# Assessing the Interplay between travel patterns and SARS-CoV-2 outbreak in
realistic urban setting

RPRohan Patil  RDRaviraj Dave  HPHarsh Patel  VSViraj M. Shah  DCDeep
Chakrabarti  UBUdit Bhatia  Discipline of Computer Science and Engineering,
Indian Institute of Technology, Gandhinagar, India  Discipline of Civil
Engineering, Indian Institute of Technology, Gandhinagar, India  Discipline of
Mechanical Engineering, Indian Institute of Technology, Gandhinagar, India
King George Medical University, Lucknow, India  Discipline of Physics, Indian
Institute of Technology, Gandhinagar, India

###### Abstract

Background The dense social contact networks and high mobility in congested
urban areas facilitate the rapid transmission of infectious diseases. Typical
mechanistic epidemiological models are either based on uniform mixing with ad-
hoc contact processes or need real-time or archived population mobility data
to simulate the social networks. However, the rapid and global transmission of
the novel coronavirus (SARS-CoV-2) has led to unprecedented lockdowns at
global and regional scales, leaving the archived datasets to limited use.
Findings While it is often hypothesized that population density is a
significant driver in disease propagation, the disparate disease trajectories
and infection rates exhibited by the different cities with comparable
densities require a high-resolution description of the disease and its
drivers. In this study, we explore the impact of creation of containment zones
on travel patterns within the city. Further, we use a dynamical network-based
infectious disease model to understand the key drivers of disease spread at
sub-kilometer scales demonstrated in the city of Ahmedabad, India, which has
been classified as a SARS-CoV-2 hotspot. We find that in addition to the
contact network and population density, road connectivity patterns and ease of
transit are strongly correlated with the rate of transmission of the disease.
Given the limited access to real-time traffic data during lockdowns, we
generate road connectivity networks using open-source imageries and travel
patterns from open-source surveys and government reports. Within the proposed
framework, we then analyze the relative merits of social distancing, enforced
lockdowns, and enhanced testing and quarantining mitigating the disease
spread.Scope Our results suggest that the declaration of micro-containment
zones within the city with high road network density combined with enhanced
testing can help in containing the outbreaks until clinical interventions
become available.

Transportation Network,

SARS CoV2,

Social Networks,

Transportation Gravity Models,

###### keywords:

Research

[id=n1]Equal contributors

## Introduction

Modern history is witness to several infectious disease pandemics which have
shaped our knowledge of their epidemiology, transmission, and management [1].
In the past 200 years, at least four strains of influenza, seven waves of
cholera, tuberculosis, and the human immunodeficiency virus (HIV) have
accounted for the deaths of nearly 100 million people [2, 3]. Mathematical
modelling of infectious diseases allows us to predict an epidemic accurately,
recognize uncertainties, and quantify situations to identify possible worst-
case situations that can guide public health planning and decision making [4,
5].

Classically, mathematical models of infectious diseases were dependent on the
classification of individuals on their epidemiological status based on their
potential ability to host and transmit a pathogen: Susceptible, Infectious,
and Recovered [SIR] [6, 7, 8]. The SIR model is the most fundamental
epidemiological model that relies on calculating the proportional burden of
each of these three classes, and the transitions between them. In the context
of an epidemic, assuming no births or deaths in a population, the only two
possible transitions are infection (movement from susceptible to infected) and
recovery (movement from infected to remission). For the sake of simplicity, it
can be assumed that the susceptibility is proportional to the prevalence of
infection or the disease burden in the community, and recoveries occur at a
constant rate [9]. Estimating transmission and recovery, the epidemic
progresses exponentially until the growth rate slows and the epidemic curve
plateaus; following which eventually over time the epidemic cannot be
sustained and is eradicated [6]. A failing of the traditional SIR model is its
inability to account for spatial aspects of disease spread. The 2001 dynamics
of foot and mouth disease in the United Kingdom were demonstrated by explicit
individualised modelling as transmission between farms that would not have
been possible by traditional SIR models alone. Such models pointed at
localised depletion of susceptible contacts as the mechanism for the slowing
of the epidemic [6, 10]. The HIV pandemic is characterised by a chronic
infective state. The transmission of such a sexually transmitted infection is
dependent on the host immune status, the infected individual’s viral load, and
multiple social aspects like sexual practices of interactions between multiple
structured risk groups within the population. In such a situation, the
variability in infectious state predicts the progression of the epidemic and
stochasticity merits more complex modelling [11]. Similarly, pandemics of
influenza or flu-like illnesses can be accounted for by age and risk-
structured models, which are explained by distinct risk groups for infection
and fatality (age, health care workers, comorbid conditions), and the inherent
nature of people to preferentially socialise with others of a similar age—a
principle known as assortativity [7, 12]. More complex modelling focuses on
the individual as a unit of the population describing individual interactions
of each person in a population, in contrast to estimating the proportions of
people with a certain disease status. The shift from a population-based model
to an individual-based model is a powerful tool that helps to account for
complex biologically and socially relevant interactions [13, 14].

Reliability of projections from individual based epidemiological model
critically depends upon the realistic estimates of human mobility, which is
often simulated using suite of agent-based simulations, network science
approaches, and data science methods [15, 16, 17]. For example, Eubank et al.
(2004) demonstrated the applicability of highly resolved agent based
simulation tools that combine census and land use data with parameterized
models to simulate the progression of infectious disease in realistic urban
social networks [15]. Balcan et al. (2009) analyzed mobility data from various
countries around the world and integrated in a worldwide structured meta-
population mechanistic epidemic model to understand the role of infection due
to multi-scale dynamic processes [18]. Ajelli et al.(2010) noted a good
agreement between highly detailed agent-based modeling approaches and
spatially structured mechanistic meta-population models. However, researchers
note that while mobility networks used in meta-population models provide an
accurate description of spreading phenomenon, detailed estimates of the impact
at finer scales is hampered by the the low level of detail contained in such
modeling schemes. On other hand, while agent-based modeling approaches are
highly detailed, gathering high confidence detailed datasets, specifically in
heterogeneous regions across the world, is a challenge [19]. In the context of
vector borne epidemics such as Zika virus (ZIKV) epidemic, researchers have
integrated the high spatial and temporal resolution real-world demographic,
mobility, socioeconomic and climatic condition datasets to estimate the
profiles of ZIKV infections [17]. In other studies, researchers have combined
the state-space models such as population based Suspected (S), Exposed (E),
Infected (I), and Recovery (R) or SEIR models in combination with Google Trend
datasets to understand the evolution of Flu trends in the United States [20].
SARS-CoV-2 epidemic spread across the Globe in early 2020, mathematical,
statistical and machine learning based models gained prominence to find out
how to slow and or stop the spread [21, 22, 23, 24] . Moreover, researchers
referred to the ongoing efforts to model the spread of SARS-CoV2 as ”war
time”research where scientists have to deal with limited data, multiple
assumptions and changing landscapes [24]. We note that despite the structural,
mathematical and procedural differences which exist in the wide spectrum of
epidemic models, realistic representation of human mobility patterns, and
social interactions play a key role in governing model performance to
understand the disease progression.

To model human mobility, travel demand models and activity based models are
typically used. While travel demand models focus on estimating aggregate road
usage in long run using aggregated zonal statistics often obtained from census
and household surveys. More recent methods try to learn about human behavior
in urban areas by using data collected from location-aware technologies
including telecommunication activity datasets and Global Position System data
archives [25].We note that while massive and passive cellphone data can
effectively generate complete urban mobility patterns [26], these datasets are
typically non-open source given privacy and security concerns. In absence of
high resolution activity data, simplified models of human mobility such as
gravity models are often used to understand the spreading of viruses and the
evolution of epidemics [19, 27, 28].

The novel Coronavirus (2019-nCoV) first identified in Wuhan, China has spread
to 213 countries as of August 2020 with more than 24 Million confirmed cases
reported worldwide [29]. While much still needs to be learnt about the virus,
its clinical characteristics, extent of inter-human transmission , and the
spectrum of clinical disease, it is established that transmission of SARS-
CoV-2 occurred to great extent through superspreading events [30]. While
accurate forecasting of spread as well as number of deaths and recoveries
require ample historical data, contact networks, and identification of zeroth
case in different regions in addition to the clinical parameters [31]. While
multiple modeling groups across the globes have proactively focused on
modeling the spread at global and regional scales [32, 24, 22], studies at
urban and city scales are limited [33, 34]. Given the diversity in
socioeconomic factors, spread characteristics, demographics, healthcare
facilities, management of 2019-nCoV needs to involve local negotiations.
Moreover, while stringent lockdowns and travel restrictions were imposed by
many nations, compliance with such preventive measures varies considerably
from one to another region [35, 36, 37]. As a consequence, regions and cities
with similar population densities and comparable socioeconomic indicators
exhibited large disparities in the trajectory of the report SARS-CoV-2 cases.

In the present study, our objectives are two-folds. First, we attempt to
understand the response of travel patterns in the city to partial lockdowns or
creation of containment zones – zones which are declared as no travel zones
due to widespread detected cases in a particular area. Using network science
based node prioritization approaches adapted from [38] and travel demand
models from [39], we model the emergence of congestion zones within the city
as a result of enforcement of containtment measures. We note that while
creation of the containment zones are typically done to contain the spread of
SARS-CoV-2, emergence of the congestion zones as a consequence of traffic
rerouting could facilitate the disease spread.

Second, we attempt to understand the relationship between drivers of intra-
city mobility and SARS-CoV-2 spread trajectory in different parts of the city.
We use generalized stochastic SEIRS infectious disease dynamic model that
allow us to model the effect of social contact network structures,
heterogeneities, and interventions, such as social distancing, testing,
contact tracing, and isolation on the spread trajectory of the disease. We
hypothesize that in addition to the population density, ease of transit within
smaller regions, which in turn is facilitated by the shorter transit/trip
times, can result in accelerated spread of the SARS-CoV-2. We simulate the
mobility patterns in different regions of the city of Ahmedabad which have
disparate road network distribution, land use patterns but comparable
population sizes. We select the period of nationwide lockdown as the temporal
window for analysis. Since non-essential inter-district and interstate travel
was prohibited during the period of lockdown, we simulate the travel patterns
confined within the regions to mimic the population mobility for essential
services.

## Region of Study and Data

We use Ahmedabad Municipal Corporation (AMC) as the study region (Figure 1).
Ahmedabad city has a population of 5.6 million (Census 2011), making it the
fifth-highest populous city in India. The city’s geographical location is
23.02250 N, 72.57140 E. In the context of the SARS-CoV-2 outbreak, Ahmedabad
city witnessed a sudden increase in the patient count since the appearance of
first case on March 19,2020 and thereafter has been classified as one of the
hotspot cities[40].

The flowchart of the methodology is shown in (Figure 2). The Road data in the
form of a shapefile is extracted from an open street map (OSM) data sets to
generate a transportation network. India’s latest census data is from the year
2011, which may not provide an accurate result due to the duration gap. To
address this, we considered the modeled population data of the year 2020
provided by the open source portal (www.worldpop.org) with a spatial
resolution of 30 arc-second (approximately 100m). To understand the city’s
traffic pattern, we use the Traffic Demand Analysis report and database which
is publicly available at [41]. The SARS-CoV-2 patient time series data with
location is collected from COVID India Tracker- an open-source repository that
compiles the patient data from various government sources in near-real time
[42].

## Methods and Materials

### Geo-spatial boundary and Population assignment

We converted the road network into an undirected network graph, which
translates the road’s geometries into nodes and links. The attributes
extracted from the data are the nodes’ location, length of links, categories
of links, and the links’ end coordinates. The travel time of the road section
is a crucial parameter to understand the behavior of traffic. We integrated
travel time considering the category and length of a road section.

To define the transportation network boundary in AMC, we used the AMC boundary
geo-spatial data and extracted the modeled population data for the same
(Figure 3(a)). To include the incoming and outgoing traffic from the city’s
outskirts, we generated a buffer radius of 1km. We considered the incoming
traffic on the boundary junction of the city network (Figure 3(c)). Then the
whole boundary of the city is split into a unique region of the Voronoi cell
(Figure 3(d)). Then we estimated the population intersecting with each unique
cell (Equation 1). Each cell population data is transferred to their
respective road junctions.

$P_{i}=\Sigma\>P_{t}\times\frac{Area\>(C_{i}\times C_{t})}{Area\>C_{t}}$ (1)

where:

  * •

$P_{i}$ = Population in each Voronoi polygon

  * •

$P_{t}$ = Modeled population

  * •

$C_{i}$ = Cell of Voronoi polygon

  * •

$C_{t}$ = Cell of population

We also incorporated the land use (Figure 3(b)) into the analysis to create
real traffic congestion scenarios for post-lock down situations in the city.
The assumption is made that around 1/4th of the population will commute from
residential to commercial for daily work, making the industrial and commercial
area more crowded compared to the residential area. The re-allocation of
population assignment indicates the highly congested pockets in the city.

### Traffic demand forecasting model

The transportation model enables travel demand forecast while taking the
population and household travel survey data into account. We generated a
commuter pattern based on a gravity model. The gravity model is a classical
trip distribution model widely used for trip distribution in an urban context.
Other approaches to model transportation networks, namely the queuing
model[43], growth factor model[44], and activity based model using human
mobility patterns [45]. These alternative approaches are not considered based
on (a) methods are data extensive and (b) some of the techniques required the
observed data, which is very difficult to gather during the lockdown traffic
patterns are completely different when compared to original traffic mobility.
We calculated the trip for road section based on the number of trips attracted
to or leaving from specific zones. These trips are distributed by the linking
of origin and destination, thus forming an Origin-Destination matrix. The
allocation of these trips is done by considering the shortest possible route.
We used the Djikstra algorithm [46] to find the shortest commuter distance.
(Equation 2)

$T_{ij}=P_{i}\times\frac{\exp{(-aC_{ij})}P_{j}}{\sum_{(i,j)\in(V_{S}\times
V_{S})}P_{j}\exp{(-aC_{ij})}}$ (2)

Where:

  * •

$T_{ij}$ = the number of trips produced in the zone i and attracted to zone j

  * •

$V_{S}$ = the set of road intersections in the region $S$

  * •

$P_{i}$ = the population of the $i^{th}$ road intersection

  * •

$C_{ij}$ = the minimum time required to go from $i^{th}$ road intersection to
$j^{th}$ road intersection.

  * •

a = adjustable parameter to be fixed after model calibration

### Network Analysis

To understand the network structure’s intricate pattern and relationship, we
carried out the network analysis using a network’s centrality measures .
Centrality provides a vital and widely used measure to analyze the network as
it helps determine the most critical nodes in the network [38]. Betweenness
centrality is an important criterion for analyzing centrality, especially in
road networks. Road networks include commuters’ flow along its edges and
assuming that people tend to optimize their commute path by taking the
shortest route. We calculated betweenness centrality for each junction in the
network and ranked according to this centrality value (Equation 3). The ranks
demonstrate the junction’s vulnerability in the network and demarcate the
tendency for high connection with other junctions and can be considered as a
measure for the potential for the junction to develop into congestion points.
We assigned the trip count calculated from the gravity model to each road
section (links) and ranked them based on the possible transition. This
provides the section with the highest vulnerability in terms of increased
traffic flow.

$Cb(v)=\sum_{u\in v}\sum_{w\in v/{u}}\frac{\sigma_{uw}(v)}{\sigma_{uw}}$ (3)

Where:

  * •

$\sigma_{uw}$ = the number of shortest paths between u and w

  * •

$\sigma_{uw}(v)$ = the number of shortest paths between u and w that passes v

### Scaling of disease spread

We use the derived transportation network to estimate the spreading of disease
in selected regions with comparable population within the cities. The spread
of disease is dependent on population density. The scaling of disease spread
to a city level helps to understand the behavior of spread in the urban
environment. Population density is used to estimate disease scaling, the
relation between them is not always linear. Moreover, this estimate holds true
when the extrapolation is done over a small region as the exposure is
dependent on the social interaction of the people. Hence, these estimations
based on a linear relationship between population density and disease scaling
work effectively when there is uniformity in the community.

On a city scale, though, the extent of such predictions is extremely limited
due to the vast diversity in living conditions and connectivity contrast in
different regions. Here we have assumed that people’s interaction in a region
is more if the connectivity inside is competently developed. Based on the
above-mentioned criteria, we calculated the scaling factor to remove a few
limitations on the population density based extrapolation. The scaling factor
for a region $S$ with respect to another region $S_{0}$, $\kappa(S,S_{0})$ is
defined in (Equation 4).

$\kappa(S,S_{0})=\frac{N_{S}\log{N_{S}}}{N_{S_{0}}\log{N_{S_{0}}}}\times\frac{A({S_{0}})}{A(S)}\times\frac{F({S_{0}})^{2}}{F(S)^{2}}$
(4)

  * •

$N_{S}$ = Population of region $S$

  * •

$A(S)$ = Area of region $S$

  * •

$F(S)^{2}$ = Interaction Coefficient

Here the F (S) is the independent interaction coefficient of the region and it
is calculated using (Equation 5).

$F(S)=\sum_{(i,j)\in(V_{S}\times
V_{S})}\frac{\exp{(-C_{ij})}P_{i}P_{j}}{N_{S}^{2}}$ (5)

Where:

  * •

$V_{S}$ = the set of road intersections in the region $S$

  * •

$P_{i}$ = the population of the $i^{th}$ road intersection

  * •

$C_{ij}$ = the minimum time required to go from $i^{th}$ road intersection to
$j^{th}$ road intersection in hours.

The route that gives the minimum time, $C_{ij}$ is calculated based on the
assumed travel speed for disparate categories of roads. The complete
calculation can be done using the transport network mentioned in the previous
subsection. The value of $F(S)$ is always less than 1 and the square of this
value gives the interaction coefficient for the region $S$. The exact details
for this construction are given in the supplementary section.

The scaling factor we have defined in the Equation 4 is a relative quantity
between region $S$ and $S_{o}$. To provide a single value that can be used to
directly quantify multiple regions and to allow direct comparison between
them, we use a base region for all calculations. The base region is the
identity region $I$, which is not physically present but an imaginary
construct such that $\frac{N_{I}logN_{I}}{A(I)F(I)^{2}}=1$. Using this
identity region as the base, for any two regions $S_{1}$ and $S_{2}$ (Equation
6)

$\kappa(S_{1},S_{2})=\frac{\kappa(S_{1},I)}{\kappa(S_{2},I)}$ (6)

For notation purpose, we define
$\kappa(S)=\kappa(S,I)=\frac{N_{S}logN_{S}}{A(S)F(S)^{2}}$

It is to be noted that the interaction factor is inversely proportional to the
scaling factor. This is because the interaction factor does not consider the
area of the region, which is separately considered in the calculation. The
construction of the scaling factor is explained in a supplementary section.

### SEIR PLUS model

SEIR PLUS epidemic spread model is applied to scaled-up data to predict the
trajectory of spread. The SEIR model is a standard compartment based model. We
used the dynamic form of the model in this study, which is used on stochastic
dynamic networks [47]. The equations governing the state of the nodes is
explained in (Equation 7).

$Pr(X_{i}=S\rightarrow E)=\left[p\frac{\beta
I}{N}+(1-p)\frac{\beta\sigma_{j\in
C_{G}(i)}\delta_{X_{j}=I}}{|C_{G}(i)|}\right]\delta_{X_{i}=S}$ (7)

$Pr(X_{i}=E\rightarrow I)=\sigma\delta_{X_{i}=E}$

$Pr(X_{i}=I\rightarrow R)=\gamma\delta_{X_{i}=I}$

$Pr(X_{i}=I\rightarrow F)=\mu_{I}\delta_{X_{i}=I}$

$Pr(X_{i}=R\rightarrow S)=\eta\delta_{X_{i}=R}$

For testing purposes and quarantining, the above equations can be modified by
adding some more compartments to account for quarantining and testing. The
details are discussed in a supplementary section.

#### The clinical parameters

Parameters such as the rate of transmission, rate of progression, recovery
rate, mortality rate depend on the cause of the epidemic. We obtained these
parameters from the clinical data [48][49]. At the same time, there is also a
dependency on the recovery rate when a positive person is detected early on or
late.

We assumed that these parameters are independent of other factors related to
social interaction and lockdown effects. A range was determined using
available clinical data for such parameters, and fixed values from this range
were used throughout.

#### Setting up the Scenarios

The interventions brought by the Government brings changes in the interaction
of people compared to daily life. At the same time, a person detected positive
will reduce the interaction with the rest of the population. To model the
different scenarios, we use Erdős-Réyni graphs with different probabilities
for edge creation. These probabilities are seen in the form $\alpha/N_{S}$ for
a region $S$.

#### Tuning of parameters

The tunable parameters are the testing rates at different periods. Also, the
SEIR model requires the number of initially infected people.

### Calibration of epidemic spread model

We now detail the steps followed during the calibration cycle:

  * •

Pick a single sub-region, say $S^{1}_{5}$, and run the SEIR plus model with
some arbitrary plausible values for the initial infected parameter, graph
$\alpha$ values, and testing rates.

  * •

Using the fitting between the observed data for infection and the resuts of
the SEIR model to fine-tune the values to be used for testing rates and
initial infected rates.

  * •

At the same time, change the $\alpha$ parameters but keep them same across all
regions.

  * •

Check the output for all $10000$ population subregions. Do further tuning of
$\alpha$ values for the different garphs and also fine tune the testing
parameters and initial infected for each region. It must be kept in mind that
the initail infected are set only for $5000$ population subregions and scaled
values are used for $10000$ population subregions.

  * •

Find a optimal way such that all plots fit with the observed values for all
$5000$ and $10000$ population subregions

## Results

Our transportation model has two indicators: Betweenness Centrality rank (BC
rank) of a road junction (Figure 4(a)) and probable trip count of each road
section(Figure 4(b)). These indicators help to identify densely connected
pockets in the city. Through plotting these results, We have demarcated the
possible traffic congestion pockets in the city. These results show the city
pockets vulnerable to spread in post-lockdown scenarios. Our analysis shows
that Ahmedabad’s central region has high BC rank, demonstrating the priority
intersection that can be vulnerable to disruption due to heavy traffic. The
road segments with the high trip count are located in the city’s central and
eastern parts, where population density is high. Also, it was seen that the
first cases of COVID -19 were observed in this part of the city, making the
transportation model useful while considering drive through testing.

We simulated targeted disruption in the road network to analyze the system’s
response, which resembles the often seen situations in SARS-CoV-2 spread. Due
to many cases in a particular region, the region might get quarantined or
declared a containment zone. The containment zones make up the proper
condition in network science that is termed as targeted disruption. The city
network is witnessing the interruption in the traffic flow with the
declaration of these containment zones as no travel zones. To analyze the
disruption, we removed one of the city’s containment zones. The possible
rerouting shows that this triggers the new potential vulnerable region to
traffic congestion and spread through our re-calibrated indicators (Figure 5).

The prediction of disease spread is difficult in the diverse living conditions
and connectivity. In that case, we hypothesize that the scaling factor derived
from the transportation network model provides a better alternative to
population density. It considers the population density and the interaction
between the community by road intersections. The travel time plays a crucial
role in determining the scaling factor. To understand the importance of travel
time, we have considered two regions with the same populations (Figure 6(a)).
The travel time changes with the amount of intersections in the region.That is
lesser number of intersections in a a region would translate to longer transit
times, on an average, in an urban setting. This eventually leads to a decrease
in the interaction coefficient and an increase in the scaling factor. Using
this concept, the population density is calculated from population data
(Figure 6(b)) and the scaling factor through interaction coefficient
(Figure6(c)) is calculated. Subsequently, we check the hypothesis by
determining the Kendall-Tau Coefficient generated between the cumulative ward
wise infected cases and the scaling factor and population density,
respectively (Figure6(d)). This analysis validates our hypothesis and shows
that the scaling factor provides a better correlation compared to the
population density (Figure 7(a)) and 7(b)).

Once the hypothesis is tested, the calibrated clinical parameters are used to
validate the prediction of infected population for the regions. The epidemic
SEIR plus model implemented on sub-regions with disparate ranges of population
such as $5000$, $10000$ and $20000$ is used to validate the model parameter
values (Figure 8(a), (b) and (c)). This approach allows us to account for the
variation that comes up with different issues of testing and variation in
actual testings when only average testing rates are known. To check the
appropriateness of the model the time observations of duration considered in
the testing are the ones, which are not considered during the calibration by
plotting calibration and testing plot simultaneously. For the region 1 with
20000 population, the SEIR plus model predicted infected population shows the
good agreement with the observed data. Region-1 shows the R2, Relative Root
Mean Square Error (RRMSE) and Relative Mean Absolute Error (RMAE) value as
0.9602, 0.688 and 0.558 respectively (Figure 8(d)). The same approach we have
implemented in region 2, region 3 and region 4 respectively (Figure 9). We
evaluate the prediction performance for the above mentioned region and it is
shown in (Table 1). The value of R2 is smaller in the region 4 as the data for
SARS-CoV-2 infected cases in the region was less compared to other regions.

The SARS-CoV-2 spread has led governments to think of different interventions
to reduce the spread by implementing various lockdown policies. These policies
can change the trajectory of disease spread. To account this into action, we
have also simulated the disparate policies in our model by selecting social
distancing and relaxation combinations.

The first policy considered is that if the government implements the lockdown
relaxation by last week of May 2020, Unlock with the strict social distancing
till mid-June 2020 and then staged back to a normal state in 50 days. The
result shows that we can expect a sudden spike of cases in the initial days
after lockdown relaxation. However, it gradually decreases in the upcoming
months, but it shows the second wave of cases forming from September 2020
(Figure 10(a)).

The second policy we consider is the lockdown reduced by May 2020 end, Unlock
with the strict social distancing followed for 15 days and then staged back to
normal state by mid-July 2020 in a staggered manner. The model result
demonstrates increased infected cases in the initial days after lockdown
relaxation and shows the fluctuation in the infected cases in the September
and October 2020 (Figure 10(b)).

Lastly, the third policy simulated is that the lockdown reduced by May end,
Unlock with the strict social distancing followed for one month and then
staged back to a normal state. The result demarcates that there is a rise in
cases in the initial few days, but after that, from the month of July-2020, it
can drop down very low (Figure 10(c)).

## Discussion

The prediction of disease spread at the city scale is often complex due to
diverse regional factors and data limitations. The road networks are the prime
sources of intracity movements. This movement pattern can lead us towards the
possible spreading of disease as most of the infectious diseases spread
through social interactions. During these scenarios, prediction of disease
spread through network-based epidemic spread models can be very helpful. This
study has presented the unique approach to model the disease spread through
transportation networks. The result of this analysis allows several meaningful
inferences, which can make a high impact on the prediction of disease spread.
First, this approach can be implemented in any congested city to determine the
interaction between the population and lead to model disease spread. Most of
the data considered during the modeling are open source or readily available.
Second, framework also provides the flexibility of understanding disparate
scenarios such as containment zone restriction. The third inference gained
from the analysis is that the different lockdown policies are highly
influenced by social interaction and can be analyzed through the network-based
epidemic spread modeling. Policymakers can choose the best possible way to
contain the spread. Although the interaction will vary from city to city based
on local conditions, we anticipate that the overall patterns will be similar
for comparable population densities and road networks. We further note that
the quality, quantity and frequency of epidemiological datasets play an
important role in establishing the correlations that we have observed in this
study. While the proposed framework can be generalized to other cities, future
efforts in this direction can greatly benefit from real-time mobility data
obtained from cellphone activity or GPS data, and high-resolution clinical and
epidemiological data with relatively longer duration of record.

## Competing interests

The authors declare that they have no competing interests.

## Author’s contributions

RP,UB, RD designed the experiments; RP, UB, RD, HP, VS performed the analysis;
RP, RD, UB, and DC wrote the manuscript.

## Acknowledgements

This work is supported by Startup Research Grant awarded to UB by Indian
Institute of Technology Gandhinagar. We are thankful to Prasanna V
Balasybramanium, Assistant Professor, Indian Institute of Technology,
Gandhinagar, for his helpful comments and suggestions. …

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## Figures

Figure 1:  Study area of Ahmedabad Municipal Corporation (AMC), located in the
central region of Gujarat in western India. Ahmedabad is selected as area of
study given its distinction as financial hub of Gujarat, disparate
socioeconomic distribution and high population density  Figure 2: Overview of
the modeling framework. The proposed methodology consist of (a) Pre-
processing, (b) Population assignment, (c) Traffic demand forecast and network
analysis, (d) Scenario generation analysis (Targeted disruption) and (e)
Epidemic spread modeling Figure 3: (a) Modeled population data extracted at 30
arc-second , (b) Land use distribution describing diverse arrangement of land
use in the city ,(c) Bound polygon to consider incoming and outgoing traffic
and (d) Assignment of number of population served by junction in a unique
Voronoi polygons.(red colour is resemblance of high population, where as green
colour is for low population Figure 4: Maps of Ahmedabad city depicting (a)
Road junction distribution according to BC Rank (Red colour nodes depicting
high BC rank) and (b) Road network link distribution according to trip count
(Red colour road sections depicting high trip count) Figure 5: (a) The road
network without disruption (normal condition), (b) Zoomed in map of road
network without disruption, (c) Targeted disruption of road network to
generate the lockdown scenario by removing containment zone and (SARS-CoV-2
spread situation) (d) Zoomed in map of targeted disruption Figure 6: (a)
Region of interest in the study area for epidemic spread model with same
population,(b) Population distribution in the city, (c) Interaction
Coefficient distribution in the city and (d) SARS-CoV-2 Patient Distribution
Figure 7: (a) Kendall Tau Correlation and (b) Kendall Tau p- value Figure 8:
Region 1 time series data set: (a) Training Data set (1-5000 Population
cumulative cases, (b) Training Data set (1-10000 Population cumulative cases),
(c) Validation Data set (1-20000 Population cumulative cases) and (d)
Prediction Data set (1-20000 Population cumulative cases)  Figure 9:
(a)Testing data sets: Region 2 Data set (1-20000 Population cumulative cases),
(b) Region 3 Data set (1-20000 Population cumulative cases) and (c) Region 4
Data set (1-20000 Population cumulative cases)

Figure 10: Time series data of patient count based on different lock down
policies (a) Lock down relaxation by last week of May, Unlock with the strict
social distancing till mid June and then staged back to normal state in 50
days , (b) Lock down reduced by May end, Unlock with the strict social
distancing followed for 15 days and then staged back to normal state by mid
July in staggered manner, (c) Lock down reduced by May end, Unlock with the
strict social distancing followed for 1 month and then staged back to normal
state

## Tables

Table 1: Error matrix of prediction depiction the model performance on region of interests Regions (20000 population) | R2 |  RRMSE |  RMAE
---|---|---|---
1 | 0.96 | 0.68 | 0.55
2 | 0.68 | 0.73 | 0.47
3 | 0.87 | 0.41 | 0.30
4 | 0.15 | 0.71 | 0.64

## Supplementary Materials

### Interaction coefficient

We constructed an interaction coefficient based on the gravity model. The
interaction between the two intersections is determined based on the two
parameters: The first one is the fraction of the population in the individual
intersections and the second is the time required to travel from one
intersection to another. $C_{ij}$ gives the information of the time needed by
a person while taking the best path possible for travel.

$C_{ij}$ is considered zero when $i$ and $j$ intersection are the same. For
which the average time required for the interaction between two persons is
believed to be very small. This holds when $C_{ij}$ is measured in terms of
hours or days. This assumption breaks when the measurement is in smaller units
as the intra-intersection interaction cannot be approximated to $0$.

Considering the constraint mentioned above, the calculation of $C_{ij}$ is
done in terms of hours. The selection of larger units such as day or month is
not considered in the analysis because for the intersection in proximity
distance, the travel time will become negligible, leading to a single
intersection. In the city environment, a person can reach any part of the city
and nearby places in the time interval of hours. The unit of time for
$C_{ij}$, in a sense, it defines the resolution at which the system is being
observed. If we consider the single interaction in the region $S$, it is
trivial to see that $F(S)=1$. In the case of two intersections, the value of
$F(S)$ will decrease as $C_{ij}\neq 0$ for all terms. The more the skewed the
distribution of population in the region, the smaller is the independent
interaction coefficient.

$F(S)$ only considers the independent interaction between two intersections.
But a commuting person may have to go from one intersection to another always
through some other intersection. For example, consider that there are $3$
intersections in a region $S$.( Equation 8)

$F(S)=\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2}+2(c_{12}\alpha_{1}\alpha_{2}+c_{23}\alpha_{2}\alpha_{3}+c_{13}\alpha_{1}\alpha_{3})$
(8)

where $\alpha_{i}$ is the fraction of population for $i^{th}$ intersection and
$c_{ij}=\exp{(-C_{ij})}$.

considering (Equation 8) the interaction coefficient is

$\begin{split}F(S)^{2}&=\alpha_{1}^{4}+\alpha_{2}^{4}+\alpha_{3}^{4}+4(c_{12}^{2}\alpha_{1}^{2}\alpha_{2}^{2}+c_{23}^{2}\alpha_{2}^{2}\alpha_{3}^{2}+c_{13}^{2}\alpha_{1}^{2}\alpha_{3}^{2})\\\
&\quad+4(c_{12}\alpha_{1}^{3}\alpha_{2}+c_{23}\alpha_{1}^{2}\alpha_{2}\alpha_{3}+c_{13}\alpha_{1}^{3}\alpha_{3})\\\
&\quad+4(c_{12}\alpha_{1}\alpha_{2}^{3}+c_{23}\alpha_{2}^{3}\alpha_{3}+c_{13}\alpha_{1}\alpha_{2}^{2}\alpha_{3})\\\
&\quad+4(c_{12}\alpha_{1}\alpha_{2}\alpha_{3}^{2}+c_{23}\alpha_{2}\alpha_{3}^{3}+c_{13}\alpha_{1}\alpha_{3}^{3})\\\
&\quad+8(c_{12}c_{23}\alpha_{1}\alpha_{2}^{2}\alpha_{3}+c_{12}c_{13}\alpha_{1}^{2}\alpha_{2}\alpha_{3}+c_{13}c_{23}\alpha_{1}\alpha_{2}\alpha_{3}^{2})\\\
&=\alpha_{1}^{4}+\alpha_{2}^{4}+\alpha_{3}^{4}+4(c_{12}^{2}\alpha_{1}^{2}\alpha_{2}^{2}+c_{23}^{2}\alpha_{2}^{2}\alpha_{3}^{2}+c_{13}^{2}\alpha_{1}^{2}\alpha_{3}^{2})\\\
&\quad+4(\alpha_{1}^{2}+\alpha_{2}^{2}+\alpha_{3}^{2})(c_{12}\alpha_{1}\alpha_{2}+c_{23}\alpha_{2}\alpha_{3}+c_{13}\alpha_{1}\alpha_{3})\\\
&\quad+8c_{13}\alpha_{1}\alpha_{2}\alpha_{3}(\alpha_{2}+c_{12}\alpha_{1}+c_{23}\alpha_{3})\\\
\end{split}$ (9)

The last term merges and accounts for the path that is passing through the 2nd
intersection. If we extend it to $n$ intersections, there will be many cross-
terms that will be extra. Also, there will be terms like $c_{12}c_{34}$, where
the question comes that how did a person jump from 2nd intersection to 3rd
intersection. A person may not always go by the shortest path but take small
detours in the way. Suppose the shortest path from 1st intersection to 4th was
direct, but the person went in 1-2-3-4 path. Then a single cross term cannot
represent this information and the cross-terms with jumps account for this to
some extent.

Calculating the population’s interaction considering multiple paths that can
be taken between two is an exponentially tricky question to solve. Taking the
square of the independent interaction coefficient provides a simple
approximation for the same.

### Scaling Factor

The inverse proportionality of the scaling factor to the interaction
coefficient comes because it captures the skewness of interaction in a fixed
population of fixed area.If we consider two regions with the same population
and area, one with a single intersection and one with two intersections. For
simplicity, in the second region, assume that the individual intersections are
associated with half the area. In that case, if there is some travel time
between them, then we cannot say that the population is using half of the area
in the region. The population are using less area then that and this depends
solely on the travel time given that both intersections have the same
population. This demarcates that more people are packed in the smaller region
and the overall interaction will fall, while the the intra-intersection
interaction will be more and the scaling of spread will increase.

Similarly, suppose only one intersection contains all the population and there
is a significant travel time between the two intersections. In that case, this
implies that most people are packed inside a small region, leading to a large
decrease in interaction factor and a shoot in the scaling factor. This odd
behavior can also be seen from the fact that the interaction factor normalizes
on the basis that every node in a region has internal interaction proportional
to the square of its population fraction. To correctly scale with the area and
population of the region, $\kappa(S)$ is multiplied with $\frac{N_{S}}{A(S)}$
which is the population density of the region.

According to the Erdős-Réyni model for random graphs, the hard threshold for
the connectedness of a graph $G(n,p)$ is $\frac{\log{n}}{n}$. The graphs
mentioned in (Setting up the Scenarios section) will require the $\log{N_{S}}$
factor for scaling the probability correctly for the graphs as interaction
coefficient and population density together quantify the net increase in
connectedness of the random graph for that region.

### SEIRS Plus with testing and quarantine

We add two more compartments, $D_{E}$ and $D_{I}$, representing the detected
exposed and detected infected. To model the quarantine, a quarantine graph $Q$
is required, where the assumption is that detected people are in quarantine,
and to model the testing, the parameters $\theta_{E}$, $\theta_{I}$,
$\psi_{E}$ and $\psi_{I}$ are added, where $\theta_{E}$ and $\theta_{I}$ are
the testing rates for exposed and infected people respectively and $\psi_{E}$
and $\psi_{I}$ are the rate of the positive test result for exposed and
infected individuals respectively. The $\theta$ are tunable parameters, while
$\psi$ are clinical parameters [47].

$Pr(X_{i}=S\rightarrow E)=\left[p\frac{\beta
I+q\beta_{D}D_{I}}{N}+(1-p)\left(\frac{\beta\left[\sigma_{j\in
C_{G}(i)}\delta_{X_{j}=I}\right]+\beta_{D}\left[\sigma_{k\in
C_{Q}(i)}\delta_{X_{k}=D_{I}}\right]}{|C_{G}(i)|}\right)\right]\delta_{X_{i}=S}$
(10)

$Pr(X_{i}=E\rightarrow I)=\sigma\delta_{X_{i}=E}$

$Pr(X_{i}=I\rightarrow R)=\gamma\delta_{X_{i}=I}$

$Pr(X_{i}=I\rightarrow F)=\mu_{I}\delta_{X_{i}=I}$

$Pr(X_{i}=R\rightarrow S)=\eta\delta_{X_{i}=R}$

$Pr(X_{i}=E\rightarrow D_{E})=\left(\theta_{E}+\Phi_{E}\left[\sigma_{j\in
C_{G}(i)}\delta_{X_{j}=D_{E}}+\delta_{X_{j}=D_{I}}\right]\right)\psi_{E}\delta_{X_{i}=E}$

$Pr(X_{i}=I\rightarrow D_{I})=\left(\theta_{I}+\Phi_{I}\left[\sigma_{j\in
C_{G}(i)}\delta_{X_{j}=D_{E}}+\delta_{X_{j}=D_{I}}\right]\right)\psi_{I}\delta_{X_{i}=I}$

$Pr(X_{i}=D_{E}\rightarrow D_{I})=\sigma_{D}\delta_{X_{i}=E}$

$Pr(X_{i}=D_{I}\rightarrow R)=\gamma_{D}\delta_{X_{i}=I}$

$Pr(X_{i}=D_{I}\rightarrow F)=\mu_{D}\delta_{X_{i}=I}$

$Pr(X_{i}=any\rightarrow S)=\nu\delta_{X_{i}\neq F}$

Where:

  * •

$\nu$: Rate of baseline birth

  * •

$\mu_{D}$: Rate of infection-related mortality for detected cases

  * •

$\sigma_{D}$: Rate of progression for detected cases

  * •

$\gamma_{D}$: Rate of recovery for detected cases

  * •

$\beta_{D}$: Rate of transmission for detected cases

  * •

$\Phi_{E}$: Rate of contact tracing testing for exposed individuals

  * •

$\Phi_{I}$: Rate of contact tracing testing for infected individuals