Predictive Mathematical Models of the Covid–19 Pandemic in Ode/sde Framework
Predictive Mathematical Models of the Covid–19 Pandemic in Ode/sde Framework
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Ode/sde 框架中 Covid-19 大流行的预测数学模型
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发表时间：
2020
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通讯作者：
Marcello Colozzo
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作者：
Marcello Colozzo
本文在常微分方程和随机微分方程框架下建立了一个病毒扩散模型（如Covid-19大流行）。分析了基于逻辑斯蒂映射的经典模型，并引入噪声项来描述否认者的行为。这种模式相当忠实地再现了当今意大利的形势。然后，我们继续进行局部分析，得出一个连续性方程，该方程与指定区域内感染者数量的密度有关。因此，我们证明了一个定理，根据这个定理，经典物流是最灾难性的预测。在现实的情况下，有必要考虑到上述密度的不可避免的波动。这意味着将初始聚类（由“患者零”生成）分割成N个不相交的子聚类。对于非常大的N，统计分析建议使用两点相关函数（更一般地，n点）。原则上，对这一函数的估计可以确定大流行病的演变。子星系团的分布可能是分形的，就像从均匀和各向同性的原始宇宙开始的星系分布一样，但物质密度有随机波动。这并不奇怪，因为由于尺度的不变性，分形具有较低的“计算成本”。因此，流行病是周期性过程的观点，即流行病的发生具有一定的周期性，仍然是可以证实的。
This article proposes a viral diffusion model (like Covid-19 pandemic) in the ordinary differential equations (ODE) and stochastic differential equations (SDE) framework. The classic models based on the logistic map are analyzed, and then a noise term is introduced that models the behavior of the so-called deniers. This model fairly faithfully reproduces the Italian situation in today’s period. We then move on to local analysis, arriving at an equation of continuity for what concerns the density of the number of infected in an assigned region. We, therefore, prove a Theorem according to which classical logistics is the most catastrophic of predictions. In a realistic scenario, it is necessary to take into account the inevitable fluctuations in the aforementioned density. This implies a fragmentation of the initial cluster (generated by “patient zero”) into an N disjoint sub clusters. For very large N, statistical analysis suggests the use of the two-point correlation function (and more generally, n-points). In principle, an estimate of this function makes it possible to determine the evolution of the pandemic. The distribution of the sub clusters could be fractal, exactly as it happens for the distribution of galaxies starting from a homogeneous and isotropic primordial universe, but with random fluctuations in matter density. This is not surprising, since due to the invariance in scale, fractals have a low “computational cost”. The idea that pandemics are cyclical processes, that is, they occur with a given periodicity, would therefore remain corroborated.