Table 2 Parameters description

M

The maximum number of individuals among all subpopulations.

\((S^{k},I^{k})\)

Two-dimensional random variable with the numbers of susceptible and infectious individuals in a subpopulation with degree k.

\(V_{k}(s,i)(t)\)

The number of nodes (subpopulations) with degree k, in which susceptible and infectious individuals are s and i at time t, respectively, s,i∈0,1,…,M.

\(p_{s,i}^{k}(t)\)

Probability that the numbers of susceptible and infectious individuals in a subpopulation with degree k are s and i at time t, respectively, that is, \(p_{s,i}^{k}(t)=P\{(S^{k},I^{k})(t)=(s,i),S^{k},I^{k}\in{0,1,\ldots,M}\}\).

\(p^{k}_{s+\Delta s,i+\Delta i}(\Delta t)\)

The transition probability from state (s,i) to state (s + Δs,i + Δi) in the interval Δt in a homogeneous Markov process.

\(\langle f(s,i)\rangle_{k}\)

The expectation of the function f(s,i), \(\langle f(s,i)\rangle_{k}=\sum_{s,i} f(s,i)p_{s,i}^{k}\). Specially, \(\langle s\rangle_{k}\) and \(\langle i\rangle_{k}\) are expectations of susceptible individuals and infectious individuals, respectively.