The COVID-19 pandemic has resulted in a wide range of spatial interventions to slow the spread of the virus. The spatial limitations of narrow public circulation spaces within informal settlements, which house over 1 billion people around the world, make it impossible for pedestrians to practice physical distancing (or social distancing). A flexible mathematical method, the Cluster Lane Method, is proposed for turning a planar circulation network of any size or complexity into a network of unidirectional lanes. This makes physical distancing possible in narrow circulation spaces by limiting face-to-face interactions. The opportunities and challenges are discussed for the implementation of this cost-efficient, low-tech solution. New notions and theorems are introduced for oriented graphs in graph theory.

A new approach based on graph theory is used to address the problem of COVID-19 contagion in the narrow public circulation networks of informal settlements. The Cluster Lane Method shows how to convert a planar circulation network of any size or complexity into a network of unidirectional lanes. This makes physical distancing possible in narrow circulation spaces by limiting face-to-face interactions between pedestrians. By involving the inhabitants of the informal settlement throughout the process, more adequate orientations of the lanes can be found.

The transformative role of pandemics and disease in changing the trajectory of
architectural thought and practice has been well documented. For example, Harriet
Jordan argued how the 19th-century movement pushing for public parks was a response
to the overcrowding and pollution brought about by industrialisation (

It is highly likely that the current COVID-19 pandemic will trigger significant
paradigm shifts in how urban infrastructure is designed in the long run. Yet, more
immediate spatial interventions that are easily deployed and implemented are needed.
In fact, urban environments have started to become more friendly to cyclists and
pedestrians with the recent implementation of physical distancing guidelines in some
of the world’s major cities (

Urban slums and informal settlements lack proper medical infrastructure,
well-enforced sanitary standards or room for physical distancing and thus pose a
significant threat to a successful response. According to ECOSOC (

The pandemic will hit the hardest the more than 1 billion slum dwellers worldwide. […] Urgent response plans are needed to prepare for and respond to outbreaks in informal settlements and slums.

The gravity of this situation is unquestionable.

This paper addresses the risk of COVID-19 contagion via airborne transmission in the narrow public circulation lanes of informal settlements. A low-tech, mathematical solution is proposed that limits face-to-face interactions among pedestrians. Thus, this strategy becomes an additional tool to combat the spread of disease.

The paper is structured as follows. Section 2.1 discusses the morphologies of informal settlements and their narrow public circulation lanes. Section 2.2 addresses the airborne transmission of COVID-19 and the role of physical distancing to reduce the spread of the disease. Section 3 explores the urban origins of graph theory and the important role of this branch of mathematics on architecture and urban planning. Subsequently, section 4 introduces the Cluster Lane Method, which makes it possible to turn a planar circulation network into a network of unidirectional lanes, thus limiting face-to-face interactions between pedestrians while limiting circulation between different clusters of dwellings. Finally, Section 5 discusses the applications of the method to real informal settlements, along with the potential difficulties and improvements of these applications. Section 6 concludes. Appendix 1 in the supplemental data online defines basic notions of graph theory; and Appendix 2, also online, describes the Cluster Lane Method in a mathematically rigorous way by introducing new definitions and theorems in graph theory.

This section explores the spatial qualities of informal settlements and their narrow public circulation lanes. Some existing evidence on the airborne transmission of COVID-19 is presented, together with the practice of physical distancing to reduce this transmission.

Dharavi in Mumbai, India, is one of the world’s densest slums and fits
close to 1 million people into an area roughly two-thirds the size of Central
Park in New York (

67% of households rely on community toilets, soap and clean water are scarce and physical distancing is impossible.

Satellite view of a small area of the Dharavi slum in Mumbai, India,
which has a population density of 354,167 people/km^{2}. The
main circulation arteries between clusters of buildings are visible, but
numerous narrow lanes are hidden under the roofs.

Photo: Google Earth.

Trailing the US with 33,165,820 confirmed COVID-19 cases, the second and third
countries with the most confirmed cases on 25 May 2021 were India (26,948,874)
and Brazil (16,194,209), according to the Johns Hopkins Coronavirus Resource
Center (

Informal settlements around the world vary in morphology, but exhibit similar
patterns which include narrow public circulation lanes. Dovey & King (

COVID-19 spreads through airborne transmission. A

scientists have demonstrated beyond any reasonable doubt that viruses are released during exhalation, talking, and coughing in microdroplets small enough to remain aloft in air and pose a risk of exposure at distances beyond 1–2 m from an infected individual.

Additionally, the WHO (

can be infected when aerosols or droplets containing the virus are inhaled or come directly into contact with the eyes, nose or mouth.

To reduce the risk of airborne transmission, ‘physical distancing’ can be practiced. This is defined as:

the practice of staying at least 6 feet away from others to avoid catching airborne diseases such as COVID-19.

(The broader term ‘social distancing’, sometimes used interchangeably
with ‘physical distancing’, means ‘staying home and away from
others as much as possible to help prevent spread of COVID-19’ (

Limiting close face-to-face contact with others is the best way to reduce the spread of [COVID-19].

Moreover, masks and other face coverings have proved to be essential to slow the spread of COVID-19. The CDC provides guidance for wearing masks in order to protect the person wearing the mask as well as others. It specifies that masks should be worn:

in addition to staying at least 6 feet apart, especially when indoors around people who don’t live in your household.

Since masks are not substitutes for physical distancing, the latter should be practiced by pedestrians in all circumstances. In the case when not enough masks are available for a given population, physical distancing becomes even more crucial to slow the spread of COVID-19.

As discussed in section 1, the public circulation spaces within informal settlements are most often outdoors but may be covered and are most often very narrow. Physical distancing should not be disregarded in these scenarios since it can be practiced both indoors and outdoors. Even though the:

odds of indoor transmission was very high compared to outdoors (18.7 times; 95% confidence interval, 6.0–57.9)

a systematic review of outdoor transmission of SARS-CoV-2 concluded that
‘there are significant gaps in our understanding of specific pathways [of
transmission]’ (

airflow, ventilation, and lack of recycled air, which all minimize the theoretical risk of aerosol transmission through smaller respiratory droplets.

However, circulation spaces in informal settlements do not possess ideal spatial qualities for airflow and ventilation, and thus outdoor transmission of COVID-19 should not be neglected.

To minimise the risk of outdoor transmission via face-to-face interactions in the narrow circulation lanes of informal settlements, and limit contagion between clusters of dwellings, the Cluster Lane Method uses graph theory. This branch of mathematics began with a problem dealing with the spatial arrangements of a city, as discussed in the following section.

Throughout history, several popular puzzles and problems relating to movement within
an urban fabric have led to significant mathematical breakthroughs. One of these
seemingly innocent problems, consisting of walks over a series of bridges, is called
‘The Seven Bridges of Königsberg’ (

The problem of ‘The Seven Bridges of Königsberg’ asked if it was possible to cross all seven bridges (marked in red) in a route without crossing any one of them more than once. Leonhard Euler proved that such a route was mathematically impossible. The white square at top right shows a graph representing the bridges as edges and the islands/banks as vertices.

Euler laid the foundations of a new branch of mathematics involving the ‘geometry of position’, which is now known as graph theory. This new approach involves simplifying spatial relationships between objects into diagrams of points (vertices) and lines (edges). Insights from the mathematical analysis of these graphs would lead to principles that would be valid for all similar graphs, of any size or complexity.

Since Euler’s solution, graph theory has become an important branch of both
pure and applied mathematics. Numerous problems in geometry, topology, among others,
have been solved with it since diverse mathematical structures can be embedded into
graphs. Famous theorems have connections to graph theory: The Descartes–Euler
Polyhedral Formula, along with the Euler characteristic applied to higher dimensions
in algebraic topology and polyhedral combinatorics, relates the number of vertices

In architecture and urban planning, graph theory has played an important role which evolved in parallel with the implementation of computation in architectural practice and theory since the 1960s. Theodora Vardouli states that:

architects turned to structural abstraction in efforts to purify their inheritance of interwar Modern architecture from stylistic doctrines and empirical conventions. The graph’s amenability both to visual depiction and to mathematical analysis furnished it with a strategic position among modern mathematical varieties: graphs made structural abstraction visible and workable. By virtue of this property, graphs proliferated in architectural theory as harbingers of a veritably modern discipline founded on rationality and geared toward ensuring functional efficiency.

Graph theory provides tools to represent the underlying structures of circulation
networks, building plans, spatial structures in cities and much more. Lionel March
discussed the educational movement of ‘new mathematics’ during the
1950s, which aimed to change the teaching curriculum to include subjects such as
graph theory, certain notions of combinatorics and set theory. These tools allowed
March & Steadman (

The mathematical aspect of this paper, thoroughly explained in Appendices 1 and 2 in the supplemental data online, focuses on oriented graphs. These are graphs for which orientations (or directions) are assigned to edges. They can represent circulation networks with both bi- and unidirectional lanes. The present method was inspired by the cost-efficient, low-tech solutions of unidirectional circulation in supermarkets and small retail environments during the early months of the COVID-19 pandemic. By using oriented graphs, the method generalises unidirectional pedestrian circulation to the scale of settlements, or any planar circulation network.

The Cluster Lane Method is a flexible mathematical method which makes it possible to turn any planar circulation network into a circulation network made of unidirectional lanes in such a way that any given location can be reached from any other location. A map of the settlement, generated with satellite images, geodata or participatory community mapping, is needed to apply the method. It is illustrated and described thoroughly, in mathematical terms, in Appendix 2 in the supplemental data online. The text and images in this section provide an overview of the method and highlight its most important notions.

In

people want to be part of a neighborly spatial cluster; contact between people sharing such a cluster is a vital function.

The Cluster Lane Method aims to adapt to the ways in which people live and is
inspired by Alexander’s discussion. Although the dense housing typologies of
informal settlements do not exactly match Alexander’s ‘house
cluster’, the idea that clusters are the ‘natural focus of neighborly
interaction’ guides the proposed method, since this also applies to informal
settlements. For the sake of this method, a

The Cluster Lane Method can be simplified as follows: An entire informal settlement,
which corresponds to a large cluster of dwellings, is subdivided into as many
smaller clusters of dwellings as desired. Given these smaller clusters of dwellings,
each lane of the circulation network is assigned an orientation. The subdivision of
the clusters of dwellings is thoroughly explained in the Appendix 2 in the
supplemental data online, as well as how the lanes are assigned unidirectional
orientations.

The unidirectional circulation networks proposed by the Cluster Lane Method are responsive to the urban fabric of informal settlements and their clusters of dwellings. An organisational hierarchy of three types of circulation lanes makes it possible for pedestrians to navigate the network. These types are:

circulation

circulation

circulation

This organisational hierarchy of lanes is present throughout the selected informal
settlement’s entire circulation network. Two important properties are
characteristic of the oriented circulation networks resulting from the Cluster Lane
Method. First, people within a cluster of dwellings can move within the cluster
without having to leave it (by using the lanes of types 1 and 2 only)
(

The images correspond to an enlargement of

This section presents the opportunities and challenges associated with the implementation of the Cluster Lane Method.

Informal settlements and slums lack the top-down organisation inherent in planned
urban developments like the Manhattan grid or the Barcelona superblocks. Their
organic growth is often dictated by self-building principles responding to local
constraints and can result in unique spatial organisations. Two of these
principles are autonomous growth and continuous development, discussed by McGill
University’s Minimum Cost Housing Group (

The spatial patterns of public circulation networks in informal settlements could be hard to discern when looking at a satellite image or plan because of their ‘organic’ or ‘free-form’ that is different from a rectilinear grid network. Orangi Town in Karachi (Pakistan), Neza in the State of Mexico (Mexico), Dharavi in Mumbai (India), Kibera in Nairobi (Kenya), Khayelitsha in Cape Town (South Africa) and Rocinha in Rio de Janeiro (Brazil) all present unique spatial qualities. Yet, regardless of the size and complexity of the circulation network, the Cluster Lane Method can always be applied, as demonstrated in Appendix 2 in the supplemental data online.

The present method corresponds to one of many applications of graph theory to
address problems within informal settlements. For example, Brelsford

The method can provide a people-centric solution to limit COVID-19 contagion in these settlements by involving their inhabitants. Further research and testing would be necessary to find a site- and culture-specific implementation of the method. An anthropological study of how people inhabit a particular informal settlement would inform key strategies for implementation, for example, whether a bottom-up or a top-down approach would be well received depending on social norms surrounding authority and discipline. Regardless of local variants, it would be important to minimise distances between homes and significant spatial nodes such as landmarks, sanitary facilities, marketplaces or public institutions. By taking these factors into account, the clusters of dwellings could be strategically defined, with the help of Appendix 2 in the supplemental data online, in order to limit traffic between different clusters. In the case of a COVID-19 outbreak in one of the clusters, the method could facilitate the isolation of this cluster to protect the inhabitants of other clusters.

The implementation of the Cluster Lane Method could present some difficult challenges. The restructuring would disrupt existing circulation patterns, which could lead to non-compliance, confusion or inconvenience on part of the inhabitants. To prevent resentment among the inhabitants, it would be important to involve the community in as many steps of the process as possible. This would help to ensure that existing circulation patterns are accommodated in the final unidirectional network while also allowing all parties to arrive at a common set of objectives and priorities. When maps are not available, mapping entire informal settlements would be necessary to apply the method. The use of drones or satellite images, as well as mapping circulation networks on foot, could help produce these maps. The participation of residents in the mapping would help integrate existing circulation patterns and could potentially win the community’s support.

Romero

Moreover, the Government of Canada recommends unidirectional circulation as part of its guidelines for the safe return to workplaces:

Due to the width of circulation areas in most office environments, unidirectional circulation patterns for corridors throughout offices and workstations should be considered where possible

as well as for lockers, coat closets and other storage areas (

As previously noted, the unidirectional restructuring would lead to a
recalibration of pedestrian density, with certain critical pathways being used
more. However, this only highlights the crucial role that capping pedestrian
occupancy limits and time allotments would play as the spacing between the
people walking along the unidirectional lanes cannot be reduced below the
minimum health guidelines. The resultant holdup and extended waiting times could
potentially frustrate pedestrians and lead them to disregard established rules
of orientation. Boo (

To apply the method on a large scale, software using the method described in this
paper could be created. Algorithms could automate the process and find optimal
solutions. Dijkstra’s algorithm, the Bellman–Ford algorithm, or
similar algorithms dealing with oriented graphs and combinatorics could be used
to minimise lengths of the most heavily used paths. Weighted graphs (those for
which each edge is assigned a weight) could be used to reflect more crucial
factors such as distances between lane intersections, the inclination of
streets, sunlight at different times of the day,

Once a circulation network has been mapped and the Cluster Lane Method used to assign orientations to each lane, this must be clearly communicated to the pedestrians. By installing colour-coded signs or by painting arrows on the floor, one could establish a consistent visual communication system that transcends any linguistic divide. An effective signage system would thus make it easier for pedestrians to follow the assigned orientation of each lane. To help people understand the logic of the circulation system as a whole, it would be beneficial to have three different types of arrows or signs associated with the three types of lanes discussed in section 4.1. Arrows could be placed throughout each lane, or simply at the intersections, marking the beginning and end of each lane. Additionally, maps with the assigned orientations of lanes could be distributed to the population and installed in key locations. This cost-efficient, low-tech solution would be fast to implement, would significantly reduce the number of face-to-face interactions between pedestrians and would allow for physical distancing in narrow public circulation spaces. By including the direction of lanes in navigation applications such as Google Maps, minimal routes could be determined for each walk. Eventually, pedestrians could get used to the new unidirectional circulation system.

To ensure universal accessibility and usage of the proposed circulation, it would be necessary to think of the passageways more holistically. The conditions of public circulation networks in informal settlements, which include irregular surfaces and various obstacles along paths, make it difficult for the visually impaired to navigate through them. Tactile paving (or Tengji blocks), accredited to Japanese inventor Seiichi Mikaye, is a revolutionary navigation aid for the visually impaired. It consists of tactile blocks on pavement which are:

intended to alert visually impaired pedestrians of upcoming dangers, like sidewalk curbs and train platform edges.

Different tiles exist (

The application of the Cluster Lane Method on a circulation network of an informal settlement could help reduce contagion within public circulation spaces by limiting face-to-face interactions between pedestrians. This is particularly useful in the context of narrow lanes with poor natural ventilation. The flexibility of the method makes it possible to adapt it to any circulation network. By studying the social behaviours within the informal settlement and by including the population in the process, a more adequate site-responsive result could be obtained.

The application of the Cluster Lane method would not reduce existing issues found in
informal settlements which facilitate contagion (dense population, lack of basic
sanitary infrastructure,

The authors thank Salmaan Craig for his comments on an earlier draft of this paper,
and for suggesting

The authors have no competing interests to declare.

Appendices 1 and 2 for this article can be accessed at: