## About this course

Biological systems are incredibly complex, characterized by nonlinear interactions and many interacting units. Biologists want to understand these systems, from explaining species coexistence in the tropics, to predicting how a novel disease will spread, to managing the conservation of endangered species. However, the complexity of these systems can make it hard to achieve such understanding through experiment alone, as it can be hard to decide which biological processes to isolate experimentally.

For example, consider trying to predict the spread of an infectious disease. At the level of an individual host, the interaction between the immune system and the pathogen determines how many pathogens the person is carrying, how long they are transmitting disease, and whether their resistance to reinfection will be permanent (as with measles), temporary (as with flu), or non-existent (as with the common cold). All of these individual-level interactions will affect the processes of transmission and recovery. But there are other considerations as well. For example, the age of the sick person will affect her contact with others: babies and the elderly don’t get out much, compared to school-age or working-age people. But working-age people are much less likely to drool/sneeze/cough on a shared crayon box than your average kindergartener. Thus the age-structure (how many people are there of each age?) and contact structure (who comes in contact with who?) of the population may also affect transmission. Which of these interactions matters, in the sense that it has some detectable impact on disease spread? Answering that question experimentally would be difficult, to say the least. It might be more appropriate to say it would be impossible. This is where mathematical modeling can be a powerful tool for gaining biological insight.

Mathematical models formalize our assumptions about the processes driving biological systems as mathematical equations, allowing you to evaluate the dynamical consequences of different assumptions. This can provide the knowledge necessary to guide experiments, to shape policy, and to facilitate understanding. In this course, you will learn commonly used approaches for analyzing mathematical models in biology. Emphasis will be placed on the translation from biology to mathematical equations, and from mathematical analysis back to biological insight. As such, the mathematical techniques will be covered through their application to particular biological problems.

This course will draw on models of a range of biological systems, but with an emphasis on questions arising out of the consideration of ecological and evolutionary systems. Biological systems that will be studied in this course (we may not get to all of these topics!):

- single-species population growth

- species interactions (competition, predation)

- infectious disease spread

- natural selection in haploid and diploid species (e.g., population genetics)

- enzyme kinetics

- dynamics of populations structured by age, stage, and space

- trait evolution (life history evolution, evolution of virulence)

- stochastic dynamics (birth-death, branching processes)

Mathematical topics that will be covered (again, we may not get to all of these!):

- linear and nonlinear single-variable differential and difference equations

- linear and nonlinear multivariate differential and difference equations

- linear stability analysis

- nondimensionalization

- sensitivity and elasticity analysis

- evolutionary invasion analysis

For example, consider trying to predict the spread of an infectious disease. At the level of an individual host, the interaction between the immune system and the pathogen determines how many pathogens the person is carrying, how long they are transmitting disease, and whether their resistance to reinfection will be permanent (as with measles), temporary (as with flu), or non-existent (as with the common cold). All of these individual-level interactions will affect the processes of transmission and recovery. But there are other considerations as well. For example, the age of the sick person will affect her contact with others: babies and the elderly don’t get out much, compared to school-age or working-age people. But working-age people are much less likely to drool/sneeze/cough on a shared crayon box than your average kindergartener. Thus the age-structure (how many people are there of each age?) and contact structure (who comes in contact with who?) of the population may also affect transmission. Which of these interactions matters, in the sense that it has some detectable impact on disease spread? Answering that question experimentally would be difficult, to say the least. It might be more appropriate to say it would be impossible. This is where mathematical modeling can be a powerful tool for gaining biological insight.

Mathematical models formalize our assumptions about the processes driving biological systems as mathematical equations, allowing you to evaluate the dynamical consequences of different assumptions. This can provide the knowledge necessary to guide experiments, to shape policy, and to facilitate understanding. In this course, you will learn commonly used approaches for analyzing mathematical models in biology. Emphasis will be placed on the translation from biology to mathematical equations, and from mathematical analysis back to biological insight. As such, the mathematical techniques will be covered through their application to particular biological problems.

This course will draw on models of a range of biological systems, but with an emphasis on questions arising out of the consideration of ecological and evolutionary systems. Biological systems that will be studied in this course (we may not get to all of these topics!):

- single-species population growth

- species interactions (competition, predation)

- infectious disease spread

- natural selection in haploid and diploid species (e.g., population genetics)

- enzyme kinetics

- dynamics of populations structured by age, stage, and space

- trait evolution (life history evolution, evolution of virulence)

- stochastic dynamics (birth-death, branching processes)

Mathematical topics that will be covered (again, we may not get to all of these!):

- linear and nonlinear single-variable differential and difference equations

- linear and nonlinear multivariate differential and difference equations

- linear stability analysis

- nondimensionalization

- sensitivity and elasticity analysis

- evolutionary invasion analysis