Logistic curve. When will the epidemic end?

A mathematical model that describes processes like the development of an epidemic is called the Verhulst equation, or the logistic equation . The equation describes an increase in a certain population in the presence of limiting its maximum. The population (for example, the number of cases) at the beginning of the epidemic increases exponentially, but the number of cases is limited by the population and growth is gradually slowing down. The curve illustrating this process is called logistic.


Unfortunately, the statistics on morbidity and mortality are heterogeneous and not always reliable. This is due to the different development of medicine by country and the lack of common standards in who is sick and, even, the cause of death. For example, in Italy, if an infected person dies, then statistics on death from a coronavirus are determined. And in Germany (and in Russia) they often diagnose death from an exacerbated chronic disease. Therefore, in Italy, mortality (4032/47021) = 8.6%, and in Germany only (45/18323) = 0.25% of the number of cases. In some countries, mass testing of residents is carried out and even asymptomatic virus carriers are included in the statistics; in other countries, only severe patients are diagnosed, and even then not always.

The official country statistics charts can be viewed at https://observablehq.com/@elaval/coronavirus-worldwide-evolution
http://shinyapps.org/apps/corona/

Diamond Princess provides good statistics.
3,500 people on board (most retired).
712 infected, of which 362 are asymptomatic.
7 deaths.
Why 80% were not infected is unknown. Some people may not be infected with this infection at all, but it’s not certain ...

From a speech by the Director-General of WHO dated 02.17.2020 . 
"More than 80% of patients the disease is mild and ends in complete recovery.
Approximately 14% of cases the disease is accompanied by severe and, in particular, pneumonia and dyspnea.
And, finally, about 5% of patients develop life-threatening disease, with symptoms such as respiratory failure, septic shock and multiple organ failure.
In 2% of cases of infection with the virus leads to death, and this risk increases proportionally with age the patient. “

It is possible that the WHO Director-General did not take into account asymptomatic infected people at all, so we divide everything in half.

So, 100% infected:
90% asymptomatic, or in a mild form;
10% - hospitalization is required, of which:
2.5% are severe, resuscitation is required (probably, mechanical ventilation?);
1% - mortality.

The exact solution to the logistic equation:
N = M * EXP (r * t) / (1 + EXP (r * t)); Where
N is the size of the population at time t;
M is the maximum population size;
r - population growth rate, daily increase (percent / 100);
t - current time in days, counted from the middle of the logistic curve.
To determine the time:
t = (1 / r) * LN (N / (MN));

The ease of use of the logistic equation lies in the fact that to solve it you need only 2 parameters - the maximum size of the population and its growth rate.
For example, let's calculate mortality in Italy. The population of Italy is 60 million people. We assume that half the population will be infected with the virus, 30 million. Alas, 1%, 300 thousand of them may die. Currently, 4032 people have died, the growth rate of the process is 15% per day. The number of cases in Italy at the beginning of the epidemic increased by 25% per day, now, with quarantine, 13%. Mortality lags behind the number of cases, they die 11 days after infection, so we believe that mortality will fall to 13%.
r = 0.13; N = 4032; M = 300,000. 
The answer is scary. t = 33. In 33 days, 150 thousand people can die from the virus in Italy. Up to 9,700 people per day will die at this time.
If there were no quarantine, the distribution rate would remain 25%, the maximum would be reached on day 17, and the daily mortality at the maximum would be 18650 people. Quarantine shifts the process and reduces the maximum. Will Italian medicine cope?

But instead of closing the industry, instead of a complete quarantine, there is a more elegant solution.
According to the Italian Higher Institute of Health, nearly 90% of deaths in Italy are over 70 years old. 
We isolate this group of people; there are plenty of tourist hotels in Italy. Full and strict quarantine. Mortality drops 10 times. Serious cases of the disease too. Even if quarantine measures are canceled for the rest, 10 times fewer people will die. 
When r = 0.2; M = 30000, the maximum will be reached after 28 days. The maximum number of new deaths per day is 1495.

If you also isolate younger people with dangerous diseases, the number of deaths can be further reduced. The most common chronic pathologies in the dead:

ischemic cardiopathy - 37.3%
atrial fibrillation - 26.5% had a
stroke - 8.2%
arterial hypertension - 76.5%
diabetes mellitus - 37.3%
dementia - 4.5%
chronic obstructive pulmonary disease - 9.7%
cancer, active over the past 5 years - 19.4%
chronic hepatopathy - 7%
chronic renal failure - 17.5%

The number of patients with coronavirus in Russia is increasing by more than 25% daily. This means 10 times in 10 days, 1000 times per month. At this rate, in a month we will have hundreds of thousands of patients. The most stringent quarantine measures will be able to reduce the rate of daily incidence growth by half, to 13% (for example, Italy). Quarantine measures bring down the economy and do not reach the goal - to reduce the burden on medicine to an acceptable level.
Meanwhile, there is an effective solution to the problem. 
90% of severe patients and fatalities are observed in people older than 70 years. Strict quarantine for the elderly and chronic patients. To do this, you need to use boarding houses and rest houses. Establish a strict regime; personnel should not be local and also should not leave quarantine. Quarantines must be guarded by the Russian Guard. The first two weeks of quarantine must be individual. 
For other residents of Russia, quarantine can be lifted, or weakened.

The forecast table for the timing and maximum number of severe patients.

	Days to the maximum epidemic	The most severe cases for this day
The status quo	39	108800
Quarantine	65	65500
Isolation of the elderly without general quarantine	37	8700

No time. If everything is left as it is, by the beginning of May we will have hundreds of thousands of only seriously ill patients requiring resuscitation. 100,000 new ones will arrive daily. General quarantine will stretch these terms by half and halve the influx of patients. 
Isolation of the risk group will reduce the number of severe patients and deaths by 10 times. 
I would like to convey this information to the authorities.

PS Sobyanin announced home quarantine for senior citizens. Italians attacked the same rake. Schools were closed and children were left in the care of the elderly. Those did not keep them at home, the children became carriers of the infection and rebranded grandfathers and grandmothers. Home quarantine is for single people only. Cottage in such a season is often an overwhelming quest for lonely sick people. And at home, we rebuild them ourselves. Only quarantine nursing homes!