Although evolution is often thought of as a slow process that proceeds on the time scale of millennia, in fact there are many very rapid evolutionary processes, often called contemporary evolution, that have profound effects on human health and welfare. For example: (1) In agriculture, plants and pests can rapidly evolve resistance to herbicides and pesticides, respectively; (2) The influenza virus, and other viruses and bacteria, often evolve within an individual host making treatment strategies difficult and/or temporary; (3) The evolution of bacteria to become resistant to most antibiotics poses a serious threat to mankind; (4) Some parasites, for example African trypanosomes, can change the proteins that they express on their surfaces and thus can become invisible to the immune system; and (5) Harvested populations may show rapid evolution in size or age at maturity, which affects both yield and recovery from depleted states.
Understanding the dynamic behavior of such problems is difficult because one is typically studying the co-evolution of two or more interacting complex systems. The mathematical challenges are daunting. On the local level (for example, the evolution of influenza within a host) mutations are driven by stochastic processes. However, one is not interested in the number of mutations per se, but in the number of successful mutations that can establish themselves in the host. This depends on the immune status of the host, including resources available to the mutant and the history of previous infections. Even when this is understood one must face the problem of transmission and spatial spread of the mutant strain in the whole population. Thus, not only is the biology very difficult, but these questions naturally involve stochastic processes and ordinary and partial differential equations on several different time scales.
Giving or not giving drugs, choosing to use or not use pesticides, or choosing when to use them, are choices that have political, ethical and economic consequences. The consequences themselves depend in many cases on changing human cultural behavior, changing technology, and climate change. Mathematical modeling, including the invention of new mathematical structures, can help us understand these rapidly co-evolving systems and thus make clear the likely consequences of various policy choices.