  Competition and collaboration are at the heart of multi-agent probabilistic
spreading processes. The battle on public opinion and competitive marketing
campaigns are typical examples of the former, while the joint spread of
multiple diseases such as HIV and tuberculosis demonstrates the latter. These
spreads are influenced by the underlying network topology, the infection rates
between network constituents, recovery rates and, equally importantly, the
interactions between the spreading processes themselves. Here, for the first
time we derive dynamic message-passing equations that provide an exact
description of the dynamics of two interacting spreading processes on tree
graphs, and develop systematic low-complexity models that predict the spread on
general graphs. We also develop a theoretical framework for an optimal control
of interacting spreading processes through an optimized resource allocation
under budget constraints and within a finite time window. Derived algorithms
can be used to maximize the desired spread in the presence of a rival
competitive process, and to limit the spread through vaccination in the case of
coupled infectious diseases. We demonstrate the efficacy of the framework and
optimization method on both synthetic and real-world networks.
