Spatial epidemic models
Ordinary epidemic models describe how numbers of susceptible, infectious and recovered individuals change through time. A spatial epidemic model also asks where those individuals are.
Why add space?
In a non-spatial SIR model, individuals are effectively treated as belonging to one well-mixed population. Real epidemics occur across towns, regions and countries. Contact rates, population density and movement can differ greatly between locations.
Two places can therefore experience different epidemic peaks even when they belong to the same epidemic.
From SIR to a spatial SIR model
A frequency-dependent non-spatial SIR model is
\[\frac{dS}{dt}=-\beta\frac{SI}{N},\qquad \frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I,\qquad \frac{dR}{dt}=\gamma I.\]Now let the compartments depend on position:
\[S=S(x,t),\qquad I=I(x,t),\qquad R=R(x,t).\]If movement is approximated by diffusion, one possible model is
\[\boxed{S_t=D_S\nabla^2S-\beta\frac{SI}{N},}\]\[\boxed{I_t=D_I\nabla^2I+\beta\frac{SI}{N}-\gamma I,}\]\[\boxed{R_t=D_R\nabla^2R+\gamma I.}\]| Part | Biological meaning |
|---|---|
| \(D_S\nabla^2S,D_I\nabla^2I,D_R\nabla^2R\) | spatial movement |
| \(\beta SI/N\) | local transmission |
| \(\gamma I\) | local recovery |
Read the infectious equation term by term
\[I_t=\underbrace{D_I\nabla^2I}_{\text{movement}}+\underbrace{\beta SI/N}_{\text{new infections}}-\underbrace{\gamma I}_{\text{recoveries}}.\]At a fixed location, infectious density can rise because infectious people move into the area or because susceptible people there become infected. It can fall because infectious people move away or recover.
What does diffusion mean here?
Diffusion is an idealised representation of undirected local movement. It is useful when movement consists of many relatively small movements and a continuous spatial approximation is reasonable.
It does not literally mean that people move like diffusing molecules. Long-distance travel and commuting often require a different spatial model.
How an infection front can move
Suppose infection begins near one location. Infectious individuals move into neighbouring locations. There they encounter susceptible individuals and create new infections. Those newly infectious individuals can then move farther.
This repeated process can create a moving epidemic front.
A model-generated infection profile
The following graph is calculated numerically from a one-dimensional spatial SIR reaction–diffusion model with an initially localised infectious population. It is not a hand-drawn curve.
Spatial spread is not necessarily a fixed travelling wave
An epidemic profile can broaden, change height and eventually decline as susceptible individuals are depleted. Therefore every spreading epidemic is not automatically a constant-shape travelling wave.
Travelling-wave analysis becomes appropriate only when the equations and parameter regime support an approximately fixed profile moving at an approximately constant speed.
Local epidemic peaks
Different locations can peak at different times. If infection starts near one region and spreads outward, locations farther away may experience later peaks.
This creates a spatially asynchronous epidemic even when the same biological parameters apply throughout the domain.
Spatially varying parameters
Transmission and movement need not be constant:
\[\beta=\beta(x,t),\qquad D_I=D_I(x,t).\]For example, population density, behaviour, interventions or transport links can make transmission and movement vary geographically and through time.
Local interventions
A spatial model can represent interventions applied only in selected regions. Reducing local contact can be represented through a lower \(\beta\); restricting movement can be represented by changing movement terms or connections between regions.
This allows questions such as whether a local intervention delays spread into neighbouring areas or merely shifts infections elsewhere.
Patch models
Continuous space is not always appropriate. We may instead divide the population into regions or patches. If \(I_i(t)\) is the infectious population in region \(i\), movement between regions can be described using rates or a mobility matrix.
A schematic infectious equation is
\[\frac{dI_i}{dt}=\text{local infections}-\text{recoveries}+\text{movement into }i-\text{movement out of }i.\]Mobility matrices
Let \(m_{ij}\) describe movement from region \(i\) to region \(j\). Then the matrix
\[M=(m_{ij})\]encodes geographic connectivity. Large entries represent strong movement links.
This framework is often more natural than diffusion when people make repeated journeys between distinct cities or administrative regions.
Network epidemic models
A network model represents locations as nodes and travel routes as edges. Infection can spread locally within nodes and geographically along edges.
Networks can capture strongly connected transport hubs and irregular geography that a simple one-dimensional diffusion equation cannot.
Commuting differs from permanent migration
A commuter leaves one region and later returns. A simple migration model may instead treat the person as having changed location permanently.
For diseases where daily travel drives transmission, explicitly representing commuting or time spent in different regions can be important.
Individual-based spatial models
At the finest scale, individual people can have explicit locations, movement rules and contact processes. Such models can represent heterogeneity in much greater detail, but they require more data and computation.
Which spatial framework should be used?
| Framework | Useful when |
|---|---|
| reaction–diffusion PDE | space is approximately continuous and movement is local |
| patch model | distinct regions are the natural units |
| mobility/network model | travel links between locations are important |
| individual-based model | individual locations and contacts matter |
Population conservation
If movement only redistributes people and there are no births or deaths, then transmission and recovery transfer individuals between compartments rather than creating or destroying people.
With suitable no-flux boundaries, the total population integrated over the whole spatial domain remains constant.
Boundary conditions
A no-flux condition such as
\[\frac{\partial I}{\partial n}=0\]means no diffusive infectious movement crosses the boundary. Other boundary conditions may represent entry, exit or connection to external populations.
Initial conditions
Spatial models require the initial geographic distribution:
\[S(x,0)=S_0(x),\qquad I(x,0)=I_0(x),\qquad R(x,0)=R_0(x).\]A small local introduction can behave very differently from infection already distributed throughout the domain.
Reproduction numbers in spatial systems
In a homogeneous non-spatial SIR model, \(R_0\) can often be written simply in terms of transmission and recovery parameters. Spatial models can require a more general invasion criterion because transmission and movement interact across locations.
In patch or structured models this is often analysed using a next-generation matrix or operator.
Deterministic versus stochastic spatial epidemics
A deterministic PDE can assign a tiny positive infectious density far from the main epidemic. In a real population, that density may correspond to less than one person and has no literal individual interpretation.
Stochastic spatial models are especially important when introductions are small, because infection can become extinct by chance rather than successfully establish in a new region.
What spatial models can help us study
They can investigate geographic invasion speed, timing of local peaks, regional hospital demand, effects of mobility restrictions, synchronisation between regions, local extinction and reintroduction, and where surveillance or intervention may be most useful.
A practical modelling workflow
Choose the spatial scale first. Decide whether continuous space, regions, networks or individuals best represent movement. Write the local epidemic model. Add movement using a mechanism that matches the data. Specify initial and boundary conditions. Estimate parameters from appropriate spatial and epidemiological data. Then analyse spread, local peaks and uncertainty rather than considering only the national total.