Fig. 3

Two-dimensional diagram illustrating different types of behaviors of the stochastic network systems. In a, the two strains have equal infection and re-infection parameters. We vary the infection parameters \(\lambda _1=\lambda _2=\lambda _s\) on the interval [0, 0.5], while the re-infection parameters \(\rho _{11}=\rho _{22}=\rho _s\) are also varied on the interval [0, 1]. The blue region represents the non-epidemic regime, the orange the epidemic regime, and the red the endemic regime. Dashed lines indicate theoretical predictions. In b, we vary the infection and re-infection parameter values. Specifically, \(\lambda _1\) and \(\rho _{11}\) are varied on the interval [0, 0.5] and [0, 1], respectively, while we set \(\lambda _2 = 0.5-\lambda _1\) and \(\rho _{22} = 1-\rho _{11}\). Seven regions are highlighted, depending on the behavior of the two strains. In Region I, strain 1 is non-epidemic and strain 2 is epidemic; in Region II, strain 1 is non-epidemic and strain 2 is endemic; In Region III, strain 2 is non-epidemic and strain 1 is epidemic; In Region IV, strain 2 is non-epidemic and strain 1 is endemic; in Region V, strain 1 is epidemic and strain 2 is endemic; in Region VI, strain 2 is epidemic and strain 1 is endemic; in Region VII, both strains are endemic