The SIR Model of Disease Modeling
“SIR what are you doing?” “Modeling diseases and changing the world DUH”
Sorry I’m not funny.
also pls excuse the redundancy of the title LOL
SIR is a calculus-based epidemiology modeling system used to predict the amount of susceptible, infectious/infected, and recovered/removed individuals of a population from a certain infectious disease. Susceptible individuals haven’t yet gotten the disease, infectious/infected individuals are infected with the disease, and can spread it. Recovered/removed individuals can be either recovered (so they are no longer susceptible due to their active immunity from infection) or removed (dead). The reason why the removed and recovered individuals are grouped is that the defining characteristic of the R’s is the inability to become infected or pass the disease.
Scientists initially developed this model in response to the influenza pandemic of the early 1900s. The most basic model is as follows:
S = number of susceptible individuals
I = number of infectious/infected individuals
R = number of recovered/removed individuals
change in amount of susceptible individuals = dS/dt
change in amount of infectious/infected individuals = dI/dt
change in amount of recovered/removed individuals = dR/dt
change in number of any given category = rate in - rate out
dS/dt = (no rate in - assuming no births/immigration/etc) - how many become infected
dI/dt = how many become infected - how many recover/die
dR/dt = how many recover/die (no rate out - assuming no emigration/return to susceptibility etc)
in mathy terms:
dS/dt = -kSI
where k is a constant
we multiply S by I because S becomes I when S interacts with I; SI is a measure of interaction likelihood, which is how infection spreads.
dI/dt = -kSI - mI
where m describes the proportion of infected individuals that either recover or die
dR/dt = mI
This model has evolved to account for various other variables, including newly-implemented health policy changes, globalization, behavioral patterns, and others.
During the COVID pandemic, advanced SIR models helped determine how to manage the supply of hospital resources (beds, ventilators, masks, other PPE, etc). SIR models provided guidance to policymakers on how to approach the pandemic. In the future, more advanced disease modeling techniques will help us better predict and reduce disease spread.
Sept 2023 update:
There are other models, including SIS, SIRV, SEIR, SEIS, MSIR, MSEIRS and many more! I wanted to provide a brief explanation of each of these, because it’s super cool to see variants of a single fundamental concept.
SIS accounts for those cases where immunity after infection is only temporary or, in some cases, nonexistent. There is thus no “recovery” group - hence, the name “SIS.” Because there is no “R,” according to this model, everyone will become infected at some point as the disease will last forever.
The “V” in SIRV stands for “vaccinated,” which helps us understand the population-wide benefits of vaccination. In effect, vaccination brings individuals to the “R” category without the intermediate infection stage.
The “E” in SEIR and SEIS stands for “exposed.” This category accounts for the intermediate stage between susceptibility and infection, where individuals have contracted the disease but are not able to pass the disease to others yet. It thus accounts for latency period (the amount of time between exposure and infectivity).
The MSIR model has an “M” which stands for “maternally-derived immunity.” This applies to diseases for which immunity (antibodies) can be transferred placentally/through milk to the baby. Members of the M group (babies) are not susceptible; they have characteristics of the “recovered” group without having been through infection. However, because this is passive immunity, they will soon move to the susceptible group once they stop breastfeeding.
Now, try to think of what the MSEIRS model does! (Also I wonder if there’s a MSEVIRS model - babies are immune, then susceptible once done with breastfeeding, then exposed, then infective, then either vaccinated or infective, and after infection is recovery, and perhaps re-susceptibility). Anyhoo, hope that was a helpful update to better understand the biological context of the model!
Here are some cool resources to learn more about SIR:
Cool COVID simulation using a more complicated SIR: https://covid19-scenarios.org/.
OSU researchers used an advanced SIR model to approach COVID: https://cph.osu.edu/news/2020/05/math-behind-modeling
Textbook chapter on the calculus of SIR: https://homepage.divms.uiowa.edu/~stroyan/CTLC3rdEd/3rdCTLCText/Chapters/Ch2.pdf
Simple video explanation of SIR in calculus: https://www.youtube.com/watch?v=-ShBluVcCTw
Nice and direct article about SIR calculus: https://www.maa.org/press/periodicals/loci/joma/the-sir-model-for-spread-of-disease-the-differential-equation-model
Scientific article on SIR: https://www.sciencedirect.com/science/article/pii/S1110016823004313
Wiki slay! https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology#The_SIR_model
