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== 马尔科夫疾病传播 ==

== 马尔科夫疾病传播 ==

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, and equal to $\beta$ and $\mu$, respectively. Equivalently, the total time $\tau_i$ that a given node $i$ remains infected is a random variable with an exponential distribution $P_i(\tau_i)=\mu e^{-\tau_i \mu}$, and that the time $\tau_a$ for an infection to propagate from an infected to a susceptible node along a given edge (the interevent time) is also exponentially distributed $P_a(\tau_a)=\mu e^{-\tau_a \mu}$.  

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, 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>.  

在经典的流行病传播中，个体间疾病传播和恢复过程被近似假设为泊松过程，为马尔科夫的疾病传播。这种泊松近似假设了单位时间内通过给定连边传播疾病的概率或针对给定感染节点恢复的概率是常数的，分别等于$\beta$和$\mu$。对应等效地，网络中处于I态的节点$i$仍然为感染态的总时间$\tau_i$是服从指数分布$P_i(\tau_i)=\mu e^{-\tau_i \mu}$的随机变量，并且一个感染态节点沿着一条边传播疾病给一个易感染节点的所需要花费的时间$\tau_a$，即事件发生间隔（the interevent time）也是服从指数分布$P_a(\tau_a)=\mu e^{-\tau_a \mu}$的随机变量。

在经典的流行病传播中，个体间疾病传播和恢复过程被近似假设为泊松过程，为马尔科夫的疾病传播。这种泊松近似假设了单位时间内通过给定连边传播疾病的概率或针对给定感染节点恢复的概率是常数的，分别等于<math>\beta</math>和<math>\mu</math>。对应等效地，网络中处于I态的节点<math>i</math>仍然为感染态的总时间<math>\tau_i</math>是服从指数分布<math>P_i(\tau_i)=\mu e^{-\tau_i \mu}</math>的随机变量，并且一个感染态节点沿着一条边传播疾病给一个易感染节点的所需要花费的时间<math>\tau_a</math>，即事件发生间隔（the interevent time）也是服从指数分布<math>P_a(\tau_a)=\mu e^{-\tau_a \mu}</math>的随机变量。