(

*a*) The open SIR model with immigration. We assume that susceptible individuals may acquire the infection not only as a result of internal transmission within their city, but also as a result of infection from an external source due to transient individual movements between cities. This formulation is the simplest way to take into account that real communities are not isolated, but spatially extended systems interconnected by migration (; ; ; ). (

*b*) The graph shows a typical stochastic realization of the SIR model. The parameters used in the simulations are

*N*=1.0×10

^{5},

*η*=1.0×10

^{−5} d

^{−1},

*δ*=5.5×10

^{−5} d

^{−1} and the mean transmission rate

*β*_{0}=1.175 d

^{−1}. Disease parameters correspond to typical measles values from and . The recovery rate

*γ* was estimated here by aggregating the exposed and the infectious phases from SEIR models with an additional exposed class. No seasonal forcing is included (

*β*_{1}=0). The model is simulated with the event algorithm of . Three different events are considered. Birth and death. Individuals in either class (

*S*,

*I*,

*R*) die at a rate

*δ*. Empty sites (

*E*) are instantaneously replenished by births of new susceptibles at a rate

*b*, where

*b*=

*δ*:

. Infection. Susceptible individuals acquire the infection at a rate

*η*+

*βI*/

*N*, the total force of infection:

. Recovery. Infectious individuals recover at a rate

*γ*:

.

## PubMed Commons