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Several countries adopted lockdown to slowdown the exponential transmission of the coronavirus disease (COVID-19) epidemic. Disease transmission models and the epidemic forecasts at the national level steer the policy to implement appropriate intervention strategies and budgeting. However, it is critical to design a data-driven reliable model for nowcasting for smaller populations, in particular metro cities.

The aim of this study is to analyze the transition of the epidemic from subexponential to exponential transmission in the Chennai metro zone and to analyze the probability of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) secondary infections while availing the public transport systems in the city.

A single geographical zone “Chennai-Metro-Merge” was constructed by combining Chennai District with three bordering districts. Subexponential and exponential models were developed to analyze and predict the progression of the COVID-19 epidemic. Probabilistic models were applied to assess the probability of secondary infections while availing public transport after the release of the lockdown.

The model predicted that transition from subexponential to exponential transmission occurs around the eighth week after the reporting of a cluster of cases. The probability of secondary infections with a single index case in an enclosure of the city bus, the suburban train general coach, and the ladies coach was found to be 0.192, 0.074, and 0.114, respectively.

Nowcasting at the early stage of the epidemic predicts the probable time point of the exponential transmission and alerts the public health system. After the lockdown release, public transportation will be the major source of SARS-CoV-2 transmission in metro cities, and appropriate strategies based on nowcasting are needed.

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), or the coronavirus disease (COVID-19) emerged in Wuhan, China and has quickly spread to most of the countries around the world. As of May 10, 2020, 3,917,366 COVID-19 cases and 274,361 related deaths were reported worldwide. At the same time, the Ministry of Health and Family Welfare, India reported 62,939 confirmed cases and 2109 deaths in India. India has 28 states and 8 union territories, out of which 26 states and 7 union territories have reported COVID-19 cases. However, a large proportion of the cases were reported from the 4 states Maharashtra, Tamil Nadu, Gujarat, and Delhi. The case-fatality rate in India remains low as compared to the global rate (7.0% vs 3.35%) [

The estimated population of the Tamil Nadu State for the year 2020 is 82.2 million and is the seventh most populated state in India. It has 37 districts and Chennai is the largest and most populated city in Tamil Nadu, and, based on the nationwide census in 2011, the projected total population of Chennai District is around 4,935,550 [

Public transportation such as trains and buses is an essential service with specific route systems. The Chennai suburban railway consists of two major networks: Chennai Suburban Railway Network and Mass Rapid Transport System; as of 2015-2016, it carried about 1.17 million passengers every day [_{0}) from a value of 2 to 3 leads to a significant amplification in the number of infected cases over subsequent generations, as shown in

Mathematical modeling plays an important role for predicting, assessing, and controlling potential outbreaks for infectious diseases such as H1N1 [

Recently, modeling approaches have been used for analysis of the transmission of COVID-19 infection with travel interventions [

When India had gone through 5 weeks of continuous lockdown during the last week of April 2020, there were 33,050 confirmed cases and 1074 deaths [

Increase in the number of cases over subsequent generations of the infection for (a) a reproductive number of 2 and (b) a reproductive number of 3.

India reported more than 100,000 cases of COVID-19 as of May 18, 2020, even after consecutive lockdown for a period of 55 days. Though the epidemic was slowed down as expected, 3 states contributed more than 58% of the total cases in the country and in each State more than 60% of the cases were reported from the respective capital cities Mumbai, Chennai, and Ahmadabad. Therefore, containment of the SARS-CoV-2 transmission in these three cities is critical in favorably modifying the transmission in India. These 3 cities share the same characteristics in terms of population structure, density, and movement of the people toward these cities for employment.

Chennai is a metropolitan city surrounded by three other districts Kancheepuram, Thiruvallur, and Chengalpattu. Based on the connectivity of the three transport systems, widespread locations of the educational institutes, and the movement of the population from these three districts into every part of Chennai, we felt it appropriate to predict the SARS-CoV-2 transmission considering all four districts as a single unit. In this study, we construct a single geographical zone “

Constructed study site “Chennai-Metro-Merge,” combining Chennai District with the three bordering districts Chengalpattu, Kanchepuram, and Thiruvallur. The estimated total population of the constructed single geographical zone by 2020 is 15,208,505.

In this study, the total reported COVID-19 cases in the constructed geographical zone

Two different models were considered for the study. First, an exponential model of the form:

The solution of equation 1 is given as:

^{rt}

Second, a subexponential model of the form:

with solution:

where,

This study uses the subexponential and the exponential models to estimate the date of transition of the epidemic, and in the field of epidemiology, these models are well suited for the study of the early epidemic growth [

where, ^{th} day. The optimization problem was solved using the MATLAB (MathWorks) programming software. The subexponential and exponential models were analyzed, and a technique for the prediction of the onset date of exponential transmission was identified. Furthermore, the developed model was simulated to approximately predict and analyze the future COVID-19 infections in these four districts.

In an enclosed environment, the number of secondary infections (_{A}

_{A}

where,

The probability of infection

where, ^{3}, ^{3}/hour,

The exponential model was further updated using the data available from May 16 to June 10, 2020, and the model was used to further predict the future number of cases up to June 30, 2020, in the considered geographical boundary. The updated exponential model output and the reported cases are shown in

The reported number of coronavirus disease cases (includes effect of intervention), and the output of the subexponential and the exponential models, shown as a function of time.

The total coronavirus disease cases in the four considered districts of Tamil Nadu predicted using the updated exponential model and the actual reported cases.

The probability of infection in a public bus with 20 passengers shown as a function of the total exposure time and the initial number of infected.

The probability of infection in a single train coach with 54 passengers shown as a function of the total exposure time and the initial number of infected.

The probability of infection in a single train coach (ladies compartment) with 36 passengers shown as a function of the total exposure time and the initial number of infected.

The number of secondary infections in the bus due to the introduction of infected individuals into the susceptible population (total population of N = S + I = 20), shown as a function of the estimated probability of infection.

The number of secondary infections in the train compartment due to the introduction of infected individuals into the susceptible population (total population of N = S + I = 54), shown as a function of the estimated probability of infection.

The number of secondary infections in the train coach (ladies compartment) due to the introduction of infected individuals into the susceptible population (total population of N = S + I = 36), shown as a function of the estimated probability of infection.

In this study, we constructed a mathematical model based on the reported cases from March 7, 2020, to April 29, 2020, to analyze the transition of the COVID-19 epidemic from the subexponential to the exponential stage in the combined Chennai metro-merge. Furthermore, the reported cases from May 16 to June 10, 2020, were used to update the exponential model to nowcast the progression of the epidemic up to June 30, 2020. Currently, five metro cities in India and several cities in South East Asian Region are facing a similar SARS-CoV-2 epidemic. The results of the modeling indicated that the transmission in all the four districts exhibited exponential transmission from the third or fourth week of the first reported case in each district. However, the number of predicted cases for this period was considerably less, and there was an opportunity until the eighth week (ie, the first week of May 2020) to favorably contain the epidemic and reverse to the subexponential transmission. On the other hand, the Government of Tamil Nadu proposed resuming both the bus and train services initially for the officials followed by the public in a phased manner. In public health, as well as the individual perspective, it is desirable to assess the risk of acquiring the SARS-CoV-2 infection while travelling for a considerable period of time in an enclosed environment. We used a probabilistic model and observed that the probability of acquiring the infection in the event of a single index case in the closed environment is lower in suburban train travel with a restricted occupancy of 50% as compared to the bus travel with the same proportion of occupancy (0.19 vs 0.07). The results also indicate that during the suburban train travel the probability of infection is higher in the ladies’ compartment as compared to the open compartment for an exposure time of 2 hours and when a single infected case is introduced (0.11 vs 0.07).

For the predictions to be reliable, the model parameters were estimated with the reported values using the optimization technique of the minimization of the sum square error between the model outputs and the reported values. In addition, standard probabilistic models were used to analyze the probability of infection in buses and trains, which are to be operated at reduced passenger loads after the release of the lockdown. Using the probability of infection due to the total exposure or travel time of the passengers and the initial number of infected individuals travelling in the bus or train coach, the numbers of possible secondary infections were estimated. The modes of SARS-CoV-2 transmission in an enclosed environment are droplet nuclei from the asymptomatic persons as well as the aerosol droplet, especially when the infected person sneezes or coughs during travel. It had been reported earlier in SARS-CoV-1 transmission that all the passengers infected during the flight travel were seated in close proximity to index cases [

The SARS-CoV-2 pandemic has gone through several continents in a short span of 12 weeks, and the length of the epidemics in various countries indicate that there is likely to be a prolonged pandemic for a period of 18-24 months as observed in the Spanish flu pandemic in the early twentieth century. Based on the R_{0} during the initial phase of the epidemic in China, it is estimated that about 60% of the population will be infected if the epidemic is not mitigated [

Disease transmission models and the epidemic forecasts at national levels provide valuable information for the policy makers to implement appropriate intervention strategies in an appropriate time. However, it is critical to design data-driven reliable models for nowcasting and for smaller populations where clustering of transmission occurs. It is a routine practice among the public health specialists to rely on mechanistic epidemic models, and the major disadvantage with these models is that there is an underlying assumption of exponential transmission during the early phase of the epidemic itself [

The major limitations of the study with reference to the predictions are that the data inputs for the study were based on the limited numbers of testing in the study districts and the limited period of predictions for only 6 weeks. In addition, when calculating the probability, we assumed the maximum possible number of initial infections in a single enclosure in a bus or train from the initial part of the journey as 3 and 4, respectively. Chennai metro services are always five times overcrowded during peak hours, and most of the enclosures are expected to be full if the restrictions on the occupancy are imposed during the initial phase of the release. With the current exponential trends, even with random contact, the passengers are likely to be exposed repeatedly during the point to point travel for a period of 2 hours.

At present, several countries are going through the early phase or the subexponential phase of the epidemic and have not yet reached the exponential phase; the methods, results, and experiences reported in this work are of high value in undertaking midcourse corrections in the implementation of the intervening strategies to contain the SARS-CoV-2 epidemic. The developed model in this study is simple and can be constructed easily in any software package using the reported infections over a period of time. Hence, this methodology can be adopted by public health specialists and epidemiologists to trace the current trend of the epidemic and to nowcast the progression of the epidemic at a small population level like in metro cities and districts.

Though all the countries are well aware of the rapid response to the epidemics, each epidemic exhibits certain challenges. There are several challenges during the current COVID-19 epidemic globally and locally. India imposed lockdown as an intervention reasonably in advance as compared to other countries. However, this epidemic has shown categorically that lockdown alone is insufficient to contain the epidemic. Lockdown provides an opportunity for the symptomatic to surface out so that the contacts are traced, quarantined, and the severe forms of the diseases or complications are identified and treated. As shown by Keeling et al [

In India, the opportunity to favorably contain the exponential transmission was missed due to inadequate testing and contact tracing, especially in the overcrowded metro cities and urban settings. Exposures in religious meetings and marketplaces resulted in several epidemic clusters in Chennai City. Now, the three major cities Mumbai, Chennai, and Ahmadabad contribute about 58% of the total cases in India, and there are claims that there are only clusters of transmission in India. The results of our study show that the country needs an exclusive containment strategy in urban areas, in particular in metropolitan cities.

The modeling outcome also forecasts the probability of the infection in the metro zone when public transports are opened up after the lockdown. The long hours of travelling in a congested metro zone enhances the exposure, even if there is a single infected person in the closed environment. Train travel appears to be safer, although the travelling time is the same for the longest travel in the constructed study area due to the architecture of the train compartment that provides more air volume for the travelers. Our model did not include random contact with the infected person to estimate the probability of the infections and the resultant secondary infection. It is desirable to apply network modeling for precise estimation of the secondary infections.

coronavirus disease

Middle East respiratory syndrome

Metropolitan Transport Corporation

_{A}

secondary infections

_{0}

reproductive number

severe acute respiratory syndrome

severe acute respiratory syndrome coronavirus 2

We express our sincere thanks to the World Health Organization for the day-to-day situation report and providing it as an open-source document. The authors also thank the Ministry of Health and Family Welfare, Tamil Nadu, India for providing the district-wise reported COVID-19 cases as open-source data. The authors thank Dr MK Surappa, Vice Chancellor, Anna University for his support.

None declared.