A mathematical model for simulating the spread of a disease through a country divided into geographical regions with different population densities

The SIR (susceptible-infectious-recovered) model is a well known method for predicting the number of people (or animals) in a population who become infected by and then recover from a disease. Modifications can include categories such people who have been exposed to the disease but are not yet infectious or those who die from the disease. However, the model has nearly always been applied to the entire population of a country or state but there is considerable observational evidence that diseases can spread at different rates in densely populated urban regions and sparsely populated rural areas. This work presents a new approach that applies a SIR type model to a country or state that has been divided into a number of geographical regions, and uses different infection rates in each region which depend on the population density in that region. Further, the model contains a simple matrix based method for simulating the movement of people between different regions. The model is applied to the spread of disease in the United Kingdom and the state of Rio Grande do Sul in Brazil.