Modeling COVID-19 Breakthrough Infections in a Vaccinated
Population
MING ZHU, EPHRAIM AGYINGI
School of Mathematical Sciences,
Rochester Institute of Technology,
84 Lomb Memorial Dr, Rochester NY 14623,
UNITED STATES OF AMERICA
Abstract: - The consequences of the COVID-19 pandemic that originated in Wuhan, China in 2019 are still
being felt globally. At the onset of the pandemic, countries had several measures in place to prevent the spread
of the virus. The development and availability of COVID-19 vaccines turned out to be one of the most effective
tools for containing the pandemic, especially in developed countries. This paper considers a model of COVID-
19 breakthrough infections, which are cases where individuals become infected with COVID-19 despite being
fully vaccinated. The model proposed is a type of the SIR model with a compartment accounting for vaccinated
individuals and is governed by a system of differential equations. We compute the basic reproduction number of
the model and use it to analyze the equilibria for both local and global stability. Further, we use numerical
simulations of the model to understand the factors that contribute to breakthrough infections such as vaccination
rates, vaccine efficacy, and virus transmission dynamics.
Key-Words: - COVID-19, SARS-CoV-2, Mathematical model, Vaccinations, Basic reproduction number
Received: September 13, 2022. Revised: April 25, 2023. Accepted: May 13, 2023. Published: June 2, 2023.
1 Introduction
The COVID-19 pandemic has affected millions of
people globally, causing unprecedented loss of life
and disrupting economies and social systems.
Vaccination emerged as one of the key tools in the
fight against the pandemic. Vaccines have been
developed and administered to people worldwide to
provide immunity against the virus, and they have
been shown to be highly effective, [1], [2].
However, there is still a risk of breakthrough
infections, where fully vaccinated individuals
contract the virus. Breakthrough infections occur
when an individual contracts COVID-19 after being
fully vaccinated, . These infections can occur
for several reasons, including vaccine efficacy, virus
variants, and individual immunity. While vaccines
are highly effective in preventing severe illness and
death, they are not 100% effective, [5]. This means
that even fully vaccinated individuals can still
contract the virus.
Virus variants, such as the Delta variant and
omicron variant, can also increase the risk of
breakthrough infections, [6]. These variants have
mutations that make them more transmissible and
resistant to antibodies, potentially reducing vaccine
efficacy, . Furthermore, individual immunity
can also play a role in breakthrough infections.
Factors such as age, underlying health conditions,
and medications can affect an individual's immune
response to the vaccine, [9]. Despite the risk of
breakthrough infections, vaccination remains critical
in the fight against COVID-19. Vaccines have been
shown to reduce the severity of illness and the risk
of hospitalization and death in breakthrough
infections, [10]. They also help to reduce the
transmission of the virus by providing herd
immunity, making it harder for the virus to spread
and mutate, .
Mathematical models have played significant
roles in epidemiological research and have been
widely used in the fight against COVID-19 since the
beginning of the pandemic,
, [19]. These models are
computer simulations that use mathematical
equations to predict the spread of the virus, the
impact of interventions, and the potential outcomes
of different scenarios. One of the key benefits of
mathematical models is their ability to provide early
warning signals of potential outbreaks, [20]. By
analyzing data on the spread of the virus,
mathematical models can predict the future
trajectory of the pandemic, [21]. Mathematical
models can also be used to evaluate the
effectiveness of different interventions, [22]. For
example, models can be used to compare the impact
of different vaccination strategies, such as
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prioritizing certain age groups or populations, to
identify the most effective approach, [23].
Several investigators have developed targeted
mathematical models for the transmission dynamics
of COVID-19 to facilitate further understanding of
the virus' characteristics and impact on humans,
. In this study, we present
a mathematical model with breakthrough infections
in vaccinated populations by COVID-19 or its
variants. We considered a SIR-type model of the
form SVIRD, where "V" and "D" represents the
vaccinated and death compartments, respectively.
The transitions between the compartments are
governed by a set of differential equations that
describe how the disease spreads over time. The
equations are based on the assumption that the rate
of transmission of the disease depends on the
number of susceptible individuals, the number of
infectious individuals, and the efficacy of the
vaccine. The model further assumes that vaccinated
populations are not immune from being infected by
the virus or its variants. Using the quantitative
results of the model, we will analyze the impact of
COVID-19 on a given population.
2 Mathematical Model
An SVIRD model is a mathematical model used to
simulate the spread of an infectious disease in a
population. It is an extension of the classic SIR
model, which stands for the Susceptible-Infectious-
Recovered model. The SVIRD model adds two
additional compartments to the SIR model: the
Vaccinated and Deaths compartments. The
compartments in the SVIRD model are defined as
follows: susceptible (S) are individuals who are
susceptible to getting infected by the disease;
vaccinated are individuals who have received a
vaccine against the disease and are thus protected
from getting infected; infectious (I) are individuals
who have been infected with the disease and are
capable of transmitting it to others; recovered (R)
are individuals who have recovered from the disease
and are no longer infectious; and deaths (D) are
individuals who have died from the disease.
We assume that the total human population at any
time denoted by is:
.
The transition rates between compartments together
with their descriptions are given in Table 1, and a
schematic diagram of the SVIRD model is given in
Figure 1.
Fig. 1: Schematic diagram of the SVIRD model.
The solid lines represent population movement from
one compartment to another. The transition from
to I or to I is a result of interaction between
individuals in the two compartments.
The system of differential equations that governs the
model is as follows:
Table 1. Description of variables and parameters in
the model, [1]
Symbols
Description
Recruitment rate into the population
Natural death rate of human individuals
The transmission rate to susceptibles
The recovery rate from infection
The transition rate from recovered to
susceptible
The transmission rate to vaccinated class
The vaccination rate of susceptible
The vaccination rate of recovered
The death rate from infection
where it is assumed that all the parameters are
positive and the initial conditions
(1)
(2)
(3)
(4)
(5)
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, and
, are all nonnegative.
Before proceeding, we note here that the fifth
equation is decoupled from the rest of the system
and thus will be neglected from subsequent analysis.
We also observe that if and , then the
model does not possess any vital dynamics. Further,
it is not difficult to establish that the right-hand side
of the system is locally Lipschitz continuous, thus
solution a with the
prescribed initial conditions exists and is unique. It
can also be easily shown that the model presented
only admits positive solutions since
,
,
,
and
,
are always positive.
3 Analysis of the Model
In this section, we state the disease-free equilibrium
(DFE) and the basic reproduction number of the
model. The DFE is defined as a steady-state solution
of the model in which the number of infected
individuals is zero. The basic reproduction number,
often denote as , is a threshold parameter used to
describe the potential of a contagious disease to
spread within a population. It is defined as the
average number of secondary infections that result
from a single infected individual in a susceptible
population. We use to show that the DFE is
locally and globally stable. Further, we establish the
existence of an endemic equilibrium (EE), a state
where the disease is present at a relatively stable
level, if .
To compute the equilibria of the model, we set
the derivatives in the first four equations, [1], to zero
and solve for the steady states in the
following equations where for simplicity we let
From above, by factoring in equation (8), it is
obvious that the DFE of the model is
Next, computing the reproduction number as the
spectral radius of the next-generation matrix was
evaluated at the DFE, we obtained,
Remark 3.1. The value of stated above can be
interpreted as a product of the sum of the
transmissibilities to susceptiles
and
vaccinated
, and the mean duration of
infectiousness
. If we assume that the
population is constant by setting the recruitment rate
, then we get
.
We now move on to establish the existence of an
endemic equilibrium that is biologically relevant,
that is, a positive equilibrium since the model does
not admit negative solutions. To achieve this
objective, we revisit equations (6), (8), and (9), and
observe that if then we get
and
, where
and .
Substituting for and in equation (7) we
obtain the following quadratic equation in the
variable :
where
(6)
(7)
(8)
(9)
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The following result guarantees that the model will
always contain at least one positive endemic
equilibrium.
Theorem 3.1. If , then the model presented
possesses at least one positive endemic equilibrium.
Proof. Let be a polynomial given by the left-
hand side of equation (10), that is,
Observe that from above. For we get
that if . Next, we see that
as . Thus, by the intermediate value
theorem, the function has a positive root on the
interval yields an endemic equilibrium.
We now consider the stability of the DFE. The
following result provides a condition under which
the DFE is locally stable.
Theorem 3.2. The disease-free equilibrium of the
model is locally asymptotically stable if the
reproduction number , otherwise, it is
unstable.
Proof. Evaluating the Jacobian matrix of the model
at the DFE gives
whose eigenvalues are
1) and . Clearly, if ,
then the eigenvalues are all negative, thus rendering
the DFE to be locally asymptotically stable. If
, then one of the eigenvalues is positive therefore
rendering the DFE unstable.
We will require the following lemma to show that
the DFE is globally stable.
Lemma 3.1. Suppose that
and
, then respectively,
and
.
Proof. From the differential equation 1, we obtain
the differential inequality.
whose solution is given as
.
Clearly,
if
. To establish the
second inequality, we use the differential inequality
which is a consequence of the differential equation
(2). On substituting for
in the above
inequality, we get
whose solution satisfies
It follows that
if
.
The next result which is stronger than the previous
theorem establishes the global stability of the DFE
by using a Lyapunov function and LaSalle's
invariance theorem, [29].
Theorem 3.3. Suppose that
and
according to Lemma 3.1. Then the disease-free
equilibrium of the model is globally asymptotically
stable if the basic reproduction number .
Proof. We consider the Lyapunov function
candidate
.
Differentiating with respect to gives
Clearly, if then we have that
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Therefore by LaSalle's invariance principle, the
DFE is globally asymptotically stable.
4 Numerical Results and Discussion
In this section, we present numerical simulations of
the model for different scenarios. The values of the
parameters used in the simulations were estimated
based on values from the published literature for
COVID-19, [30], [31], [32], [33], [34], and other
parameter values were assumed to demonstrate the
predictions of the model. All populations were
normalized to the total population . We also
assume that the recruitment rate so that the
population remains constant. The simulations
reported in this paper were carried out using the
initial conditions ,
and . We
remark here that several simulations were carried
out with small perturbations or different sets of
initial conditions and the long-term behavior of the
model's predictions were unchanged.
Fig. 2: Model simulations demonstrating a disease-
free equilibrium for parameter values
, , and .
We start by confirming numerically the analytic
results established in the previous section. The
simulations in Figure 2 demonstrate numerically
both the existence and global stability of the
disease-free equilibrium. The introduction of an
infected individual leads to a wave of COVID-19
propagating within the population until it reaches a
peaking. After cresting, the wave continues to die
out till there are no more infectious individuals left
in the population.
Fig. 3: Model simulations demonstrating an endemic
equilibrium. The parameter values are the same as in
Figure 2 with the exception that .
This result appears to reflect a pattern so far
observed in COVID-19. That is the emergence of a
variant that eventually dies out or is displaced by a
more competitive variant. Recall that we only
established the existence of an endemic equilibrium
in the previous section without discussing its
stability. The result in Figure 3 represents an
endemic equilibrium for the chosen parameter
values. The results demonstrate COVID-19
persisting in the population after a wave has crested,
however, at a stable level.
Fig. 4: Model simulations of COVID-19 infections
demonstrating the efficacy of a vaccine.
Here we simulate the number of cases by varying
the transmission rate to vaccinated individuals. All
other parameter values are as stated in Figure 2.
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Fig. 5: Model simulations of COVID-19 infections
demonstrating the effect of the vaccination rate on
the trajectory of COVID-19. Here we vary , set
0.025 and all other parameter values are as
stated in Figure 2.
Next, we turn our attention to simulating the
efficacy of COVID-19 vaccines. We do so by
comparing the number of COVID-19 infections for
vaccinations having different degrees of
effectiveness. We assume that a vaccine is more
effective if the transmissibility rate to vaccinated
individuals approaches zero, and that
effectiveness is achieved if . The results
presented in Figure 4 show the number of COVID-
19 cases for each vaccination regiment. A vaccine
having a high transmissibility rate will
lead to COVID-19 being endemic while one with a
much lower transmission rate will
eventually lead to the eradication of the virus. It is
important to observe that in the simulations
provided, it takes a while before a disease-free
equilibrium is attained. This underscores the
necessity of public health officials to recommend
additional measures needed to combat the pandemic.
A mildly effective vaccine is better than none.
Though an effective vaccine is crucial in reducing
the overall incidence of COVID-19, the results in
Figure 5 do show that the behavior of the population
plays an important role. No matter how effective a
vaccine is, if only a small proportion of the
population is getting vaccinated, then COVID-19
will persist in the population. However, a higher
vaccination rate will shift an endemic state to a
disease-free state. Put together, the simulations in
Figures 4 and 5 demonstrate that to move the
population towards herd immunity, the vaccine has
to be very effective and the rate of vaccine uptake
should also be very high.
Fig. 6: Model simulations of COVID-19 infections
demonstrating the longevity of immunity provided
by the vaccine.
Here, we set and vary the vaccine
waning rate . We consider the effect on the
number of cases ranging from months of
protection. All other parameter values are as stated
in Figure 2.
The efficacy of a vaccine is also dependent on
the longevity of the immunity it provides. To
investigate this, we slightly modify the first two
equations of the model by subtracting a vaccine
waning term from the vaccination compartment
and adding it to the susceptible compartment. With
this modification, the DFE of the model is
. A corresponding
basic reproduction number is
, where we have set (see
Remark 3.1). Clearly, if we recover the
previous DFE and . Here, we take to be the rate
at which the COVID-19 vaccine is waning. The
results given in Figure 6 consider different cases
where the protection granted by COVID-19 ranges
from months. We can see that, the shorter
the duration of afforded protection, the more the
number of COVID-19 cases, thus necessitating the
need for more frequent vaccination or boasting.
Finally, the mean duration of infectiousness, that is,
how long it takes infectious people to recover and
stop being a threat to others plays an important role
in determining the trajectory of each COVID-19 or
its variants. The results in Figure 7 simulate the
trajectory of COVID-19 for infectiousness periods
of 7 and 14 days for vaccination and non-
vaccination regiments. The results indicate that
COVID-19 or a variant that has an infectiousness
period of up to 14 days is likely to remain endemic
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even in a vaccinated population. On the other hand,
a variant with a shorter infectiousness period when
combined with vaccination will be quickly
eradicated.
Fig. 7: Model simulations of COVID-19 infections
by comparing different vaccinations status and
recovery rates .
We let represent no vaccine uptake and
for vaccine uptake. The recovery rates are
set at and , indicating an infectiousness
period of 7 and 14 days, respectively.
5 Conclusion
In this paper, we have developed an SVIRD model
for COVID-19 and used it to study breakthrough
infections in a vaccinated population. The model is
based on the assumption that COVID-19
vaccinations do not provide full immunity to
vaccinated individuals. The model is formulated as a
system of differential equations and the existence of
a disease-free equilibrium that is globally stable
when the basic reproduction number is less than one
is established. The existence of an endemic
equilibrium is also ascertaining if the basic
reproduction number is greater than one and
numerical simulations do show that it is globally
stable. We used simulations of the model to study
different vaccination strategies and the predictions
do provide early warning signals of potential
outbreaks. A key prediction of the model is that an
effective vaccine will reduce the number of
infections in a short amount of time as long as there
is a high vaccination rate. Considered alone, a very
effective vaccine, or one that protects for more than
a year, combined with low vaccine uptake may
eventually eradicate the disease, however, only after
a very long time. The model also predicts that given
any vaccine that affords some immunity, if there is a
high vaccination turnout, then the virus will still be
eradicated in a shorter amount of time. Further
predictions of the model suggest that knowing the
infectious period of COVID-19 or its variants is
very important. Variants with shorter duration of
infectiousness are easily eradicated while those with
long durations are likely to become endemic within
a population. The predictions made in this study are
mostly limited to the assumptions made and the
values of the parameters used. The model does not
account for all the factors that can influence the
spread of the virus, such as human behavior and
social dynamics. In conclusion, while breakthrough
infections do occur, vaccination remains an essential
tool in the fight against COVID-19. Vaccines are
highly effective in preventing severe illness and
death, and they can help to reduce the spread of the
virus. Additional measures, such as booster shots,
masks, and social distancing, can also help to reduce
the risk of breakthrough infections and protect
individuals who are not yet vaccinated. Individuals
must continue to follow public health guidelines and
get vaccinated to help bring an end to the pandemic.
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The authors equally contributed in the present
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Conflict of Interest
The authors have no conflict of interest to declare.
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WSEAS TRANSACTIONS on SYSTEMS
DOI: 10.37394/23202.2023.22.59
Ming Zhu, Ephraim Agyingi
E-ISSN: 2224-2678
592
Volume 22, 2023
