005), and show that a simple
rescaling can be used to broaden the results of a sensitivity and
uncertainity analysis. Based on available estimates, we predict a
condom preventability of approximately 95% is necessary to ensure
control of the epidemic. A further simplication of the model,
replacing the standard incidence with a bilinear infection term and
assuming the demographic timescale is much slower then the disease
timescale, allows an estimate of the peak size of the epidemic.
 CHANDRA PODDER, University of Manitoba, 342 Machray Hall, 186 Dysart Road,
Winnipeg, R3T 2N2
Mathematical Analysis of a Model for Assessing the Impact of
Antiretroviral Therapy, Voluntary Testing and Condom Use in
Curtailing HIV
[PDF] 
This paper presents a deterministic model for evaluating the impact of
several antiHIV strategies, namely the use of antiretroviral drugs
(ARVs), voluntary HIV testing (using standard antibody test and a new
DNAbased test) and condom use. The model is rigorously analysed,
showing the existence of a globallystable diseasefree equilibrium
whenever a certain epidemiological threshold, known as the
effective reproduction number (R_{eff}), is less
than unity, an endemic equilibrium whenever R_{eff} > 1.
Simulations, using plausible parameter values, show that for
reasonably small testing and treatment rates, as well as modest condom
compliance (70%) and efficacy (87%), the use of condoms is the most
effective single intervention for reducing HIV burden, followed by the
use of ARVs and then voluntary HIV testing. If the testing and
treatment rates are increased (by 10fold) further, the use of ARVs
can offer better longterm benefit than any of the other
interventions. It is shown that the combined use of voluntary testing
methods and condom use can lead to significant reduction in HIV burden
than the singular use of ARV treatment if the testing and treatment
rates are low. Although it is shown that the use of ARVs is the most
effective control strategy, in the long run, for modestly high
treatment and testing rates, the lack of widespread availability of
these drugs call for the consideration of other affordable
interventions. This study shows that the combined use of voluntary
testing and condoms can be a costeffective means of combatting the
global spread of HIV.
 TIMOTHY RELUGA, Los Alamos National Laboratory
Population Games for Epidemiology
[PDF] 
In recent years, game theory has gained attention as a method to
explain behavior and evolution. In this talk, I'll describe how the
combination of classic epidemiology models with Markov decision
processes can be used to formulate population games and study public
health policy problems. Example applications to influenza vaccination
and polio will be discussed.
 OLUWASEUN SHAROMI, University of Manitoba, 342 Machray Hall, 186 Dysart Road,
Winnipeg, R3T 2N2
Mathematical Analysis of the Transmission Dynamics of HIV/TB
Coinfection in the Presence of Treatment and Condom Use
[PDF] 
This paper addresses the synergistic interaction between HIV and
mycobacterium tuberculosis using a deterministic model, which
incorporates many of the essential biological and epidemiological
features of the two diseases. In the absence of TB infection, the
model (HIVonly model) is shown to have a
globallyasymptotically stable diseasefree equilibrium whenever the
associated reproduction number is less than unity; and has a
unique endemic equilibrium whenever this number exceeds unity. On the
other hand, it was shown, using Centre Manifold theory, that the model
with TB alone (TBonly model) undergoes the phenomenon of
backward bifurcation, where the stable diseasefree equilibrium
coexists with a stable endemic equilibrium when the associated
reproduction threshold is less than unity.
The full model, with both HIV and TB, is also rigorously analysed.
Its simulation shows that the use of a treatment strategy that targets
only one of the two diseases not only results in significant reduction
of new cases of the disease being targeted for treatment, but also
induces an indirect benefit of reducing the number of new cases of the
other disease. Further, although treating individuals with TB only
(and those with dual HIV/TB infection treated for TB) always results
in more cases of TB prevented than that of HIV, the treatment of
people with HIV (including those with dual infection treated for HIV)
results in more cases of TB prevented than cases of HIV prevented.
Finally, the study shows that the universal treatment of individuals
infected with both diseases is more beneficial compared to treating
individuals infected with a single disease only.
 NAVEEN VAIDYA, York University, 4700 Keele Street, Toronto, ON, M3J 1P3
Modeling, Analysis, and Contol of the HIV Epidemics in Far
Western Nepal
[PDF] 
We present a dynamic transmission model of the HIV epidemics in Far
Western Nepal, where high rate of seasonal migration to India has been
the most threatening HIV risk factor. In addition to some analytical
and simulation results, we discuss the optimal control strategy based
on the results of our model.
 JAMES WATMOUGH, University of New Brunswick
The final size of an epidemic
[PDF] 
The early disease transmission model of Kermack and McKendrick
established two main results that are still at the core of most
disease transmission models today: the basic reproduction number,
R_{o}, as a threshold for disease spread in a population;
and the final size of an epidemic. As models become more complex, the
relationships between disease spread, final size and R_{o}
are not as clear; yet R_{o} remains the main object of study
when comparing control measures. In this talk I review the final size
relation for a simple epidemic model and discuss its form in more
complex models for treatment and control of influenza and HIV.
 MATTHIAS WINTER, Brunel University, Mathematical Sciences, Uxbridge UB8 3PH,
UK
Spikes for Biological Systems: The Role of Boundary Conditions
[PDF] 
We consider the shadow system of the GiererMeinhardt system in a
smooth bounded domain W Ì R^{N}:

ì ï ï í
ï ï î

A_{t} = e^{2} DA  A + 
A^{p}
x^{q}

, 
 
tW x_{t} = W x+ 
1
x^{s}


ó õ

W

A^{r} dx, 
 



with Robin boundary condition
e 
¶A
¶n

+ a_{A} A = 0, x Î ¶W, 

where a_{A} > 0.
The positive reaction rates (p,q,r,s) satisfy
1 < 
qr
(s+1)(p1)

< +¥, 1 < p < 
æ è

N+2
N2

ö ø

+

, 

the diffusion constant is chosen such that e << 1 and the time
relaxation constant such that t ³ 0.
We rigorously prove the following results on the stability of spiky
solutions:
(i) If r=2 and 1 < p < 1+4/N or if r = p+1 and 1 < p < ¥ then for a_{A} > 1 and t sufficiently small the
interior spike is stable.
(ii) For N=1 if r=2 and 1 < p £ 3 or if r = p+1 and
1 < p < ¥ then for 0 < a_{A} < 1 the nearboundary spike is
stable.
(iii) For N=1 if 3 < p < 5 and r=2 then there exist a_{0} Î (0,1) and m_{0} > 1 such that for a Î (a_{0},1) and m = [(2q)/((s+1)(p1))] Î (1,m_{0}) the nearboundary spike
solution is unstable. This instability is not present for the
Neumann boundary condition but only arises for Robin boundary
condition. Further we show that the corresponding eigenvalue is of
order O(1) as e® 0.
These results imply that some patterns may become more robust at the
expense of others which turn unstable. Results of this type are
important to understand the role of the boundary conditions in pattern
selection. For some biological applications such as the modelling of
skeletal limb development Robin (mixed) boundary conditions are more
realistic than Neumann (zeroflux) boundary conditions which are used
in most models.
This is joint work with Philip K. Maini (Oxford) and Juncheng Wei
(Hong Kong).
