Posts Tagged Gaussian
At the height of the novel Coronavirus infection the transmission rate is about one to three. Meaning that one person infects three. When that rate drops from three to less than one then the virus is losing ground and will fade away. When no new cases happen then the virus has no way to reproduce and is gone (yea!).
An individual who contracts the virus (sorry) the illness lasts about two weeks. A small percent (less than 2%) may die. The good news is that surviving the virus means the body fought it off leaving lots of antibodies floating around in the blood to prevent reinfection with the same virus (at least for several years).
The initial fear was that the virus might mutate (change it’s structure) quickly so that a person’s antibodies would not be effective for very long. Fortunately for humans, the little devil does not seem to mutate quickly (needs more observation to be sure). Surviving persons cut the transmission rate since they don’t catch the virus.
That brings us back to the original question: exactly when will the pandemic end? We can only guess because the answer is up to the virus and how well people avoid each other. Effective drugs against this virus or immunizations are simply not available now and almost certainly will not be available for at least a year (perhaps in 2021).
Did you say guess? Yes. So let’s make an educated guess. China was a huge experiment. In that country the virus went away (with great effort) in about sixty days. That’s roughly what to expect in the United States.
The Chinese experience revealed the course of the virus followed a ubiquitous mathematical progression called a Gaussian curve, otherwise known as the “bell-shaped curve”. The number of new cases goes up, hits a peak and then declines. The mathematical equation for the Gaussian curve is a little complicated:
y = a * exp( 0 – (x – b)^2 / (2 * c^2))
where y it the number of cases on the vertical axis
x is the day on the horizontal axis (1,2,3,4,5…)
a is the height of the peak
b is the day where the peak happens
c is the width of the bell.
Once some of the actual data is known (e.g. the numbers of new cases) a curve fitting program can figure out a,b and c. Here is an example for the State of Colorado in the United States at the time this post was written: (see updated graphs at end of post)
In the graph the blue dots are the number of new cases each day and the red line is the Gaussian curve fitted to the available data. The best “guess” is that new cases will stop at the end of April where the red curve hits zero. Of course, the medical havoc from the virus in those final few people infected would last for another two weeks. The peak of new cases happens at about April 5th. Unfortunately, the peak of deaths occurs about a week after the peak of new cases.
The end of new cases for the United States overall is more complicated than China since the virus started in the various states at different times. The sum of all the bell-shaped curves from each State may create a US curve that shows several small delayed peaks or just a skewed curve with a longer tail on the right side — time will tell.
Once the virus has subsided in one area it is possible a flare-up could happen due to travel of infected persons into an area that had many non-infected people. If that happens, the State health department should quickly quarantine the area — another mini bell-shaped curve will happen in that area.
Whether the virus will come back later this year or next year or never is unknown. If it does, many people will be immune and laboratories may have a greater ability to test for it. Hopefully pharmaceutical companies will manufacture an immunization. Is this the last pandemic? NO. We must do a better job of preparation and acting on the warning signs. Will humans remember this lesson? (no answer).
Updated graph of cases per day in Colorado, USA as of 4-14-2020
Update for Colorado, USA as of 4-21-20
Observation: as more testing is done more asymptomatic cases are being found. This has the effect of hiding stay-at-home measures with an artifactual bump in numbers if new cases. The aberration should be less with time and with lower number of cases. At this point many analysts believe the rate of hospitalizations may be the best indicator of disease activity. Steve Goodman, Stanford Professor of Epidemiology & Medicine, gave an interview to KPIX, a local TV station 3/25/2020, supporting the importance of hospitalization data:
A graph of current COVID-19 cases is below and now includes hospitalizations (and a 4-point smoothing curve).
Update for Colorado, USA as of 5-1-2020
Today the state will begin to allow some workers to return to work and stores to offer limited (curbside) service. The new case and new hospitalization data have reached a variable level that is much lower than most models (without stay-at-home orders) predicted. The stay-at-home strategy appears to have reached the goal of “flattening” the curve. Unfortunately, restrictions of movement are being lessened while the virus is still at peak activity. An additional concern both nationally and in Colorado is the late reporting of cases which has muddied the waters.
Some modeling experts predict a resurgence is in the offing. Below is the latest graph without the Gaussian prediction for new cases and with the use of a 7-day smoothing (necessary due to the erratic reporting). The “dump” of “unreported” deaths confuses the overall picture — such deaths probably happened over several weeks, not on one day.
The success of stopping the virus in New Zealand, Australia, Vietnam and China was due to forceful stay-at-home orders. In Colorado and the US as a whole, such force of law and will is lacking and stems from a huge concern about economics. The sentiment among retired people is to stay-at-home to minimize risk; consequently, many new deaths will likely be in members of the workforce.
At the start of this post the “assumption” was the US would try to stop the virus. That made predicting an “end” a reasonable endeavor — now that is no longer the case and some time will need to pass for a new pattern to emerge. Only when hospitalizations are near zero will we feel Colorado is close to the end of the pandemic.