gravityWithDistance: Gravity with distance model

In SEEG-Oxford/movement: Modelling and Analysing Human Movement Data

Gravity with distance model

Description

In order to obtain more accurate results, following Viboud et al. 2006 we implement a nine-parameter form of the gravity law, in which short and long trips are fitted separately. Similarly to the gravity model we fit each parameter (equation 1) using a Poisson regression:

T_{ij} = \theta \frac{ N_i^{\alpha} N_j^{\beta} }{d_{ij}^{\gamma}}

where \theta is a proportionality constant and the exponents \alpha and \beta respectively, tune the dependence of dispersal on donor and recipient population sizes (N), and the distance between the two communities d_{ij}^{\gamma}. By taking the logarithm of on both sides this becomes:

\ln(T_{ij}) = \ln(\theta) + \alpha \ln(N_i) + \beta \ln{N_j} - \gamma \ln(d_{ij})

Viboud et al. show that below 119km, the population exponents are relatively high and larger for the destination population. Therefore we allow the flexibility to adjust based on a distance cutoff for the model.

Usage

gravityWithDistance(theta1 = 0.01, alpha1 = 0.06, beta1 = 0.03,
  gamma1 = 0.01, delta = 0.5, theta2 = 0.01, alpha2 = 0.06,
  beta2 = 0.03, gamma2 = 0.01)

Arguments

theta1	Model parameter with default value and the limits theta = [0, Inf].
alpha1	Model parameter with default value and the limits alpha = [-Inf, Inf].
beta1	Model parameter with default value and the limits beta = [-Inf, Inf].
gamma1	Model parameter with default value and the limits gamma = [-Inf, Inf].
delta	Model parameter with default value and the limits delta = [0, 1].
theta2	Model parameter with default value and the limits theta = [0, Inf].
alpha2	Model parameter with default value and the limits alpha = [-Inf, Inf].
beta2	Model parameter with default value and the limits beta = [-Inf, Inf].
gamma2	Model parameter with default value and the limits gamma = [-Inf, Inf].

Value

A flux model object with the gravity with distance flux function and a set of starting parameters.

Note

Limits 0 and Inf will be changed internally to the numerically safe approximations 0 -> sqrt(.Machine$double.eps) and Inf -> sqrt(.Machine$double.xmax), respectively.

References

Viboud, C. et al. (2006). Synchrony, waves, and spatial hierarchies in the spread of influenza. Science, 312, 447-51

See Also

movement, originalRadiation, radiationWithSelection, uniformSelection, interveningOpportunities, gravity

SEEG-Oxford/movement documentation built on April 17, 2023, 4:17 p.m.