Box 1: Spatial models of pathogen transmission
Partial Differential Equations (PDEs) describe the size of the infected class over continuous space and time. Theirhost density models are continuous, and often homogeneous (but see Garlick et al. 2011, Hefley et al. 2017). Themobility model1 is a spatial diffusion rate and a corresponding functional form. Outputs include existence, structure, and speed of traveling epidemic waves and spatially explicit times to epidemic peak. Assumptions:animals move according to the kernel, which is often isotropic and independent of environment; transmission occurs locally.
Semi-spatial and static network models allow pairwise interactions within local neighborhoods. The host densitymodel is implicit, but relies on discrete units with corresponding disease states. The mobility model is defined through pairwise coupling coefficients between the “locations”, along with a specified “neighborhood” with which each location interacts.Outputs are usually derived from a master equation or simulation. Assumptions: known network structure and disease status; a priori definition of “neighborhood” (depending on analytical approach).
Metapopulation models track disease dynamics at physical locations coupled with one another across space. The host density and mobility models mirror those of semi-spatial models, though locations are now spatially explicit, and mobility can include explicit functions of geographic distance. Outputsinclude spatial spreading rate, spatial synchrony among subunits, and individual- and patch-level reproductive numbers.Assumptions: a priori knowledge of system connectivity.
IBMs allow movement and transmission to emerge organically from predefined rules applied to a set of actors. Inputs are individual-level attributes and parameters that govern them. Thehost density model can be continuous or discrete. Themobility model typically allows an individual’s state and environment to interact through a set of movement rules.Outputs range from a simple wave front of disease spread to each individual’s spatiotemporally explicit contribution to reproductive numbers. Assumptions: depend on model specifics.
Spatially embedded social networks describe disease dynamics across multipartite networks whose nodes correspond explicitly to locations in space. Inputs are bipartite networks linking individuals to locations of different types (households, peer groups, etc.), The host density model is a set of spatial centroids associated with each group, and mobility models can be distance-, gravity-, or radiation-based. Outputsinclude estimates of R0, total epidemic size, and spatial and temporal patterns of transmission.Assumptions: constant connectivities; central-place space use patterns.
1The contact process is often subsumed into a constant transmission rate or absorbed into themobility model .