Bacteria, viruses, or cancer cells, by means of mutation and replication, are sometimes able to escape the selective pressure exerted by treatment. This is called the development, or evolution, of resistance.
This dissertation is a study of some of the mathematics underlying a model of resistance put forth by Iwasa, Michor, and Nowak (IMN) [48, 49] (2003, 2004).
In the IMN model the pre-treatment phase is modeled as a determinist dynamical system using Eigen and Schuster’s quasispecies theory of evolution . It is assumed that at the start of treatment the system has reached an invariant distribution: the quasispecies equilibrium eigenvector.
The equations of the quasispecies theory can be viewed as projections of linear differential equations onto hyperplanes and their asymptotic behavior can be understood via Birkhoff’s Projective Contraction Theorem , which is related to the Perron-Frobenius Theorem. An understanding of Birkhoff’s contraction theorem requires an understanding of the Hilbert Projective Metric and so we develop an extensive collection of useful related results, some novel, about cones, hyperplanes, and the Hilbert Projective Metric.
In the IMN model, the post-treatment phase is modeled as a stochastic multi-type branching process on the various mutant types. The key calculation is the vector of extinction probabilities: the ith entry of the vector being the probability that a process, starting with a single mutant of type i, will eventually go extinct (under the selective pressure of treatment). The techniques for calculating these extinction probabilities involve the use of multi-type probability generating functions (PGF’s).
We prove results about the existence of continuous multi-type PGF’s and branching processes. Our proofs involve customizing techniques from the theory of differential equations in complex vector spaces, and then applying results from the theory of several complex variables. We also develop a method to numerically calculate the vector of extinction probabilities.
The pre and post-treatment models are fitted together and the probability of a successful treatment is numerically calculated using a combination of standard techniques from numerical analysis together with insights gained from our examination of the mathematical aspects of the model. Our investigation leads to a phenomena somewhat reminiscent of Eigen’s error catastrophe theory.