We have created a spreadsheet with a simple model for Covid-19 infections and deaths under different reopening policies. The model uses R, the virus reproduction number (how many get infected by a single infectious person). We find it useful to view four forms of R:
(1) starting R₀ at time zero when everyone was suceptible since no one was immune from having been infected;
(2) lockdown R₀, which was around 1.0 in the US, meaning the curve was flattened and the death rate stable;
(3) Target R₀, which reopening will result in, the value depending on the level of relaxation and monitoring - the focus of this model; and
(4) Actual R_t, which differs from Target R₀ when a significant fraction of the population gets infected or vaccinated.
We make no claim that the model fits US data to date or will in the future. Rather, our goal is to provide any interested person a simple tool with which to understand R, and how small changes in Target R₀ can mean big differences in deaths. The spreadsheet shows what may happen when R₀ changes to different values under different reopening policies. For anyone interested in the math, the various formulas are completely visible and explained below the graphs, but there is no need to understand the math to use the tool.

We term "Cuomo" a policy that opens up the economy gradually, with aggressive test/trace/isolate, to increase R₀ only slightly (to 1.06 in our illustration) to keep the national death rate close to or below the current 1000 a day. We term "Trump" a laissez faire policy (no test/trace/isolate) to increase R₀ significantly (to 1.2 in our illustration, a 20% increase in exposure, still small compared to the near tripling to the level before the pandemic). We term "Merkel" an aggressive policy that pushes R₀ down (to 0.7) to eliminate the virus in a few months even without a vaccine (emulating the German experience).

While "Trump" and "Cuomo" are US references and the numbers used in the illustration are representative of the US situation, the spreadsheet can be used for any place by altering the values in yellow.

The model expresses the following understanding:
(a) Each infected person is infectious for 15 days.
(b) The number likely to be infected by one infectious person is termed Target R₀ (to convey that it is a planned or unplanned target of governmental reopening policy), but is effectively smaller since the infected and (efficaciously) vaccinated are immune, so the Target R₀ is multiplied by the susceptible (ie uninfected and unvaccinated) fraction of the population to get the Actual R_t.
(c) The Actual R_t may be further reduced in the first few cycles by the warm weather (there is no hard evidence to support this speculation).
(d) The number of deaths during any cycle is a fixed fraction, termed case mortality fraction (0.01 ie 1% in our illustration), of the new infections in the previous cycle.
(e) While deaths from Covid-19 take place over a wide range of days following infection, all deaths from an infection are put in the next 15-day cycle.
(f) When a vaccine becomes available, vaccination is instantaneous (to keep Excel formulas simple).

[For viewing ease, the graphs obstruct much of the data they depict, but can be moved out of the way to view the numbers in the tables changing as you alter any parameter.]

We hope our simple model, while easy to understand and explore with our spreadsheet, is consistent with the better, more complex models. Our spreadsheet tool is meant to help understand the basic principles of virus propagation and the likely consequences of different government policies as well as vaccine availability.

All comments and criticisms are appreciated -
Sekhar Ramakrishnan sekhar@caa.columbia.edu,
Pediatrics & CTSA, Columbia University, NYC
Janak Ramakrishnan janakdaniel@gmail.com, Google Inc., NYC

There are five simulations shown: Trump, Cuomo and Merkel all without a vaccine; Trump and Cuomo with a vaccine available in 8 months (the date can be changed). The take-aways:
(1) While a Target R₀ of 1.06 and 1.2 may not seem so different and the death counts start off not very different, the curves diverge quickly, pointing to the absolute importance of keeping the curve flat or tending down.
(2) Deaths under Trump are many times the deaths under Cuomo;
(3) The vaccine is of little help to Trump because most of his deaths take place before vaccine availability, and the virus disappears due to herd immunity from nearly everyone getting infected;
(4) By keeping the curve flat, Cuomo saves many lives when a vaccine becomes available, and the virus disappears due to herd immunity from vaccination;
(5) Deaths under Cuomo are still many times the deaths under Merkel, which wipes out the virus even without a vaccine, with a fraction of the deaths under the other policies (and the reason why there is no simulation of Merkel with a vaccine);
(6) The effect of lowering Trump's Target R₀ is dramatic: if 1.2 is changed to 1.1, the deaths are reduced by half, but are still many more than the deaths under Cuomo (this last point highlights the value of the spreadsheet tool).

The seven major parameters to play with are:
a) specific population to look at: US or any state;
b) the Target R₀, ie the R₀ you expect a policy to result in (1.2 for Trump and 1.06 for Cuomo in our illustration),
c) the number of initial warm (15-day) cycles (8),
d) the factor by which Target Rt is scaled down during warm cycles (0.9; a value of 1 means no warmth effect),
e) vaccine availability date (8 months),
f) fraction that gets vaccinated (1.0 ie 100%),
g) vaccine efficacy (0.5 ie 50%), and
h) the case mortality fraction (0.01 ie 1%).

The case mortality fraction for the US is uncertain in the absence of reliable mass screening data on prevalence; we find that changing it results in a proportionate change in total deaths, without affecting policy comparisons.

The six parameters (b to g) are set to our best guesses; you are encouraged to try values you think reasonable or worth exploring.

The total population (331m in our illustration), number of deaths prior to Reopen (100000), and the 15-day death count at Reopen (15000) are set at well-established values for the US and also for all states in the US (not perhaps R₀), and can be altered for other nations (or for any reason).

The model is simple (perhaps simplistic) and may fail to fit actual data for a variety of reasons:

1. Most importantly, we assume a single R₀, and a single infected fraction, for an entire region. There is obvious heterogeneity in these two and other parameters, addressed, for instance, in the work of Prof. Shaman's group at Columbia:
http://columbia.edu/~jls106/yamana_etal_reopening_projections.pdf

2. Target R₀ remains fixed for a given policy. In practice, if the death rate goes up significantly under "Trump" (our model predicts a rise from 1000 to 7000 a day in under 5 months), GOP leaders may be forced to move in the Cuomo direction. Our model doesn't account for such developments.

3. There may be factors other than the above two that are not part of the standard virus propagation model, due to which the graphs generated by the model don't correspond to what actually transpires.