Fig. 1

Conceptual diagram of the model. The population was divided into four subpopulations: women (w), MSMW (\(\overset{\sim }{\mathrm{m}}\)), MSME (\(\overline{\mathrm{m} }\)) and MSW (m). The subpopulation i individuals are born per unit time in \(B{\gamma }_{i}\) and die at a rate of \({\mu }_{i}\), where \({\gamma }_{i}\) represents the proportion of subpopulation \(i\) at the time of birth (\({\gamma }_{\mathrm{w}}+{\gamma }_{\overset{\sim }{\mathrm{m}}}+{\gamma }_{\overline{\mathrm{m}} }+{\gamma }_{\mathrm{m}}=1\)). δ is the rate of infertility or stillbirth. The number per unit time of sexual contacts between individuals in subpopulation i and individuals in subpopulation j is assumed to be \({f}_{ij}\) (white arrows in both directions). Sexual transmission occurs at a rate of \({\beta }_{i\to j}\), and mother-to-child transmission occurs at birth at a rate of \(\alpha\) (black arrows)