  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.
