The spread of disease with birth and death on networks
  In this paper, we introduce a modified epidemic model on regular and scale-free networks respectively. We consider the birth rate $\delta$, cure rate $\gamma$, infection rate $\lambda$, $\alpha$ from the infectious disease, and death rate $\beta$ from other factors. Through mean-field analysis, we find that on regular network there is an epidemic threshold $\lambda_{c}$ dependent on the parameters $\delta$, $\gamma$, $\alpha$, and $\beta$; while for power law degree distribution network epidemic threshold is absent in the thermodynamic limit. The result is the same as that of the standard SIS model. This reminds us the structure of the networks plays a very important role in the spreading property of the infectious disease. 
Jingzhou Liu, et al