Could a partial differential equation like the diffusion equation be considered as a model equation for predicting immigration/emigration rates?

The diffusion equation, after assumptions and simplifications, can be derived from the Boltzmann transport equation, which describes the time evolution of a distribution function. The time dependent diffusion equation can be written as:

d (phi)/dt = div (D grad(phi)) + S

where, D = diffusion coefficient, phi = property to be modeled (Temperature, Electric Potential) and S = sources/sinks.

The flux (J) can be expressed as D grad(phi) .

Immigration/Emigration mainly depends on the disparities in the per-capita GDP and the health of an economy of the immigrant and emigrant country. So an ideal candidate for ‘phi’ in an immigrant model would be the per-capita income.

The phase-space would be the different regions (states, countries or continents) under consideration.

**Determining the diffusion coefficient (let’s call it the immigrant coefficient)**

The immigrant coefficient could be a function of

- immigrant region’s immigration/employment policies (liberal vs strict, caps on immigration)
- sectorial (agriculture, industry, services) distribution of the GDP. An agriculture and industry based economy would employ more people compared with a services-based economy. On the other hand, a services based economy would have a higher demand for educated immigrants.
- exchange rates between the immigrant and emigrant countries.
- age distributions
- unemployment rates, inflation
- existing immigrant populations

A Curve fitting/regression analysis could be done to fit the immigration rates of the previous years to estimate contributions from each factor. Events which led to any big migration e.g. second world war probably need to be excluded from the regression analysis. A bigger challenge would be determining the multi-immigrant coefficient (a generic coefficient for a single country wrt all countries).

**Sources/sinks (time-dependent)**

- investments (private & public)
- expenditure on military, health-care, social-security etc.

Needless to say, such a model would be too simplistic, but perhaps a good starting point. But what would be the use of such a model? Financial institutions such as the world bank or the IMF could look at how an investment would affect the immigration/emigration rates.

…time to do a literature survey and brush up my matlab skills.