论文标题
传染病的随机SIR模型中的关键性
Criticality in stochastic SIR model for infectious diseases
论文作者
论文摘要
我们讨论了传染病随机SIR模型中的关键性。我们采用路径综合形式主义来传播易感性，感染和删除个体之间的感染，并执行扰动和非扰动分析，以评估基本复制数量$ {\ cal r} $的关键价值。在扰动理论中，我们计算了初始时间附近传染个体数量的平均值和差异，并发现关键值$ {\ cal r} _ {\ text {c}} = 1/3 $ - $ 2/3 $应被采用，以抑制感染的随机传播。在非扰动方法中，我们通过整合随机波动，并获得有效的Euler-Lagrange方程来获得有效潜力，以获取有效的Euler-Lagrange方程，以易于易感性，感染性和去除的个体的时间进化。从渐近行为长期以来，我们发现临界值$ {\ cal r} _ {\ text {c}} = 2/3 $应用于足够的感染收敛。我们还发现，流行状态可以通过传统的SIR模型中不存在的随机波动动态生成。这些分析表明，与通常已知的临界值$ {\ cal r} _ {\ text {c}} = 1 $相对于通常已知的临界值$ {\ cal r} = 1 $时，基本复制号的临界值应小于一个。
We discuss the criticality in the stochastic SIR model for infectious diseases. We adopt the path-integral formalism for the propagation of infections among susceptible, infectious, and removed individuals, and perform the perturbative and nonperturbative analyses to evaluate the critical value of the basic reproduction number ${\cal R}$. In the perturbation theory, we calculate the mean values and the variances of the number of infectious individuals near the initial time, and find that the critical value ${\cal R}_{\text{c}}=1/3$-$2/3$ should be adopted in order to suppress the stochastic spread of infections sufficiently. In the nonperturbative approach, we derive the effective potential by integrating out the stochastic fluctuations, and obtain the effective Euler-Lagrange equations for the time-evolution of the numbers of susceptible, infectious, and removed individuals. From the asymptotic behaviors for a long time, we find that the critical value ${\cal R}_{\text{c}}=2/3$ should be adopted for the sufficient convergence of infections. We also find that the endemic state can be generated dynamically by the stochastic fluctuation which is absent in the conventional SIR model. Those analyses show that the critical value of the basic reproduction number should be less than one, against the usually known critical value ${\cal R}_{\text{c}}=1$, when the stochastic fluctuations are taken into account in the SIR model.
