All right. Hello, one. So I've written down the table. You're given Andi about a two lines s. So the first thing to do is to take the difference between the number of cases in one period on bond that of the previous period. Eso, for instance, here, uh, 220 to minus 95 gives you 100 27 on that basically gives you the number of cases. Total number of cases between the two intervals. Eso again here 417 minus 2022 gives you two and 48 and so on. But now the question asked you the average number of cases in each interval on. But then, to do that, you have to divide by the number of days in each interval and the first interval has five days and all the other ones have seven days. So you then divide here 127 by five to obtain 25.4 on That basically tells you that on average and you have 25.4 new cases in the first five days and then the following seventies eso between the fifth day and the 12 day you have 35.4 Sorry. 35.4 cases on average a zoo divide two and 48 by seven. And then you continue for all the different values. Eso Exactly. It's the same thing. You divide this number. So, for instance, this number here during eight by seven, 2, 15, 44. so I've just written down values. Um, I stopped a bit before because it basically is. It's the same thing for later values, Onda. That gives you three answer for question A on now, regarding the next question. Um, so when did the PMO just have evidence to locate the rate of new cases began to slow? Well, either you then simply have to take the average rate on DSI. When that starts to decline on approximately, it's more or less here. Okay? Because so at T equals 19. So, unfortunately, we don't have a day by day data. Uh, we only have every weekly data set. Um, but we can say that approximately after 20 days theme, the rate of inflection started to slow down as the average number of cases per day source to decline. Um, Okay, Right. So why would an expansion exponential model be inappropriate. So the two ways of seeing this firstly well, as you can see, it slows down. Whereas an exponential model, uh, simply grows forever on grows out of increasingly fast rate eso. In a way, the exponential model works quite well. In the early phases of this, um would fit quite well the data in the first few days on board. That's generally the case. And that means that is the case for logistic models, logistic growth models. Um, but as you can see, the number of cases slowly starts to decline, and then at the end, it becomes quite stable, which is definitely not the case for an exponential growth model. On. Additionally, there is some form of absurdity which arises through most exponential models is it's because exponential models grow to infinity on bond. Um, it would imply that at some point P would take the value, Let's say 10 billion and there are no there. There aren't 10 billion people on this earth. So, um, there is some form of of impossibility through expansion holds. Yeah. Um, Okay. So, um, getting so then we're told that a logistic model fits the data quite well. Um, And now regarding the inflection point, we know that the inflection point is the point at which the rate of growth starts to decline on. But that is basically equivalent to the question be, that is, as you can see again, growth here starts to decline. AT T equals 19 eso. That is our inflection point. And when it comes to limiting value of P, um, you know that the limiting value off P, um, which is the carrying capacity also of this logistic growth model, eyes twice the value taking at the inflection point. Now again, we don't have a precise, precise data set. A, we can say is that it is approximately a T equals 19. So what you do is you double the value taken by P. AT T equals 19, which is 800. So the model would predict more or less a total number infection off 1600. Now again, if we have a date today data, we would maybe see that this increase carries on the YouTube 20 40th or 21st day, and we'd have a slightly higher number on when you look at the the limiting value it seems to converge towards 1750 something. Okay, so we're not far off. Um, okay. And so were given a precise, uh, precise logistic function which fits this data in some sense. Andi, you asked for limiting value of team. So let me write down P, which is a function of time. Okay, 50 is equal to 1600 divided by one plus 17.53. Sorry. I eat to the power of minus zero point 14. 0, 80. Okay, Now, if you already familiar with realistic group logistic girls, you know that this year is the carrying capacity. Andi, that, uh, dysfunction p converges towards this point or another way of seeing this is to calculate this limit explicitly on. Do you know that e to the power of minus something which goes to infinity converges to zero. So this everything here converse to zero, and you're again left to left with 7 1760 divided by one. So 1760 which again fits the data quite well. We know that after 87 days, we're at 1000 and 55 the number of cases really seems to die out at this point. Thank Andi. Yes, that's that's it.