Real networks are not lattices; they have varied degree distributions. The famous result of Marián Boguña and Alessandro Vespignani shows, that for scale-free networks (which closely resemble the networks we see in the real world), the epidemic threshold could be arbitrarily low: https://arxiv.org/pdf/cond-mat/0208163.pdf.

I have also published some research among these lines: https://arxiv.org/pdf/1403.5815.pdf