SIR Model

00:00 Hello, in this video, we will tackle that all important topic of modeling.

00:05 It's a complicated topic and a scary one, but the basics are pretty easy to understand.

00:10 In this video, we'll talk about the SIR and SEIR models.

00:15 The assumptions, and weaknesses, and strengths underlying those models.

00:18 We'll talk about the contact rate and the recovery rate.

00:21 These very important quantities that we use to compute the reproduction number.

00:26 We'll introduce the idea of Farr's law, which you might have heard of.

00:29 Farr's law tells us that the rise of an epidemic resembles the fall of an epidemic.

00:34 And lastly, we'll finish off by talking a little bit about the difference between forecasting models and dynamic models.

00:42 I hope you enjoy.

00:44 So, the basic transmission model for propagating infectious diseases is called the SIR model. Sometimes SEIR.

00:55 S stands for Susceptible, Infected, and Removed.

00:58 Sometimes the Remove is called Recovered.

01:01 Either way, it's SIR.

01:02 And we sometimes put an E in there for Exposed.

01:07 The SIR model was developed by Ross and Hamer in the early 20th century.

01:11 It's the foundation for most of the new epidemic models that have some more nuances, and bells, and whistles.

01:19 But the SIR model has some assumptions.

01:22 Most importantly, that there is a large closed population, like the population of a nation with closed borders.

01:30 This is because the SIR model bears a strong similarity to some of the fluid flow models in engineering and physics.

01:40 Where various compartments are observed to have fluids flowing in between them within an enclosed body.

01:47 Another assumption underlying the SIR model is that no new people are added.

01:52 So, no new kids are born in this population.

01:56 And no people are immigrating from other parts of the world.

02:01 And there are no natural deaths.

02:03 So the only people who die or are removed from this population do so because of the disease being studied.

02:10 And no one is emigrating either.

02:12 They are all staying in this population.

02:16 We also assume that recovery from the illness confers immunity.

02:22 So, once you get it, you can't get it again.

02:25 And we assume that people in the population are homogeneously distributed had engaged in mass action.

02:32 What does that mean? It means that everyone in the population has an equal probability of interacting with everybody else in the population, which we know is not how the real world works.

02:43 Most people have a set number of contacts and are unlikely to see anyone beyond those contacts.

02:49 But for the purposes of modeling, these assumptions can be allowed.

02:56 Initial challenge when thinking about infectious disease modeling, is to differentiate between linear growth and exponential growth.

03:04 And to my mind, this is the greatest challenge for the layperson in understanding, how infectious diseases explode out of control? Exponential growth has essentially two phases: the slow, almost linear phase in the beginning, and the explosive, almost vertical phase, later on.

03:24 By the time you're in the vertical phase, you need to be investing an extraordinary public health infrastructure to control the situation. I liken it to investing.

03:36 If there is a bit of stock that you want to buy, that we know is going to be worth a lot of money because of exponential growth.

03:45 If you invest a little bit of money early on, you get a big payout several months later.

03:52 If you wait later, until the stock is clearly doing well, you have to invest a lot more money to get the same outcome.

04:00 So acting early, allows you to act less strenuously when controlling exponential growth in an explosive disease.

04:10 So, it's important from a public health perspective, to act early and possibly to act hard.

04:16 So early on, a disease growing with exponential growth may not seem as dangerous or as impressive as a disease growing with linear growth.

04:26 But there comes a time when exponential growth equals then exceeds the seriousness the case growth of a linear growth disease as shown by these two curves.

04:40 As noted, the SIR model contains three compartments, essentially.

04:45 They are those who are susceptible, which is everybody in the population, because they've never seen the disease before.

04:52 They all have an equal probability of getting the disease and no one is immune.

04:57 Then there are the infected people.

05:00 So, as the susceptible individuals are exposed to the infection and get the disease, they move into the infected compartment.

05:10 And gradually, those who are infected either recover or they die.

05:15 In either case they are removed from the compartment.

05:23 So, over time, eventually everybody in the susceptible category will enter the infected compartment.

05:30 And over time everyone who is infected will either recover or will die.

05:35 That is the assumption for the SIR model, the flow of people from compartment to compartment.

05:42 As I noted, sometimes we add some additional compartments among them, the E.

05:48 So, first we have a population of susceptible people.

05:52 No one has seen the disease before, no one is immune.

05:55 Then gradually, people become exposed to the disease.

06:00 And over time, they become infected and infectious.

06:04 Over time, they become removed as well.

06:08 The thing about the SIR and SEIR models is that they don't account for several things like incubation period.

06:16 Or the heterogeneity of people's interactions, or the fact that in reality, not everybody is susceptible.

06:24 Some people are more likely to get diseases than others.

06:27 So, it's a very simplistic way of thinking about population health dynamics.

06:32 But again, a model doesn't have to be perfect.

06:35 It just has to be useful.

06:39 So in the SIR model, there are several characteristics, several mathematical aspects that we care about.

06:46 There is the function of change of people in the susceptible compartment, right, that's given by S of T, S as a function of time.

06:56 We have I, which is the number of infected individuals, and I changes over time.

07:01 And R, the number of removed individuals, those who either recover or die.

07:06 And that changes over time.

07:08 We can divide each of these by N.

07:10 The total number of people in the population to give this thing called the fraction, the susceptible fraction, the infected fraction, or the fraction of people removed from the population.

07:24 At any given time T however, the sum of s(t) + i(t) + r(t) = 1.

07:32 Why is that? Because we have a closed system.

07:36 There is a set number of people at all times.

07:39 And if you're not susceptible, it means you're either infected or you've been removed.

07:44 If you're not infected, it means you're still susceptible or you've been removed.

07:47 And if you're not been removed yet, it means you're either still susceptible or infected.

07:52 So, at all given times, if you live in this population, you're in one of these compartments.

08:01 So, some more assumptions about the ideal SIR model.

08:05 First everybody gets removed, eventually.

08:08 It's enough time goes by.

08:10 Everybody gets moved through the machine and either recover or die.

08:15 The duration of infection is the same for everybody in this model.

08:19 Clearly, that's not true in real life.

08:21 But again, a model doesn't have to be perfect, just useful.

08:25 And once recovered from infection, the person is immune and can no longer infect anyone else.

08:32 They cannot get the disease again.

08:36 Only a fraction of contacts with the disease actually cause infection in these models.

08:41 And as I mentioned, the system is closed.

08:43 Meaning that the total of those in the susceptible compartment, those in the infection compartment, and those in the recovered compartment does not change.

08:55 So, here's a sample population of 500 people.

09:00 And the red curve shows us the number of people in the S, or Susceptible category.

09:10 As you see, everybody starts out susceptible and over time that compartment dwindles to zero as they become infected.

09:20 The green curve shows us the infection curve.

09:23 The number of people in the infection compartment.

09:27 And that's really the curve that most people look at when they look at the SIR model, because it shows us the path of the disease through the population.

09:39 So at Time 0, nobody is infected.

09:41 But as more and more susceptible people become infected, that I curve increases, until finally it starts to diminish because not enough people are still susceptible.

09:53 This is a natural infection curve, by the way.

09:56 There's no vaccine at work here.

09:58 There's no interventions being apply at any point in time.

10:01 As a result, this decline in infection is caused by people becoming immune or resistant naturally over time.

10:12 And this blue curve, that's the recovered curve.

10:15 Zero is how many people have recovered or removed at Times 0 because no one's got the disease yet.

10:23 But over time, as more people become susceptible and become infectious, more people end up exiting from the other end of this either recovering or died until finally 100% of the population is in the R compartment.

10:38 Here's the example of an SIR model made for Spain in the early days of the COVID-19 infection.

10:46 So, the red curve shows us the path of infected individuals.

10:54 It was expected that infections that is detected infections would arise up until the end of April and then start to decline thereafter.

11:05 Because the number of recovered people would be meeting saturation.

11:10 But if you look at the blue curve, the death rate was just starting to climb as well.

11:15 So, that's how they hope to get a handle on the disease by putting forth an SIR model to better understand the timelines at play.