  In this paper, we propose an optimization framework to design a network of
positive linear systems whose structure switches according to a Markov process.
The optimization framework herein proposed allows the network designer to
optimize the coupling elements of a directed network, as well as the dynamics
of the nodes in order to maximize the stabilization rate of the network and/or
the disturbance rejection against an exogenous input. The cost of implementing
a particular network is modeled using posynomial cost functions, which allow
for a wide variety of modeling options. In this context, we show that the
cost-optimal network design can be efficiently found using geometric
programming in polynomial time. We illustrate our results with a practical
problem in network epidemiology, namely, the cost-optimal stabilization of the
spread of a disease over a time-varying contact network.
