Publications

The importance of being atomic: ecological invasions as random walks instead of waves, by Timothy Reluga.
In preparation, January, 2015.
Preprint PDF.

Resource distribution drives the adoption of migratory, partially migratory, or residential strategies. By Timothy Reluga and Allison Shaw.
In submission, July, 2014. Revised, January 2015.
This is a beautiful little analysis of a spatially explicit seasonal migration model that shows how seasonality, movement costs, and resource heterogeneity can conspire to drive co-existence, and potentially, speciation. A number of results can be obtained by hand, but there are also some open mathematical questions.
Preprint PDF.

A mixed-strategy game theoretical approach for infectious disease prevention by social distancing. By Jing Li and Timothy C Reluga.
In submission, January, 2015.
This is a nice theoretical example of how diversity in believe and action within a community can lead to better health than homogenized communities without any diversity.
Preprint PDF.

Population viscosity stops disease emergence by preserving local herd immunity. By Timothy Reluga and Eunha Shim.
Proceedings of the Royal Society B , 281(1796), 20141901, October 22, 2014.
Covered in media by Johanna Ohm for CIDD and by Tanya Lewis for Livescience.com. This is related to my earlier paper on risks from virus reservoirs.
DOI:10.1098/rspb.2014.1901, Preprint PDF.

Optimal migratory behavior in spatially-explicit seasonal environments. By Timothy Reluga and Allison Shaw.
Discrete and Continuous Dynamical Systems - Series B October 9, 2014, 19(10), 3359-3378. (Special issue in honor of Chris Cosner on the occasion of his 60th birthday, Submitted July, 2013).
I think this kind of model need some further attention. For example, I've never been particularly happy with the economics idea of discounting, and my work in this paper and my early paper on the discounted reproductive number haven't asuaged my concerns. Somebody can probably do a good job improving things with some density-dependent simulation models.
DOI:10.3934/dcdsb.2014.19.3359, Preprint PDF

The systems theory of community health and infectious disease. By Jing Li, Darla V. Lindberg, Rachel A. Smith, and Timothy C. Reluga.
submitted, 2013.
Preprint PDF.

A reduction method for Boolean networks proven to conserve attractors. By Assieh Saadatpour, Reka Albert, and Timothy Reluga.
SIAM Journal on Applied Dynamical Systems, November, 2013, Volume 12, Issue 4, pp 1997-2011.
DOI:10.1137/13090537X, Preprint PDF.

Equilibria of an Epidemic Game with Piecewise Linear Social Distancing Cost By T. Reluga.
Bulletin of Mathematical Biology October 2013, Volume 75, Issue 10, pp 1961-1984.
DOI:10.1007/s11538-013-9879-5, Preprint PDF.

Games of age-dependent prevention of chronic infections by social distancing. By T. Reluga and J. Li.
Journal of Mathematical Biology, 2012.
DOI:10.1007/s00285-012-0543-8, Preprint PDF.

A general approach to population games with application to vaccination. By T. Reluga and A. Galvani.
Mathematical Biosciences, 230 (2): 67-78, April, 2011. (received the 2013 Bellman Prize for best biannual paper)
DOI:10.1016/j.mbs.2011.01.003, Botched Pubmed, Preprint PDF.

Erratic flu vaccination emerges from short-sighted behaviour in contact networks. By D. M. Cornforth, T. C. Reluga, E. Shim, C. T. Bauch, A. P. Galvani, , and L. A. Meyers.
PLOS Computational Biology, 7 (1): e1001062, 2011.
DOI:10.1371/journal.pcbi.1001062, Botched Pubmed, Preprint PDF.

Game theory of social distancing in response to an epidemic. By T. Reluga.
PLOS Computational Biology, 6 (5): e1000793, 2010.
The papers used differential game theory to find the equilibrium behavior during an epidemic.
DOI:10.1371/journal.pcbi.100079, Botched Pubmed, Preprint PDF.

Branching processes and non-commuting random variables in population biology. By T. Reluga.
Canadian Applied Math Quarterly, 17 (2): 387, 2009.
Link, Preprint PDF.

An SIS epidemiology game with two subpopulations. By T. Reluga.
Journal of Biological Dynamics, 3 (5): 515-531, 2009.
See Cressman 1996 for some earlier discussion of related stability ideas.
DOI:10.1080/17513750802638399, Preprint PDF.

The discounted reproductive number for epidemiology. By T. Reluga, J. Medlock, and A. Galvani.
Mathematical Biosciences and Engineering, 6 (2): 377-393, 2009.
This paper uses M-matrix theory, non-negative matrices, and Perron-Frobenius theory to establish some useful results regarding next-generation matrixes for population biology.
DOI:10.3934/mbe.2009.6.377, Botched Pubmed, Preprint PDF.

Analysis of hepatitis C virus infection models with hepatocyte homeostasis. By T. Reluga, H. Dahari, and A. S. Perelson.
SIAM Journal on Applied Mathematics, 69 (4): 999-1023, 2009.
This paper provides a complete bifurcation analysis of Harel's homeostasis hypothesis for hepatitis C treatment responses, including formulas that can used to predict clearance, partial infection, and bistability. The analysis has been subsequently discussed by DebRoy, Bolker, and Martcheva, 2009.
DOI:10.1137/080714579, Botched Pubmed, Preprint PDF.

Backward bifurcations and multiple equilibria in epidemic models with structured immunity. By T. Reluga, J. Medlock, and A. Perelson.
Journal of Theoretical Biology, 252 (1): 155-165, 2008.
One of the issues that bugged me when first learning mathematical epidemiology was that it completely ignored the internal state of the hosts changed because of immune responses. How did we know that theories which ignored the complexities of the immune response within individuals were adequate to explain population-scale dynamics? This paper is a step forward in resolving this by constructing some specific hypotheses and conditions. (update 2012-08: Our results are nicely complementary to those in an earlier paper by Hethcote, Yi, and Jing, 1999, which we were un-aware of in 2008.)
DOI:10.1016/j.jtbi.2008.01.014, Botched Pubmed, Preprint PDF.

Optimal timing of disease transmission in an age-structured population. By T. Reluga, J. Medlock, E. Poolman, and A. Galvani.
Bulletin of Mathematical Biology, 69 (8): 2711-2722, 2007.
This paper studies how age-dependent virulence can lead to a a social-distancing game with two different Nash equilibria - one that maximizes transmission and one that minimizes transmission. This is closely related to the concept of ``endemic stability'' from veterinary science. Polio is used as an illustrative example.
DOI:10.1007/s11538-007-9238-5, Preprint PDF.

Reservoir interactions and disease emergence. By T. Reluga, D. B. Walton, R. Meza, and A. Galvani.
Theoretical Population Biology, 72 (3): 400-408, 2007.
This paper provides a modelling framework for the study of disease-emergence pathways. This is a concrete approach to risk-assessment associated with emergence patterns like those proposed by Wolfe et al.. It's model analysis contains some useful discussions of reducible branching processes and the multivariable form of L'Hopital's rule. L'Hopital's rule is particularly useful for multivariable generating functions because critical processes are sure to have a double root.
DOI:10.1016/j.tpb.2007.07.001, Botched Pubmed, Preprint PDF.

Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum. By A. Galvani, T. Reluga, and G. Chapman.
Proceedings of the National Academy of Sciences, 104 (13): 5692-5697, March 27 2007.
DOI:10.1073/pnas.0606774104, Pubmed, Preprint PDF.

Resistance mechanisms matter in SIRS models. By T. Reluga and J. Medlock.
Mathematical Biosciences and Engineering, 4 (3): 553-563, July 2007.
This paper provides a resolution to a question of the time as to why different models of immunity seemed to yield contradictory results.
DOI:10.3934/mbe.2007.4.553, Link, Preprint PDF.

Evolving public perceptions and stability in vaccine uptake. By T. Reluga, C. Bauch, and A. Galvani.
Mathematical Biosciences, 204: 185-198, 2006.
DOI:10.1016/j.mbs.2006.08.015, Preprint PDF.

A model of spatial epidemic spread when individuals move within overlapping home ranges. By T. Reluga, J. Medlock, and A. Galvani.
Bulletin of Mathematical Biology, 68 (2): 401-416, February 2006.
This paper uses an Ornstein-Uhlenbeck process to describe spatial movement and obtains some asymptotic results for the speed of spatial spread of an epidemic. It provides a resolution to the problem of whether spatial epidemic spread is governed by distributed contacts or distribution of infected, without resorting to Baysian melding approaches. A more recent paper exploiting this idea to a different conclusion is by Kenkre and Sugaya (2014).
C++/Linux Code.
DOI Link, Preprint PDF.

On antibiotic cycling and optimal heterogeneity. By T. Reluga.
Mathematical Medicine and Biology, June 2005.
This paper studies generalizations of the Meissner equation to show how changes in antibiotic use may increase or decrease resistance prevalence through resonance phenomena. The same results apply to general habitat switching problems in genetics (see Salathe 2009 and Gaal, 2010 )
DOI Link, Preprint PDF.

Nonequilibrium thermodynamics of a nonlinear biochemical switch in a cellular signaling process. By H. Qian and T. Reluga.
Physical Review Letters, 94: 028101, January 2005.
DOI Link, Preprint PDF.

Simulated evolution of selfish herd behavior. By T. Reluga and S. Viscido.
Journal of Theoretical Biololgy, 234 (2): 213-225, 2005.
There's now a PNAS paper by Pearce et al. making a much bigger deal out of the same basic idea that Steve and I were working on 10 years ago.
C++/Linux Code.
DOI Link, Preprint PDF.

Stochasticity, invasions, and branching random walks. By M. Kot, J. Medlock, T. Reluga, and D. B. Walton.
Theoretical Population Biology, 66 (3): 175-184, 2004.
DOI Link, Preprint PDF.

A two-phase epidemic driven by diffusion. By T. Reluga.
Journal of Theoretical Biology, 229 (2): 249-261, July 21 2004.
This paper shows how a double-epidemic might emerge from a bioterrorism attack. In an important special case, travelling epidemic waves can emerge even when only driven by a bioterrorism agent in 1 or 2 dimensions. This is a pretty strange effect, and yet another example of how 3 dimensions are special. This also provides an explanation of anomolous travelling wave speeds found by J. Cooke.
DOI Link, Preprint PDF.

Analysis of periodic growth-disturbance models. By T. Reluga.
Theoretical Population Biology, 66 (2): 151-161, September 2004.
DOI Link, Preprint PDF.