Vector difference equations, substochastic matrices, and design of multi-networks to reduce spread of epidemics

We start with the SIR model (susceptible, infected, removed) on a network. See, e.g., Cooper, Mondal & Antonopoulos, in Chaos, Solitons & Fractals 2020. Since the goal is to make I = 0 a (Lyapunov) stable equilibrium, we linearize the discrete-time SIR model to obtain difference equations of the form I_new = I(1 + aS - b) at each node before including infections derived from other nodes. We assume S equal to its initial value at that node. Here a depends upon the infectivity and contact rate, b = 1/(duration of infectivity) and the traditional R_t = aS/b (R_t < 1 ↔ aS < b). This yields a vector difference equation I_new = MI. The entries of M may vary in time, even discontinuously as flows between nodes are turned on and off. The column sums of M may be interpreted as generalizations of (scalar) growth rates. Theorem. If the matrices M are column-substochastic, then the I=0 equilibrium is stable. 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.