Lyapunov methods, Differential equations, Global stability, Mathematical biology.
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
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
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.
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
Au (2006) - Maple implementation of compound
matrix techniques and construction of Lyapunov functions for linear
time-dependent differential equations.
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
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.
2002 PhD Global Stability in Epidemiological Models.
1996 MSc Bendixson Criteria for Difference Equations.