Math theory predicts fire, disease spread
Copyright 2001
United Press International
July 13, 2001
By MIKE MARTIN, UPI Science Correspondent
WASHINGTON, July 13 - A Russian mathematician has developed a complex proof that may shed light on percolation processes -- not those associated with morning coffee, but with mathematical models that describe the random spread of forest fires, orchard blight, infectious diseases and the way that hard-to-tap oil moves through porous rock.
Stanislav Smirnov's unique approach to the complexities of percolation predicts how changes take place across the individual squares of a 2-dimensional lattice -- essentially a checkerboard -- as the lattice grows or shrinks.
Since fires spreading through forests and oil flowing through rock may be modeled by complex lattices, Smirnov's research sheds light on how things move and how discrete clusters -- pockets of fire or disease -- coalesce.
"There is a huge class of models that can be formulated discretely that may or may not converge to a continuum. My research deals with models of this sort," Smirnov, a senior lecturer at Royal Institute of Technology in Stockholm, Sweden, told United Press International from the Kennedy Center in Washington, where he is the recipient of the prestigious 2001 Clay Institute Research Award for his work.
Percolation theory, a field of mathematics which had its beginnings in the early 1950's, is used to answer many questions. If you immerse a large porous stone in a bucket of water, what is the probability that the center of the stone gets wet? Suppose a rat is released in a maze -- what is the probability the rat gets out?
Smirnov's research describes the behavior of mazes, porous rocks, and other lattices as they grow to infinity or shrink to zero. His theory -- which completes the research of British physicist John Cardy -- is important because it permits researchers in a variety of fields to make predictions, even with limited data, over much larger or much smaller scales.
A researcher studying the spread of AIDS, for instance, only needs the data from a small village and Smirnov's theory to build a computer model detailing how the dreaded disease spreads globally -- moving from host to host through the random maze of towns, cities, countries, and continents.
"Smirnov's work also has applications in phase transitions -- it deals with the behavior of individual atoms and molecules as they coalesce from being discrete entities into something more continuous -- a gas becoming a liquid, for instance," Princeton University physicist Ed Witten told UPI from Washington. "There are many still mysterious things that go on right at the point of phase transitions that Smirnov's research may help to resolve."
"A perennial charm of percolation theory is the beauty and apparent simplicity of its open problems," Cambridge University mathematician Geoffrey Grimmett said in a recent paper. "Percolation theory is a source of fascinating problems of the best kind for which a mathematician can wish."
Examples of two such problems are those faced by a blight-plagued orange farmer and a harried fire fighter faced with a blazing forest.
The farmer tells the mathematician that blight is destroying his trees. He needs to know how the disease will spread, and how to prevent it next season.
The mathematician maps half an acre of the orchard using an imaginary lattice. She examines the trees at the vertices of the grid -- the corner points where the many "squares" of the checkerboard meet. She then supposes there is a certain probability that a healthy tree will be infected by a blighted tree. The probability is determined by the distance between the trees.
With this half acre of data, she uses Smirnov's theory to extrapolate the behavior of the blight as it spreads through the trees that populate the hundred-acre orchard.
To prevent a single blighted tree from endangering the whole orchard, the mathematician tells the farmer he must space his trees -- the vertices of his latticework -- far enough apart so that the probability one tree will infect another is smaller than the critical probability of blight percolating between the lattice spacing. Using percolation theory, she calculates both the probability and the exact spacing, saving the farmer acres of land and millions of dollars.
The mathematician next races to a burning forest where a fire chief must know where the fire is heading, and whether it will threaten a nearby town.
The mathematician lays out a small grid near the fire's origin. She notes three types of trees -- alive; burning; and burned, and that trees destroyed by fire can't threaten their neighbors. She supposes the length of time for a tree at a vertex to burn after catching fire and, using a percolation model on her laptop computer, predicts which trees will burn -- drawing a small-scale map of the process.
Using Smirnov's theory, she extrapolates the data to a million acre region, quickly predicting that the fire will not reach the nearby town but will burn in precisely the opposite direction.
The firefighters redirect their efforts and extinguish the blaze in record time.