 Solution

The cross-infection terms transfer susceptible individuals into the infected class: $\beta S\hat{I}$ is male incidence caused by infected females and $\hat{\beta }\hat{S}I$ is female incidence caused by infected males. The terms $\nu I$ and $\hat{\nu }\hat{I}$ are recovery flows back to susceptibility. Adding each susceptible-infected pair shows that $N$ and $\hat{N}$ are conserved.