Evolution has long been understood as the driving force for many problems of medical interest. The evolution of drug resistance in HIV and bacterial infections is recognized as one of the most significant emerging problems in medicine. In cancer therapy, the evolution of resistance to chemotherapeutic agents is often the differentiating factor between effective therapy and disease progression or death. Interventions to manage the evolution of resistance have, up to this point, been based on steady-state analysis of mutation and selection models. In this paper, we review the mathematical methods applied to studying evolution of resistance in disease. We present a broad review of several classical applications of mathematical modeling of evolution, and review in depth two recent problems which demonstrate the potential for interventions which exploit the dynamic behavior of resistance evolution models. The first problem addresses the problem of sequential treatment failures in HIV; we present a review of our recent publications addressing this problem. The second problem addresses a novel approach to gene therapy for pancreatic cancer treatment, where selection is used to encourage optimal spread of susceptibility genes through a target tumor, which is then eradicated during a second treatment phase. We review the recent in Vitro laboratory work on this topic, present a new mathematical model to describe the treatment process, and show why model-based approaches will be necessary to successfully implement this novel and promising approach.

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