Abstract

With only 536 COVID-19 cases and 11 fatalities, India took the historic decision of a 21-day national lockdown on March 25, 2020. The lockdown was first extended to May 3 soon after the analysis of this article was completed, and then to May 18 while this article was being revised. In this article, we use a Bayesian extension of the susceptible-infected-removed (eSIR) model designed for intervention forecasting to study the short- and long-term impact of an initial 21-day lockdown on the total number of COVID-19 infections in India compared to other, less severe nonpharmaceutical interventions. We compare effects of hypothetical durations of lockdown on reducing the number of active and new infections. We find that the lockdown, if implemented correctly, can reduce the total number of cases in the short term, and buy India invaluable time to prepare its health care and disease-monitoring system. Our analysis shows we need to have some measures of suppression in place after the lockdown for increased benefit (as measured by reduction in the number of cases). A longer lockdown from 42–56 days is preferable to substantially ‘flatten the curve’ when compared to 21–28 days of lockdown. Our models focus solely on projecting the number of COVID-19 infections and thus inform policymakers about one aspect of this multifaceted decision-making problem. We conclude with a discussion on the pivotal role of increased testing, reliable and transparent data, proper uncertainty quantification, accurate interpretation of forecasting models, reproducible data science methods, and tools that can enable data-driven policymaking during a pandemic. Our software products are available at covind19.org.

Keywords: basic reproduction number, coronavirus, credible interval, India, intervention forecasting, SIR model

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1. Introduction

Four months since the first case of COVID-19 in Wuhan, China, the SARS-CoV-2 virus has engulfed the world and has been declared a global pandemic (World Health Organization [WHO], 2020b). The number of confirmed cases worldwide stands at a staggering 1,930,780 (as of 9:20 a.m. EST April 14, 2020, Microsoft, n.d.). Of these, 10,815 confirmed cases are from India (Figure 1), the world’s largest democracy with a population of 1.34 billion (compared to China at 1.39 billion and the United States at 325.7 million) (World Bank, n.d.). India has been vigilant and early in instituting strong public health interventions, including sealing the borders with a travel ban/canceling almost all visas, closing schools and colleges, and diligently following up with community inspection of suspected/exposed cases with respect to adherence of quarantine recommendations (Table 1). On March 24, India took the historic decision of a 21-day national lockdown starting March 25, when it had reported only 536 COVID-19 cases and 11 fatalities. In the subsequent days, we have seen a steady growth in the number of new cases and fatalities, with growth rates slower than other affected countries, but in 21 days, the curve has not yet ‘turned the corner’ or showed a steady decline in the number of newly diagnosed cases (Figure 2). All forecasting models in this article use data up to April 14 with the premise of a 21-day lockdown in place.

While India seems to have done relatively well in controlling the number of confirmed cases compared to other countries in the early phase of the pandemic (Figure 2), there is a critical missing or unknown component in this assessment: ‘The number of truly affected cases,’ which depends on the extent of testing, the accuracy of the test results and, in particular, the frequency and scale of testing of asymptomatic cases who may have been exposed. The frequency of testing has been low in India. According to the Indian Council of Medical Research (ICMR; 2020), only 229,426 subjects have been tested as of April 14 (<0.03% of the population). When there is no approved vaccine or drug for treating COVID-19, entering phase 2 or phase 3 of escalation will have devastating consequences on both the already overstretched health care system of India, and India’s large at-risk subpopulations (see Appendix Table A1). As seen for other countries like the United States or Italy, COVID-19 enters gradually and then explodes suddenly.

Table 1. Timeline of COVID-19 Interventions in India

^{Source: https://www.pharmaceutical-technology.com/news/india-covid-19-coronavirus-updates-status-by-state/}

We provide a table listing other highly affected countries along with their first reported case, initial interventions, crude fatality rates, and active case counts in Appendix Table A2 for reference. The estimated capacity of hospital beds in India is 70 per 100,000 people (World Bank, 2020), which is an upper bound on treatment capacity. Given an average occupancy rate of 75%, only a quarter of these are available (Sindhu et al. 2019). Moreover, critically ill COVID-19 patients (about 5 to 10% of those infected) will require ICU beds and ventilator support. India has only 35,000–58,000 ICU beds, with very high occupancy rates and at most one ventilator per two ICU beds (Times of India, 2020). In order to roll out interventions and plan for health care infrastructure, robust projection models for outcomes of interest are necessary. There are many outcomes that are of potential interest to policymakers, for example: How many infected cases will be hospitalized? How many will be admitted to the ICU? How many patients will need ventilators? And, finally, what will be the mortality due to COVID-19 infections? We focus on the number of active cases as our target of prediction due to the limited data from India on the other outcomes. From other nations we know that roughly 20% of infections will probably need hospitalization (Root, 2020), 5% will need ICU admission (Guan et al., 2020), and case-fatality rates vary from 1 to 5% of those hospitalized(Oke & Henegan, 2020). This may provide crude estimates of other outcomes from case-count predictions.

At the time of this writing, there exist several models that have been used to analyze the COVID-19 case-count data from India. The approaches for modeling the disease transmission and then forecasting the number of cases at a future time can be broadly categorized into two genres: exponential/Poisson type models, and compartmental epidemiologic models. For instance, Ranjan (2020) and Gupta and Shankar (2020) used the classical exponential model, S. Das (2020) used a Poisson regression model, while Deb and Majumdar (2020) used an auto-regressive moving-average model to analyze incidence pattern over time. The compartmental epidemiologic models include variations of the susceptible-infected-removed (SIR) model, which is guided by a set of differential equations relating the number of susceptible people, the number of infected people (cases), and the number of people who have been removed (either recovered or dead) at any given time. This simple SIR model has been used by Ranjan (2020) and Dhanwant and Ramanathan (2020). Singh and Adhikari (2020) used an age-structured SIR and social contact model, where an SIR model is assumed in each age category. Another extension of the SIR model is the susceptible-exposed-infectious-recovered (SEIR) model, which incorporates an additional compartment of truly exposed people that is a latent variable. Mandal et al. (2020), Chatterjee et al. (2020), Sardar et al. (2020), and Senapati et al. (2020) used one or the other variation of the SEIR model. For example, Sardar et al. (2020) used an extra compartment for lockdown to capture in-home isolation and study the effects of lockdown on future case counts. Appendix Table A3 compares and summarizes these existing models for India.

In this article, we apply a Bayesian extension of the SIR model, the extended susceptible-infected-removed (eSIR) model, to explore two primary forecasting objectives: (a) forecasting future case counts (short term and long term) with different forms of suppression measures in place (post-lockdown) and (b) studying the relative impact of length/duration of a lockdown on our predictions of cumulative COVID-19 infections. We carry out extensive sensitivity analysis to assess the robustness of our forecasting models. We conclude with a discussion regarding the need for reliable case-count data, increased testing, the importance of uncertainty quantification of the projected case counts, and transparent data science methods that can inform and influence policymaking during a pandemic. Our data science products include three articles on media studying pre- (Ray et al., 2020), during (Salvatore, Wang, et al., 2020), and post- (Salvatore, Ray, et al., 2020) lockdown effects, providing critical information for policymakers and having an extensive reach (Reuters [Ghosal, 2020], Times of India [P. Das, 2020], The Guardian [Ellis-Petersen, 2020], The Economic Times [Noronha, 2020]), an interactive and dynamic R Shiny app that daily updates forecasts as new case counts are reported, and publicly available codes for reproducible research.

The rest of the article is organized as follows. In Section 2 we describe the structure of the eSIR model, our parameter choices, and the Bayesian computational algorithm. In Section 3 we present results from analyzing the data from India that include a sensitivity analysis. We assess how our forecasting model updates itself with more accrual of data over time. In Section 4 we provide an itemized discussion of some of the salient data and data science issues related to intervention forecasting and case-count projections. Section 5 presents a brief conclusion.

2. Methods and Notation

2.1. Study Design and Data Source

We used the current daily data on number of COVID-19 infected cases, recoveries, and deaths in India to predict the number of infected and removed cases at any given time (L. Wang et al., 2020). We obtained the data (up to April 14) from the 2019 Novel Coronavirus Visual Dashboard operated by the Johns Hopkins University Center for Systems Science and Engineering (JHU CSSE) and from covid19india.org (covid19india, 2020; Dong et al., 2020; Johns Hopkins University Center for Systems Science and Engineering, n.d.). Some of our testing data came from https://ourworldindata.org/coronavirus.

2.2. Our Statistical Model for Predictions

Overview. The standard SIR model was recently extended to incorporate time-varying transmission rates or time-varying quarantine protocols and is known as the eSIR model (L. Wang et al., 2020). When using the eSIR model with time-varying disease transmission rate, it can depict a series of time-varying changes caused by either external variation like government-initiated macro-isolation measures, community-level protective measures and environment changes, or internal variations like mutations and evolutions of the pathogen. To implement the eSIR model, a Bayesian hierarchical framework is assumed. Using the current time-series data on the proportions of infected and the removed people, a Markov chain Monte Carlo (MCMC) implementation of this Bayesian model provides not only posterior estimation of parameters and prevalence of all the three compartments in the SIR model, but also predicted proportions of the infected and the removed people at future time points. The R package for implementing this general model for understanding disease dynamics is publicly available at https://github.com/lilywang1988/eSIR.

Mathematical Framework of the eSIR Model. The eSIR model works by assuming that the true underlying probabilities of the three compartments follow a latent Markov transition process, and that we observe only the daily proportions of infected cases and removed cases. First, let us establish some notation. Assume that the observed proportions of infected and removed cases on day t are denoted by $Y_{t}^{I}$ and $Y_{t}^{R}$, respectively. Further, denote the true underlying probabilities of the S, I, and R compartments on day t by $\theta_{t}^{S}$, $\theta_{t}^{I}$, and $\theta_{t}^{R}$, respectively, and assume that for any t, $\theta_{t}^{S} + \theta_{t}^{I} + \theta_{t}^{R} = 1$. Assuming a usual SIR model on the true proportions (Appendix Figure A1-A), we have the following set of differential equations:

Here, $\beta > 0$ denotes the disease transmission rate, and $\gamma > 0$ denotes the removal rate. The basic reproduction number $R_{0} ≔ \frac{\beta}{\gamma}$ indicates the expected number of cases generated by one infected case in the absence of any intervention and assuming that the whole population is susceptible. At this stage, for the observed infected and removed proportions, we assume a Beta-Dirichlet state-space model, independent conditionally on the underlying process:

Further, the Markov process on the latent proportions is built as:

where $\mathbf{\theta}_{\mathbf{t}}$ denotes the vector of the underlying population probabilities of the three compartments, whose mean is modeled as an unknown function of the probability vector from the previous time point, along with the transition parameters; $\mathbf{\tau =}(\beta,\gamma,\mathbf{\theta}_{\mathbf{0}}^{T}\mathbf{,}\lambda,\kappa)$ denotes the whole set of parameters where $\lambda^{I},\ \lambda^{R}$ and $\kappa$ are parameters controlling variability of the observation and latent process, respectively. The function $f(.)$ is then solved as the mean transition probability determined by the SIR dynamical system, using a fourth-order Runge-Kutta approximation.

Priors and the MCMC Algorithm Setup of the eSIR Model. The prior on the initial vector of latent probabilities is set as

$\mathbf{\theta}_{\mathbf{0}}\sim Dirichlet\ (1 - Y_{1}^{I} - Y_{1}^{R},\ \ Y_{1}^{I},\ \ Y_{1}^{R}) , \theta_{0}^{S} = 1 - \theta_{0}^{I} - \theta_{0}^{R}.$

The prior distribution of the basic reproduction number is$R_{0} \sim LogNormal(0.582,0.223)$ so that $E\left( R_{0} \right) = 2$ and $\text{SD}\left( R_{0} \right) = 1$, where $E$ and $\text{SD}$ denote the mean and standard deviation, respectively. The prior distribution of the removal rate is $\gamma \sim LogNormal( - 2.955,0.910)$ so that $E\left( \gamma \right) = 0.082$ and $\text{SD}\left( \gamma \right) = 0.1$. The prior mean of the removal rate $\gamma$ indicates an average infectious period of 12 days, which is originally set using estimates from the SARS outbreak in Hong Kong (Mkhatshwa & Mummert, 2010) due to the similarity between the two viruses; this value also aligns well with several recent studies on COVID-19 in China (Chen et al., 2020; Li et al., 2020; Ryu & Chun, 2020). The prior mean of the basic reproduction number, 2.0, is approximately the average of estimates from many other COVID-19 studies on the Indian population (S. Das, 2020; Deb & Majumdar, 2020; Ranjan, 2020; Sardar et al., 2020; Singh & Adhikari, 2020). We have conducted a sensitivity analysis to evaluate how robust the model is toward the prior settings using Indian population COVID-19 data. The sensitivity issue can be minimized with more observed data of a longer exponentially increasing period and stronger intensities by focusing on cities or states that are highly exposed. Note that the prior mean of the distribution of the transmission rate $\beta$ equals $\gamma R_{0}$. For the variability parameters, the default choice is to set large variances in both observed and latent processes, which may be adjusted over the course of the epidemic with more data becoming available:

Denoting $t_{0}$ as the last date of data availability, and assuming that the forecast spans over the period $\lbrack t_{0} + 1,T\rbrack$, our algorithm is as follows.

0. Take $M$ draws from the posterior $\lbrack\mathbf{\theta}_{\mathbf{1:}\mathbf{t}_{\mathbf{0}}},\mathbf{\tau|}\mathbf{Y}_{\mathbf{1:}\mathbf{t}_{\mathbf{0}}}\rbrack$.

1. For each solution path $m \in \{ 1,\ldots,M\}$, iterate between the following two steps via MCMC.

i. Draw $\mathbf{\theta}_{\mathbf{t}}^{\mathbf{(m)}}$ from $\left\lbrack \mathbf{\theta}_{\mathbf{t}} \middle| \mathbf{\theta}_{t - 1}^{\left( m - 1 \right)},\mathbf{\tau}^{\left( m \right)} \right\rbrack,\ t \in \{ t_{0} + 1,\ldots,T\}$.

ii. Draw $\mathbf{Y}_{\mathbf{t}}^{\mathbf{(m)}}$ from $\left\lbrack \mathbf{Y}_{\mathbf{t}} \middle| \mathbf{\theta}_{t}^{\left( m \right)},\mathbf{\tau}^{\left( m \right)} \right\rbrack,\ t \in \{ t_{0} + 1,\ldots,T\}$.

Modeling Intervention. We model the effect of interventions by assuming that the intervention will result in a decrease in the transmission from the S compartment to the I compartment. We do so by decreasing the effective rate of transition (or, equivalently, the chance of interaction between members of S and I), by introducing a time-varying transmission rate modifier $\pi\left( t \right) \in \lbrack 0,1\rbrack$. This updates the flow between the three compartments (Appendix Figure A1-B) via a set of differential equations as follows:

The reproductivity is thus modified by the intervention over time as $R_{0}\pi\left( t \right)$. To better understand the introduction of this effect modifier, we follow an example given by L. Wang et al. (2020). Suppose at a time $t$, $q^{S\left( t \right)} \in \left\lbrack 0,1 \right\rbrack\$is the chance of an at-risk person being in home isolation, and $q^{I\left( t \right)} \in \left\lbrack 0,1 \right\rbrack$ is the chance of an infected person being in hospital quarantine. Consequently, the chance of disease transmission when an at-risk person meets an infected person is $\text{β\ }\left\{ 1 - q^{S\left( t \right)} \right\}\theta_{t}^{S}\ \left\{ 1 - q^{I\left( t \right)} \right\}\theta_{t}^{I}\ : = \beta\ \pi\left( t \right)\ \theta_{t}^{S}\theta_{t}^{I}\$, with $\pi\left( t \right)\ : = \ \left\{ 1 - q^{S\left( t \right)} \right\}\left\{ 1 - q^{I\left( t \right)} \right\} \in \lbrack 0,1\rbrack$. In effect, this $\pi(t)$ modifies the chance of a susceptible person meeting with an infected person, which is termed as a transmission modifier. In this article, the functional form of $\pi(t)$ is a continuous function that reflects a combination of steadily increased community-level awareness and responsible quarantine and preventive measures, and the country-wide lockdown measures initiated by the government. This predefined transmission modifier can be smoothly incorporated into the differential equations as well as the MCMC algorithms. Its functional form can be quite flexible in reflecting the changing pattern of human intervention that affects the transmission rate of the epidemic within the population.

Implementation of the eSIR Model. We implemented the proposed algorithm in R package rjags (Plummer et al.) and the differential equations were solved via a fourth-order Runge–Kutta approximation. To ensure quality of the MCMC, we set the adaptation number to be $10^{4}$, thinned the chain by keeping one draw from every 10 random draws to reduce autocorrelation, set a burn-in period of $10^{5}$ draws to let the chain stabilize, and started from 4 separate chains. Thus, in total, we have ${2 \times 10}^{5}$ effective draws with about ${2 \times 10}^{6}$ draws discarded. One could reduce the computation time, but consequently might risk the quality of data. This implementation provides not only posterior estimation on parameters and prevalence of all three compartments in the SIR model, but also predicted proportions of the infected and the removed cases at future time points. To obtain predicted case counts from the predicted prevalence, we used 1.34 billion as the population of India, thus treating the country as a homogeneous system for the outbreak (World Bank, 2017).

Uncertainty Quantification. One major advantage of a Bayesian implementation is that uncertainty associated with all parameters and functions of parameters can be calculated from exact posterior draws without relying on large-scale approximation or the delta method. The credible intervals (CrI) for the prevalence are computed using the posterior distribution of proportions given the observed confirmed and removed prevalence, that is, $Y_{{(t}_{0} + 1):T}^{I}|Y_{1:t_{0}}^{I},\ Y_{1:t_{0}}^{R}$ and $Y_{{(t}_{0} + 1):T}^{R}|Y_{1:t_{0}}^{I},\ Y_{1:t_{0}}^{R}$, where $t_{0}$ denotes the last observed date, and $T$ denotes the last forecast date. More specifically, suppose we want to compute the 95% posterior CrI for the observed proportion of confirmed cases on the first day of forecast, that is, a CrI for the random variable $Y_{t_{0} + 1}^{I}$. Then, from the $M$ solution paths of the posterior, we have the draws $\left\{ Y_{t_{0} + 1}^{\text{I\ }\left( m \right)},\ 1 \leq m \leq M \right\}$. We construct a 95% posterior CrI for $Y_{t_{0} + 1}^{I}$ by simply computing the 2.5th upper and lower percentiles from this set of M draws. The cumulative prevalences are sums of the draws from the I and R compartments at a given time and thus the confidence interval for the sum can be calculated in a similar way. Case counts can be obtained from prevalences by using population size. Similar techniques apply to $\theta_{t_{0} + j}^{I}$ for any $1 \leq j \leq T - t_{0}$ and transmission parameters like $\beta$ and $\gamma$. For instance, a 95% posterior CrI for $\beta$ can be constructed by calculating the 2.5th upper and lower percentiles of $\left\{ \beta^{(m)},\ 1 \leq m \leq M \right\}$. Therefore, we could simply define $R^{(m)} = \frac{\beta^{(m)}}{\gamma^{(m)}}\ \forall\ 1 \leq m \leq M$, and compute the 95% posterior CrI for the effective reproduction number $R$ from $\left\{ R^{(m)},\ 1 \leq m \leq M \right\}$.

2.3. Parameter Choices for Short-Term Forecasts

We made projections of the cumulative number of cases over a time horizon to assess the short-term impact of lockdown as well as the long-term impact of lockdown and post-lockdown activities. For the short-term forecast on April 30, we assumed lockdown is implemented until April 14 with either a 1- or a 2-week delay in people’s adherence/compliance to lockdown restrictions. We compared these projections with two hypothetical scenarios: (A) no non-pharmaceutical intervention (i.e., a constant disease transmission rate over time since the first case was reported in India), (B) a moderate intervention with social distancing and travel bans only (i.e., a decreased transmission rate compared to no intervention). The prior mean for $R_{0}$ (the expected number of cases generated by one infected person assuming that the whole population is susceptible) was set at 2.0. This was estimated based on the early phase data in India and is consistent with other models (S. Das, 2020; Deb & Majumdar, 2020; Ranjan, 2020; Sardar et al., 2020; Singh & Adhikari, 2020). For the no intervention and the moderate intervention scenarios, we chose the transmission rate and the removal rate such that the means for the prior distribution of the basic reproductive number $R_{0}$ are 2.0 and 1.5, respectively (SD = 1). The change in $R_{0}$ from 2.0 to 1.5 as an effect of intervention was created based on what we saw regarding the effect of interventions and the relative reduction of $R_{0}$ in Wuhan (C. Wang et al., 2020). Given the similar population size and comparable population densities in China and India, the assumption on similar effect of interventions on the pandemic across the two countries does not seem too restrictive. For the current scenario of lockdown, our chosen mean for the prior of $R_{0}$ starts with 2.0 during the period of no intervention, drops to 75% of its original value or 1.5 during the period of moderate intervention, and further drops to 0.8 during the 21-day lockdown period, and moves back up to 1.5 after the lockdown ends as described in Figure 3 (assuming a gradual, moderate resumption of daily activities). The drop in $R_{0}$ from 2.0 to 0.8 during lockdown represents a 60% reduction, which is proportionally slightly less than the ~65% drop estimated in $R_{0}$ from the COVID-19 outbreak in Wuhan following the introduction of cordon sanitaire, or restriction of movement of people (Lin, 2020; Pan et al., 2020).

2.4. Parameter Choices for Long-Term Forecasts

For the longer term forecast until June 15, we considered three hypothetical post-lockdown scenarios: (i) people return to normal activities due to the urgent desire for reconnecting after lockdown, (ii) people return to moderate activities as they did during the period with social distancing and travel ban intervention, and (iii) people make a cautious return out of fear for the coronavirus and partake in subdued activities. For these three scenarios, we assume the prior mean on $R_{0}$ moves back up from 0.8 to 2.0, 1.5 and 1.2, respectively, 3 weeks after lockdown ends on April 14. We compared these post-lockdown scenarios with another hypothetical scenario involving perpetual social distancing and travel ban without any lockdown (we fixed the prior mean on $R_{0}$ at 1.5 over the entire forecasting interval). The time-dependent changes to $R_{0}$ values across our simulation scenarios are depicted in Figure 3.

2.5. Parameter Choices for Duration of Lockdown Analysis

To assess the long-term impact of lockdown duration, we considered four scenarios: 21-, 28-, 42-, and 56-day lockdown periods. In all scenarios, we assume the prior mean on $R_{0}$ remains at 0.8 for the duration of the lockdown. Post-lockdown, the prior mean on $R_{0}$ gradually returns to a value of 1.5 over a span of 3 weeks (analogous to the ‘moderate return’ scenario). The changes to $R_{0}$ values across our simulation scenarios for studying length of lockdown are depicted in Appendix Figure A7.

2.6. Open-Source Software

We are committed to data transparency and reproducible research. Daily updates of our India projections, based on cases, recovered, and deaths reported the day before by covid19india.org, a crowd-sourced database using state bulletins and official handles, can be found in our interactive and dynamic Shiny app (covind19.org). Apart from the scenarios described in this article, anyone can create predictions under other hypothetical scenarios with quantification of uncertainties. Open-source code underlying this app are available at https://github.com/umich-cphds/cov-ind-19.

3. Results

As we interpret the results from our model, let us use caution in not overinterpreting the numbers. Any statistical model is wrinkled with many assumptions. Similarly, the predictions themselves have large uncertainty (as reflected by the large upper-credible limits). A rigorous quantitative treatment often allows us to analyze a problem with clarity and objectivity, but we recommend focusing more on the qualitative takeaway messages from this exercise rather than concentrating on the exact numerical projections or quoting them with certainty.

3.1. Short-Term Forecast of Cumulative Case Counts in India

Under national lockdown (March 25–April 14), our predicted cumulative number of COVID-19 cases in India on April 30 are 19,625 and 19,503 (upper 95% CrI 130,326 and 129,422) assuming a 1- or 2-week delay (i.e., either a quick or a slow adherence), respectively, in people’s adherence to lockdown restrictions and a gradual, moderate resumption of daily activities post-lockdown (Figure 4, Appendix Figure A3). In comparison, the predicted cumulative number of cases under “no intervention” and the “intervention involving social distancing and travel bans without lockdown” are 222,000 and 53,000 (upper 95% CrI of nearly 1.4 million and 0.3 million), respectively. Under quick adherence, these figures correspond to a relative reduction of 91% and 63% of cases due to lockdown with moderate return compared to “no intervention” and “social distancing and travel ban.” The relative reduction in cases between two scenarios (often from the least to the most intense intervention) is calculated as the difference between an estimate (on a particular day, e.g., April 30) under the social distancing and travel ban scenario and under the lockdown with moderate return scenario and then divided by the estimate under the social distancing and travel ban scenario.

We are reporting only the upper credible limit here and elsewhere since the lower credible limits are very small and uninformative due to the large uncertainty in our predictions arising from many unknowns. We also believe that our point estimates are at best underestimates due to potential surveillance bias (underreporting and/or misdiagnosis of case counts) and our model not taking into account the population density, age-sex composition, and regional contact network structure of the whole nation. Increase in testing and community transmission may lead to a spike in a single day and that may increase the projections substantially upward. Regardless of the exact numbers, it is clear that the 21-day lockdown will likely have a strong relative effect on reducing the predicted number of cases in the short term when compared to weaker interventions.

3.2. Long-Term Impact of Lockdown on the Outbreak in India

We took a close look at what might be coming in the next 2 months, based on what we have seen in other countries and an epidemiologic model that has been gainfully employed to assess the effect of interventions in Hubei province (L. Wang et al., 2020). We estimate that roughly 388,000 (upper 95% CrI 2.4 million), 7.5 million (upper 95% CrI 104 million), and 18.5 million (upper 95% CrI 196 million) cases are prevented on May 15, June 15, and July 15, respectively, by instituting a 21-day lockdown with quick adherence and a cautious return compared to perpetual social distancing and travel ban (without lockdown) (Figure 5). This corresponds to relative reductions in cases of 93%, 96%, and 91%, respectively, compared to perpetual social distancing.

Without some measures of suppression after lockdown is lifted, the impact of lockdown in bringing down the case counts (the now ubiquitous term, ‘flattening the curve’) can be negated by as early as the first week of June. In fact, in Figure 5a, the preintervention (‘normal’) curve first passes the social distancing and travel ban curve on June 5. In particular, if people immediately go back to preintervention (‘normal’) activities post-lockdown, a surge in the predicted case counts is expected in the long term beyond what we would have seen if there were only social distancing and travel ban measures without any lockdown (27 million when post-lockdown activity returns to preintervention levels versus 26 million under social distancing and travel ban without a lockdown period on July 31; Figure 5). Longer lockdowns would delay this crossover, but a normal (preintervention) return post-lockdown would surpass social distancing and travel ban (if these scenarios continued perpetually). Long-term forecasting under slow adherence (2-week delay) can be seen in Appendix Figure A4.

We present posterior density and trace plots for the underlying model parameters$\ \beta,\ \gamma,$ and $R_{0}$, posterior distributions for the predictions $\text{Y\ }$and the latent proportions $\theta$ for the I and R compartments over time, and estimates and posterior distribution of the daily prevalence of active cases over time or $\frac{d\theta_{t}^{I}}{\text{dt}}$ . These are contained in Appendix Figure A5.

3.3. Relative Impact of Duration of Lockdown on Predicted Case Counts

We took the quick adherence epidemiologic models and compared the 21-day lockdown with hypothetical 28-, 42-, and 56-day lockdown scenarios (Figure 6). When comparing a 21-day lockdown with a hypothetical lockdown of longer duration, we find that 28-, 42-, and 56-day lockdowns can approximately prevent 733,000 (upper 95% CrI 6.8 million), 1.4 million (upper 95% CrI 9.8 million), and 1.6 million (upper 95% CrI 10.3 million) cases by June 15, respectively. These numbers correspond to a relative reduction in cases of 45%, 86% and 96%, respectively. A 28-day lockdown does not appear to have a substantial impact on cumulative case counts when compared to a 21-day lockdown. From an epidemiologic perspective, there appears to be some evidence that suggests a 42- or 56-day lockdown would have a more meaningful impact on reducing cumulative COVID-19 case counts in India. Our models suggest that some form of post-lockdown suppression (e.g., extension of social distancing measures, limits of gathering size, etc.) is necessary to observe long-term benefits of the lockdown period. We note that longer lockdown periods are also accompanied by increasing costs to individuals, such as economic costs, mental health issues, and other public health exacerbation costs and must be considered in policymaking.

Lockdown duration study under the slow adherence (2-week delay) scenario can be found in Appendix Figure A6. The implied $R_{0}$ plots can be found in Appendix Figure A7.

3.4. Sensitivity Analyses

We did explore some alternative assumptions and conducted thorough sensitivity analysis before settling on the models presented above. In one example, we assumed that there are actually 10 times the number of reported cases to date to reflect potential underreporting of cases due to lack of testing. We note that our predictions of case counts indeed go up and the effect of underreporting of cases is more palpable with long-term projections (see Table 2, underreporting). In another scenario, we assumed these cases occurred in metropolitan areas to reflect a potential intensification of case clustering. In our primary analyses, we assumed that the cumulative case counts across the country represent equal contributions from all the regions, and using the whole population of India as a scaling factor to compute initial inputs for $Y_{1:t_{0}}^{I}$ and $Y_{1:t_{0}}^{R}$. This may lead to extremely small proportions, which may in turn yield underestimated outputs from the eSIR model. Changing the total population size from that of India to that of representative (large) cities from the hub states (the states of Kerala, Maharashtra, and Karnataka, and the national capital region of Delhi) is a simple but intuitive way to potentially do away with the aforementioned underestimation. We note that this substantially reduces the width of the credible intervals (see Table 2, case-clustering). In yet a third scenario, we hypothesized that the prior mean of $R_{0}$ is set at 3.0 or 4.0 instead of 2.0 (i.e., a single infected individual would infect 3 or 4 susceptible individuals, on average, instead of 2). In most of our analyses (Table 2), the posterior mean for $R_{0}$ was seen to be from 1.8 and 2.4, irrespective of whether a higher/lower starting (prior) mean was used. We observe that a prior mean of 4.0 for $R_{0}$ sways the posterior $R_{0}$ estimate substantially (posterior mean 3.38). As more data accumulate, we will expect the effect of the prior on the posterior estimates to diminish. The posterior distributions of the prevalence in each compartment and latent proportions under these changing scenarios are available in Appendix Figure A8.

Table 2. Comparison of Estimated Projections and Posterior Estimates of Model Parameters Across Different Sensitivity Analysis Scenarios Under 21-Day Lockdown With Moderate Return, Using Observed Data Until April 14

^{Note. Prior SD for} $R_0$ <sup>is 1.0.
* Observed case counts are multiplied by 10, prior mean for</sup> $R_0=2$
^{** Assume that the cases happen in metro hotspots, use population size N=32 million instead of national population 1.34 billion, prior mean for} $R_0=2$

In summary, these sensitivity analysis scenarios did not appreciably change our conclusions in broad qualitative terms, though the exact quantitative projections for case counts are quite sensitive to such choices. We note that the estimate of basic reproduction number $R_{0}$ is more robust to underreporting issues because counts in all compartments of our eSIR model are assumed to be underreported. Since underreported case counts affect all our hypothetical intervention scenarios in a similar way, the relative comparison of interventions and the associated conclusion remain valid in a qualitative sense. In all cases, our confidence in these projections decreases markedly the farther into the future we try to forecast. It is extremely important to update these models as new data arise.

3.5. Model Calibration

To check the calibrating properties of our model, we truncated the data to certain dates and tried assessing the quality of the case-count predictions with essentially adding one more week of data for predicting active cases at a future date (Table 3 and Appendix Figure A9). We do notice the projected case counts change significantly with more data and improve (become closer to the observed) with more data. Our projections always underestimate the observed counts. This phenomenon is also due to more testing being done each week. However, the observed number of infected cases is always within the 95% prediction credible interval provided by our model. This again reveals the large uncertainty in our predictions.

Table 3. Comparison of Model Projections Using Observed Data up to Different Dates Assuming a 21-Day Lockdown With Moderate Return

^{Note. All prediction scenarios assume a prior mean of} $R_0=2$

4. Discussion

Our projections using current daily data on case counts until April 14 in India show that the lockdown, if implemented correctly in the end, has a high chance of reducing the number of COVID-19 cases in the short term and buying India invaluable time to prepare its health care and disease-monitoring system. In the long term, we need to have some measures of suppression in place after the lockdown is lifted to prevent a massive surge in the number of cases that can quickly overwhelm an already overstretched Indian health care system resulting in increased fatalities. Specific vulnerable populations will be at higher risk of severity and fatality from COVID-19 infection: older persons and persons with preexisting medical conditions (e.g., high blood pressure, heart disease, lung disease, cancer, diabetes, immunocompromised persons) (Centers for Disease Control and Prevention [CDC], 2020; WHO, 2020a). Appendix Table A1 provides a description of the approximate number of individuals in these high-risk categories in India. Beyond the fragile population characterized by health and economic indicators, we must remember that health care workers and first responders at the frontline of this pandemic are among the most vulnerable (C. Wang et al., 2020). Though we have focused on forecasting and modeling of COVID-19 case counts in this article, we recognize that this is only one component of the problem. Long-term surveillance and management of the COVID-19 crisis is needed with not just public health in mind but also to take care of the economic, social, and psychological impact that it will have on the people of India.

4.1. Limitations

Our statistical modeling and forecasts are not without limitations. We have limited number of data points and a wide time-window to extrapolate for the long-term forecasts. The uncertainty in our predictions is largely due to many unknowns arising from model assumptions, population demographics, the number of COVID-19 diagnostic tests administered per day, testing criteria, accuracy of the test results, and heterogeneity in implementation of different government-initiated interventions and community-level protective measures across the country. We have neither accounted for age-structure, contact patterns, or spatial information to finesse our predictions (Klein et al., 2020; Mandal et al., 2020; Singh & Adhikari, 2020) nor considered the possibility of a latent number of true cases, only a fraction of which are ascertained and observed (C. Wang et al., 2020). Increase in frequency and scale of testing, and community transmission of the SARS-CoV-2 virus may lead to a spike in a single day and that can shift the projection curve substantially upward. COVID-19 hotspots in India are not uniformly spread across the country, and state-level forecasts (S. Das, 2020) may be more meaningful for state-level policymaking. We are assuming that the implementation and effects of public health interventions and policies are the same everywhere in India by treating India as a homogeneous unit.

The eSIR model treats the entire group of people within a single compartment as homogenous and exchangeable. We also assume that all subjects who were not infected are susceptible. Certainly, this overlooks the possibilities of people moving between states and different subsets of infected and susceptible populations having greater or lesser likelihood of coming into contact with one another. To account for all such potential factors, we need more-nuanced modeling. One potential way is to break up the interaction component into further subcompartments; however, sparsity of current data in each subcompartment is an issue. Another way that has been pursued in a recent study is to inform the SIR modeling via introducing contact networks at the initial stage (Bhattacharyya et al., 2020). However, it is worth noting that any such approach would need more-granular and reliable data containing individual details of the confirmed cases, including their location and travel history. Even though such data are available from some self-reporting–based repositories (Kaggle, 2020), the quality and detail of the information provided are quite heterogeneous and, thus, how to best utilize such data remains a question. We see the tremendous role of data and its transparency of collection and reporting in finessing our predictions. Accurate and consistent reporting of case counts and deaths due to COVID-19 are extremely critical. Future opportunities for improving our model include incorporating contagion network, age-structure, and test imperfection and estimating SEIR model and true fatality/death rates. Future research would benefit from easily accessible hospitalization data, accurate recording of deaths in death records, and availability of ecological-level covariate data. Regardless of the caveats in our study, our analyses show the impact and necessity of lockdown and of suppressed activity post-lockdown in India. Though the exact numerical projections are perhaps far from the truth, the qualitative inference on the relative effects of the interventions are still valuable and valid since all projections are subject to similar biases.

One ideological limitation of considering only the perspective of controlling COVID-19 transmission in our model is the inability to account for excess deaths due to other causes during this period (chronic disease and mental health–related diseases in particular), or the flexibility to factor in reduction in mortality/morbidity from some other infectious or flu-like illnesses, traffic accidents, or the health benefits of reduced air pollution levels. A more expansive framework of a cost–benefit analysis is needed as we gather more data and build an integrated landscape of changes in population-attributable risks due to various disease categories.

4.2. Testing

A reviewer of this article suggested giving a sense of the testing data from India and how that may affect our conclusions. India’s priorities in testing have changed multiple times over the past few weeks. On March 17, India proposed testing all people who recently traveled internationally and developed symptoms (fever, cough, difficulty in breathing, etc.) of COVID-19 within 14 days of return, all symptomatic contacts of laboratory-confirmed positive cases, or all symptomatic health care workers managing respiratory distress. On March 20, India revised this testing strategy to include all symptomatic health care workers, all hospitalized patients with severe acute respiratory illness (fever and cough and/or shortness of breath), and asymptomatic direct and high-risk contacts of a confirmed case to be tested once between day 5 and day 14 of coming into contact. The testing strategy was again revised on April 9 to include testing of all symptomatic people in hotspots/clusters and in large migration gatherings or evacuee centers. These testing strategies are obtained from the Indian Council of Medical Research (https://www.icmr.gov.in).

We looked a little deeper into the issue of testing bias using data from Our World in Data (Hasell et al., n.d.). Appendix Figure A10-A shows that, even though the number of tests in India has increased in recent days, the proportion of daily discovered positive cases still remain stable (at about 4%) and do not yet show an obvious increasing trend like in other countries such as the United States and United Kingdom. We also plot Iceland and South Korea on this figure as they have performed remarkably in administering a large number of tests per detected case and serve as examples for the world. We also looked at the proportion of the population tested in 61 countries around the world (Appendix Figure A10-B). While most advanced countries have tested around 1 to 3% of the population, India has tested roughly 0.06% population and it will take weeks, if not months, for India to reach testing 1–3% of the population. In the absence of rapid and large-scale testing, informative proxies or surrogates can be tracked through syndromic surveillance, temperature reporting, and monitoring hospital admissions and medical claims due to respiratory and flu like illnesses. This additional information will strengthen the prediction models. In absence of these models, we rely on sensitivity analysis for underreporting as presented in Table 2.

4.3. Our Data Science Product

Finally, in our strong commitment to reproducibility and dissemination of our research, we have made the code for our predictions available at GitHub and created an interactive and dynamic R Shiny app to visualize the observed data and create predictions under hypothetical scenarios with quantification of uncertainties. These forecasts are updated daily as new data come in. We hope these products will remain our contribution and service as data scientists during this tragic global catastrophe, and the model and methods will be used to analyze data from other countries.

5. Conclusion

Our epidemiologic and mathematical calculations make a convincing case for enforcing national lockdown in the largest democracy in the world, acting early, before the growth of COVID-19 infections in India starts to accelerate. We observe the public health benefit of extending the lockdown by 3 to 5 weeks in our projections. Measures of suppression are needed post-lockdown to acquire maximal long-term benefits from the lockdown. We also illustrate the critical role of epidemiologic forecasting in aiding policy decisions through this modeling exercise. We highlight the importance of conducting model checks, sensitivity analysis, and uncertainty quantification.

We realize that these draconian public health measures come at a tremendous price to social and economic health that can last for months or even years after the restrictions on social mobility are lifted. Our general message to the public is to proceed with prudence and caution and adhere to effective public health interventions until there are rapid and reliable home testing kits; there are none yet (Food and Drug Administration [FDA], 2020), FDA approved drugs (WHO, 2020c), and a vaccine (Craven, 2020).

Disclosure Statement

No conflicts of interest to declare. The research was supported by the University of Michigan Precision Health Initiative, University of Michigan School of Public Health, University of Michigan Institute for Health Care Policy and Innovation, Michigan Institute of Data Science, and the University of Michigan Rogel Cancer Center.

Acknowledgments

We the authors would like to thank the University of Michigan Advanced Research Computing Services, in particular Professor Ravi Pendse, for enabling daily updates to our models and allocating us abundant computational resources. We would also like to thank Professor Matthew Fox from the Boston University School of Public Health for his valuable comments on our R Shiny App. This work originated at a time when the authors in this study team came together to provide their service as quarantined data scientists. We thank our families and friends for their support during these unprecedented times. The initial report made a profound impact through media outlets; we remain grateful to journalists across the world who reported our findings. In the middle of a global pandemic, we found inspiration in the heroic battles by frontline health workers, we found hope in science, we found strength in the power of common people and we found solace in the magic of human kindness. We wish the government and the people of India the very best in their fight against the coronavirus. We are all in this together.

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Appendix

Table A1.  Proportion of Population in Specifically Vulnerable Subgroups at Potentially High Risk of COVID-19 Severity Risk in India

^{† based on 2020 est. of 1.38 billion from UN Department of Economic and Social Affairs}

^{* age-standardized; ** risk; *** defined as within 5-kilometer distance of home or work}

^{Abbrev.: COPD, chronic obstructive pulmonary disease; IDF, International Diabetes Federation; NICPR, National Institute of Cancer Prevention and Research}

Table A2. Intervention Landscape of Countries Severely Affected by COVID-19

<sup>¶ Date of 1st case data obtained from JHU CSSE time series data on COVID-19 (except for China & South Korea)
† Microsoft bing COVID-19 tracker (https://bing.com/covid, as of 1:00 PM EST April 10, 2020)
* https://www.livescience.com/first-case-coronavirus-found.html
**  https://www.worldometers.info/coronavirus/
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3 https://www.nytimes.com/article/coronavirus-timeline.html
4 https://en.wikipedia.org/wiki/2020_coronavirus_pandemic_in_Germany
5 https://www.gov.uk/government/publications/coronavirus-action-plan/coronavirus-action-plan-a-guide-to-what-you-can-expect-across-the-uk
6 https://www.axios.com/italy-coronavirus-timeline-lockdown-deaths-cases-2adb0fc7-6ab5-4b7c-9a55-bc6897494dc6.html
7 https://www.reuters.com/article/us-health-coronavirus-spain/shellshocked-spain-reports-record-832-new-coronavirus-deaths-idUSKBN21F0NR
8 https://www.nytimes.com/aponline/2020/03/15/world/middleeast/ap-ml-virus-outbreak-mideast.html?searchResultPosition=10</sup>

Table A3. Comparison of Types of Infectious Disease Models, Specifically Those Used to Study COVID-19 in India

^{Figure A5. Trace plots and posterior density plots for the underlying model. Parameters}$\text{\ β}$ ^(a), $\gamma$ ^{(b), and} $R_{0}$ ^{(c), posterior distributions for the predictions} $\text{Y\ }$^{and the latent proportions} $\theta$ ^{for the I (d), and R (e) compartments over time, and estimates and posterior distribution of the daily prevalence of active cases over time or} $\frac{d\theta_{t}^{I}}{\text{dt}}$ ^{(f). These plots correspond to the 21-day lockdown with moderate return scenario under quick adherence.}

^{Figure A8. Posterior distributions of the projected case counts and latent proportions under sensitivity scenarios.}

^{Figure A10. Daily testing patterns in selected countries (A); Testing numbers and proportions for 61 countries around the world affected by COVID-19 (B).}

This article is © 2020 by Debashree Ray, Maxwell Salvatore, Rupam Bhattacharyya, Lili Wang, Jiacong Du, Shariq Mohammed, Soumik Purkayastha, Aritra Halder, Alexander Rix, Daniel Barker, Michael Kleinsasser, Yiwang Zhou, Debraj Bose, Peter Song, Mousumi Banerjee, Veerabhadran Baladandayuthapani, Parikshit Ghosh, and Bhramar Mukherjee. The article is licensed under a Creative Commons Attribution (CC BY 4.0) International license (https://creativecommons.org/licenses/by/4.0/legalcode), except where otherwise indicated with respect to particular material included in the article. The article should be attributed to the author identified above.