Vaccination Prioritization
Strategy with SIR Models
AM-115 Project1
Author: Shaoyi Li, Pedro Duarte Moreira, Astrid Zhao, Mofei Wang
01
02
Introduction
Models
03
Analysis
04
Discussion & Conclusion
1. The Introduction
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Vaccination is an effective way to reduce cases in an epidemic as well as
to accelerate the time to reach herd immunity.
Prioritization for certain groups is a commonly employed vaccination
strategy to optimize the use of vaccines.
The original SIR model mentioned in class is simple and reasonable but
it fails to capture the effect of vaccination.
We are going to use two modified SIR models to represent and
compare two possible vaccination strategies:
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One strategy revolves in vaccinating the entire population at the
same time;
The second one is based on the prioritization of people with higher
infection risk but with a lower vaccination speed.
2. Models
Original SIR model (Model 1) & SIR model with vaccination (Model 2)
Transition from S to R (vaccination speed) will be at the capacity at the
beginning and then slows down when S becomes small. When it starts to
slow down the speed is proportional to S.
● Model 1 is Model 2 with c=0.
● Model 2 represents one simple vaccination strategy.
★ We assume the vaccine is 100% effective.
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2. Models
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Two susceptible groups: S1 and S2.
We assume the two groups have different infection rates and vaccination speed
capacities: a1>a2>0 and c2>c1>0.
Vaccinate Group S1 before Group S2.
Vaccination speeds will start at their capacity and then slow down when S1 or S2
becomes small. When the speeds start to slow down until they are proportional
to S1 or S2. (The vaccination for S2 starts as soon as the speed for S1 slows down.)
SIR model with vaccination and two S groups (Model3)
2. Models
Rationale to have 2 S Groups in Model 3
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We treat S1 as the high infection rate group and S2 as the regular group.
In reality, S1 represents the group with seniors or people with
pre-existing conditions. In general, at the vaccination sites, the process
usually moves slower for these people because:
○ It usually takes longer for them to line up, walk to certain spots, fill
out the forms, etc.
○ The health care workers also tend to spend longer with them right
before and after the shot to make sure vaccines don’t increase their
risk of severe illness due to the side effects.
It’s possible some other high infection rate groups have no slower
vaccination speed than the regular group. We can intuitively tell
these groups should be prioritized. There’s no need of
mathematical models for tradeoff analysis.
2. Models: Parameters and Relationship
How Do We Compare Model 2 and 3 Apple-to-Apple?
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We make a as initial population weighted average of a1 and a2;
c as initial population weighted average of c1 and c2
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We of course assume that S1(0)+S2(0) = S(0) as they represent
the same system.
In addition, here’re the default parameters for the analysis:
a1=8, a2=4, b=2, c1=0.1, c2=0.15, S1(0)=240, S2(0)=720, I=30, R=10
The unit of t is assumed to be one month
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3. Analysis: Model Comparisons
How Do the Models Output Look like?
3. Analysis: Model Comparisons
Comparing I(t) and R(t) among the three models:
● We can compare by plotting the curves from models in the same figure
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I peak reduction: max I value in Model 2 minus max I value in Model 3.
Herd immunity delay: time reaching herd immunity in Model 3 minus
time reaching herd immunity in Model 2
I peak reduction is our primary measurement for our study.
3. Analysis: Sensitivity Analysis
Parameters
When parameters increase, the I peak reduction:
The ratio of c2/c1
Decreases
c
Decreases
a1
Increases
a2
Increases until a2 = 2.7, then decreases
3. Analysis: Sensitivity Analysis
Parameters
The ratio of c2/c1
When parameters increase, the herd immunity delay:
Decreases until c2/c1 = 1.8, then increases
c
Decreases
a1
Increases
a2
Decreases until a2 = 6, then nearly constant
3. Analysis: Strategy choices
When the vaccination prioritization strategy will be useful?
The prioritization strategy is the best when the people in the risk groups are very at risk, and not much
slower speed than the other group in getting the vaccine. Also useful when there is vaccine shortage.
4. Discussion & Conclusion
● Three models:
SIR, SIR with vaccination,
SIR with prioritized vaccination.
● If we do not have enough vaccine, or the
risk of S1 is much higher, or the speed of S2
group getting vaccination is much quicker,
prioritization strategy is useful.
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Vaccination prioritization strategy is
useful in certain scenarios.
The result aligns with our intuition.
For most values that would be
observed in real societies and diseases,
the best strategy is to use the
prioritization strategy.
OTHER SOLUTION TO DECREASE THE PEAK OF THE EPIDEMIC:
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try to implement social distancing and lock down to decrease R0
try to acceleration the production of vaccine
Reference
[1] Smith , David, and Lang Moore. “The SIR Model for Spread of Disease - The Differential
Equation Model.” The SIR Model for Spread of Disease - The Differential Equation Model |
Mathematical Association of America, 2021,
www.maa.org/press/periodicals/loci/joma/the-sir-model-for-spread-of-disease-the-differential-equa
tion-model.
[2]Hannah Ritchie, Esteban Ortiz-Ospina, Diana Beltekian, Edouard Mathieu, Joe Hasell, Bobbie
Macdonald, Charlie Giattino, Cameron Appel, Lucas Rodés-Guirao and Max Roser (2020) "Coronavirus Pandemic (COVID-19)". Published online at OurWorldInData.org. Retrieved from:
'https://ourworldindata.org/coronavirus' [Online Resource]