The modeling framework presented in the previous sections is mostly based on the Poisson approximation <ref>Tijms H C. A first course in stochastic models[M]. John Wiley and sons, 2003.</ref> for both the transmission and recovery processes. The Poisson approximation assumes that the probabilities per unit time of transmitting the disease through a given edge, or recovering for a given infected node, are constant,<math> </math> and equal to <math>\beta</math> and <math>\mu</math>, respectively. Equivalently, the total time <math>\tau_i</math> that a given node <math>i</math> remains infected is a random variable with an exponential distribution <math>P_i(\tau_i)=\mu e^{-\tau_i \mu}</math>, and that the time <math>\tau_a</math> for an infection to propagate from an infected to a susceptible node along a given edge (the interevent time) is also exponentially distributed <math>P_a(\tau_a)=\mu e^{-\tau_a \mu}</math>. | The modeling framework presented in the previous sections is mostly based on the Poisson approximation <ref>Tijms H C. A first course in stochastic models[M]. John Wiley and sons, 2003.</ref> for both the transmission and recovery processes. The Poisson approximation assumes that the probabilities per unit time of transmitting the disease through a given edge, or recovering for a given infected node, are constant,<math> </math> and equal to <math>\beta</math> and <math>\mu</math>, respectively. Equivalently, the total time <math>\tau_i</math> that a given node <math>i</math> remains infected is a random variable with an exponential distribution <math>P_i(\tau_i)=\mu e^{-\tau_i \mu}</math>, and that the time <math>\tau_a</math> for an infection to propagate from an infected to a susceptible node along a given edge (the interevent time) is also exponentially distributed <math>P_a(\tau_a)=\mu e^{-\tau_a \mu}</math>. |