Influence of f₁ and f₂ on the epidemic thresholds β_1c and β_2c with fixed p = 0.005, where the “squares” and “uptriangles” represent the numerical simulations and “circles” and “downtriangles” represent the corresponding theoretical results from Eqs. (2)–(5). (a) β_1c versus f₁ with f₂ = 1, where the “squares” and “uptriangles” represent the cases of β₂ = 0.005 and 0.003, respectively. (b) β_2c versus f₁ with f₂ = 1, where the “squares” and “uptriangles” represent the cases of β₁ = 0.003 and 0.0014, respectively. (c) β_1c versus f₂ with f₁ = 1, where the “squares” and “uptriangles” represent the cases of β₂ = 0.005 and 0.003, respectively. (d) β_2c versus f₂ with f₁ = 1, where the “squares” and “uptriangles” represent the cases of β₁ = 0.003 and 0.0014, respectively. The results are averaged over 1000 realizations.