SIR Model: Flattening the curve....!

Dawn of Math and Infectious Diseases

Let's keep up with this!

It doesn't matter whether you are a Physics Major or Biology Major or Mathematics Major when it comes to facing one heart-throbbing name "The Bernoullis"! Can you imagine 8 members from the same family with immense contribution to mathematics? This name occurs so frequently in science and mathematics that sometimes it's pretty hard to track down which Bernoulli contributed to Fluid mechanics and which one rubbed an ink for the Probability theory and Economics and which one was possessed by solving the Brachistochrone curve.

Fig 1. Daniel Bernoulli [1]

One of the prominent names that I recently came across is Daniel Bernoulli or just Dan to make it look more familiar. He is known for his work in Fluid mechanics and Principles of Superposition. Apart from achieving success in the field of Physics as well as Economics, he was the first one to lay the foundation for mathematical modeling of Infectious disease epidemics. 

Smallpox was one of the nightmares during his time that caused mortality in 1 out of every 14 individuals. Daniel Bernoulli, in his seminal paper in 1766, described the first mathematical model of the smallpox epidemic [2]. Later on, many mathematicians and theoretical biologists came up with their own models to explain disease dynamics in population. Somewhere, I think we miss this name when we discuss epidemiological models in theoretical biology. On the other hand, he had no idea of what bacteria and viruses were. Until the late 19th century, i.e., even after bacteria had become known as infectious agents, it was commonly believed that epidemics die out because the infectious agent becomes less infectious and less virulent.

One of the well known infectious disease models was published by a Scottish biochemist William Kermack and a Scottish military physician Anderson McKendrick in their three seminal papers [3-5] in 1927, the SIR Model. So let's start with SIR Model. It's quite popularised on media these days due to the ongoing pandemic of COVID-19.

What is the SIR Model  ????

Before introducing this model, it's essential to understand that mathematical modeling in biology begins with setting up straightforward rules and parameters. The behavior of the system (In this case, epidemics) is then observed, adding more complexities layer by layer. This helps us to understand which parameter has the most significant effect on the behavior of the system.

The beauty of the SIR Model lies in its simplicity. SIR stands for Susceptibles, Infected and Removed (Being more of an optimist, I would rather call it Recovered). It's unidirectional in a sense: S -----> I -----> R. Before we jump into the model, let me mention several assumptions to make it more understandable later:

• R (or Removed) includes those who recovered during the epidemic and immune to getting re-infected. In short, they are not included in the S (or Susceptibles). This also includes those who actually died during an epidemic. Hence this model shows unidirectionality in disease dynamics.
• The population is fixed. The transmission rate and recovery rate is much greater than birth rates and death rates. Hence, this model does not take into consideration of births and deaths (here deaths apart from epidemic).
• Each individual is equally susceptible to the disease. No age, sex or racial factors are considered to be affecting differentially in this model.

Fig 2. SIR Model [6]

Let's start the game...!

Mathematical models require some initial conditions and the kind of behavior that the system is predefined with as in (Fig 2). In short bear with some differential equations. Since this model is "Deterministic" meaning that the epidemic model can be differentially determined w.r.t. time (t). So, we know that if S in contact with I develops the disease, it is included in the I compartment and with time if it gets recovered (or dies ) it is included in R. Hence, numbers and identities of individuals in each compartment changes with time. So the derivatives for this system will be:

Let N be the total population. So at time t:

N(t)= S(t) + I(t) + R(t)              (Eq. 1)

Since the population remains fixed with time. Even the dead ones are counted. (Remember the rules!!)

𝜕N/𝜕t = 𝜕S/𝜕t + 𝜕I/𝜕t + 𝜕R/𝜕t = 0              (Eq. 2)

Now if you are familiar with the Law of mass action, we assume here individuals in the population are homogeneously mixed just like molecules in an ideal solution. 

This seems to be stricter for an assumption but this is how simplicity brings an advantage of not missing any gold from a huge heap of sand. 

The real paradigm of Mass-Action assumption in epidemiology brings us more Nonlinear ODEs. For those familiar with Enzyme kinetics it would be easier to understand and those who are not, don't worry feel safe. Keep in mind (Fig 2) and follow.

For the rate of change of S with time,

𝜕S/𝜕t = -𝞫SI                  (Eq. 3) 

Here, 𝞫 is the "Transmission rate" or "Infection rate". It states how fast the infection spreads. More the infected individuals I, more will be the contacts within the population causing more Susceptibles S to convert to being Infected I over a time period. Minus sign suggests that with time the number of Susceptibles S is going to decrease and get converted to I. Faster the rate, Faster will be the conversion of S -----> I. So, 𝞫 needs to be multiplied to the product of S and I. 

For the rate of change of I with time,

𝜕I/𝜕t = 𝞫SI - 𝜸I                 (Eq. 4) 

In the first part of the (Eq. 4) i.e. 𝞫SI suggest that the rate at which S decrease over time in (Eq. 3) is the same rate at which I increase over time. The positive sign of 𝞫SI indicate the conversion of S -----> I. 

The second part of the (Eq. 4) i.e. -𝜸I suggest the conversion of I -----> R. Here, 𝜸 is the "Recovery rate" or the rate at which I turn into R. Minus sign reflects the decrease of the Infected individuals once they recover or die.

For the rate of change of R over time,

𝜕R/𝜕t = 𝜸I                 (Eq. 5) 

Since over a while, all Infected individuals will be identified into Recovered (either dead or immune), the term 𝜸I remains positive. Of course, higher the value of 𝜸 faster will be the increase in R. This defines the filling up of the R compartment I -----> R.

Adding (Eq. 3, 4 and 5), leads you to (Eq. 2) that adds to ZERO. 

Trust me in science the most fabulous moment is the one when your equation adds to ZERO, it means something is conserved. Here individuals instead of mass are conserved.

It looks quite boring, isn't it? Bunch of equations and some naive assumptions. That's it? Why is all this screaming, FLATTEN THE CURVE.. !! in the ongoing COVID-19 pandemic? What curve is that?

The task is not complete yet. From the (Eq. 3, 4 and 5) we can just know the rate of change of Susceptibles, Infected and Recovered w.r.t. time respectively. We still cannot know HOW MANY of them lie in S, I and R categories. To know this we need to come up with the solution of all the three highlighted differential equations (Eq. 3, 4 and 5).

Thanks to the mathematicians and computer programmers to develop software that can make our task a bit easier. To demonstrate the behavior of these equations, I have used Geogebra which is quite handy to use. You can also go for other tools that are available online. Or if you are a coding geek, you can go for Python or MATLAB.

Before jumping to the simulation, let me come up with some important remarks from the equations:

1. The number of Susceptibles in a population will always decrease with time. That's what negative sign is about.
2. 𝞫SI suggest " Number of newly infected cases " and 𝜸I suggest " Number of recovered cases ".
3. Please don't forget that R includes dead (due to the epidemic) as well as recovered individuals.

Time to play with the curve...!!

Click here to open the simulation of the SIR model made by me. Well, Thank you! But apart from flaunting myself, I would say you can search online many cool videos on the SIR model.

Fig 3. SIR Model- Geogebra file 

If you have opened up the Geogebra file, you will find 3 different curves (Red, Green, and Blue) with 3 sliders which you can use to see the behavior of these curves.

Blue curve: 𝜕S/𝜕t = -𝞫SI  which shows how the number of susceptible individuals in the population change with time. It shows S reduces with time.

Red curve: 𝜕I/𝜕t = 𝞫SI - 𝜸I show that number of Infected individuals I initially show the increase and after reaching the maximum peak, it starts to decrease.

Green curve: 𝜕R/𝜕t = 𝜸I shows the number of the Recovered (or Removed) R population that increases with time.

MaxTime slider is set to 30 which is arbitrary. I have assumed a total duration of say 30 weeks till which I want my model to run.

𝞫 and 𝜸 sliders represent the arbitrary range of "Transmission rate" and "Recovery rate" respectively. You can move these sliders to see how all the three curves are affected. 

The total population N is set to 1 for the simplicity of the graph.

Initial conditions of the model are

At time t=0:

S(0)= 0.99 (or 99% are susceptibles)

I(0)= 0.01 (or 1% of the Infected people)

R(0)= 0 (Since initially no one is Recovered)

Below the sliders, you see an interesting parameter R0  (R naught) which changes as you move 𝞫 and 𝜸 sliders. Don't confuse R0 with R(0). Let me give you a brief about it.

The witchy number: R0

During a pandemic, epidemiologists are behind cranking down the R0 value. Every health professional, politician, economist, scientist have a role to play while dealing with this number. You can consider it as a demon against which all of mankind come together to defeat it. 
R0 is Basic Reproduction Number which simply means the "Average number of healthy individuals that can be infected by one infected person." But it also tells us whether to panic or chill during an epidemic. This expression also helps us formulate policies to bring down the epidemic. Let's see how.

From the SIR model, one can derive an expression for R0:

R0 = (𝞫/𝜸)*S(0)               

(Note: S(0) represents the initial number of susceptible i.e 0.99 in my GeoGebra file)

If R0 > 1: It means New cases are on an exponential rise. So PANIC.
If R0 < 1: It means New cases are decreasing exponentially. So CHILL.

Now equations should mean something, not just thrown as jargon to prove some sort of intellect. 

So, How to reduce R0 to the chilling point of < 1 with this ongoing pandemic.

𝞫 (Transmission rate) can be reduced with frequent Hand-washing, wearing a mask and Physical distancing. 

S(0) can be reduced by vaccinating the population and making them Non-susceptible by immunizing them. For that intensive funding to R&D will help. (Clickbait to politicians for not to search for water when the fire is out already. Water is still easier to find but not vaccines)

𝜸 (Recovery rate) can be increased by treating people and isolating them until the disease wanes off from them.

That's it...!!

Before leaving hope you would love to do some Quarantine Exercise:

1. Click here to open the model. See which parameter (𝞫 or 𝜸) one affects better to flatten the Red curve of Infected individuals?

2. Explain the peak (call it as Imax)of Infected individuals in the Red curve.

3. How do the exponential curves demonstrated in https://www.worldometers.info/coronavirus/ correlate with the SIR model?

4. How do the SIR model look like (Fig. 2) if following complexities are added:

• If the viruses are mutating (Hint: Mutation Rate to the flow diagram of SIR model)
• If vaccination is started (Hint: Vaccination rate to the flow diagram of SIR model)  

5. If you are a math geek try, How R0 decides the exponential growth or decay of infected individuals? 

You can answer in the Comment section below. 

One of the best videos I came across for my blog:

https://www.youtube.com/watch?v=NKMHhm2Zbkw by Dr. Tom Crawford (Tom Rocks Maths)
https://www.youtube.com/watch?v=gxAaO2rsdIs by Grant Sanderson (3Blue1Brown)
https://youtu.be/k6nLfCbAzgo by Numberphile (Ben Sparks)

References

1. E-Pics Bildarchiv online http://doi.org/10.3932/ethz-a-000046381
2. https://onlinelibrary.wiley.com/doi/abs/10.1002/rmv.443
3. Kermack WO, McKendrick AG. Contribution to the mathematical theory of epidemics. Proc R Soc Lond, A Contain Pap Math Phys Character. 1927;115:700–21. DOI: 10.1098/rspa.1927.0118
4. Kermack WO, McKendrick AG. Contributions to the mathematical theory of epidemics, part II. Proc R Soc Lond. 1932;138:55–83. DOI: 10.1098/rspa.1932.0171
5. Kermack WO, McKendrick AG. Contributions to the mathematical theory of epidemics, part III. Proc R Soc Lond. 1933;141:94–112. DOI: 10.1098/rspa.1933.0106
6. https://commons.wikimedia.org/w/index.php?curid=88814805