Differential equations

Added: 2026-05-28

[00:00:00]Hello, my group consists of Johan Sebastian Herald, David Fernando Gonzalez and Michael Steven.

[00:00:11]This is the differential equations thought by Diego at the University of America.

[00:00:19]What do differential equations study?

[00:00:21]Differential equations study problems that represent the real world in motion.

[00:00:25]For example, they can involve problems related to velocity and position, population issues, temperature issues and many others problems of the real life. What do it mean to model change?

[00:00:41]When we refer to modeling change, we want to make reference to tell a situation in which a situation is changing with the passage of time. We are going to model our structure in a mathematical way. It is a way in which we can predict or get closer to a real value of that situation in t time. What is a dynamic system? A dynamic system describes how the state of a physical biological or economic systems change.

[00:01:09]This is a shift from rules or equations that govern how it state variable change at each instant. They alter quitative properties. If it is stable or tends to get out of control, if it repeats patterns periodically, if small change leads to completely different behaviors but with no asbes or even if it enters a cotic behavior where the systems is unpredictable in the long term despite following defined rules.

[00:01:42]What is a state variable? A state variable is a quantity that depending on the context in which it is being applied is represent with certain symbols to correctly indicate what it is measuring.

[00:01:55]What makes these quantities special is that they contain all the necessary information to know how the systems is going to behave in the future.

[00:02:05]These quantities can vary with respect to time what is the temporal evolution.

[00:02:11]Continuing with the previous idea to complement temporal evolution in a dynamic systems means the change that state variables experience with the passage of the time. This is because the variables do not always remain constant nor do they follow the same trajectory but rather they respond to the inputs of the systems and to its own internal dynamics relation with the OD dynamic systems and ordinary differential equations are closely related and that is because ODEs are precisely the mathematical tool that is used to describe how the state variables of a dynamic system evolved with the passage of time. That is to say, when we want to express mathematically, we do it through ordinary differential equations. So, one would not make much sense without the other. What is an ordinary differential equation? An ordinary differential equation is an equation that involves a single unknown function of a single variable and a number of its derivatives. So that the relationship contains the independent variable, the function and its derivatives. The order of an OD is the order of the highest ranking derivative that appears in the question. In ordinary differential equations, we find two types of variables, the dependent and independent ones. A name indicates the dependent variables are those that depend on the other to take a certain course.

[00:03:47]Meanwhile, the independent variables are those that do not depend on another to take a value. The easiest way in which this can be seen apply is in pro that include time since it is the most common example. Time acts as an independent variable because putting it in logical terms, we are never going to be able to control time. So, it takes it values on it own. The independent variable changes according to the problem. On the other hand, y is the independent variable since its value depends on the independent variable. A basic example of an ordinary differential equations is dy by dx equal y. It is ordinary d to the fact that it relates a single function with a single independent variable which in this case is x. The order of this OD is first order due to the fact that the highest order derivative that appears in the equations is the first derivative.

[00:04:48]Now then the solution to this equations is y equ= to cer power to x where c is a constant that is going to depend on the initial conditions of the problem. This means that once we know those conditions we can determine exactly how the function behaves.

[00:05:10]All right, hypothetically speaking, you have engineered rapids. Nothing personal, just use science.

[00:05:16]One infected person and other a billion potential victims, you know, completely unaware.

[00:05:24]So the real question is, does your virus take over the world or does it just, you know, fizzle out quietly embarrassingly?

[00:05:36]Turns out that's not afterlook. It comes down to a single number and that number come from a differential equation.

[00:05:43]This is the math that decides if your virus wins.

[00:05:48]Okay, the real world is messy. People travel, hospital exist, some people are immune. Forget all of that. We're going to build a simpler world. A world with only three types of people. First, the susceptible. They are healthy. They have no idea what's coming. In our model, we'll call them S.

[00:06:06]Then the infected. That's your virus at work. They carry it. They spread it. We call them I. And finally, they removed.

[00:06:15]They either recover and can get sick again or they didn't make it. You know, either way, they are out of the game. We call them error. And well, in this world, people only move in one direction. You start susceptible. You might become infected. and eventually you end up removed uh not going back.

[00:06:33]This is simple and brutal and quite surprisingly accurate for our purposes.

[00:06:41]So we have our three groups, but here's the real question. How fast does the infection actually grow? Let's think about it. Don't memorize anything. Just think if there are more infected people or there are more cars walking around.

[00:06:53]Does the virus spread faster or slower?

[00:06:56]Faster, obviously. So the rate of new infections grows with the number of infected. Let's try that. But infected people need someone to infect. If everyone is already sick, there is nobody alive to catch it. So the spread also depends on how many susceptible people are still out there. Not infected, more susceptible. Multiply them. That's your transmission. Now infected people don't stay infected forever. They recover. They leave the group. Remember the faster they recover, the faster I shrink. So we subtract that. Gamma is just recover rate. High gamma people recover faster. Low gamma they stay they stay longer. And that's it. That's the whole story of the infected group writing in one line. The left side d over dt is just asking how fast is the infected group changing right now. The right side gives the answer. Two forces one pushing up and one pulling down. Your virus wins when the pushing beats the pulling.

[00:07:50]We have our equation. But here's the honest truth. This system has no clean solution. You can just integrate it and get a neat solution. The three variables are all tangled together. But we don't need a formula. We need to ask a different question. Instead of asking where does the system end up, let's ask at any given point, which way is it moving. Think of it like wind. You don't need to know where every air particle will be in an hour. You just need to know right now which way is it blowing.

[00:08:19]And my little dark scientist, that's exactly what a directional file does.

[00:08:24]Our world has two moving part S and I.

[00:08:27]Error just follows. So let's put S on one axis, I on the other. Every point on this plane is a possible state of our epidemic. And that every single point or equation tell us exactly which way the system is moving. We draw that as an arrow. Now you see this little Y though that is our starting point. One infected person, almost everyone susceptible. And now watch this still follows the arrows the file it has no choice.

[00:09:01]Okay we have already seen the threshold line that's exactly the point where the epidemic tips where infection stop winning and start losing. Ask a different question. What happened at the very beginning when the virus is brand new when almost nobody is infected yet but at that moment s is basically one then interpolution is susceptible let's modify our original equation into this way which means everything depends on the sign of the thing beta minus gamma if beta is bigger than gamma the infection grows if gamma is bigger it dies instantly only one so let's write that more cleanly take beta and divide it by gamma that the number that and with that we can make a little change in our equation. R not greater than one your virus wins the opening move. R Z less than one it never gets started. So R is simple. How many people does one infected person infect before they recover? One person, one number. That's the whole game. Watch what happens when we move it.

[00:10:02]Some equations, same structure, one number change and the entire personality of the system flipped. That's not a coincidence. Air not is the skeleton key. Every epidemic that has ever existed has an air not.

[00:10:15]Different ceases, different worlds in question about air zero. The epidemic picks exactly when their mind susceptible population drop below the threshold.

[00:10:26]R0 isn't just a formula. It's a moment where mathematics earned its keep.

[00:10:32]There is no close for a solution to the steer equation. You cannot write a clean formula. Not tight the expression gives you I of T. The system is coupled, nonlinear. It refused to be solved the polite way. So we do something more honest. We let a machine the path step by step. This a numerical integration, not an approximation, not a The same logic the universe uses. Given where you are now and the direction of change, find where you'll be next.

[00:11:03]And with the little tree, we've got the susceptible population. It falls slowly at first, then fast, then it low again.

[00:11:10]It never reaches zero. Red, the infected, it rises, it peaks, it falls.

[00:11:14]That shape, that arc, it is the entire story of an epidemic in one single curve. Green recovered. It climbs as the other resolve quietly inevitably recurr.

[00:11:35]But look what happens just before the peak. The susceptible population crosses a threshold. The same line we found in the directional file and eventually here I reaches zero. Not because the vir ran out of pins because the remaining susceptible fell below the critical threshold the fire ran out of field it could reach. Now watch what zero does to all of this. Save model works. We've seen but it made promises it can keep.

[00:12:00]Let's look at what it assumed quitly.

[00:12:02]Ris assumption. The population is close.

[00:12:05]Nobody is born during the epidemic.

[00:12:06]Nobody dies from anything else. Second assumption. Recovery is permanent. One you live high and enter earth. You stay.

[00:12:13]No re infection. Third but is constant.

[00:12:16]The transmission rate never changes. No season, no lockdowns, no behavior change. People spread the virus at the same rate on day one as on the day 2004.

[00:12:24]There is no space, no network. Everybody is equally likely to infect everyone else. A farmer in Montana and a subet community in Tokyo living the same imaginary soap. These are minor technical complaints. Each one is a door. Will the first one open at burst and deaths and s don't refill the epidemic becomes endemic. It never fully leaves the second. Let air flow back to S and you get reinfection models. Wave after wave make beta a function of time and suddenly interventions work. A lockdown is just better dropping. A super spur event is better speaking at an exposed class. People infected would not get infections and the S model becomes there. The incubation period enters the model. The curve shifts in time. Each crack is not a failure. It's a coordinate. It tells you exactly where the next equation lives. The same language, more words. Here was the first weapon. Just these are the others. We started with a question. One infected person, 8 billion potential victims.

[00:13:25]Does your virus win or silently die? Now you know the answer. It's a look. It is a number. One dimension is number that decides everything. Where the arrows in the file point up or down, where the rises or collapses, where the epidemic earns its name. Not equals beta over kama.

[00:13:46]But here's what I didn't tell you at the start. Not is not fixed. Beta can be engineered. How easily the virus binds, how long it survives on surfaces, how silently it spreads before the symptoms appear. Push better upgra climb. And we've only been working with the simplest mode. The real models are darker.

[00:14:10]They have incubation periods. Host who spread before they know they are sick.

[00:14:15]They have network structure. Super spreads would touch a thousand people while others touch her. They've got waiting immunity. A record population that is slowly forgets. Each fet is a lever. Each lever moves zero.

[00:14:29]You engineered a virus. You choose beta.

[00:14:32]You choose gamma. You choose the incubation window. The transmission network. The immune escape. So let me ask the question again. The one from the beginning. Does your virus win?