Mathematical Ecology and Epidemiology
An introduction to the applications of ordinary, delay and partial differential equations to ecology and epidemiology.
(i) Review of simple ODE models in ecology such as the logistic and Lotka-Volterra models. Extensions of such models such as the use of the Holling functional responses. Phase plane analysis of such models.
(ii) ODE models in epidemiology. The Kermack McKendrick model. Higher dimensional models that include, for example, an exposed compartment, or which incorporate treatment, vaccination or quarantining. Analytical techniques useful in the linearised analysis of high dimensional systems, such as the Routh Hurwitz conditions. The calculation of the basic reproduction number and its importance in epidemiological modelling.
(iii) Age structured models and their reformulation into delay differential equations or renewal integral equations. The study of the characteristic equations resulting from the linear stability analysis of such models. Use of such equations in ecology and epidemiology, to include the Ross Macdonald model of malaria transmission. The basic reproduction number for models with delay.
(iv) Reaction-diffusion equations. Travelling wave solutions; applications to ecology and epidemiology.
J.D. Murray: Mathematical Biology: I. An Introduction. Springer (2002).
- N.F. Britton. Essential Mathematical Biology. Springer (2003).