In the present coronavirus pandemic the new cases are somewhat larger or somewhat smaller than 10% of the total cases each day. Assuming that the daily increase is about 10%, we get about 7 days as the doubling time, i.e., 1.17=1.9487 is about 2. In 28 days the number of cases grows by the multiplier 24=16. The average time of an ill person either to die or to recover completely seems to be about 28 days. We can assume that the deaths or recoveries are approximately normally distributed around the mean of 28 days.
Assuming that N days have passed since the start of the epidemic in a country, the total number of infected is A=1.1N. If N sufficiently much larger than 28, say 2*28, the number of people who either died or recovered is about B=1.1N-28=A/16. We get the proportion of dead from dead+recovered from German figures for a selected day: 230 dead, 3960 recovered. Thus, p=230/(3960+230)=0.0549 is the proportion of dead. The death rate as long as the number of infected grows geometrically each day is C=p/16=0.00343. As these figures are very approximate, we round the number to 0.004. This means that 0.4% of infected die in the geometrically growing stage, provided that the health care is as good as in Germany. If the health system becomes congested, as happened in Italy or Spain, the number of deaths increases to a higher value, but part of this high value is caused by testing only more seriously ill people. From the percentage 0.34% we can estimate that in Germany there should be 230/0.00343=67,000 infected people. The number of tested cases in Germany in that day was 40,500. It shows that not all infected cases were found. Indeed, here is the problem: how can we know how many cases there really were as mild cases are not all recorded?
South Korea tested still more extensively and has managed to get the epidemic in control. The epidemic in South Korea started February 15. 2020 and did not continue as a geometric series for about 2*28 days. Our simple model only describes the exponential growth phase. Therefore the result is not likely to give an exactly correct number, but it can be tried. The geometric growth continued to about March 5 2020. In that day there were 6284 infected and 42 dead. As 42/6284=0.0067 and not 0.004 our model is not perfect, but reasonably good. It does not seem that there are very many mind cases that go unnoticed. South Korea population is 51 million and the number of infected may stay under 10,000, that is, about 1/5000 infected. Obviously, European countries cannot reach a so good result. It is still possible that the epidemic can be stopped in European countries, but if not, then the virus will infect about half of the population.
Finland currently has 5 deaths and the number of confirmed cases is 958. Finland does not test all suspected cases, thus the actual number of infected is larger, but how much larger? Finnish health authorities have guessed that the real number could be 20-30 times larger. We can use our model and give an estimate of 5/0.004=1250 infected cases. Thus, it is unlikely to be as high as 20-30 times 958, unless the number of deaths starts to grow fast in the coming days. My argument that it is not 20-30 times as high as observed is that Germany and especially South Korea have tested many people and still get about 0.004. However, in Italy the real number of infected may well be 20-30 times the known number of infected. In one day the number of dead in Italy is about 10,000 and the number of infected about 100,000. The explanation probably is that the real number of infected is 10,000/0.004=2.5 million. This would make sense since in a logarithmic scale both the number of infected and the number of dead in Italy seem to converge smoothly to a finite limit. Such smooth convergence usually means saturation. Limitations of infection rate do not produce smooth convergence, they only change the linear growth rate in a logarithmic scale. Thus, infections in Italy are slowly becoming saturated. As 16 million are in quarantine in Italy, it means that the real number of infected is already approaching 16 million, like it can be 2.5 million but it cannot be 100,000.
Donald Trump has said somewhere that the death rate of Covid-19 is under 1%. As I derive to 0.4% to the phase of geometric growth, I must agree with him: so it looks like if the number of deaths is divided with the number of infected. Though he does not look all that intelligent, I remember that he tweeted being very intelligent with the IQ of 150. Well, maybe he is. You never know. Some Israeli Nobel prize winner, who then must by definition be very smart (or not?) estimated that in Israel will die not more than 10 people. Currently 8 Israelis have died. My estimate just for the geometric phase is (1/2)*8.7*0.004=17,000 (unless the epidemic is stopped), yet I agree with the Nobelist that it will not be hundreds of thousands (that is, I think the epidemic will be stopped in some way, if half of the population gets infected and infected include old and ill, then the death ratio in the end will be p as every infected person either recovers or dies.)
After the geometric growth phase the epidemic must stop growing and eventually stop. The final death rate will be p, that is, 16 times the death rate calculated in the geometrically growing phase. Thus, it is about 16*0.4%=6.4%. German figure actually gives p=5.49%, so it is around that range, or 4.5% as in China (or 1.38% in China according to the latest estimates, for middle aged people it is clearly higher), something of that scale, unless hospitals get too full, doctors and nurses die and there are not enough respirators. Under these latter conditions the death rate can be much higher.
If we assume that a person infects in average for 10 days, after this average time he notices being ill and stays at home or in the hospital and does not infect others any more, then we can deduce that if the number of infected cases increases by less than 7.177% each day, the epidemic dies out. That is, 1.07177 to power 10 is 2, one of these two is the ill person, who is not infecting any more, so there remains 1 to infect. Before the ten days there was one to infect, after 10 days there is one to infect. If the daily increase is less than 7.177%, then the number of people who can infect decreases every day and the epidemic stops. This may be the case for Finland. It is the case for Italy and South Korea.
Thus, consider the case that the daily number of infected grows by 6%. Then 1.06 to power 12 is about 2. The doubling time is 12 days, but we are interested in the increase in 10 days. It is 1.06 to power 10, that is 1.7908. The infected person is at home or hospital and does not infect others. There remains 0.7908 persons who infect for the average of 10 days. After 10 days we have 1 person at home, another 0.7908 person at home and 0.625 persons who do not know they are infected. The series of infected people is 1+0.7908+… This series of powers of 0.7908 sums to 1/(1-0.7908)=4.78. That means that if the daily increase is reduced to 6%, the epidemic reaches the value 4.78 times the original number without the population becoming saturated (we do not need to take into account the reduced number of people who are not yet infected). The series does grow exponentially (and not logistically, as a saturated curve), but the growth exponent is less than 1 and the series sums. Based on this, I hope that in Finland the current 1,300 infected grows towards 6214 infected and stops before reaching this value. If so, the number of dead should be about 1/16 of this value, some 388. But so far there are more than 6% infected each day in Finland. It seems to be more like 8-11%. There is need for still stronger restrictions. The figure should be under 7% for the epidemic to die out.