An idealised representation of disease transmission and recovery

Mathematically orientated models of the spread of a infectious disease have a long pedigree, dating back to Bernoulli. The complex underlying dynamics of intra-host infection, immune response and inter-host transmission are coarse-grained into a limited number of disease states for host individuals. The simplest common choices for disease state are 'susceptible to infection' (S), 'currently infected' (I), and if recovery imparts long-term future immunity 'Removed from infection' (R). Epidemic spread can then be modelled as a stochastic dynamical system on the space of possible population disease states.

Classical homogeneity assumptions on the of host population and their location/interactions in space lead, in the limit of large numbers of possible hosts, to the well-known SIR model of epidemic dynamics. The possible relaxations away from this classical picture are too numerous to include in their entirety, but some major areas of active and interconnected research are:

• Stochastic vs deterministic models. Characterising the negative correlation between susceptibles and infecteds. Role of stochastic forcing.
• Spatial spread. Introducing appropriate sense of locality. Understanding and quantifying Human and Animal behaviour and mixing patterns.
• Biological realism and non-Markovian dependence on history. Incorporating variation between host immuno-responses and during the life-cycle of the pathogen. Understanding interaction between seperate pathogen strains.
• Vector-bourne Disease. The role on intermediary vector hosts.

A schematic of forcing a suspension of particles down a channel