A model does not determine or influence the real-life progression of the observed data -it reflects it. Real-life events influence the progression. In the case of Total Covid Cases, disruptive events can be planned initiatives (lockdown, containment, mass movements), or unplanned reckless social behaviour. It is known that the effects of such events show up in observed Total Cases with a lag of 1 to 2 weeks. The change in the nature of progression should show up in the output of the mathematical model as well. This does not happen automatically -what happens is that estimates thrown-up by the model begin deviating significantly from observed data. This is the signal that the underlying pattern has changed, and so the model must be revised to reflect the new underlying pattern. Thus, over the life of this pandemic, we expect to see different models in different phases, as the real-life nature of progression changes.
Conversely, when a model continues to present a close fit between model-output and actuals (high R 2 ) for a long time, then it means that the model is well-constructed and robust enough to absorb small deviations in observed data. Such a model, stable over time, gives confidence when used to make forecasts. The current model in our study has remained unchanged for the last 30 days with R 2 >=0.998 throughout (reporting date 8 th May, with data upto 7 th May).
Assuming that the model continues unchanged, we forecast numbers of Covid19 cases in India. Separately, we also examine Fatalities, Recoveries and Active Cases, and derive conclusions. The study is not important for the forecasts it makes. The study is directionally important to identify trends and trajectories. But most of all the study is a comparator to measure the consequences of policy action. If actual cases are lower than forecasted numbers, then we are doing well.
Data on Daily Infected Cases (Fresh Cases), Daily Deaths, Daily Recoveries and Daily Active Cases have been consistently sourced from the daily 8am update of the MoHFM (1) from 3 rd March 2020 (Day 1) -data reported at 8am 4 th Mar -from the time the 4 th case of Covid19 was recorded. [Initial 3 diagnosed/recovered in Jan/Feb.] A 2-day Moving Average (2dMA) of Fresh Cases was done to smoothen random delays in Test sample processing, Results Announcement, etc. The daily 2dMAs were cumulated to give the daily Total Cases. Modelling commenced from Day 13 (15 th March), when the total cumulated 2dMAs (Total Infected Cases) reached 100.
Three models have been successively developed so far to depict the changing pattern of Total Cases progression. Table 1 When and how will the model be updated? When real-life events are significant enough to change the nature of the growth curve, this will show up as actual numbers of Covid Infections . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)
The copyright holder for this preprint this version posted May 18, 2020. . https://doi.org/10.1101/2020.05.11.20097865 doi: medRxiv preprint deviating from the forecasts. As in the earlier phases of the pandemic, when the deviations lead to a fall in R 2 (robustness of predictability) -typically R 2 <0.98 -a fresh model will be fitted spanning the period when deviant observations started. A significant event that can alter the shape of the curve can happen any time, or may have already happened and we are in the lag period. Whenever it does, the model needs to be updated. Some models may be short-lived, actuals soon beginning to deviate again. The update is then repeated afresh. When a model is stable -high R 2 for at least 14 observations -it can be used as a basis for making further forecasts.
That the current model has remained valid for 30+ days means that newsworthy events (of spikes and troughs) have not been statistically significant. In plain English, they have "cancelled out" and been "absorbed" within the predictive ability of the model. This means that without a disruption greater than in the last 29 days, this graph would continue on its own pattern and trajectory -curves, lines, plateaus or whatever. That this graph would continue on its own trajectory unless interrupted by a significant-enough intervention, makes it an important comparator. For example, "highest ever cases on a day" is not a shock if the graph were forecasting a similar number. Similarly, a particular containment strategy can only be claimed to have worked if the shape of the graph changes and reaches a point lower than where it would have naturally gone.
Of course, this graph will eventually change shape, and a new model will be required for the new pattern. Significant-enough events that could make this change include medical events (vaccine, cure or acquired immunity), climatic triggers, other acts-of-God, and of course, planned policy interventions. For example, the strategy of actively testing large numbers, tracing and quarantining, may shortly lead to a reduction of numbers (say, if all prospects traced and isolated). The relaxations during Lockdown 3.0, planned inter-state migrations and further relaxations are other policy interventions that may change the change the shape of the graph. The last set of interventions have the potential to make the pattern steeper than at present.
We hear regular announcements on "doubling time" of Total Infected Cases, and how this time has increased (to a figure of 12 days, currently). The joy that accompanies such announcements is, unfortunately, totally misplaced. Increase of the doubling time is an inherent part of the pattern of the model of virus spread. For the graph remaining unchanged, the current doubling time of 12 days, would automatically increase to 16 days by the end of May, and to 19 days by mid-June. This reiterates one of the most important arguments of this studythat a mathematical model is a comparator, and successes occur only when the outcomes are lower than forecasts by the model. Conclusions from this study show later that doubling time must soon rise to 38+ days for the pandemic to be under control.
A standard disclaimer that must accompany all uses of this model for forecasting is to repeat that disruptive events on-ground can/will change the model with some lag, that near-term forecasts are more plausible than longer-term ones (statistical errors of forecasting multiply for longer ones), and that even small and acceptable statistical off-targets may have huge physical implications -a 3% forecasting error, statistically acceptable, may involve 3000 human beings. Any forecast always assumes that the model remains stable.
. CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint this version posted May 18, 2020. . https://doi.org/10.1101/2020.05.11.20097865 doi: medRxiv preprint Subject to this disclaimer, the near-term forecasts by this model are that in the weeks 4 th -10 th May, 11 th -17 th May, 18 th -24 th May and 25 th -31 st May, the fresh cases added will exceed, respectively, 22000, 25000, 33000 and 38000. Thus, by May-end, the total number of Covid19 cases recorded would exceed 150,000
The current model of Total Infected Cases has no inherent "pull-back" element which can lower its trajectory in the foreseeable future. That is, its trajectory will keep growing at progressively slower speeds. The import of this statement is that there is no point in the foreseeable future where growth of new cases will zeroise. Strictly as per the current model, the country would cross a million recorded cases by early-August.
Current announced measures of fatality rates are flawed because they do not reflect the disease progression. These measures divide the number of recorded deaths by the number of recorded cases to arrive the Fatality rate. This measure is incorrect.
Disease progression of Covid19 has been extensively studied. The report by Lord (2) (April 2020) concludes that the mean time from diagnosis to death for non-survivors is 18.5 days. The same report, computes mean time from diagnosis-to-cure as 17 days, and average ICU stay, for those who need it, as 15 days.
For our purpose, we will take the mean time diagnosis-to-death is 15 days. Any measure of fatality rate must naturally against the cohort who were patients at the same time. In other words, if average time diagnosis-to-death is 15 days, the number of deaths occurring today must be compared against those who were patients 15 days ago. We define a term Ongoing Fatality Rate (OFR) -OFR on any day is the cumulative number of deaths divided by the total infected cases 15 days ago. This is a more realistic approximation of fatality rate. To appreciate this better, visualise how the current (flawed) computation of Fatality Rate will change as Fresh Cases go down and become zero, though a few hospital patients still pass away -Fatality Rate will increase and that is illogical. Our definition of OFR formalises the fact that death-rate cannot be measured out of those who joined as patients within the last 15 days, because they have yet to live out their disease process. . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint this version posted May 18, 2020. . https://doi.org/10.1101/2020.05.11.20097865 doi: medRxiv preprint  Conceptually, until testing and new-case identification reaches steady-state, OFR will continue to decline as more and more cases get rapidly added. In due course, OFR should stabilise close to the eventual CFR.
The Ongoing Fatality Rate (OFR) in India has dropped sharply from ~40% in early-April to ~9% now, as testing has speeded up. It has remained steady between 8.73% and 9.71% over D57 to D65 (7 th May, last data date), averaging 8.93% over last 6 days, and declining. Since the inception of the pandemic, 8.73% has been the lowest value of OFR.
There is no way to predict where OFR in India will reach -that is, what will be the eventual Fatality Rate. Total deaths divided by total infected patients worldwide on 6 th May, stands at 7.02% (3) . By country, these rates vary widely, between 12%+ and sub-1%, for various reasons like demography, healthcare readiness, inherent immunity, etc. Public perception has been created that fatality in India will be 3%-4%. As already explained, this uses a flawed metric. The closest assumption to Fatality Rate in India is OFR as we have defined it, 9% at present.
OFR will India will decline further for two reasons, both related to low testing rates so far. One, because of low testing many late (advanced) cases are being diagnosed, who expire earlier than the assumed average 15 days from diagnosis. Thus the number of deaths in the OFR calculation is contaminated by numbers less than 15 days old, who are not part of the denominator. Second, as testing rates increase, they will identify more asymptomatic or mild cases, who will go through their 15-day disease process and survive -again lowering the OFR.
If OFR in India were to decline, what eventual figure (Case Fatality Rate) might it reach? Predicting this figure is out of the remit of mathematical modelling, and belongs at best to medical science. Irrespective of questionable data, China could be an analogy, since it is closest to India by demographic profile. China's Fatality Rate presently announced is 5.5% (3) , and this is likely to stay that way since fresh cases are few.
We make the conscious assumption that India's OFR, currently 9%, will reduce by 1 percentagepoint every week, starting from Monday 11 th May, as testing approaches steady-state. India's CFR will eventually stabilise at 5.5% from 1 st June.
Subject to the usual disclaimer, the model predicts that India will have nearly 5000 deaths by May 31, may reach a total of 50000 deaths by mid-August and 100000 by mid-Sept. . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint this version posted May 18, 2020. . https://doi.org/10.1101/2020.05.11.20097865 doi: medRxiv preprint
Since we have assumed that CFR in India may be 5.5%, the final Recovery Rate (Case Recovery Rate) will be 94.5%.
The physical significance of these Ratios is expressed by the following statements:  On any day D, the Total Number of Deaths will be OFR x Total Infected Cases on D-15.  On any day D, the Total Recovered Cases will be ORR x Total Infected Cases on D-15  Therefore, on any day D, Total Active Cases will be Total Infected Cases on day D, less Total Deaths on day D, less Total Recovered on day D.
Since the current ORR of 75% will eventually rise to 94.5% (1-CFR), for ease of computation we make the following assumptions --ORR=75% till 10 th May, 80% from 11 th -31 st May, 85% in June, and 90% thereafter.
Using the relations above, we computef Total Active Cases. As shown in Fig 5, reports that India has 400,000 isolation beds in hospitals. 50% of all available beds would be in use by early July, if current hospital protocol continues. Since hospital beds are location-specific, many regions would be near their maximum limits at this stage. We expect some policy changes by end-June. Either, self-isolation of mild cases without hospitalisation -this change would make availability of hospital facilities a non-issue till end-August. Also additional hospital facilities would be created. Creating the infrastructure for such facilities is not the problem --the challenges here are the availability of doctors, para-medical staff, PPE, medicines and consumables. Implementation plans would have to begin by mid-April, if this option were to be chosen.
. CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint this version posted May 18, 2020. . https://doi.org/10.1101/2020.05.11.20097865 doi: medRxiv preprint Another aspect of healthcare shortage -that of ICU beds and ventilators -is not explored as all available indications are that ventilator needs would be addressed by a combination of planned production/imports and innovative solutions from several agencies. It is assumed that hospital beds can be repurposed into ICUs if required.
The conclusion from this exploration of active cases is that though hospital facilities are stretched by early July, hospital capacity is not an unsurmountable problem, if policy decisions are taken early and associated software issues planned for.
The forecasts generated by the model in the foregoing sections are summarised in Table 2 . 15 th Sept (4 months away) has been chosen as the end-date of forecasts on the assumption that some medical solution would become visible by then, even if not implemented. 
The curve appears "flat enough" for healthcare infrastructure proposed, though shortage of people and consumables would remain a challenge.
But the curve is NOT FLAT ENOUGH from the point of human cost -100000 lives lost by the next 4 months, possibly the highest figure worldwide. Such heavy human casualties must be avoided. So how flat should, and can, the curve be?
We set a limit of 50,000 deaths by Sept 15, and derive backwards the model of pandemic spread that reaches this figure, which by the by the current model is reached on 13 th Aug, That is, the proposed change delays the outcome by about 32 days. This slowing-down limits the spread after Sep 15 th as well.
To reduce the number of deaths reached by a given date, we can either reduce the Ongoing Fatality Rate (OFR), or the numbers infected. We have already factored-in OFR reduction to 5.5% from June 1, and there is no basis to assume further reductions Our option is to reduce the growth of numbers infected.
We make realistic assumptions about when one can implement policy interventions to bring growth below present rate of growth. Considering further relaxations after Lockdown 3.0 and massive inter-state migration and repatriation plans, it is unlikely that tougher regulations can . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity.
The copyright holder for this preprint this version posted May 18, 2020. . https://doi.org/10.1101/2020.05.11.20097865 doi: medRxiv preprint come into effect before June 1. Thus our task is to find a pattern and growth trajectory of Total Infected Cases between Jun 1 and Sep 15 (106 days) that leads to a Total Deaths figure of 50,000 in contrast to the figure of nearly 100000 reached by the current trajectory.
The required model has been determined,. Fig 6 shows the graph of the current model, and the graph of the desired model, naturally much flatter. The current model has an overall doubling rate of 23 days in the period Jun 1 -Sep 15. In contrast, the desired model that limits fatalities should have a doubling rate of 38 days. This will call for massive changes in social behaviour and regulations. This study cannot speculate what such measures can bemandated protective equipment, more severe containment strategies, modified forms of lockdown, or any other. Table 3 is the final summary that shows Total Cases, Total Deaths, Total Active Cases and Doubling Time under two situations. The first, the model that the growth of Total Cases has steadfastly followed for the last 30 days with exceptionally high R 2 . The second, the desired model that Total Cases must follow -in order that Total Deaths are limited to 50,000 by Sept 15 th , with the added restriction that no change in pattern should be expected before June 1.
. CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review)
The copyright holder for this preprint this version posted May 18, 2020. . https://doi.org/10.1101/2020.05.11.20097865 doi: medRxiv preprint Table 3 : Comparison of key parameters between Current and Desired Models
At the end, we reiterate the objectives and results of this study. The study has presented the current model of Covid19 cases in India over time. Based on the stability of the model, while elaborating the events that would cause the model to change, the study has extrapolated Total Cases, Total Deaths and Active cases till Sept 15 th . In doing so, the study has argued that the current publicly-reported Fatality and Recovery Rates are flawed, and derived more realistic estimates of these parameters. The forecasted numbers provide a basis for comparison with the results of planned policy interventions -if the actual observed data is less than the forecasts, we are doing well. This is the more important reason to use the forecasts, rather than as astrological predictions of the shape of things to come! While exploring the number of Active Cases forecasted by the model, the study has found that Indian healthcare infrastructure has the capacity to deal with number of cases projected, more or less. In that sense, the Indian curve is flat enough. However, the current curve of Covid19 cases is NOT flat enough with respect to expected fatalities, which by the current growth curve would reach 100,000 by Sept 15 th . The study set an arbitrary limit of 50000 Total Deaths by Sep 15, and retro-fitted a mathematical model that was acceptably flat. The key parameters of this desired curve -Total Cases, Total Deaths and Active Cases -have been tabulated fortnightly, as also done for the current model.
While we look forward to Public health initiatives to tackle the pandemic, the outputs of this study provide the reference points against which the success or failure of public health initiatives can be judged.