STATISTICS and DEMOGRAPHY


The Bridge of Life
Sovereigns may sway materials, but not matter,
and wrinkles (the damned democrats) won't flatter.

And Death -- the sovereign's sovereign, though the great
Gracchus of all mortality, who levels
With his agrarian laws, the high estate
Of him who feasts and fights and roars and revels
To one small grass-grown patch (which must await
Corruption for its crop) with the poor devils
Who never had a foot of land till now --
Death's a reformer, all men must allow.
-- Lord Byron, Don Juan
"The Bridge of Life". Painting commissioned by Karl Pearson.



Summary of my research

Since July, 2001, I have been working to understand the connection between population level data and individual-level processes, particularly with regard to aging and mortality.

Attempts to understand aging in a broad biological context, in particular evolutionary models, have suffered at times from antiquated mathematical technology.  I have been working to extend and reanalyze existing models, and to develop new models, based on modern mathematical (particularly stochastic-process) methods.

A major effort, of late, has been to develop more flexible models of mutation-selection balance, a fundamental component of evolutionary explanations for aging.

A generalized model of mutation-selection balance with applications to aging (Joint with Steven N. Evans and Kenneth W. Wachter).  Advances in Applied Mathematics 35:1 (2005), pp. 16--33.

This has been extended to a (quite different) limit process that incorporates recombination. This has just appeared as a monograph in the Memoirs of the AMS...

... which has been applied to computing ageing profiles connected to various standard models in
Vital rates from the action of mutation accumulation (Joint with Steven N. Evans and Kenneth W. Wachter)
        Journal of Population Ageing.

Our recent work on the biological implications is a theoretical analysis of "walls of death" (infinite mortality at finite ages) and an age-specific generalisation of Haldane's principle.
        The Age-Specific Force of Natural Selection and Walls of Death (Joint with Steven N. Evans and Kenneth W. Wachter)

We have also published some ideas about how these models might be linked up with information about genetic variation existing in the human genome.
       
Evolutionary shaping of demographic schedules
(Joint with Steven N. Evans and Kenneth W. Wachter)

Random matrix products are used to represent age- (or stage-) structured populations whose growth and survival depends on randomly changing environments. I worked with Shripad Tuljapurkar and Carol Horvitz to compute stochastic elasticities: How does the growth rate change when the distribution of environments changes? Derivatives of the stochastic growth rate Theoretical Population Biology 80:1 (2011), pp. 1--15. Preprint here.

An account of some not-yet mathematised ideas about modeling the aging process, written
jointly with Lloyd Goldwasser, may be found as
Aging and Total Quality Management:
Extending the reliability metaphor for longevity. In Evolutionary Ecology Research 8 (2006), pp. 1445--59.

One of the most exciting developments in ageing research is the extension of theories of senescence down to protozoans.  An analysis of the evolutionary consequences of unequal inheritance of damage in fissioning organisms may be found as:
Damage segregation at fissioning may increase growth rates: A superprocess model.
Theoretical Population Biology 71:4 (2007), pp. 473-90.

So far, I have written several papers on modeling mortality plateaus. Two papers, written jointly with Steven Evans, look at the implications of convergence to quasistationarity for Markov models of mortality. “Markov mortality models” describes the implications for mathematical modeling. “Quasistationary distributions” is a mathematical paper, proving some of the results referred to in “Markov mortality models”.  The results have recently been extended in joint work with Martin Kolb.

Markov mortality models: Implications of quasistationarity and initial distributions (Joint with Steven N. Evans). Theoretical Population Biology 65:4 (June 2004).

Quasistationary distributions for one-dimensional diffusions (Joint with Steven N. Evans).  Transactions of the American Mathematical Society 359:3 (March 2007), pp. 1285--1324.

Quasilimiting behavior for one-dimensional diffusions with killing (Joint with Martin Kolb). Annals of Probability 40:1 (Jan 2012), pp 162--212.

Some recent work has concerned statistical issues in fitting lifetime models to small data sets.
"Validated analysis of mortality rates demonstrates distinct genetic mechanisms that influence lifespan.” (Joint with Kelvin Yen and Charles Mobbs).  Experimental Gerontology, 43:12 (2008), pp. 1044--51.  (published version doi:10.1016/j.exger.2008.09.006)

I have also been working on mathematical and statistical questions connected with a demographic explanation of mortality plateaus.  If the initial population were composed of subpopulations with differing initial robustness, this would produce an appearance of flattening mortality at advanced ages, even if individual hazards kept increasing.

"Reevaluating a test of the heterogeneity explanation for mortality plateaus".  (Experimental Gerontology 40:1-2 (Jan/Feb 2005), pp. 101-13) considers some statistical issues that arise in fitting plateaued Gompertz curves to small-population survival data.  It reanalyzes data from an experiment which purported to show that there was no heterogeneity effect in Drosophila mortality plateaus.


Understanding mortality rate deceleration and heterogeneity  (Joint with Kenneth W. Wachter).   (Mathematical Population Studies, 13:1 (2006), pp. 19--37). This paper uses an Abelian theorem to clarify the connection between high initial robustness and the mortality plateau. Qualitative behavior of the heterogeneity model is linked to human and invertebrate mortality data.




Presentations on Aging


For the more mathematically inclined

Santa Fe Institute review of theories of ageing

Lecture notes on mortality in heterogeneous populations from Second Annual Stanford Workshop on Formal Methods in Demography:
  1. Mortality plateaus and fixed frailty models
  2. Changing frailty models (slides)


Last updated 7 November, 2014

Home page