In the wake of the novel coronavirus Covid-19 pandemic that is currently sweeping the planet, there is increasing concern over the impact on large urban slums in the developing world. This concern primarily stems from the nature of dwelling arrangements in developing cities, where large proportions of the population live in densely populated slums and shantytowns [1] .
Broadly, slums are defined as "communities characterized by insecure residential status, poor structural quality of housing, overcrowding, and inadequate access to safe water, sanitation, and other infrastructure" [2] . This definition emphasizes the fact that slums house the poorest and most vulnerable populations in cities. UN-Habitat estimated that over a third of the urban population lived in slums in 2012-13, with significant geographical heterogeneitythe proportion was 62% for sub-Saharan Africa and 35% for southern Asia, and 25% for Latin America [1] . The sheer scale of slums is further exacerbated by the density of population in such settlements. Table 1 presents some statistics on the density of living in some of the largest metropolises of the developing world and shows that these cities have high average population densities (and high slum populations), but individual slum neighbourhoods even within these cities often show population densities an order of magnitude higher, suggesting significant intracity heterogeneity in densities of living. It is important to remember that the high population densities in developing cities are being attained without building vertically (unlike cities like New York City, Seoul, or Tokyo), with typical living conditions in slums described as small single room shacks (~10 2 ) with around 5 people living in them, situated adjacent to one another, and with up to 10 families sharing a water tap and a pit latrine [3] . High population density achieved under such conditions therefore creates an environment rife for epidemic spread through air or water. Our specific concern relates to the spread of disease through such urban slums, which represent a critical feature of urbanization in developing nations [1] , especially in the context of infectious disease outbreaks like Covid-19 where viral transmission is aided by increased population density, manifested as more frequent person-to-person contact, crowded housing, and unsanitary environments [4, 5] .
In this work, we use Covid-19 caseload data at a sub-city level (ward or neighbourhood or local government level) to empirically characterize the spread of the epidemic across urban neighbourhoods in six developing world metropolises, specifically to understand the nature of infectious spread at fine-grained levels in contexts where slums are a salient feature of the urban landscape. Based on this characterization, we explore the differential impacts of Covid-19 across slum and non-slum neighbourhoods in these cities. In order to explore whether the observed epidemy outcomes could be representative of other similar epidemics in developing urban contexts, we create a network model to propagate the stochastic dynamics of the spread of infection through an urban system (city) and estimate the differential impacts on slum and nonslum populations. Finally, we discuss the results obtained in the context of cities in the developing world.
We focus our attention on six specific cities (Table 1 ) because they are amongst the primate cities of the global south; are severely impacted by Covid-19; and have made available data at the required level of local granularity to enable this fine-grained analysis. However, even for many of these cities, data at the sub-city level is not released regularly and is only available occasionally. We discuss all sources of data and constraints in Appendix 1. Table 2 provides greater detail on the sub-city units we consider for the analysis. When we explore the distribution of Covid-19 caseload across states or provinces within the countries containing these six cities, we again find that ~0.70, which is very similar to theindex observed for caseload distribution within these cities. The distribution of cases across the states of India, states of Brazil, states of Nigeria, provinces of South Africa, and districts of Bangladesh yield -indices of 0.77, 0.65, 0.75, 0.70, and 0.76 respectively (we were unable to find province level data for Philippines). Therefore, the distribution of caseload across states/provinces in nations mirrors the distribution across neighbourhoods in cities, indicating self-similar behaviour across scales.
Given this unequal distribution, we now explore the characteristics of neighbourhoods that have the highest caseloads. Our current understanding of Covid-19 suggests that physical proximity is an important determinant of local spread. Therefore, we study caseloads across neighbourhoods in all six cities, with a focus on differential impacts of Covid-19 on high density neighbourhoods with slums, and other neighbourhoods. In order to do this, we first map large slum settlements in these cities to the appropriate sub-city unit; it is important to note that given the high levels of The only exception here is Rio de Janeiro, where we find that neighbourhoods with slums have a lower caseload per capita than non-slum neighbourhoods; this should however be seen in light of the many concerns have been raised about testing and measurement of Covid-19 cases in Brazil's favelas [7, 8, 9] .
Overall, this finding suggests that density of living in neighbourhoods is a critical mediator of the dynamics of infectious spread in a city, and that the urban poor in slums are starkly worse off in terms of epidemy outcomes. Given this empirical context, we attempt to model these differential impacts by simulating the spread of a typical respiratory epidemic in a network representing a city-system composed of slum and non-slum neighbourhoods, and test whether the observed intra-city epidemic outcomes from Covid-19 are representative more generally of respiratory epidemic spread in cities with slums.
We model a network of a city consisting of nodes, with each node representing an agent in the city, and neighbourhoods amongst which agents are distributed. While we lack empirical data on the structure of real networks of physical proximity in cities of the developing world, there is a growing body of work indicating that highly connected nodes or 'super-spreaders' are [10, 11, 12] . Therefore, we propose to explore the dynamics of transmission on a scale-free Barabási-Albert (BA) network [13] . The Barabási-Albert (BA) network with nodes is generated by attaching new nodes with neighbours, such that the links of a new node show preferential attachment for existing nodes with high degree. We also test the robustness of model outcomes for sensitivity to network type in Appendix 3.
Our interest is in studying differential impacts across slum and non-slum neighbourhoods described by a wide variation in population density, and we simulate such density differences as differences in the average degree of nodes in each neighbourhood of the network. That is, we model connectedness of a neighbourhood as the average degree of nodes in a neighbourhood.
We construct neighbourhoods in the network by ordering all agents based on node degree and then allotting each of them in order to the neighbourhoods in the city system, such that the first neighbourhood is filled with the first set of ordered / agents, followed by the second neighbourhood and so on, until the final neighbourhood is filled with the last set of / agents. Creating neighbourhoods in this way ensures that average degree of nodes across neighbourhoods shows significant heterogeneity. It also means that nodes in neighbourhoods with higher average degree are connected to many nodes both within and outside of their neighbourhoodsthis is meaningfully representative of the urban poor in cities of the developing world, who live in densely populated slums and have high interconnectedness (unavoidable physical proximity) within the slum, but work largely in other non-slum neighbourhoods, including as essential services workers such as sanitation and health workers.
This algorithm also means that neighbourhoods with high average degree correspond only to high-density slum neighbourhoods, and not high-density neighbourhoods in generalfor instance, neighbourhoods that well-off and where high densities are obtained by building vertically are represented in our model as neighbourhoods with lower average node degree (lower connectedness), which is a more likely representation of their daily contact networks. We also vary the algorithm to populate neighbourhoods, while still ensuring reasonable heterogeneity in average degree of neighbourhoods, and find that model outcomes are robust to these changes (Appendix 3).
We use the three compartment Susceptible-Infected-Recovered (SIR) model as the basis for an agent's progression through the duration of the epidemic [14] . Agents start out in the Susceptible ( ) compartment until the time they are infected, at which point they fall into the Infected ( ) compartment. After spending a specified duration of time being infected, when they spread the infection in the network, they move to the Recovered ( ) compartment, at which time they are immune -neither infective nor susceptible to the infection again. At = 0 days, we have one random node that is infected ( ), while the remaining − 1 nodes are susceptible ( ).
At each time step , the dynamics of infectious spread in the network are modelled as follows:
First, each infected ( ) agent spreads the disease to each of its susceptible ( ) neighbours in the network with transmission probability . Given a node with neighbours (or a contact rate of ), the average daily infections caused by this node, or its daily transmission rate ( ), is the product of the transmission probability and the contact rate of the node, = . Second, each infected agent moves into the recovered ( ) compartment if it has spent 1/ days in the infected ( ) compartment. is defined to be the recovery rate and remains constant through the dynamics.
We propagate these dynamics over a period of = days and study the distribution of cases across the neighbourhoods over time, as well as the current and cumulative caseloads across neighbourhoods over time. Table 3 provides the complete set of parameter values and initial conditions for the simulations.
Populationnumber of network nodes, 10,000
Number of edges from new node to extant nodes, (for Barabási-Albert graph) 50
Probability of node to node transmission, 0.004
Number of simulations 100
Number of susceptible nodes, (0) 9,999
Number of infectious nodes, (0) 1
Number of recovered nodes, (0) 0 
The evolution of cumulative fraction of caseload across neighbourhoods clearly shows that the rate of case growth increases with population density (Fig. 3a) . This is in keeping with the empirical finding that once epidemy dynamics are underway and the infection has reached higher density neighbourhoods, caseload per capita is higher in high density neighbourhoods. For instance, at day 10 of the dynamics, the densest neighbourhood in our network (with average node degree, = 421) has a cumulative caseload of 4.7% (as a fraction of its population), while the lowest density neighbourhood (with average node degree, = 50) is at 0.5%, and all other neighbourhoods with densities in between these extremes show caseloads between 0.5% and 4.7% (Fig. 3a) . The corresponding caseloads on days 20 and 30 are 66% and 96% for the densest neighbourhood, and 24% and 71% for the lowest density neighbourhoods. k=50  k=52  k=53  k=55  k=56  k=58  k=61  k=63  k=66  k=69  k=73  k=77  k=82  k=88  k=97  k=107  k=122  k=146  k=194 To explore these dynamics analytically, consider a neighbourhood with ℎ nodes, each with degree . Given and , at = 0, the average daily transmission rate is = . At the end of a time interval , let ( ) be the fraction of population still susceptible and ( ) the fraction that has ever been infected until , such that ( ) = 1 − ( ). ( ) is given by (Eq. 1):
The effective transmission rate of the epidemic in the neighbourhood, ( ), is the average number of people infected by an individual in the neighbourhood at time :
Using this simple construct, we consider two neighbourhoodsa slum with average degree 1 and a non-slum with average degree 2 ( 2 < 1 ) -with probability of transmission and a single node infected at = 0. The evolution of ( ) shows that the slum has a much higher effective transmission rate in the early part of the dynamics due to higher (Fig. 3b) . This results in sharp increase in caseloads in this period, causing a simultaneous sharp decline in ( ) due to the coevolution of susceptible and infected populations. The non-slum neighbourhood has a lower effective transmission rate to begin with and shows a more gradual increase in cases. The overall effect is that higher density results in higher caseloads per capita in the slum as against the non-slum (Fig. 3b) , which offers a possible explanation for the empirical observations from developing world cities where case density increases with neighbourhood population density (Figs. 2b and 2c).
We also study the distribution of cases across neighbourhoods and find that, just as observed empirically, there is an unequal distribution of caseload across neighbourhoods during the dynamics (Fig. 4a) . However, as the epidemic runs its course, the inequality in distribution progressively reduces - Fig. 4a plots the distribution of caseloads at different points in time and we see that inequality in distribution of cases is greatest at = 10 when = 0.62, following which there is continuous reduction in inequality until = 50 when = 0.51, at which point the epidemic has ended. For the epidemic to end, it infects as much of the population as is required for the effective transmission rate to summarily decline below 1; therefore, even as dense slum neighbourhoods see their caseloads rise steeper and peak earlier (Fig. 4b) , thus yielding higher inequality in case distribution, lower density non-slums are not immune to the epidemy and will see delayed but increasing caseloads resulting in declining inequality in the distribution towards the end of the epidemic (Figs. 3a, 3b, and 4a ). Our empirical findings from
Mumbai, Cape Town, and Lagos conform with this modelled outcome, though Rio de Janeiro, Dhaka and Manila do not (Fig. 1) . Our model suggests that both in terms of cumulative caseload outcomes at the end of the epidemic ( Fig. 3a) , as well as (higher and earlier) peak caseloads during the epidemic (Fig. 4b) , slum neighbourhoods are much worse off than non-slums in an epidemic. We find that the nature of outcomes described here are robust to a wide range of change changes in model parameters such as population of the city system ( ), probability of transmission ( ), and number of neighbourhoods ( ), as well as changes in network structure and mechanism of populating neighbourhoods (Appendix 3).
In summary, our modelled outcomes are in broad agreement with the empirical observations,
suggesting that the nature of these outcomes is more generally reflective of epidemic spread in cities with slums. 
We study the evolution of the Covid-19 epidemic across neighbourhoods within a city, for a set of metropolises in the developing world. We find an unequal distribution of cases, with a small number of the most densely populated neighbourhoods containing a significant proportion of total caseload across all cities, as illustrated by a -index ~0.70 across these cities. This finding appears to hold across scales, with national case distribution across these states/provinces also displaying similar inequality in case distribution. We also find that neighbourhoods with the highest case densitiesboth in terms of population and area -contain the largest slums in these cities, and that consequently the urban poor in slums are at the highest risk in this epidemic.
Using a simple network model, we simulate the emergence of differential outcomes for slums and non-slums in a city. Model outcomes replicate both unequal distribution of cases as well as higher case densities in high-density neighbourhoods, suggesting that these outcomes are reflective of outbreaks in general for cities with slums. In addition, simulations also predict that as the epidemic progresses, distribution of cases across neighbourhoods becomes less unequal, and that both peak caseloads and cumulative caseloads are worse for slum neighbourhoods vis-àvis non-slum neighbourhoods in cities.
Urban slums reflect increased demographic growth, migration, population densities, and poverty, which are the main processes found to be linked with prevalence of infectious diseases [15] .
There is evidence to suggest that slum populations scale super-linearly with city size [16] , meaning that larger cities have more than proportionally larger slums. It is anticipated that there will be over 40 megacities in the world by 2030 and most will be located in the developing world [17] . The evolution of larger slums and higher population densities will mean that slums will continue to at the forefront of epidemics, both in terms of public health and socioeconomic outcomes.
In the immediate or near term, health departments in developing countries must prepare specific guidelines for physical distancing in high density settlements that are clearly communicated and can be implemented by slum dwellers, so that their exposure risks are minimized [18] . However, the long-term solution to this lies in ensuring that slum settlements, which often house a large proportion of the urban population, are provided with functioning environmental infrastructure for piped running water and private sanitation, waste management, and electricity, in addition to basic health infrastructure such as primary healthcare facilities [19, 20, 21] . Creating cleaner, more sanitary environments will make it possible to more systematically counter the easy spread of infections.
Author Contributions: AS and HJJ conceived the methodology, wrote the analytical description, reviewed and edited the manuscript. AS conceived the research, collected the data, performed the analysis, programmed the simulation and wrote the draft manuscript.