We use a deterministic model to study two competing viruses spreading over a two-layer
network in the Susceptible–Infected–Susceptible (SIS) framework, and address a central
problem of identifying the winning virus in a “survival-of-the-fittest” battle. Almost all existing
conditions ensure that the same virus wins regardless of initial states. In the present paper,
we ask the following question: can we systematically construct SIS bivirus networks with an
arbitrary but finite number of nodes such that either of the viruses can win the survival-of-the …