How the reversible change of contact network affects the epidemic spreading Xincheng Shu12Zhongyuan Ruan1

2025-05-06 0 0 5.66MB 22 页 10玖币

侵权投诉

How the reversible change of contact network affects

the epidemic spreading

Xincheng Shu,1,2Zhongyuan Ruan,1∗

1Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou, 310023, China

2Department of Electrical Engineering, City University of Hong Kong, Hong Kong, 999077, China

∗Correspondence: zyruan@zjut.edu.cn

The mobility patterns of individuals in China during the early outbreak of the

COVID-19 pandemic exhibit reversible changes — in many regions, the mobil-

ity ﬁrst decreased signiﬁcantly and later restored. Based on this observation,

here we study the classical SIR model on a particular type of time-varying net-

work where the links undergo a freeze-recovery process. We ﬁrst focus on an

isolated network and ﬁnd that the recovery mechanism could lead to the resur-

gence of an epidemic. The inﬂuence of link freezing on epidemic dynamics is

subtle. In particular, we show that there is an optimal value of the freezing

rate for links which corresponds to the lowest prevalence of the epidemic. This

result challenges our conventional idea that stricter prevention measures (cor-

responding to a larger freezing rate) could always have a better inhibitory

effect on epidemic spreading. We further investigate an open system where a

small fraction of nodes in the network may acquire the disease from the “en-

vironment” (the outside infected nodes). In this case, the second wave would

appear even if the number of infected nodes has declined to zero, which can

arXiv:2210.12670v1 [physics.soc-ph] 23 Oct 2022

not be explained by the isolated network model.

Introduction

During the early prevalence of COVID-19 (coronavirus disease 2019) in mainland China, the

Chinese government implemented a series of containment policies aiming to mitigate the spread

of the disease (1–5). These policies led to substantial changes in human mobility patterns (6, 7)

[see Fig. 1 (a) and (b)]. Figure 1 (a) shows the intracity travel intensity (deﬁned as the ratio of

the number of individuals traveling in the city to the number of people living in it (8)) for each

city in Zhejiang province in China as a function of time from 1 January to 22 March, 2020.

We observe that mobility was considerably reduced at ﬁrst, and after a turning point (around 8

February, 2020), it started to restore and ﬁnally returned to the pre-pandemic state, displaying

a clear reversible process. Since mobility directly affects the average number of contacts per

person, it is expected that the contact patterns would change correspondingly: we assume that

the links (connections) between individuals would experience a freeze-recovery process (the

frozen links are equivalent to being removed from the contact network temporarily), as shown

in Fig. 1 (c).

Many studies have focused on how the time-varying contact networks may affect the epi-

demic dynamics (9–21). For instance, it has been shown that adaptive networks (individuals

avoid contact with the infected) could give rise to rich dynamics like hysteresis and ﬁrst-order

transitions (13). Apart from this, some research focused on the situation in which the contact

structures evolve independently of the dynamical process (16–21). For example, it is demon-

strated that temporal heterogeneities in contacts can slow down spreading (16). Besides, a

notable ﬁnding shows that in the activity-driven networks, the epidemic threshold does not de-

pend on the time-aggregated network representation, but is a function of the interaction rate of

the nodes (17).

2020-01-16 2020-02-08 2020-03-01















(c)(a)

(b)

Freeze

Recover

Figure 1: (color online). (a) Evolution of the intracity travel intensity for all 11 cities in Zhejiang

province in China from 1 January to 22 March, 2020. (b) Visualization of the intracity travel

intensity for each city in Zhejiang province on three typical dates. (c) Reversible change in

the contact network. The travel restriction during the COVID-19 disease outbreak led to many

people cutting their physical connections to others over a period of time. After that, these links

started to recover, becoming active again. The intracity travel intensity data are collected from

Baidu Map Smart Eye.

Despite these progresses, the recovery process of the contact networks has not been fully

emphasized yet. As the containment measures during an epidemic can have severe conse-

quences (or costs) on a region’s society and economy (though they may effectively hinder the

spread of diseases) (22,23), it is of paramount importance to address the questions such as when

and how to restore the social activity so that the costs can be minimized. To solve these prob-

lems, we need a comprehensive understanding of how the reversible change in contact networks

can inﬂuence epidemic dynamics.

Here, we analyze the conventional Susceptible-Infected-Removed (SIR) model (24, 25) on

a particular type of time-varying network which presents a reversible change in its structure.

Speciﬁcally, the contact network would ﬁrst freeze, with links becoming inactive with rate γ.

After a time t∗, the frozen links begin to recover and become active again with rate η. We show

that all the three parameters γ,ηand t∗are signiﬁcant in the spreading dynamics. In particular,

we ﬁnd that the recovery of the network may markedly alter the epidemic shape, leading to

the resurgence of the epidemic. This effect consequently brings about a counterintuitive result:

there exists an optimal value of freezing rate which corresponds to the smallest epidemic size.

We further extend this model (which is an isolated network) by letting the nodes interact weakly

with the outside environment. The new model has the potential to explain the observational data

which shows that in some regions, the epidemic could resurge even if the infected cases have

dropped to zero.

Epidemic spreading in an isolated network

Let us begin by considering the SIR model deﬁned on an isolated Erd˝

os-R´

enyi (ER) random

network with average degree hki. Nodes in the network are divided into three compartments:

infected (I), susceptible (S), and removed (R), corresponding to different states of individuals

in the contagion process. Let St,It, and Rtbe the fraction of susceptible, infected, and removed

nodes at time t, respectively. We have

St+It+Rt=1.(1)

At each time step, every susceptible node is infected with probability βif it is connected to one

infected node (hence, the node is infected with probability kβif it is linked to kinfected nodes).

At the same time, each infected node becomes removed with probability µ.

We suppose the underlying contact network varies with time as follows: Initially, all links

in the network are in an active state (meaning that a disease could normally transmit through

these links). As the epidemic spreads, the system enters into a freezing stage, where each link

freezes (or we can say that these links become inactive) with probability γat each time step.

Until time t∗, the system starts to recover, and the frozen links become active again with rate η.

Denote atas the fraction of active links at time t. The dynamics are governed by the following

set of differential equations

dSt

dt =−βhkiatItSt,(2)

dIt

dt =βhkiatItSt−µIt,(3)

dRt

dt =µIt,(4)

with

dat

dt =(−γat,t≤t∗

η(1−at),t>t∗.(5)

Assuming that only a very small fraction of nodes are infected at the beginning, the initial

conditions are S0≈1, I0≈0, R0=0 and a0=1. From Eq.(5), we obtain that

at=(e−γt,t≤t∗

1−(1−e−γt∗)e−η(t−t∗),t>t∗.(6)

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

HowthereversiblechangeofcontactnetworkaffectstheepidemicspreadingXinchengShu,1;2ZhongyuanRuan,11InstituteofCyberspaceSecurity,ZhejiangUniversityofTechnology,Hangzhou,310023,China2DepartmentofElectricalEngineering,CityUniversityofHongKong,HongKong,999077,ChinaCorrespondence:zyruan@zjut.edu.cnThemob...

展开>> 收起<<

How the reversible change of contact network affects the epidemic spreading Xincheng Shu12Zhongyuan Ruan1.pdf

共22页,预览5页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

How the reversible change of contact network affects the epidemic spreading Xincheng Shu12Zhongyuan Ruan1

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: