There’s a plot that has been stuck in my brain for about 10 days:
The plot is based on an infection model that was published by a team of researchers in the Seattle area who projected future Covid-19 cumulative infections for King and Snohomish (I’ll use the term Sno-King) over a four week interval commencing on 11 March. (Klein et. al., 2020)  I wanted to understand how our communities are doing in the struggle to slow the rate of new infections.
We do not yet know which scenario best represents current conditions in King and Snohomish counties, but previous experience in the region with weather-related social distancing and in other countries suggests to us that current efforts will likely land between baseline and 25% reduction scenarios.(Klein et. al., 2020) on 11 March 2020
I believe there’s enough data (as of 25 March 2020) to claim that the Sno-King population has achieved the 25% reduction level due to social distancing. That’s certainly an improvement over two weeks but there’s still more work to be done. I’ll present some results from a very simple model I made and then discuss how I made this model, what its limitations are and suggest some improvements.
In Figure 1 below, the blue curve is a projection using an exponential model of the baseline for new infections for 28 days ending on 7 April 2020. The red curve is a projection of a 25% reduction in infection rate. The intent of these curves was to reproduce the model curves of Klein et. al. Finally, the yellow curves plots the number of cases in Sno-King. The underlying data, model and plots are available on a Google Sheet I created.
Now, I realize that it’s difficult to see much from this curve. I looked at the change in new daily infections, I believe that’s a better metric to visualize how we are improving. Figure 2 provides this visualization with the yellow data points show the daily increase in new cases compared to the baseline (blue) and 25% reduction (red) lines. I realize that there’s not very much data yet but we are seeing new data being published daily (and I will update this analysis sheet daily). Still, this made me realize that we are likely achieving 25% reduction level due to social distancing in Sno-King!
Building the Exponential Model
Since I’m an engineer, the first thing I did was build my own model and then start feeding in data to measure how we are doing.Of course, the devil is in the details, and I’m not an epidemiologist. What I did was reproduce the results of Klein et. al.. 2020 using the facts they provided: a doubling time for the epidemic of 6.2 days, one large transmission cluster, four week duration, and less than 1% (25,000 of the 3 million infected at the end of the four weeks.
Let me explain the simple, deterministic model I used. To follow along, you may find it helpful to look at the underlying Google Sheet I developed. I am including underlying cell references in parentheses. I used an exponential increasing model that starts with 267 infections (cell L2) at day zero. For a time series of 28 days (cells F2:F30), I calculated the total number of infections using a baseline doubling rate parameter of 6.2 days (cell L3). For the entire 28 days, the baseline I calculated was 24,425 cumulative cases (cell G30). For comparison, the multiple simulations performed by Klein et. al. estimated 25,000 cumulative cases. Thus, my baseline model estimate is about 2% lower than Klein et. al.
For the the social distancing intervention representing a 25% reduction, I increased the baseline doubling rate parameter by a factor of 1.25 to 7.8 days (cell L4) For the 25% reduction case, I calculated was 9,899 cumulative cases (cell H30). For comparison, the multiple simulations performed by Klein et. al. estimated 9,700 cumulative cases. Thus, my model estimate is about 2% higher than Klein et. al. in the the 25% reduction case.
Next, I used the graphic “COVID-19 in Washington State” published daily by the Seattle Times to determine the cumulative number of cases in King (C3:C16) and Snohomish (D3:D16) county for each day and then computed the Sno-King total cases (E3:E16). I have included a tab labeled “DataSource” that lists the URL for each day’s data. Some days, the actual case data is suspect, for example, there was disagreement between King County and Washington State regarding cases on 17 March 2020. The cumulative infections are plotted for the baseline, 25% reduction, and actual cases in the chart labeled “Cumul25March”
I computed the first order difference in infections, the change in number of daily infections, for each condition: baseline (cells J3:J16), 25% reduction (cells K3:K16) and actual cases (cells I3:I16). The change in number of daily infections are plotted for the baseline, 25% reduction, and actual cases in the chart labeled “1stDiff25March”
Limitations and Utility of this Model
This is a very simple model of a complex epidemic. I don’t think it will have much validity beyond 7 April 2020. At that point, a more rigorous model that models the underlying epidemic using ordinary differential equations is in order. Also, the underlying parameters will be better understood. Finally, once more than 1% of the Sno-King population is infected, the simple exponential model will become less accurate.
I think this model will be a useful way for people to visualize the impact of social distancing efforts in the short-term. There’s enough data here to say that social distancing is making an impact. Also, in a small way, this simple model and analysis confirms that our current effort will result in 25% reduction due to social distancing. Maybe the next two weeks will show additional improvement?
Thanks to the Seattle Times and Klein et. al. making me think and look at the data. Also, hats are off to all the essential workers in government, grocery stores, pharmacies and the news media for keeping us safe, healthy and informed.
I welcome suggestions for improvement, corrections and any words of wisdom from anyone reading this. Stay safe.