We construct a stochastic process model for cascading families of time series descriptive of the initial development and then controlled evolution of a general epidemic/pandemic phenomenon. A distinguishing feature of the model is the effect of the spatial position of individual infections amongst other appropriate characteristics. The model naturally reproduces regime transitions representative of shifts from full community to localized contagions as phase transitions along time. More specifically, the model is defined as a growing family of renewal processes with an exponential time proliferation. The core part of the model consists of a renewal process of non-independent events that has been shown (Velázquez and Robledo (2011)) to be analogous to a statistical–mechanical thermal system capable of undergoing thermodynamic phase transitions. An external control parameter, akin to temperature, affects the spread of contagions, as quarantine is enforced on agents. When the thermal analog of the model is particularized to the so-called Hamiltonian Mean Field Model the space and time properties of the stochastic process can be solved explicitly in full detail. We show quantitative agreement with time series data from the recent COVID19 pandemic. We discuss additional modeling that could make this idealized construction perhaps closer to useful applications.

我们为级联的时间序列家庭构建了一个随机过程模型，描述了最初的发展，然后控制了一般流行/大流行现象的演变。该模型的一个显着特征是个体感染的空间位置以及其他适当特征之间的关系。该模型自然再现了政权转型，代表了随着时间的推移，从完整的社区到局部传染的转变。更具体地说，该模型定义为具有指数时间扩散的增长中的更新过程家族。该模型的核心部分包括一个非独立事件的更新过程，该过程已被证明（Velázquez和Robledo（2011））类似于能够经历热力学相变的统计-机械热系统。类似于温度的外部控制参数会影响传染性的传播，因为对代理商实施了隔离。当将模型的热模拟具体化为所谓的汉密尔顿平均场模型时，可以明确，详细地解决随机过程的时空特性。我们显示了与近期COVID19大流行中的时间序列数据的定量一致性。我们讨论了其他建模，这些建模可能会使这种理想化的结构更接近有用的应用程序。我们显示了与近期COVID19大流行中的时间序列数据的定量一致性。我们讨论了其他建模，这些建模可能会使这种理想化的结构更接近有用的应用程序。我们显示了与近期COVID19大流行中的时间序列数据的定量一致性。我们讨论了其他建模，这些建模可能会使这种理想化的结构更接近有用的应用程序。