Vector Difference Equations, Substochastic Matrices, and Design of Multi-Networks to Reduce the Spread of Epidemics

Cities have long served as nucleating centers for human development and advancement. Cities have facilitated the spread of both human creativity and human disease, and at the same time, efforts to minimize the spread of disease have influenced the design of cities. The purpose of this paper is to explore the dynamics of epidemics on networks in order to help design a multi-network city of the future aimed at minimizing the spread of epidemics. In order to do this, we start with the SIR model (susceptible, infected, removed) on a network in which nodes represent cities or regions and edges are weighted by flows between regions. Since the goal is to stabilize the zero infections steady state, we linearize the discrete-time SIR model yielding difference equations for the dynamics of infections at each node and then include flows of infections from other nodes. This yields a vector difference equation for the spread of infections. We then generalize the concept of stochastic matrix in order to quantify the dynamics of this update equation. The entries of the update matrix $M$ may vary in time, even discontinuously as flows between nodes are turned on and off. This may yield useful design constraints for a multi-network composed of weak and strong interactions between pairs of nodes representing interactions within and among cities.