CoViD-19 Michigan Spring '20

The corona virus SARS-CoV-2 has been spreading here seriously now since the begin of March. Without discussing how this pandemic affects the USA and what measures are being taken to reduce its impact, I just follow the daily postings of the total number of confirmed infections and the daily additions of new cases in Michigan. The data are displayed in a semilogarithmic plot where an exponential change with time appears as a straight line, both increasing or decreasing. From the slope of this line the infection rate can be calculated in terms of the time it would take to double the number of infections, the doubling time, if the infection mechanism doesn't change. For daily new cases the numbers can also become smaller than the previous day so that a "halfing time" can be calculated that describes how fast the pandemic subsides. 

The cumulative number of covid-19 cases, and the daily additions, are plotted in this graph. 

Confirmed cases of Covid-19 in Michigan after Monday, 4/9/20. (Last data point from Monday, 7/6/20.) Blue circles are for cumulative numbers with the vertical scale on the left; orange triangles represent the daily cases added (omitting values for the first seven days) for which the scale on the right applies. The four regimes in which the cumulative cases increase exponentially are identified with least-square-fitted lines. They are described by exponential growth rates of 0.444, 0.257, 0.0297, and .00876 per day.  The daily new cases increase initially also exponentially (day 10 to 25) with a rate of 0.157/d.  Between days 21 and 28, the new cases begin to decrease with an exponential decay rate of  –0.0326 per day; the least-squares fitted line is calculated from the new cases of days 23 to 65. Between days 63 and 70 a change in the spreading rate occurs described by a faster decline of –0.0500 per day. After day 98 the number of daily new cases increases again at an exponential rate of 0.0378 per day.                                         Data from: https://www.michigan.gov/coronavirus/

The table shows the data for the least-squares fits of the slopes. The doubling or halfing times are calculated from the slopes of the linear least square fits to the natural logarithms of the cases as a function of the time in days from ln(2)/slope; a negative number indicates the time to reach half the number of cases.  

The growth regime over which the cumulative number of cases increases exponentially with a constant growth rate reflects a particular mode of interaction of the people who get infected, and the way testing procedures reach the infected persons. The initial growth rate corresponds to a doubling time of cases of only 1.6 days. The transition between regimes 1 and 2 occurred rather abruptly around day 13, continuing with a doubling time of 2.7 days. Between regimes 2 and 3 the doubling time improves substantially; the transition appears to follow a quadratic function in the semilog plot as indicated by the fitting curve labeled "quadr.fit" in the graph, covering the range of ~17 ... ~33 days. During this period, the daily decay rates, over a 3–day running average, appeared to decrease linearly (correlation factor 0.975) giving rise to the curvature.  The regime 3 which then follows this transition increases somewhat linearly again (in the semi-log plot) but with substantially lower rate of 0.0297 per day, more than a factor of 8 lower and equivalent to a doubling time of 23 days. The fourth regime continues this levelling off for cumulative cases with a doubling time of 79 days and more beyond day 80. This is more clearly seen by looking at the table's "new cases" rows which show a transition between halfing times from (-)21 to (-)14 days. This improvement is indicative of a change in the interaction patterns of Michiganders. However, after the strict 'stay–at–home' rule was lifted with day 84 (6/1/20) a flattening of the decrease is observed, followed by an exponential increase in new cases with a doubling time of 18 days, see fit regime lsqft-d-3 row. The delay time between cause and effect is roughly two weeks. 

It is interesting to speculate what kind of change in the social interactions of Michigan's population gave rise to such a change-over characteristic, particularly in light of the observation that after day 30 the new rate appears to be somewhat constant again for about 3 weeks. Of course, the restrictions on assembling, working, and leaving the house, issued by Governor Whitmer around that time, initiated the change over. The effect appeared to set in after day 16 and took a little less than two weeks to get to a clear exponential decline in new cases. This is similar to the delay time after lifting some of the restrictions after day 84 so this may be characteristic of the spreading mechanism and the behavior patterns of people in their understanding of the measures implemented.