 Imagine an influenza epidemic spreads through a population rapidly at a rate that depends on two factors. So when it comes to infectious disease, whether it's the common cold, the common flu, the coronavirus, whatever, when it comes to the spread of disease, there's sort of two factors to determine how quickly this disease will spread. The more people that have the flu, the more rapidly it spreads. So the more people that are infected, the more opportunity there is to spread from those who are infected. But also, the more uninfected people there are, the more rapidly it spreads. So if you have a lot of people who are infected and a lot of people who are not infected, then there is great opportunity for the sick people to spread the disease to those who are not sick. If you have a lot of people who are uninfected but like hardly anyone has the disease, then there's not an opportunity to spread because the vector transmission, there's only a few of those. But on the other hand, if you have like a zombie apocalypse where everyone except for Will Smith is a zombie, right, then there's not a lot of opportunity for the zombie disease to spread anymore because there's only one guy hiding out that can get infected, right? So for massive spread, we need to have a lot of sick people and a lot of healthy people. It's sort of like this combination. And so because of these two factors playing together, the logistic model actually is a very good model to measure growth right here because the logistic model has two horizontal asymptotes built into its formula. And so the graph would look something, something like you see right here where at first it looks like it grows really slowly because you although have tons of healthy people, you have very few sick people. So the opportunity to spread is very small. Then sort of in the middle, you're going to see this like explosion of spread, where there's enough sick people and healthy people that start spreading around. But then eventually the opportunity to spread might die off over time. That is to say there's no more uninfected people. Maybe that is you hit some type of like herd immunity. So the opportunity to spread kind of is done with. And so logistic model is a really, really good way of measuring the spread of a communicable disease here. So let's remind ourselves the formula, right? So the amount, the amount here, let's say a people that currently have the disease is going to look like C over 1 plus, whoops, 1 plus a times e to the negative KT here, where C is the carrying capacity. It's the maximum population. You can't have more people sick than the people who are in the population. And then K is the rate in which it spreads. So that'll depend on the disease, right? Some diseases might be more infectious than others. And so that depends disease to disease to disease. So with this flu strand, we'll see some data to talk about how quickly it spreads in just a second. All right. And then this number A here, this is going to look like the carrying capacity minus the initial population. We'll call that A sub zero divided by A sub zero. So this is our relative elbow room. So dependent on how much room do we have to spread at the very beginning of this exercise? Okay. So let's, let's look at some specific numbers. So for this, for this, this flu strain, right? Let's say that at time equals zero at the start of this thing, there's one person infected. So A naught equals one. And the initial population of just like a small town is 1000. So we might actually think this is like a school. So, you know, we could change this to be like one person has the Corona virus at the local elementary school of 1000 kids or something like that. That's kind of big for an elementary school, maybe like a high school, whatever. So you have one person infected at the beginning of 1000 students total. So let's say that's how things start off with this logistic growth. And so for a specific, for this specific virus, let's say that it spreads at a rate of 60%. So 0.6030. Let's say that it's, that's the growth. And I'm not saying this is for the Corona virus, this is just for a fictitious strain of the flu. So if these were the parameters for our growth, what would the formula be? Well, we'd see that A equals the carrying capacity of 1000. You can't have more than 1000 sick kids. I should say infected. And so I'm not, I'm not necessarily worried about like those who are currently infected, just those who were infected. So you have 1000 as the total plus A. Well, A here, you're going to take, you're going to take 1000 minus one, which is going to give you 999 over one. So this gives you 999 e to the negative 0.6030 t. So this right here is going to be the function that we use to model the spread of this flu for our community here. And again, we can see there's a school or small town, whatever. We just have some community where these things will kind of spread amongst each other. So we might ask the question. All right. So we have one infected. So we have case zero right here. So how many, how many say infections after 10 days? How long does it take? After 10 days, how many people will be infected? I didn't write down the number. Try that again. After 10 days, there you go. That's a complete sentence now. And so what we need to do is we need to basically compute what is A of 10, right? So what's A of 10? This, this growth rate I should mention was per day attached to the rate always needs to be a time stamp of some kind. So it was growing logistically at 60% per day. So A of 10, we're going to take 1000, which is carrying capacity over one plus 999 the initial elbow room times e to the negative 0.6030 times 10, right? Which times it's something by 10 moves the decimal place by one. So that's pretty easy to do without our calculator. Negative 0. I guess we moved it by one. So we're going to get 6.03 right there. Now after that moment, there's really not much more we can do without using a calculator. So feel free to consult your calculator at that moment, plug the thing in there. And you're going to end up with approximately, approximately 293.85. So, you know, this has got to be a person, right? These were counting people here. So we're right up to the nearest person 294. This is just a model after all. It's not going to be perfect, but we would anticipate that after 10 days, there'll be 294 cases of this flu throughout the school. Right? There's going to be 294 people that were infected by that. So 10 days is not necessarily a long time, right? If we think about the school, they only meet Monday through Friday, right? That's basically in two weeks. We're going to have almost 300 people infected by this strain on the flu virus. Okay. We could ask then, at the rate in which this thing spreads, you know, predict, predict when, you know, we'll reach, predict when, let's say, predict when 90% are infected. Okay. So what we're trying to figure out here is when is 90% of the population going to be infected? Well, since there are 1000 people total, 90% of that is going to be 900. So we have to solve the equation. Solve the equation 900 equals 1000 over 1 plus 999 e to the, I'm just going to say negative t right here. We have to solve for t. And this is the trickiest part of using a logistic function, but at the same time, the process is exactly the same. The arithmetic's never going to be changed. You could actually pay attention to what we're going to do right here. So think of this number on the left as a fraction. We're going to cross multiply and we end up with 900 times 1 plus 999 e to the negative kt is equal to 1000. We divide both sides by 900 and we end up with 1 plus 999 e to the negative kt equals 100. If we subtract 900 from both sides, excuse me, we divided by 900. So it's be 1000 divided by 900. So we end up with 10 ninths right there. Now we're going to subtract, we're just going to subtract one from both sides. So you can take away 9 ninths away from it. And you end up with 999, so many 9s here, e to the negative kt equals 1 ninth, like so. So we're going to divide by 999, 999, 999. Okay, and now we're not multiplying, we're dividing by 999. So that right there is going to give us e to the negative kt is equal to 1 over 9 times 999. For which that's going to be 8991 in the denominator here. Take the natural log of both sides to get rid of the base e. Take the base e, natural log right there, base e. So we end up with negative kt is equal to negative the natural log of 8991. For which then to finish, we need to divide both sides by negative k. And we see that the answer is going to be t equals negative the natural log of 8991 over negative k, right? Which then makes it a double negative. So you get the natural log of 8991 over k, which remember was 0.6030. In which case then we want to estimate this thing and then put that in your calculator. You're going to get about 15.0978. So in about 15 days, right, 15 days, which for our hypothetical school right here, that's three business weeks, right? So about three weeks, you expect 90% from one case to 90% of everyone has the disease after that. So remember that because we're dealing with a virus, we cannot predict with certainty the number of people infected, right? This is just a model. The model only approximates the number of people infected and will not give us the exact or actual values. These are just estimates. But when someone studies infectious disease, like for example, the coronavirus COVID-19, it spread in the year 2020. Admittedly, the models they use are much more sophisticated when we're doing this example. But this can give you an idea of how one can make predictions about case counts and deaths and things like this. That we can use the parameters, we use the statistics and function models to make predictions about how these things are going to go. And sometimes the predictions are very bad. And that's because it's based upon assumptions, right? Our assumption here was that the disease spreads fast when there's a lot of people infected and a lot of people unaffected. That's sort of an assumption. There could be difference assumptions about the disease, which would affect things. And like I said, real-life problems get much more complicated and the assumptions become much more critical. And so bad assumptions can lead to bad models. But the best mathematical models out there, we can see that they have the good data. They have good assumptions and therefore the predictions turned out to be very accurate in the end.