  In this paper, a stochastic dynamic control strategy is presented to prevent
the spread of an infection over a homogeneous network. The infectious process
is persistent, i.e., it continues to contaminate the network once it is
established. It is assumed that there is a finite set of network management
options available such as degrees of nodes and promotional plans to minimize
the number of infected nodes while taking the implementation cost into account.
The network is modeled by an exchangeable controlled Markov chain, whose
transition probability matrices depend on three parameters: the selected
network management option, the state of the infectious process, and the
empirical distribution of infected nodes (with not necessarily a linear
dependence). Borrowing some techniques from mean-field team theory the optimal
strategy is obtained for any finite number of nodes using dynamic programming
decomposition and the convolution of some binomial probability mass functions.
For infinite-population networks, the optimal solution is described by a
Bellman equation. It is shown that the infinite-population strategy is a
meaningful sub-optimal solution for finite-population networks if a certain
condition holds. The theoretical results are verified by an example of rumor
control in social networks.
