Daniel Archambault

Max Sondag Sondag

Cronfa at Swansea University

Lines: 66What is R? A graph drawer’s perspective?Max Sondag1 , Cagatay Turkay2 Sibylle Mohr3 , Louise Matthews3 , KaiXu4 , and Daniel Archambault11 Swansea University, Swansea, UK{m.f.m.sondag,d.w.archambault}@swansea.ac.uk2 Warwick University, Warwick, UKCagatay.Turkay@warwick.ac.uk3 University of Glasgow, Glasgow, UK{Sibylle.Mohr,Louise.Matthews}@glasgow.ac.uk4 Middlesex University, Middlesex, UKK.Xu@mdx.ac.uk1 Introduction1In 2020, the worldwide pandemic of Covid-19 had a profound impact on society.2One of the most important metrics that is being used to investigate the effec-3tiveness of various interventions to control the spread of the pandemic is R, the4effective reproduction number. It is talked about extensively in the news, and5a wide-ranging array of different interventions are put in place by governments6with the aim of getting R below 1 to curb the spread of the disease. In this7abstract, we will investigate what R means from a graph drawer’s perspective8and aim to open up interesting and relevant research avenues.92 Defining R10To define R, we start with a temporal network [2] G of the contacts between11people, for example all contacts within a city or country. Each node u represents12a person, and a temporal edge et = (u, v) represents that person u was in contact13with person v at time t. We overlay the disease we are interested in on G. Some14nodes will be index cases, the initial cases where the disease emanates from.15These nodes are exposed to the disease from outside the network (i.e. a pangolin16or international travel) A person v who has become infectious at time t, has a17chance to expose their neighbors over edges with a time greater than t until v is18no longer infectious. After an incubation time, these neighbors become infectious19in turn, and can propagate the disease further. This propagation through the20network creates an infection map: A set of rooted directed trees T , where each21node v represents a person and has a value e(v) which indicates when this node22was exposed to the disease. Using this infection map, we can define R.23? This work was supported by UKRI EPSRC grant EP/V033670/1 and EP/V054236/12 M. Sondag et al.(a)1,2,33,4,53521,2,3464,61,22,53120535(c)(b)02553155012453t Yt Xt R∗t2101310020136 021553210Fig. 1. Example calculation of Rt. (a) Con-tact graph with highlighted paths of infec-tion. Values within nodes show exposuretime e(v). (b) The resulting infection map isa single infection tree. Nodes with the sameexposure time have the same color. (c) Rtcalculation table. Rt is the average over the 7values of R∗t : (2+0+3+0+0+0+0)/7 ≈ 0.71There are two different R’s that24are of interest to epidemiologists.25R0 (known as, “R naught” or “ba-26sic reproduction number”), which27indicates the capacity for the dis-28ease to spread when the entire29population is susceptible and no30interventions are in place [1], and31the R commonly used by the me-32dia: Rt (“R” or “effective repro-33duction number”), which takes in-34terventions and immunity into ac-35count [3,4]. Rt The t in Rt is the36time point of interest, often a day37or a week, but is ultimately de-38pendent on the characteristics of39the disease and available data. In40graph-theoretical terms, we can41define Rt as follows. Let Yt denote42the nodes with e(v) = t. Let Xt43be the set of children of Yt in T .44The Rt value for a single point in45time is then: R∗t =|Xt||YT | . Due to46pragmatic issues such as report-47ing issues and weekly updates for48policymakers, Rt is often averaged over a period of time (usually 7 days for49COVID) to give the final Rt value. This calculation of Rt is shown in Figure 1.503 Open problems in real and simulated data51We briefly examine various (open) problems that are encountered when using52this graph-based data. We first consider problems in the temporal network. In53real data, both edges and nodes are missing from the graph. The time of contact54could be incorrect, and is at best an approximation (often at day accuracy) even55for simulated data. Continuing with problems in determining the spread of the56disease over the network, the directionality of the edges is generally unknown.57It is also not typically known for certain whether one nodes infected another, or58if there was an outside influence. The time of exposure is often an estimation,59as testing requires time, and the results might even be incorrect. For both real60and simulated data, these data characteristics lead to the graph structure of the61infection map being a large forest of small trees, with a few larger trees. While62not all of these problems and characteristics are unique to epidemic graphs, we63believe that there are nevertheless a number of interesting research questions64present in this setting, and research from the Graph Drawing community could65help in assisting with the current pandemic and mitigating future pandemics.66What is R? A graph drawer’s perspective 3References671. Anderson, R.M., May, R.M.: Infectious diseases of humans: dynamics and control.68Oxford university press (1992)692. Holme, P., Saramäki, J.: Temporal networks. Physics Reports 519(3), 97 – 12570(2012)713. Nishiura, H., Chowell, G.: The effective reproduction number as a prelude to statis-72tical estimation of time-dependent epidemic trends. In: Mathematical and statistical73estimation approaches in epidemiology, pp. 103–121 (2009)744. Wallinga, J., Lipsitch, M.: How generation intervals shape the relationship between75growth rates and reproductive numbers. Proceedings of the Royal Society B: Bio-76logical Sciences 274(1609), 599–604 (2007)77

What is R? A graph drawer’s perspective?

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