Relating Eulerian and Lagrangian spatial models for vector-host diseases dynamics through a fundamental matrix

We explore the relationship between Eulerian and Lagrangian approaches for modeling movement in vector-borne diseases for discrete space. In the Eulerian approach we account for the movement of hosts explicitly through movement rates captured by a graph Laplacian matrix $L$. In the Lagrangian approach we only account for the proportion of time that individuals spend in foreign patches through a mixing matrix $P$. We establish a relationship between an Eulerian model and a Lagrangian model for the hosts in terms of the matrices $L$ and $P$. We say that the two modeling frameworks are consistent if for a given matrix $P$, the matrix $L$ can be chosen so that the residence times of the matrix $P$ and the matrix $L$ match. We find a sufficient condition for consistency, and examine disease quantities such as the final outbreak size and basic reproduction number in both the consistent and inconsistent cases. In the special case of a two-patch model, we observe how similar values for the basic reproduction number and final outbreak size can occur even in the inconsistent case. However, there are scenarios where the final sizes in both approaches can significantly differ by means of the relationship we propose.