Fig. 4

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 colourless area, no type-reproduction numbers are well defined. In the top three figures, homosexual and heterosexual transmission rates are the same, whereas in the bottom three figures, homosexual transmission is twice as high as heterosexual transmission. The ratio of MSMW and MSME is set as (a,d) \({\gamma }_{\overset{\sim }{\mathrm{m}}}/{\gamma }_{\overline{\mathrm{m}} }=3\), (b,e) \({\gamma }_{\overset{\sim }{\mathrm{m}}}/{\gamma }_{\overline{\mathrm{m}} }=1\) and (c,f) \({\gamma }_{\overset{\sim }{\mathrm{m}}}/{\gamma }_{\overline{\mathrm{m}} }=1/3\) (set \({\gamma }_{\overset{\sim }{\mathrm{m}}}+{\gamma }_{\overline{\mathrm{m}} }=0.02\))