Critical value in a SIR network model with heterogeneous infectiousness and susceptibility.

Using the technique of edge-based compartmental modelling (EBCM) for the spread of susceptible-infected-recovered (SIR) diseases in networks, in a recent paper (PloS One, 8(2013), e69162), Miller and Volz established an SIR disease network model with heterogeneous infectiousness and susceptibility. The authors provided a numerical example to demonstrate its validity but they did not perform any mathematical analysis of the model. In this paper, we resolve this problem. Using the nature of irreducible cooperative system in the theory of monotonic dynamical system, we prove that the dynamics of the model are completely determined by a critical value : When > 0, the disease persists in a globally stable outbreak equilibrium; while when < 0, the disease dies out in the population and the disease free equilibrium is globally stable.