Fig. 3

The phase plane of the STI survival region and smallest type-reproduction numbers. The region above the black bold curve indicates the survival region of STIs (i.e., \({T}_{\mathrm{w}},{T}_{\mathrm{m}},{T}_{\overset{\sim }{\mathrm{m}}},{T}_{\overline{\mathrm{m}} }>1\)), and the four colours indicate which type-reproduction numbers are the smallest. The black dashed curves indicate that \({T}_{\overset{\sim }{\mathrm{m}}}\) is not well defined above them. In the cases of a and c the dashed curves are identical to the bold ones. In the colourless area, no type-reproduction numbers are well defined. The other parameters are set as \(({\gamma }_{\mathrm{w}},{\gamma }_{\mathrm{m}},{\gamma }_{\overset{\sim }{\mathrm{m}}},{\gamma }_{\overline{\mathrm{m}} })=(\mathrm{0.5,0.48,0},\mathrm{01,0.01})\)