A new paper, Competition Between Transients in the Rate of Approach to a Fixed Point, SIAM J. Appl. Dyn. Syst. 8, 1523 (2009) by Judy Day, Jonathan Rubin and myself appears today. The official journal link to the paper is here and the PDF can be obtained here. This paper came about because of a biological phenomenon known as tolerance. When the body is exposed to a pathogen or toxin there is an inflammatory response. This makes you feel ill and initiates the immune system to mount a defense. In some cases, if you are hit with a second dose of the toxin you’ll get a heightened response. However, there are situations where you can have a decreased response to a second dose and that is called tolerance.
Judy Day was my last graduate student at Pitt. When I left for NIH, Jon Rubin stepped in to advise her. Her first project on tolerance was to simulate a reduced four dimensional model of the immune system and see if tolerance could be observed in the model . She found that it did occur under certain parameter regimes. What she showed was that if you watch a particular inflammatory marker, then it’s response could be damped if a preconditioning dose is first administered.
The next step was to understand mathematically how and why it occurred. The result after several starts and stops was this paper. We realized that tolerance boiled down to a question regarding the behavior of transients, i.e. how fast does an orbit get to a stable fixed point starting from different initial conditions. For example, consider two orbits with initial conditions (x1,y1) and (x2,y2) with x1 > x2, where y represents all the other coordinates. Tolerance occurs if the x coordinate of orbit 1 ever becomes smaller than the x coordinate of orbit 2 independent of what the other coordinates do. From continuity arguments, you can show that if tolerance occurs at a single point in space or time it must occur in a neighbourhood around those points. In our paper, we showed that tolerance could be understood geometrically and that for linear and nonlinear systems with certain general properties, tolerance is always possible although the theorems don’t say which orbits in particular will exhibit it. However, regions of tolerance can be calculated explicitly in two dimensional linear systems and estimated for nonlinear planar systems.
 Day J, Rubin J, Vodovotz Y, Chow CC, Reynolds A, Clermont G, A reduced mathematical model of the acute inflammatory response II. Capturing scenarios of repeated endotoxin administration, Journal of theoretical biology, 242(1):237-56 2006.