Oxford Mathematician explains SIR Disease Model for COVID-19 (Coronavirus)

Hello math fans, the only silver lining to the current Covin 19 pandemic is that I now have a lot more free time to focus on making videos, so I thought I'd start by explaining some of the mathematical

model

s we use to

model

the spread of

disease

s infectious. the first is the si R model in the SI R model, the total population is divided into three categories or three components, so first we have s, which is equal to susceptible, so they will be people who could potentially contract the

disease

, then we have AI. these are called infectious so they will be people who currently have the disease and can infect others and then R means eliminated and this is the group of people who have already contracted the disease and have now recovered from the disease or have died.

With all mathematical models we have to make several assumptions to simplify real world phenomena because things are too complicated to express everything in a set of simple equations, so here the first assumption we make is that the epidemic is short enough as for it doesn't last that long, so we can assume that the total population remains constant. The second assumption of our model relates to the way the disease is transmitted and we assume that the rate of increase in infectious x' is proportional to the contact between susceptible and infectious x' and we assume that this occurs at a constant rate.

Now our third assumption relates to the elimination rate and this category is so we're going to again assume that there is a constant rate, this could be a mortality rate or a recovery rate, but again I'm going to assume that that is constant. Now that we have made our assumptions, we can begin to write the equations that will govern our model, so remember that we are interested in whether and how many of the susceptible are infectious and the population eliminated, so if we start with the susceptible , the rate of change over time of the number of susceptible, then based on our assumptions, we expect that of the number two to decrease as people become effective and therefore the rate of change. of the number of susceptible is going to be equal to minus because the contact rate is decreasing ah that is an assumption in the number two and we said that it was proportional to the number of infectious x' and the number of susceptible so these two letters being here together symbolizes a contact between the number of infectious x' and the number of susceptible and the R here is this rate of contact or transmission between them now for the troops we have a similar equation, so we want to know the rate of change of time and this will now grow according to the people who go from susceptible to infectious, so now we have our eyes, so the same term is what we had for the first equation for the rate of change of susceptible, but now it is an advantage because the susceptible are moving to become infected and now we also have, of course three, that the infectious czar covers or dies at a constant rate, so if you are infectious then you go to the third category, the R or the eliminated category, so here we have minus this constant rate. which I'm going to label as multiplied by the number of infectious lose. from the infective category they move to the eliminated category here at the same rate, we now have three differential equations for three categories of people within the population, so for the susceptible the number of susceptible will decrease according to the number of contacts between infectious. x' and susceptible i is the number of troops and this will increase due to contact between people and will decrease if people recover or die as a result of the disease and, finally, the category of people who can no longer contract the disease is eliminated because ' If you have recovered or died, this will increase at this constant rate depending on how many infectious --zz there are, so we have differential equations.

If you are familiar with them, you will know that you need some initial data before you can solve the problem. system of equations and then the way we do this is we define the initial number of susceptible people in the population, then we say that it will be equal to s zero, then we say that the initial number of infectious x' will also be specified, let's call it I nothing and al beginning of the outbreak we don't expect there to be anyone in this deleted section because no one has recovered or died yet from the disease, so the initial value of R will be zero now we still have to talk about the assumption. one in the context of our model in the context of our equations, so if we go back to this it says that the population must remain constant during the epidemic, then what that really means is that the rate of change of susceptible more infectious x' more eliminated or added must be zero because the total population is given by s plus I plus R, we can actually go one step further with this first assumption and we can solve this equation because we know the initial conditions for the population, so if the total population does not change with respect to time and that says that it is the same constant value for all possible values ​​of time, so we simply take the initial value as a starting point, which is the value of the population at the beginning, but then as time progresses.

It cannot change because it has a constant value, so it is always equal to that initial value. Now that we have formed our differential equations which together form our model, we can start asking some interesting questions, so the first question might be: the disease spreads, so we have an initial number of infected people given by zero at the beginning of the epidemic and what we want to know is whether it will increase, because if the number of infected people begins to grow, then there will be a spread of a disease through a population, then what interests us will be the race of change of the number of infectious --zz but before we do that we want to start with the question DS by DT because this tells us that the rate of change of the number of susceptibles is equal to a negative value because R is a positive constant it is a transmission rate I is a number in a population like s is then all these three things are positive and therefore the change of s is the rate of change of s is always negative, so this tells us that s must always be less than its initial value and this Of course it makes a lot of sense, I think in the context of a disease because at the beginning of the outbreak, in theory, everyone in the population is susceptible to the disease, especially with something new like kovat 19 that has never been seen before , so if s is always going to decrease because its rate of change is negative, then this tells us that s must be less than or equal to its initial value.

We can't take anything. this value is zero and plug it into our di by DT equation, so now we have an inequality in our rate of change for the number of infected and we said that an epidemic will occur if the size of I increases from the initial value of I. 0 So to answer our first question, will the disease spread? It all comes down to the sign of this particular constant, so if this constant is positive, then there will be a spread of the disease. What that means is that if s nothing is greater than a divided by R, then disease will correct this relationship here.

It's actually a little easier to think about a/R if you turn it around and consider what we call Q, where Q is equal to R divided by a and is called the contact ratio. is the fraction of the population that comes into contact with an infected individual during the period in which they are infectious. We can also rearrange this inequality to get a slightly different version of the same condition about whether or not an epidemic will occur, so if we multiply by R and then divide by a we create a new parameter our zero divided by a and this is called the basic reproductive number or basic reproductive ratio and this condition here tells us that we will have an epidemic if it is greater than 1 The zero R number or the basic reproductive ratio or the basic reproductive number is something that you may have heard about in the context of Co vid 19 because This number represents the number of secondary infections in the population caused by an initial primary infection, in other words, if a person has the particular disease, then the zero R value will tell you how many infections on average that person will cause and how many other people will transmit the disease to them within the population for seasonal flu.

The R zero value is between 1.5 and 2 while for Kovan 19 it is estimated to be more like 3 to 4, so the exact numbers are obviously still being determined because this is an ongoing outbreak that we have never seen. seen before, but the number is certainly much higher than the 1.5 or two seen for seasonal flu and this tells us that for every person infected with the disease it is transmitted to three or four other people, which is why is spreading so quickly throughout the world, another question we could ask ourselves. I am interested in knowing the answer to what the maximum number of infected will be at a given time because knowing the number of infected people is very useful when planning how to distribute the health resources, so we want to create an equation for I that is in terms of various parameters that we know within our system of equations and what we do this time is actually combine the DS by DT equation with the DI by DT equation, so we take these two together because if I do di times DT divided by s times DT I end up with an equation di times D s so if we simplify this a little bit these two terms will cancel perfectly because they are exactly the same so we get a minus 1 and then the second term here I'll do it we cancel on both terms to get more a over R and then with an S at the bottom, thinking about our answer to question one about the spread of the disease, we introduced this parameter Q which was equal to R divided by a, so if we rephrase that final term in terms of Q we have negative 1 plus 1 divided by Q times s and this equation di times D s equals negative 1 plus 1 over Q s is something that we can now directly integrate and solve now too, of course. we have initial conditions and that is what will form the right side of my equation, so our final equation is given in the blue box at the bottom and it is just in terms of I is this contact relation Q which is fixed by the model and the initial conditions I 0 and s naught, although we have this equation for I in terms of s and the parameters of our model we have not yet found I max, the maximum number of infected at a given time, which is what we want to answer to our second question now normally we are thinking about maxima and minima of functions, you would differentiate the function but fortunately we already have the derivative of combining these first two equations and we can see that this is when s is equal to 1 over Q because if s is 1 over Q, you have minus 1 plus 1 equals zero, so the maximum value of I IMAX will actually occur in this equation when s equals 1 over Q, substituting this value into our equation and rearranging 4i we get the value of I. max , so this is our final expression for I max and by writing it like this we can see a little more what is happening, so it says that the maximum number of troops is the answer to the question we are interested in.

The maximum number of people who will have the disease at any given time, the effective maximum is equal to the total population, so to begin with it is almost everyone, but then we remove something here that turns out to be positive, now the positive number depends a lot . in this parameter Q this contact proportion because it is zero for a disease like COBIT 19 for example, it is everyone because this is the first occurrence of this disease and therefore the entire population is in fact susceptible to the disease initially, so nothing is anything very large fixed number, but the interesting thing here is what happens when q varies if we consider this as a function, let's call it f of looks f from X and then just Remember we're subtracting f from f of X and What does this mean in practice?

So the key parameter here is the Q value which I have plotted as Kovan 19 this value of Q is actually very high. Because the disease is very easy to transmit, many people contract it and many people come into contact with those who suffer from it, especially due to this long incubation period in which symptoms may not appear, so, inultimately for our model it means that Q is very very large for the

covid

19 outbreak and therefore looking at our graph, if Q is large or X in our graph, then f of X is actually very small, so we're here at the opposite end of the graph, so the value of F is actually quite small and what this means for our maximum number of infectious x' is that the maximum number of people who can have the disease in a given time is equal to the total population minus this function where our function is now quite small. small, so this is very, very bad news for an outbreak that has a large Q value like kovat 19 because it says that the maximum number of people who can get the disease at any given time is actually the majority of the population .

The third and last question. What we might want to ask about the spread of our disease is how many people in total will end up getting the disease and to answer this question we need to go back to assumption 1 and this idea that the total population is constant, we first need to think about what it means that the disease end because if we want to know the total population number of people who contracted the disease, we need the actual spread of the disease to have ended first, this means that the number of troops must be reduced to zero, so if we call this point in the future just the end of the outbreak, so what we can do is look at our total population equation and rearrange it to find the size of the component removed at the end of the outbreak because the number of people who got the disease or died from it, That is, all the people in the removed or our component of the model will actually give you the total number of people who have contracted the disease, so what we can do now is rewrite this equation in the yellow box to see what it should look like when end of the epidemic. so here everything is known except s end so we have the total population at zero plus s zero minus the number of susceptible people left at the end of the epidemic to find the value of s end what we do is go back to our equation from the question to that of this blue box that emerged from integrating I by D and now we let time progress until the end of the epidemic, so we solve this equation to obtain the number of susceptible people left at the end of the epidemic. and then we plug that value into this equation to get the number of people eliminated or the size of the population eliminated at the end of the epidemic and this is exactly the answer to how many people contracted the disease during the outbreak like we did before.

To get an idea of ​​what's going on here, let's consider the graph of this function, so if I plot the end s as the Y value up here and then on the x axis, I'm going to put Q as our contact ratio because , as I saw in question 2 the maximum number of x' infectious that was really key to controlling the behavior of the disease outbreak, so here we have Q, so what I'm really plotting is that I have Y minus 1 over the log of Y is equal to a load of constant values ​​minus 1 over we looked in question 2. that the value of Q here, this contact is very, very high and therefore for a large value of Q, this will have a very small value here, so SN will actually be quite small and that, again, is bad news in terms of answering our question because our final, the total number of people who get the disease, remember, is equal to the total population at zero plus s zero and then we subtract this final value of s, but for a large value of Q s final is small, so we are not actually subtracting much from the total population, so, in short, quite a bit, if not the vast majority of the population, will contract the disease if the value of Q is large enough.

So what does all this mean for Covid 19? I've said throughout this video that this contact relationship Q is really key to determining this behavior and we can see it very clearly in these three answers to our important questions about this spread of disease, so what we're seeing here is that if Q It is big then, first of all, the disease will spread, an epidemic will occur, of course, it is too late to say this now that there is already a pandemic all over the world. The second question tells us that the maximum number of infectious x', so the maximum number of people who have the disease at a given time is the same for everyone: this function of Q and we saw that the function of a large value of Q, so in the case of Kovan 19 the maximum efficacy at any given time is almost equal to the entire population and then in question three, we ask how many people will get Kovin 19 disease again.

Q This proportion of contact is large, so this tells us that basically again the majority of the population or the vast majority will contract the disease. for a large value of Q now, of course, we knew that this is all an epidemic and that most of them are going to get infected, but what this model tells you and I must emphasize that this is possibly one of those basic disease models What you can do, but the power of mathematical models is that they not only tell you things that may seem obvious, but they also tell you how to alter things and control them and get them back under control and in your favor.

What we can see here from our A simple model is the importance of this contact relationship signal that appears in the answers to our three key questions, so although we can no longer stop the spread, that has already happened, what we can do is look at questions two and three because we can see that if we want to reduce the number of people who have the disease at any given time IMAX, then what we need to do is make the value of F as large as possible and we saw earlier in our graph that this happened when Q was small and similarly for question three, the total number of people who get the disease we want to make G of Q, we want to make it as large as possible so that the total number of people is much smaller and again that happened for smaller value of Q, so what can we do to reduce the value of Q?

Remember it is the contact proportion, this is the fraction of the population that comes into contact with an infectious individual during their period of infectivity, which is exactly why we are currently told to wash our hands. If you wash your hands, even if you have been in contact with someone who has the disease, you are much less likely to contract it yourself. Social distancing. That's why we have new measures that tell us to stay away from people, because if you stay away from others. people, they are reducing their probability of coming into contact with someone who has the disease and therefore they are reducing this Q value, so we must do everything we can to reduce the q value and all the current measures tell us to do exactly That's right, if you weren't already convinced, I hope you are now, let's keep washing our hands, let's keep practicing social distancing and ultimately let's try to reduce the value of the contact relationship Q because we've seen here in the SAR model The lower the value of Q, the fewer people get the disease, thank you all for watching, subscribe to my channel if you liked this video and I will be back soon and remember to keep improving mathematics.