构建多尺度空间流行病模型的方法_Methodology for Multiscale Pandemic Modelling

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"这篇技术报告是 Brunel University London 和 Brandenburg Technical University Cottbus-Senftenberg 的合作成果，主题是从流行病到流行病建模，作者 Shannon Connolly, David Gilbert, 和 Monika Heiner。报告发布于2021年3月30日，旨在介绍一种系统性扩展流行病模型的方法，以适应多层、多尺度的时空大流行病模拟。" 本文详细探讨了如何将基本的SIR（易感者-感染者-康复者）模型扩展到包含人口分层和地理空间信息及旅行连接的多层面、多尺度的流行病大流行模型。SIR模型是一种常用于理解和预测疾病传播的经典理论模型，但其通常假设一个均匀混合的人口，忽略了个体间的差异和空间分布。报告提出的方法基于彩色随机和连续的Petrinets（一种形式化建模工具），这种工具能帮助以组件为基础的方式扩展模型，同时保持模型的清晰度和可读性。Petrinets 是一种强大的建模语言，可以表示复杂的系统行为，包括随机性和并发性。通过引入人口分层，模型能够更准确地反映不同群体（如儿童、老年人、有基础疾病的人等）对疾病的敏感性和传播能力。同时，利用图论来表示地理空间信息和旅行连接，可以模拟人员流动对疾病传播的影响，这对于理解和预测大流行病的地理扩散至关重要。该方法的另一个优点是其可扩展性和可重用性。无论是在确定性还是随机性框架下，都可以轻松阅读和理解这些模型。报告中提到的PetriNuts是一个公开可用的平台，支持这种方法，为研究人员和政策制定者提供了构建和分析这些复杂模型的工具。这份技术报告提供了一种系统性的建模方法，对于理解和控制如COVID-19这样的全球大流行病具有重要意义。它不仅增加了模型的复杂性以更真实地反映现实世界，还确保了模型的实用性和易用性，为未来流行病研究和防控策略的制定提供了有力的理论支持。

From Epidemic to Pandemic Modelling

2.4 Pandemic models

The distinguishing characteristic of a pandemic model is the concept of multiple

intercommunicating epidemic models in a spatial (geographic) context. In the

following we preﬁx standard epidemic model names by ‘P’ for pandemic, thus

obtaining PSIR, PSIQR, etc.

Two countries, linked by travel. Figure 4 (Top) shows a PSIR model com-

prising epidemic SIR models in each of two countries, with population movement

permitted between the countries for all compartments. As for the age strata

model, we can fold it into a coloured version (Figure 4 (Bottom)). The process

of colouring is very similar to that described for SIR-S

age

(see Methods section).

N countries, linked by travel. Generalising this idea to more than two coun-

tries, the travel connectivities between countries need to be deﬁned. For example

when considering surface travel, not all countries are directly connected to each

other. We assume that the connectivity is described by a graph (Figure 5 (Top

left)); this could be directly encoded as a standard Petri net (Figure 5 (Top mid-

dle)), which we fold for convenience into a coloured Petri net (Figure 5 (Top

right)), see Methods section for details. Equally the graph could include air

and/or sea connections; even then, connectivity could still be less than fully con-

nected.

The coloured Petri net representation of the PSIR over four countries (Fig-

ure 5 (Middle)) is derived by adjusting the colour deﬁnitions in the SIR for two

countries (Figure 4 (Bottom)) according to the connectivity graph, and getting

the initial marking right.

The generalisation to connectivity graphs representing diﬀerent geographical

relationships is straightforward, merely requiring adjusting the two related colour

deﬁnitions (Countries, Connections) and the initial marking. This example illus-

trates how our modelling approach supports reusability in an easy to use manner.

In general the unfolded PSIR model for n countries with overall m mutual con-

nections comprises 3n places, 2n + 6m transitions and 6n + 12m arcs, and the

ODEs generated consist of 3n equations.

We provide a number of PSIR pandemic models in the Supplementary Mate-

rial, including Europe (P

SIR – 10 countries, 15 mutual connections, P

SIR –

48 countries, 85 mutual connections), China (P

China

SIR – 34 provinces, 71 mu-

tual connections) and the USA (P

USA

SIR – 50 states, 105 connections), and the

guidelines how to obtain the corresponding ODEs using our tool platform.

30 March 2021 8

From Epidemic to Pandemic Modelling

FISE

10 10

1010

10,000

10,00010,000

10,000

10,00010,000

10,000

Figure 6: P

SIR-S

age

model. (Top left) Map of 48 European countries cre-

ated with https://mapchart.net. (Top right) Undirected graph representa-

tion of the map where edges represent two-way surface connections. Blue edges

stand for road connections, orange dashed edges for sea connections. (Bottom)

Unfolded version of the coloured net in Figure 5 (Bottom); layout automatically

generated. Iceland appears as an isolated subnet because it is not connected by

surface to the rest of Europe, and is thus similar to the SIR net with 20 strata in

Figure 3 (Bottom). Model size: 1,440 places; 10,380 transitions.

30 March 2021 11

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