Spatial epidemic models
Spatial epidemic models allow infection risk and population state to vary by location.
Reaction–diffusion form
A spatial SIR-type model may include
\[\frac{\partial S}{\partial t}=D_S\nabla^2S-\beta\frac{SI}{N},\]\[\frac{\partial I}{\partial t}=D_I\nabla^2I+\beta\frac{SI}{N}-\gamma I.\]Diffusion terms represent movement, while reaction terms represent infection and recovery.
Other spatial approaches
Models may instead use geographic patches, mobility matrices, commuting networks or individual-based movement.
Why space matters
Spatial structure can alter epidemic speed, local peaks, synchronisation between regions and the effects of local interventions.
Key idea. Spatial epidemic models combine transmission dynamics with movement or geographic structure.