In a previous post I discussed a paper of Lev Bar-Or on the dynamics of the interactions between T cells and macrophages. More precisely, the unknowns in the dynamical system used in that work are concentrations of cytokines produced by those cells. The object of study is a four-dimensional dynamical system depending on eighteen parameters. In the paper only a very few aspects of this system are studied. Two phase portraits are given which are presumably the result of numerical simulations. Some graphs of time evolutions are also shown. There are no theorems. I have now written a paper where I tried to prove as much as possible about the solutions of this system. My motivation for doing so is twofold. On the one hand I am interested in immunology and the contributions which mathematics can make to that field. On the other hand I am interested in exploring the role of rigorous proofs in applications of mathematics. To what extent can proofs be obtained and what insights can they, and the process of obtaining them, give into the original problem?
All the available results indicate that the dynamics of solutions of this system is simple, with all solutions converging to stationary solutions at late times. I was able to prove that for a large open subset of the parameter space there is a unique stationary solution which acts as an attractor for all solutions. Apart from this results were only obtained under strong restrictions on the parameters. There are cases where there are two stable stationary solutions. These correspond biologically to Th1 and Th2 dominance. No evidence was found for cases with a larger number of stable steady states. Numerical experiments also gave no indications of dynamical behaviour more complicated than what has just been described for any values of the parameters.
In studying this system I ran up against barriers which I was not able to overcome. It remains to be seen to what extent these barriers are in some sense intrinsic to the mathematical problem and to what extent they are a consequence of my inexperience with this type of dynamical system. This is the first time I have written a paper in the area of mathematical biology and so it is a personal landmark for me.
Everything I found is consistent with the idea that the inclusion of such a large number of parameters in the model is superfluous. All types of qualitative behaviour shown by the general system already seem to occur for simpler models with a greatly reduced set of parameters. Moreover switching off some of the major qualitative effects included in the model does not seem to make a big difference. This includes switching off antigen presentation or even leaving out the macrophages completely.