The spreading dynamics of infectious diseases is determined by the interplay between geography and population mixing. There is homogeneous mixing at the local level and human mobility between distant populations. Here I model spatial locations as a type and the population mixing by intra- and inter-type mixing patterns. Using the theory of multi-type branching process, I calculate the expected number of new infections as a function of time. In 1-dimension the analysis is reduced to the eigenvalue problem of a tridiagonal Teoplitz matrix. In d-dimensions I take advantage of the graph cartesian product to construct the eigenvalues and eigenvectors from the eigenvalue problem in 1-dimension. Using numerical simulations I uncover a transition from linear to multi-type mixing exponential growth with increasing the population size. Given that most countries are characterized by a network of cities with more than 100,000 habitants, I conclude that the multi-type mixing approximation should be the prevailing scenario.
Transition to multi-type mixing in d-dimensional spreading dynamics