# Research

**Key Words**

**Lyapunov methods, Differential equations, Global stability, Mathematical biology.**

**Main Areas**

**Mathematical Biology** - Mostly, I study population models, including epidemiology and ecology. Given a
disease in a population, can we determine the long-term
population-level behaviour of the disease? Will it die out or
persist? Oscillate periodically, go to a constant level or behave
chaotically?

In ecology, mathematical modelling can be used to gain insight into the dynamics of systems such as predator-prey interactions, competition for resources, mimicry of poisionous frogs by non-poisonous frogs.

These questions lead to interresting mathematical problems. The first issue is to determine what mathematical model might give useful insight into the biological system. Then, what can you do to analyze the model?

**Functional Differential Equations** - Many disease systems can
be modelled with delay differential equations or differential-integral
equations. In either case, the system is infinite-dimensional and
must be handled carefully. Nevertheless, progress can be made and
it is often possible to show global stability by using Lyapunov
functionals.

**Global Stability** - I am
interested in techniques for applying stability theorems including
Lyapunov methods and compound matrices. Given a differential
equation, numerical simulations can provide evidence of stability,
leading one to believe that a system is stable. However, this
approach does not *prove* that the system is stable. On the
other hand, applying stability theorems generally involves producing a
Lyapunov function that can be used to prove stability.
Unfortunately, there are very few constructive techniques for finding
Lyapunov functions. So, I am interested in constructive
techniques for just this purpose. How do you find a Lyapunov
function for a particular system - that seems to be stable, but has not
yet been proven to be stable.

**Ordinary Differential Equations** - Qualitative analysis. Given x'=f(x), how can you use
information about the vector field f to determine information about
solutions to the differential equation? Particularly, what
information about solutions can be found, when you are unable to find
the actual solutions?

**Matrix Analysis** - How
can we determine bounds on the eigenvalues of a matrix (or family of
matrices) given limited information. For example, if all we know
about each entry of a matrix is its sign, can we determine whether or
not all of the eigenvalues have negative real part? Given a
family of matrices A(t), can we determine the stability of the system
x'(t)=A(t) x(t)?

**Other Stuff** - Tuberculosis Models, Lozinskii Measures, Invariant Manifolds

# Some Current Projects

Modelling nosocomial infection (with Pierre Magal - Université de Bordeaux 2)

Modelling disease transmission within a transit/transportation system (with Fei Xu and Ross Cressman - WLU)

Age Structure and delay in the Chemostat

Lyapunov functions and functionals for disease models

Attractivity of Coherent Manifolds in Metapopulation Models (with David Earn - McMaster University)

Constructing discontinuous Lyapunov functions

# Student Supervision

Typically,
students that want to work with me will have some exposure to
differential equations. Projects could be theoretical or
applied. Applied projects would typically have a biology theme, however a biology background is not necessary.

### MSc:

Mihaela Popescu - current - *Global stability for the chemostat with delay*

Julie Nadeau - 2011 - *Mathematical models for feline infectious peritonitis*

Kazi Rahman - 2007 - *Modelling the
spread of HIV/AIDS in India: The role of transmission by commercial sex workers*

### NSERC summer students:

Chad Wells (2007) - *Synchronization in Mathematical Biology*

Adley
Au (2006) - *Maple implementation of compound
matrix techniques and construction of Lyapunov functions for linear
time-dependent differential equations.*

### Honours Thesis:

Amanda Vincent - 2010 - *Predator-Prey Models*

Katie Shepley - 2010 - *The Historical Development of Calculus*

Devin Glew - 2009 - *Mathematical Modeling of Disease Spread*

Mark MacLean - 2008 - *Calculus of Variations and Its Application to Physical Problems*

Mario Ayala - 2007-8 - *The Butler-McGehee Lemma*

Anne Coppins - 2007 - *An
Introduction to the Use of Difference Equations in Modeling Epidemics*

### Sanofi-Aventis BioTalent Challenge:

Michael Xu - 2010 - *Webs of Infection: Using Networks to Model Epidemics*

# Publications

**Papers in Refereed Journals**

22. C. McCluskey, D. Earn (2011).* Attractivity of coherent manifolds in
metapopulation models*, Journal of Mathematical Biology. 62:509-541.

21. C.
McCluskey (2010). *Delay versus age-of-infection – global stability,* Applied Mathematics and Computation.
217:3046-3049.

20. C.
McCluskey (2010). *Global
stability of an SIR epidemic model with delay and general nonlinear incidence,*
Mathematical Biosciences and Engineering. 7:837-850.

19. J.
Arino, C. McCluskey (2010).* Effect of a sharp change of the incidence function on the dynamics of a
simple disease,* Journal of Biological Dynamics. 4:490-505.

18. C.
McCluskey (2010). *Global
stability for an SIR epidemic model with delay and nonlinear incidence. *Nonlinear Analysis - Real World
Applications. 11:3106-3109.

17. P.
Magal, C. McCluskey, G. Webb (2010). *Liapunov functional and global asymptotic stability for an
infection-age model, *Applicable Analysis. 89:1109-1140.

16. C.
McCluskey (2010).* Complete
global stability for an SIR epidemic model with delay – Distributed or
discrete,* Nonlinear Analysis - Real World Applications. 11:55-59.

15. C.
McCluskey (2009).* Global
stability for an SEIR epidemiological model with varying infectivity and
infinite delay, *Mathematical Biosciences and Engineering. 6(3):603-610.

14. C.
McCluskey (2008).* On using
curvature to demonstrate stability,* Differential Equations and Nonlinear
Mechanics. Online:7 pages. doi:10.1155/2008/745242.

13. C.
McCluskey (2008). * Global
stability for a class of mass action systems allowing for latency in
tuberculosis,* Journal of Mathematical Analysis and Applications.
338:518-535.

12. A.
Gumel, C. McCluskey, P. van den Driessche (2006). *Mathematical study of a staged-progression HIV model with
imperfect vaccine,* Bulletin for Mathematical Biology. 68:2105-2128.

11. A.
Gumel, C. McCluskey, J. Watmough (2006). *An SVEIR model for assessing
potential impact of an imperfect anti-SARS vaccine, *Mathematical
Biosciences and Engineering*. *3:485-512.

10. C.
McCluskey (2006). *Lyapunov Functions for Tuberculosis Models
with Fast and Slow Progression,* Mathematical Biosciences and Engineering.
3:603-614.

9. C.
McCluskey (2006). *Equivalent Embeddings of the Dynamics on an
Invariant Manifold,* Journal of Mathematical Analysis and Applications.
317:277-285.

8. C.
McCluskey (2005). *A strategy for constructing Lyapunov
functions for non-autonomous linear differential equations,* Linear Algebra
and its Applications. 409:100-110.

7. M.
Ballyk, C. McCluskey, G. Wolkowicz (2005). *Global Analysis of
Competition for Perfectly Substitutable Resources with Linear Response*,
Journal of Mathematical Biology. 51:458-490.

6. C.
McCluskey, E. Roth, P. van den Driessche (2005). *Implication of Ariaal
Sexual Mixing on Gonorrhea,* American Journal of Human Biology. 17:293-301.

5. C.
McCluskey, P. van den Driessche (2004). *Global Analysis of Two Tuberculosis
Models*. Journal of Dynamics and Differential Equations. 16:139-166.

4. J.
Arino, C. McCluskey, P. van den Driessche (2003). *Global Results for an
Epidemic Model with Vaccination that Exhibits Backward Bifurcation,* SIAM
Journal on Applied Mathematics. 64:260-276.

3. C.
McCluskey (2003). *A model of
HIV/AIDS with staged progression and amelioration,* Mathematical
Biosciences. 181: 1-16.

2. C.
McCluskey, J. Muldowney (1998). *Stability
Implications of Bendixson's Criterion,* SIAM Review. 40: 931-934.

1. C. McCluskey,
J. Muldowney (1998). *Bendixson-Dulac Criteria for Difference
Equations,* Journal of Dynamics and Differential Equations. 10: 567-575.

**Papers in Refereed
Conference Proceedings**

2. C.
McCluskey, J. Muldowney (2004). *Stability Implications of Bendixson
Conditions for Difference Equations*, in 'New Progress in Difference
Equations' (editors B. Aulbach, S. Elaydi, G. Ladas). pp. 181-188.

1. C.
McCluskey (2003). *Stability for
a Class of Three-Dimensional Homogeneous Systems,* in 'Dynamical Systems and
Their Applications in Biology' (editors S. Ruan, G. Wolkowicz, J. Wu), Fields
Institute Communications. 36: 173-177.

**Theses**

2002 PhD *Global
Stability in Epidemiological Models*.

1996 MSc *Bendixson
Criteria for Difference Equations*.