Developing a Markov-Modulated Process Model for Mobility Processes

Abstract

A major factor in the spreading of viruses is human-to-human transmission, and human mobility is clearly linked to the spreading process of epidemics. If we hope to understand the evolution of an epidemic, then we must also understand the underlying mobility process and the interaction between the two.

We propose the Markov modulated process (MMP) model as a tool for modeling mobility processes. We take a link-based approach to the MMP model, where modulating actions are a joint combination of adding and removing a number of links. We demonstrate that for such an approach, the states of the Markov process must encode not only a modulating action, but also the current number of links in the graph. We further show that in this case, the number of states will increase with O(N⁶) where N is the number of nodes. We refer to this as the "unaggregated" MMP model, and we introduce the concept of state aggregation to create the "aggregated" MMP model which only requires O(N²) states.

We demonstrate that both the aggregated and unaggregated MMP models are able to match the average number of links in the simulated mobility process, capturing the long-term dynamics of the mobility process with high accuracy. Both the aggregated and unaggregated MMP models are also able to match the average number of links added and removed between two time steps, capturing the short-term dynamics with high accuracy. Finally, we demonstrate that for certain mobility processes, the aggregated MMP model is able to produce results which are indistinguishable from the unaggregated MMP model while requiring just O(N²) states compared to O(N⁶).