The Impacts of Vaccination on Controlling Infections
by Byungmin Choi and Dongwoong Seo
Abstract
This paper suggests an SIVR model, as an extension of the classic SIR model for testing vaccination
strategies. With this model, we calculate the basic reproduction number and corresponding herd
immunity threshold. We simulate multiple scenarios with different sets of parameters. As a result,
we find that the higher vaccination rate and efficacy of vaccination not only reduces the peak level of
infection but also helps reach the HIT sooner. These results imply that it is important to implement
fast and effective vaccinations in facing the contagious diseases like COVID-19.
1
Introduction
In the emergence of an epidemic, a vaccination is a way to control it by achieving immunity without
actual infections. How can we control a disease and minimize its negative impacts? First of all, we may
be able to lower the peak of infections with vaccination. This is crucial in reducing damages of infectious
diseases. Due to limited medical resources, controlling the level of infections low is especially required for
the pandemic, as we recently experienced. Also, this might help us reduce the overall deaths caused by
the epidemic. French et al. (2021) analyzed the impact of excessive strain on the local healthcare system
during the pandemic. They argued that lack of capacity for intensive care units by COVID-19 may have
significantly increased the number of expected deaths.
For vaccination strategies, we first consider the speed of vaccination. There are some factors that
can affect the vaccination rates. At the beginning of the recent pandemic, many countries suffered from
delayed vaccinations due to the shortage of vaccination supplies. The hesitancy of people to get vaccinated
can also affect the speed of vaccination. On top of that, various vaccines have different efficacy. According
to the U.S. Food and Drug Administration, the efficacy of three vaccines available in the U.S. varies from
67%(JJ) to 95%(Moderna and Pfizer-BioNTech). Hence, adopting a different sort of vaccination can be
considered as an valid way to enhance the level of immunity in a society.
In this paper, we try to model an epidemic with vaccination strategies. Section 2 describes our
model and its key parameters. Then, we derive equilibria of the model and their stability conditions
mathematically and numerically. Section 3 conducts sensitivity analysis with different sets of parameters.
Then we discuss the implications of the results in section 4. Finally, we conclude this paper with some
remarks.
2
2.1
Model
Model Description
We start from the classic SIR model devised by Kermack and McKendrick (1927). In the classic SIR
model, a population is divided into three compartments: susceptible S(t), infected I(t), and recovered
R(t). Our SIVR model extends the SIR model by introducing a new compartment, vaccinated V (t) to
describe vaccination strategies. For vaccination strategies, we consider the speed of vaccination(δ) and
its efficacy(λ).
dS
S
= αN − βI − δS − µS
dt
N
dI
S
V
= βI + (1 − λ)βI − γI − µI I
dt
N
N
dV
V
= δS − (1 − λ)βI − µV
dt
N
dR
= γI − µR
dt
1
(1)
We also include constant birth(α) and death rates(µ, µI ). Birth and death rates are typically set to be
the same to keep the total population constant. Since we do not assume a constant population, it is not
necessary for these to be the same Also, the death rate of the infected(µI ) can be different from the usual
death rate(µ) to reflect the mortality of the disease and to evaluate its impacts. Other parameters just
follow the definitions of the original SIR model. Table 1 summarizes all the variables and parameters.
Symbols
S(t)
I(t)
V (t)
R(t)
N
α
µ, µI
β
γ
δ
λ
Definitions
the number of susceptible individuals at time t
the number of infectious individuals at time t
the number of vaccinated individuals at time t
the number of recovered individuals at time t
total population
the birth rate
the death rate (of infected people)
the background transmission rate
the recovery rate of infected individuals
the rate of the susceptible getting vaccinated
the efficacy of vaccination
Values (range)
0 ∼ 1000
0 ∼ 1000
0 ∼ 1000
0 ∼ 1000
1000
0.0001
0.0001, 0.0002
1.5
0.1
0.05 ∼ 0.20
0.85 ∼ 0.95
Table 1: Variables and Parameters
2.2
dI
dt
Derivation of the Herd Immunity Threshold
We can find two equilibria by solving a series of equations,
= 0 gives us two possible cases. That is,
dS
dt
=
dI
dt
=
dV
dt
=
dR
dt
= 0. Notice that
S
V
dI
= βI + (1 − λ)βI − γI − µI I = 0
dt
N
(N
I=0
=⇒
I>0
At I = 0, we can derive the following equilibrium(E0 ), called a disease-free equilibrium.
αN
αδN
, 0,
,0
E0 (S0 , I0 , V0 , R0 ) =
µ+δ
µ(µ + δ)
(2)
To determine the stability of the equilibrium rigorously, we need to find its Jacobian matrix and
check whether its eigenvalues have all negative values at the point (Roussel, 2005). Here we suggest a
simpler way to determine the stability, which gives the same result. Near the equilibrium, we can solve
the differential equation in terms of I as follows.
S
dI
V0
0
= β
+ (1 − λ)β
− γ − µI I
dt
N
N
S
V0
0
=⇒ I(t) = C exp [β
+ (1 − λ)β
− γ − µI ]t
N
N
This converges to I0 = 0 only if
S0
V0
+ (1 − λ)β
− (γ + µI ) < 0
N
N
α
αδ
=⇒ β
+ (1 − λ)β
− (γ + µI ) < 0.
µ+δ
µ(µ + δ)
β
This leads us to get the basic reproduction number as the stability condition of the equilibrium.
R0 =
β{α + (1 − λ)αδ/µ}
<1
(µI + γ)(µ + δ)
(3)
This basic reproduction number(R0 ) is defined as the average number of secondary infections produced
when one infected individual is introduced into a host population where everyone is susceptible (Hethcote,
2000).
2
What if I > 0? The other equilibrium is called an endemic equilibrium because it implies that
the disease never perishes. When R0 > 1, the endemic equilibrium becomes stable (Brauer, Van den
Driessche, Wu, & Allen, 2008). Instead of a mathematical derivation, we use a numerical method to show
this. Figure 1 describes the number of infection changes over time, with different R0 ’s. We can verify
that I(t) does not converge to 0 when R0 > 1.
Figure 1: Steady States of the Number of Infections Depending on R0
By definition, the herd immunity refers to the state that enough fraction of population has acquired
immunity so that an introduction of a one infected person cannot make the disease spread (Brauer et al.,
2008; Hethcote, 2000). Let’s denote p as the fraction of the immunized. Then, the basic reproduction
number reduces to R0 (1 − p) and hence the stability condition of the disease-free equilibrium is as follows.
R0 (1 − p) < 1 =⇒ p > 1 −
1
R0
(4)
Therefore, the herd immunity threshold can be represented as 1 − R10 . For example, the reproduction
number of the alpha variant of COVID-19 is estimated as 4 so that its HIT = 75%.
Note that R0 = µ(µβα
when there’s no vaccination. (i.e. δ = 0) This leads us to see how the basic
I +γ)
reproduction number changes with higher vaccination rates in the following section. We also analyze
whether the vaccination can shorten the time for the share of immunized population(λV + R) to reach
the HIT.
3
Sensitivity Analysis
In this section, we analyze how different vaccination strategies affect the peak of the infections and
related mortalities with varying sets of parameters. We further investigate when the population reaches
HIT depending on the vaccination parameters. We assume that the vaccination start with the influx of
the disease. We set R(0), V (0) zero, I(0) one, and S(0) is 999.
First, we analyze the peak of infection and how many people die from varying δ and λ. We set the
time 52 weeks as one year and test how the variables change as time goes by. Figure 2 shows the process
of vaccination and infections. The susceptibility decreases exponentially as people get vaccinated and
infected by the disease. The peak of Infected is 0.327, which means that at that time about 32% of the
total population is infected with the disease.
We investigated the peak of Infected and Infected Death influenced by δ and λ. Figure 3 shows how
many people die from the disease depending on δ and λ. For each column in Figure 3, an increase in δ
lowered the peak of Infected and related death. Similarly, an increase in λ reduced the peak of Infected
and Infected Death. These results are consistent with our expectation that an increase in vaccination
speed lowers the infected death and higher efficacy lowers the death from the disease.
3
Figure 2: Dynamics of SIVR Model
Figure 3: Peak of Infected Depending on δ and λ
4
Second, we analyze when the population achieves herd immunity depending on the vaccination speed
and the vaccine efficacy. The basic reproduction number and the herd immunity threshold of our model
are defined just the same as in the previous section. Figure 4 shows the benchmark dynamics of our
model and when the population reaches the HIT. As there are two types of immunity, V (t) and R(t),
we compare R(t) + λ × V (t) with the HIT, considering the efficacy of the vaccine. Figure 5 shows the
HIT with given parameters and how long it takes for the population to achieve HIT when the speed of
vaccination and its efficacy change. Each label indicates the time to achieve herd immunity. In the first
column of Figure 5, an increase in δ does not reduce the time to reach the HIT. It takes 15 weeks to
achieve the HIT. In the second column of Figure 5, where λ is 0.90, an increase in δ lowers the time to
achieve the HIT. Lastly, the HIT is not defined in the third column of Figure 5. From Equation (3),
when λ is 0.95, the HIT value becomes negative. These results coincide with common sense that faster
vaccination and effective vaccines reduce the time to achieve the HIT.
Figure 4: Dynamics of Immunization to reach the HIT
5
Figure 5: Herd Immunity and Achieving Time Depending on δ and λ
4
Discussion
We have analyzed how the vaccination can affect the dynamics of infections. With our benchmark
parameters, the infections seem to keep decreasing as the immunity level of population rises in Figure 2.
However, this is only a part of the story. As we mentioned with Figure 1, the infection never goes away
and remains as an endemic with R0 > 1. This sounds paradoxical in that the immunity level reaches the
herd immunity level. In this section, we would like to take a deep dive into the concept of herd immunity.
The term, herd immunity, has been widely used since the outbreak of COVID-19. It is often suggested
as a goal to end the infectious diseases. Recall that the basic reproduction number denotes the state that
one new infection is introduced to a population where all individuals are susceptible. It is determined
which equilibrium the population will converge to at the beginning. Thus, achieving the HIT during an
infection doesn’t affect whether the infection entirely ends in disease-free equilibrium.
This problem may be specific to our model which assumes constant parameters. Our model also
depends on an assumption that the population is well-mixed at random. That is, each individual is
equally likely to make contact with others and get vaccinated (Fine, Eames, & Heymann, 2011). If we
further consider the heterogeneity of population, such as ages, the concepts of basic reproduction number
and herd immunity threshold would be more complicated. Moreover, infection of disease is stochastic
process rather than deterministic one. There are various attempts to set up stochastic models for it
(Tornatore, Vetro, & Buccellato, 2014).
Lastly, the deaths of infected are underestimated. Our model does not reflect a strain on the health
care system when the number of infection increases because we set the death parameters(µ) constant.
6
Randolph and Barreiro (2020) explains how the death rate changes depending on the strain on a health
care system.
5
Concluding Remarks
In this paper, we set up a new epidemic model by adding a vaccination compartment to the classic
SIR model. It was designed to describe two vaccination strategies: the speed of implementation and the
efficacy of vaccine.
We derived two equilibria of our model and the basic reproduction number as their stability condition.
We found that introducing vaccination parameters can significantly decrease the R0 given our benchmark
parameters. This implies the vaccination is an effective tool to achieve the herd immunity of population.
Further, we conducted sensitivity analysis with different choices of vaccination parameters, δ and λ. The
results verified our expectation that faster implementation of effective vaccines can help us to control the
infectious diseases. Lastly, we discussed how the mathematical concept of herd immunity differs from
what we commonly consider to be herd immunity.
References
Brauer, F., Van den Driessche, P., Wu, J., & Allen, L. J. (2008). Mathematical epidemiology (Vol. 1945).
Springer.
Fine, P., Eames, K., & Heymann, D. L. (2011). “herd immunity”: a rough guide. Clinical infectious
diseases, 52 (7), 911–916.
French, G., Hulse, M., Nguyen, D., Sobotka, K., Webster, K., Corman, J., . . . others (2021). Impact of
hospital strain on excess deaths during the covid-19 pandemic—united states, july 2020–july 2021.
Morbidity and Mortality Weekly Report, 70 (46), 1613.
Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM review , 42 (4), 599–653.
Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics.
Proceedings of the royal society of london. Series A, Containing papers of a mathematical and
physical character , 115 (772), 700–721.
Randolph, H. E., & Barreiro, L. B. (2020). Herd immunity: understanding covid-19. Immunity, 52 (5),
737–741.
Roussel, M. R. (2005). Stability analysis for odes. Nonlinear Dynamics, lecture notes, University Hall,
Canada.
Tornatore, E., Vetro, P., & Buccellato, S. M. (2014). Sivr epidemic model with stochastic perturbation.
Neural Computing and Applications, 24 (2), 309–315.
A
Attribution of Effort
• Byungmin Choi: coding, plotting, write up in latex. (sensitivity analysis, limitation)
• Dongwoong Seo: literature review, model setup, and write up in latex. (abstract, introduction,
model description, and discussion)
B
Code Appendix
Please refer to APM115 group project group3.ipynb.
7