1. Introduction
COVID-19 has not been properly controlled for over two years, and the number of new infections remains among the highest ever [
1]. According to the World Health Organization (WHO), as of 21 June 2022, there were 187,108,697 confirmed cases globally, with 3,841,225 deaths. Global public health and economic problems are at risk due to the COVID-19 outbreak. In the United Kingdom, South Africa, and Brazil, SARS-CoV-2 variants Alpha (VOC 2020/12/01), Beta (501Y.V2), and Gamma (P.1) have been found. In numerous nations, COVID-19 is spreading faster due to its higher transmission rate [
2]. The early stages of COVID-19 were fought with non-pharmaceutical intervention tactics, such as contact tracking, social distancing, isolation, treating sick people, and lockdowns. These restrictions, however, disrupt people’s lives and significantly impede economic development. Consequently, when COVID-19 outbreaks slowed, many countries relaxed these efforts to strengthen their economies. The spread of COVID-19 has not been stopped as a result. To prevent COVID-19’s spread and minimize its impacts on the economy, effective vaccines must be invented and utilized. In order to control COVID-19 effectively and reduce its effects on economic development, people are looking forward to developing and using effective vaccines. Vaccination effectively controls epidemic spread. Several vaccines have been approved for use through the unremitting efforts of all parties, bringing hope that the spread of COVID-19 can be completely controlled [
3,
4,
5,
6].
Vaccinations have received relatively little attention in the study of COVID-19 spread. In [
7], a mathematical model was used to investigate the impact of a hypothetical ineffective vaccination on COVID-19 control in the United States. A SIRV model was proposed in [
3] to predict and model the spread of the COVID-19 outbreak in the presence of vaccination. Reference [
8] presents a mathematical model that analyzes the effects of medication (vaccination with complete efficacy) and drug-free prevention strategies on the spread of COVID-19. A wide range of literary studies have been published on COVID-19 models, including review papers, special issues, and books [
9,
10,
11]; however, deterministic modeling of COVID-19 and vaccination is very limited [
6]. In mathematical epidemic models, time delays are investigated to better understand and characterize the dynamics of infectious diseases; for example, see [
12,
13,
14,
15]. Further, time delays may lead to periodic solutions via the Hopf bifurcation; see, e.g., Reference [
16] and references therein. COVID-19 spreads primarily during latent and incubation periods, as with other viral diseases.
Mathematical models can be used to predict and simulate the spread of epidemics and provide a theoretical basis for developing epidemic prevention strategies [
4,
17,
18,
19,
20,
21,
22,
23]. Many mathematical models have been developed to simulate COVID-19 spread, including those in [
24,
25,
26,
27,
28,
29,
30]. Infectious disease research utilizes classical mathematical models with integer-order derivatives, see [
31,
32,
33,
34]. Fractional derivatives, however, are advantageous in mathematical models due to their non-locality or long-memory properties [
35,
36]. COVID-19 dynamics in Lagos, Nigeria, for instance, are described in [
37]. The use of fractional derivatives and fractional integrals in epidemiological modeling is important because they can be used to describe memory and hereditary characteristics of various materials and processes [
38,
39,
40,
41]. When comparing the fractional-order derivative to the integer-order derivative, it is evident from the literature that fractional operators provide more accurate and deeper results while representing real-life scenarios [
42,
43,
44]. The model used in this work is called the SIR model; it divides the population into three groups, namely susceptible (S), infected (I), and recovered (R). The extensions of this basic model have been applied to COVID-19 in recent times. The main difficulty with this type of model during a pandemic is in determining the model’s parameters. A reasonable estimation of constant and time-dependent parameters can be made using the conventional least squares method. Using a fractional-order mathematical model, we investigated the effects of COVID-19 vaccinations on the dynamics of the disease in a population. The proposed model, in this paper, addresses two new issues: (1) the effects of time delay on infection transmission rates; and (2) vaccination rates in the infected population. We illustrate how the fractional derivative order affects the dynamics of state variables by using the fractional-order differential equations numerical solver in MATLAB. As a result of the study, government, and public health authorities may be able to develop new strategic plans for reducing the spread of COVID-19 outbreaks in the future. The suggested model is examined by fitting real observations in the UAE.
This paper has the following structure.
Section 2 presents a delayed fractional-order mathematical model for COVID-19 vaccine efficacy. We provide sufficient conditions for the positivity, boundedness, and uniqueness of mathematical model solutions in
Section 2.1.
Section 3 provides details of the model’s dynamic analysis.
Section 4 provides numerical simulations of the fractional-order vaccination model and the impact of changing the fractional derivative order. Conclusions are found in
Section 5.
2. Preliminaries and Mathematical Formulation of the Model
The fractional derivatives have several definitions. The Caputo-type fractional derivative is the most common and is utilized in real-world applications. The fractional derivative of a function
with the Caputo type is defined as [
45]:
where
n is the first integer greater than
, i.e.,
,
is the gamma function.
The
order Riemann–Liouville integral of a function
is expressed as [
45]:
The one-parameter and two-parameter forms of the Mittag-Leffler functions are defined as [
38]:
where
.
Numerous researchers have examined various types of models in order to understand the dynamics of COVID-19 using case studies of various specific nations. In order to minimize the likelihood of infection in a susceptible population, vaccination is one of the most effective approaches. Despite the lack of COVID-19 vaccination at birth, the inclusion of it in our theoretical study does not affect the conclusions because some analytical/numerical modeling results are independent of the type of vaccination program used [
46]. A graphic representation of the impact of the model settings on initial disease transmission is used to reach this conclusion. COVID-19 has reappeared after numerous waves and strains, and vaccines are still being developed with lower age groups in mind. Since vaccination can involve vaccinating individuals from birth, our proposed strategy is proactive. In order to control epidemics, vaccinations are essential. Globally, several COVID-19 vaccines are in use. In this section, we extend the model proposed by Torku et al. [
47] on COVID-19 vaccinations to determine if the disease can be contained by solely relying on the vaccine. The proposed model is governed by a simple system of ODEs.
At any time
t,
represents susceptible individuals,
represents infected individuals, and
represents recovered individuals. The first equation presents the rate of change of susceptible individuals in the ordinary model (
1).
represents the transmission rate,
represents the efficacy rate, and
represents the vaccination rate. The second equation presents the rate at which infected individuals change, while
represents the rate at which infected individuals recover. Assume that
, and
are the initial conditions for the model discussed above. The population is assumed to be homogeneous and to have equal chances of becoming infected. This study only considers the human-to-human transmission of COVID-19.
represents the human population at time
t based on the disease status of people. The
population is divided into three subpopulations:
susceptible individuals,
infected individuals, and
recovered individuals. Based on the vaccination regime,
is assumed to remain constant. Since the population is homogeneous, the standard incidence is
. Parameter values are given in
Table 1.
During the course of a disease, time delays occur spontaneously and are significant factors. As a representation of the latent period of the intervention strategies, we include a discrete time-delay
in system (
1). In system (
1), human behavior is adapted to intervention tactics. Due to poor knowledge about the disease, people are more likely to be infected when a new infectious disease is discovered. Further, as the number of infected individuals increases and the disease becomes more serious, psychological factors lead people to change their behaviors and implement appropriate measures/interventions to reduce the chances of infection. During infectious disease modeling, delays in intervention processes are significant. In [
51], for instance, the length of time for people to react to the reported infection, as well as the delay in reporting, were noted. In addition, the fractional derivative is highly effective in modeling epidemic transition systems since it takes into account the memory effects and the system’s universal features, which are important for deterministic systems. The fractional operator has this memory effect property, making it particularly useful in modeling the COVID-19 model since its future state is dependent on its current state. By substituting the Caputo fractional derivative with the first derivative, we can incorporate past historical or hereditary features into the model. The graphical representation of the interactions between the populations in the proposed model is shown in
Figure 1. Thus, the time delay fractional-order differential equations system can be generalized as follows:
Although the time-delay system (
2) is simple, it provides complex dynamics. For its solution, we should provide initial history conditions:
defined in terms of space
where
,
is the Banach space of all continuous functions in the interval
into
, and considering the feasible region of system (
2) as
and
is the interior of
Thus, the region is positively invariant with respect to system (
2), which means that all solutions of model (
2) are contained within the above region for all time
t, and those outside are ultimately attracted to it. In this sense, system (
2) has been posed appropriately from an epidemiological perspective.
Remark 1. Is it possible to completely eliminate COVID-19 through vaccination? A fractional-order model with time delay is used here to analyze it. As the infectious disease outbreak spreads, fractional-order models are very useful for evaluating the efficiency of several interventions, such as vaccination and lockdown. In this study, the primary objective is to evaluate the efficacy of COVID-19 infection vaccination strategies using the fractional-order model with the Caputo-type derivative. Memory is accounted for by the fractional order.
2.1. Positivity, Boundedness, and Uniqueness of the Solution
Positivity is necessary for biologically feasible model solutions, whereas boundedness indicates that solutions are finite.
Lemma 1. (Positivity) If , then the solution of the fractional-order model (2) remains non-negative for all positive times t. Proof. Since , one obtains for any Thus, remains non-negative In the same manner, are all non-negative. This completes the proof. □
Our next step is to demonstrate that the model system solutions (
2) enter a bounded region.
Lemma 2. (Boundedness) All of system (2)’s solutions with non-negative initial history conditions are bounded. Proof. To show that system (
2) is bounded, the population growth can be expressed as
It can be seen from (
6) that
where
. Since the human population
is positive, by solving Equation (
7), the total human population satisfies the following equation:
The solution is given by
where
is the Mittag-Leffler function. The Mittag-Leffler function
is asymptotic in nature; knowing that, the asymptotic behavior of the Mittag-Leffler function is as follows:
In particular,
for
, i.e., the exponential function. In light of this, all of system (
2)’s solutions with non-negative initial history conditions remain bounded. The proof is completed. □
Lemma 3. (Uniqueness) For every there exists a unique solution of system (2), where Proof. Based on the Banach space of all continuous and differentiable functions from , we demonstrate that is Lipschitz continuous with Lipschitz constants using the fundamental fixed-point theorem. Based on a triangle inequality and the Chebyshev norm, let be the second solution.
Consider the contraction mapping
Then, for any vectors
,
where
Thus, with the agreement in (
10) and Lemma 5 in [
52],
satisfies the Lipschitz condition in its second argument with the Lipschitz constant
, then model (
2) has a unique solution
□
2.2. Equilibrium Points (Disease-Free and Endemic)
In this subsection, we explore the existence of equilibrium points. According to (
2), the stability analysis of model (
2) is carried out to determine the disease-free and endemic equilibrium point. Every equation in (
2) needs to be equated to zero in order to establish the equilibrium points, or
, achieved as follows:
Then the equilibrium point of
is determined. The equilibrium where the number of infected individuals is zero is the so-called disease-free equilibrium. When COVID-19 is not spreading, the conditions that define the disease-free equilibrium are met, which means
. Using (
11), we obtain
. Therefore, the disease-free equilibrium points for the COVID-19 vaccination model are:
. By taking into account the scenario in which
is positive, we can identify the endemic equilibria of the model. Endemic equilibrium points are used to predict whether a disease will continue to spread because populations
under endemic conditions when the disease is spreading. Solving for
and
in Equation (
11), the endemic equilibrium points for the vaccination model were determined:
Then, the COVID-19 vaccination model equilibrium points of the endemic are:
2.3. Basic Reproduction Number
The basic reproduction number,
, is an epidemiologically significant threshold value that predicts the probability that infectious disease will spread throughout a population. The matrix generation method is used to determine the basic reproduction number
. Using Equation (
2), we determine
The compartments of model (
2) consist of
classes if we take
; we now want to write the infection subsystem in the following form:
which is equivalent to
Let
, system (
15) can be rewritten in the following form:
where
, and
d are defined as
The following differential equations are satisfied by the remaining variables:
The expected number of secondary cases produced by a single infected person over the course of his/her infectiousness in a population that is totally susceptible is known as the basic reproduction number
. Then from (
17), using the approach of matrices generation method, we obtained the basic reproduction number
of (
2) as
is denoted as the basic reproduction number in the without-vaccination cases. In a vaccination scenario, the current reproduction number
is defined as the reproduction number with respect to time. It is calculable as
depends on a time-varying recovery rate
and transmission rate
. The effective reproduction number
is defined as
where
is the number of susceptible people,
is the population density of a certain location, and
is the basic reproduction number at a given point in time.
3. Stability and Hopf Bifurcation Analyses
This section focuses on the local stability and bifurcation analysis of model (
20). For the local asymptotic stability analysis, let us reduce system (
2) by discarding the last equation as
does not appear in the first two equations of model (
2). If we study the qualitative or dynamic behaviors of
, then the dynamic behaviors of
are also obtained from the dynamic behaviors of
. Here is the simplified fractional system:
The equilibrium points of model (
20) are defined in
Section 2.2. The method of linearization entails taking a nonlinear function’s gradient with regard to each variable and converting it into a linear representation at that point. It is necessary for some analyses, including stability analysis, Laplace transform solutions, and putting the model into a linear state-space form. Consider the differential model (
20). The right-hand side of the model can be linearized at any steady-state
using a Taylor series expansion, which involves only the first two terms.
Next, we take the Laplace transform on both sides of (
21) to obtain
where the Laplace transforms of
and
are
and
, respectively. Then, Equation (
22) can be written as
where
(
) is the characteristic matrix of system (
20) at
The characteristic equation of (
20) at the disease-free equilibrium
is represented by
Now, as observed is the fact that if
then the above characteristic equation becomes
which has two roots
When all coefficients of the characteristic Equation (
26) are positive, both the roots in (
27) will be negative. Therefore, the disease-free equilibrium
is locally asymptotically stable. In case
, the characteristic Equation (
25) has the following roots
and the equilibrium is locally asymptotically stable. The stability conditions of the infection-free steady-state are presented in the following theorem.
Theorem 1. The characteristic equation at the disease-free equilibrium has two negative roots, when , and . Then the disease-free equilibrium is locally asymptotically stable.
Remark 2. Eventually, the disease will disappear if is less than 1. If is greater than 1, severe effects will result. When , the disease is spreading steadily and persistently. Increasing (decreasing) the parameter δ (β) leads to an increase in the basic reproduction number from (18). A small change in any of these factors can result in a large variation in the reproduction number . Remark 3. All of the roots of Equation (25) have negative real parts due to . The disease-free equilibrium is locally asymptotically stable when . If , Equation (25) has a positive root. As a result, the disease-free equilibrium is unstable. Now we will look at the local stability of an endemic equilibrium
. Because
and
, then the following characteristic equation is obtained at endemic equilibrium
or
Let us proceed with
, then (
29) takes the following form
When
, it is important to note that the Routh–Hurwitz criterion provides both sufficient and necessary conditions for roots of (
31) to have negative real parts and
,
are the conditions. Therefore, if
, the equilibrium point
is locally asymptotically stable.
It will then be verified that
does not have any pure imaginary roots for any
. The fact is testified by contradiction. Assume that there exists a pure imaginary root
for (
30), where
is a real positive number. When
, we substitute the pure imaginary root
in the Equation (
30), obtaining
Separating the real and imaginary components of (
32) results in
Using Cramer’s rule to solve (
33) and (
34), one obtains
Let
and
where
If the condition
holds, according to
for all
. It is evident that
Then if
, then (
37) has at least one positive real root. Therefore, (
32) has at least one pair of purely imaginary roots.
If
. Then (
32) becomes
where
If the condition
holds, then all the roots
of (
40) satisfy
Therefore, we can deduce that the endemic equilibrium point
of (
20) with
is locally asymptotically stable.
Lemma 4. The characteristic Equation (37) of model (20) has no pure imaginary roots if and . If , then the characteristic matrix of model (20) has at least a pair of purely imaginary roots. If and there exists a positive constant such that the derivative , then the characteristic Equation (37) has at least two pairs of purely imaginary roots. Proof. In light of
and
, then
for all
and
Then we already know that (
37) has no positive real root. Therefore, (
32) has no pure imaginary root. In light of the condition
,
is not the solution of the characteristic Equation (
32). Therefore, the characteristic Equation (
32) does not have any root with a zero real part. Evidently,
and
. Then (
37) has at least one positive real root. Therefore, Equation (
32) has at least one pair of purely imaginary roots. Moreover, we know that
and some positive constant
, then the derivative
and
, then there exists two constants,
and
, such that
, which implies that (
37) has at least two positive real roots. Then the characteristic Equation (
32) has at least two pairs of purely imaginary roots. This completes the proof of Lemma 4. □
We suppose that (
37) has a positive real root
. By (
35), we obtain
where
. Then
is the pair of purely imaginary roots of (
32) when
, denoting
and
Lemma 5. If we write (38) as where , , if holds and , then (32) has no root with zero real parts when If and , then (32) has a pair of purely imaginary roots when , where is the unique positive zero solution of the function We give the following assumption
where
Lemma 6. If is the root of (32) near , satisfying and , then the following criterion holds Proof. By condition
, one has
The proof is now completed. □
Theorem 2. Based on the discussion for system (2), if and hold, then the endemic equilibrium is locally asymptotically stable when and a Hopf bifurcation appears around when We derived certain conditions in the preceding section under which system (
20) experiences the Hopf bifurcation at
In this section, we assume that when
, system (
20) experiences a Hopf bifurcation at the zero equilibrium, which is from the zero equilibrium, a family of periodic solutions bifurcates. In the following, we use the normal form theory and center manifold reduction from [
53] to find the Hopf bifurcation direction, which determines whether the bifurcating branch of the periodic solution occurs locally for
or
, and we identify the features of these bifurcating periodic solutions, such as the center manifold stability and period. It is important to assume in the following that
. For convenience, let
and
where
is defined in the above section and
, then system (
20) can be written as the functional differential equation in
as
where
,
and
are, respectively, given by
and
where
.
According to the discussions above, if
, system (
45) experiences a Hopf bifurcation at the zero equilibrium, and the associated characteristic equation of system (
45) has a pair of pure imaginary roots
.
According to the Riesz representation theorem [
54], there exists a bounded variation function
for
such that
Moreover, one can choose
where
is defined by
For
, define
and
The functional differential system (
45) is then equivalent to
where
for
For
, the adjoint operator
of
is defined by
and a bilinear inner product for
, and
is defined by
where
Then,
and
are adjoint operators, as
From Lemma 4, we know that
are eigenvalues of
. As a result, they are also eigenvalues of
. Suppose that
is the eigenvector of
corresponding to the eigenvalue
. Then,
From (
46) and (
49), one has
According to the definitions of
and (
47), where
which gives that
Similarly, it can be demonstrated that
is the eigenvector of
corresponding to the eigenvalue
where
. Then,
To ensure that
, the value of
D must be determined. By (
45), (
46), and (
54), one has
Thus, in order to ensure can be chosen as
Next, we compute the coordinates describing the center manifold
at
using the same notations as before. Let
be the solution of (
45) when
Then, define
In the center manifold
, we obtain
where
where
K and
are local coordinates for the center manifold
in the direction of
and
Notice that
H is real if the solution (
45)
is real and we consider only real solutions. For the solution
in the center manifold
and
, one has
This equation is rewritten as
where
From (
57), we have
and
and then
It follows from the definition of
that
when we compare the coefficients to (
61), we have
In order to calculate
, we must first compute
and
. By (
51) and (
57), one has
where
and
denotes
at
In light of (65), one obtains
On the other hand, notice that on the center manifold
This, along with (
58) and (
66), equals
From (
64), we know that for
When the coefficients are compared to (
65), it is revealed that
From (
49), (
68), and (
70), we can obtain
We know that
, one has
where
is a constant vector. Similarly, we can obtain from (
68) and (
70)
where
is also a constant vector.
Following that,
and
will be determined. Based on the definition of
L and (
68),
and
where
. By (
64) and (
65), we have
When the coefficients of (
76) and (
65) are compared, the result is
and
By noticing that
then form (
72) and (
77), we obtain
where
which leads to
Similarly, by substituting (
73) and (
78) into (
75), we can obtain
and we have
where
Further, we can also calculate the values listed below:
that establishes the number of bifurcating periodic solutions on the center manifold at
. Then the following results are obtained
Theorem 3. - (i)
The stability of the bifurcating periodic solution is determined by : when , the bifurcating periodic solutions are stable or (unstable).
- (ii)
The direction of the Hopf bifurcation is determined by : when or , the Hopf bifurcation is supercritical (subcritical), and for , bifurcating periodic solutions exist.
- (iii)
The period of the bifurcating periodic solution is determined by : when , the period increases (decreases).
4. Simulation Results and Discussion
This section numerically investigates the local stability and Hopf bifurcation of the COVID-19 vaccination model, exhibiting our findings from
Section 4. Moreover, we simulate how vaccinations impact COVID-19 prevention and control. Finally, we investigate the impact of time delay on the epidemic, and we make reasonable suggestions for effectively reducing the COVID-19 epidemic. All numerical computations were carried out in MATLAB R2020b and Maple 2013 numerical computing environments using the Adams–Bashforth method. Calculating the parameters of the model is difficult because the COVID-19 scenario changes frequently and from nation to nation. The parameters are likely to change over time as new policies are implemented on a daily basis. As a result, in order to simulate the COVID-19 vaccination model (
2), we use certain model parameters from the literature and estimate or assume the rest based on actual conditions. For other assumed values, the model is stable and can provide the model results under reasonable conditions. The most extensively used vaccinations currently available have efficacies of 95% (Pfizer) for the COVID-19 mRNA vaccine BNT162b2, 94.1% (Moderna) for the mRNA-1273 vaccine, 78% for Sinovac, and 70.4% (AstraZeneca) for the ChAdOx1 nCoV-19 vaccine/AZD1222, according to reports from appropriate departments. We take
by considering the efficacy of various vaccines.
In
Table 1, all parameter values are displayed. These factors are used to calculate
without vaccinations, as shown in
Table 2 and
Figure 2c. This shows that during the infection period, a COVID-19-infected individual can cause disease, on average, 2 to 3 susceptible individuals. In this instance, COVID-19 is spreading quickly. In this model, stability can be achieved and valid conclusions can be drawn with reasonable parameter assumptions.
In
Figure 3, the vaccination model (
2) is presented without time delay. The effect on the population’s susceptible
, infected
, and recovered
cases can be seen; whereas
Figure 3d–f show that the proportion of susceptible individuals,
, declines as vaccination rates,
, rise, and the proportion of recovered individuals,
, increases. With the chosen fixed vaccination efficacy rate
, two significant points were obtained, one is model (
2) with the vaccination percentage
; it is evident that
Figure 4d shows that the percentage of the spread of the disease decreases. In model (
2) with
, it is evident that
Figure 4f shows that the percentage of the spread of the disease further decreases. This implies that the virus will stop spreading more quickly with a higher vaccination rate.
Taking the initial state value as
, we analyze the behaviors of the numerous different infected populations
without time delay with a fixed efficacy rate
within 300 days in the two vaccination cases (with and without), as shown in
Figure 5. It can be seen that the level of infectiousness decreases as the vaccination rate
increases from
to
. This demonstrates that COVID-19 can be effectively contained through vaccination.
Figure 5a shows the changing trend of the infected population with fractional-order
Without vaccination, the infected population rises at around the 30th day and there are about 220 infected individuals. The peak of the infected population
in the case of the vaccination rate
occurs around day 15, and there are roughly 135 asymptomatic individuals. In
Figure 5b,c the peak of the infected population
in the case of the vaccination rate
occurs around day 12 and day 8, and there are roughly 137 and 139 asymptomatic individuals, respectively. That is, a 7-day delay in the peak of
will significantly lessen the burden that COVID-19 is placing on medical resources, allowing more people to access timely medical care and lowering mortality.
In
Figure 6, the vaccination model (
2) is presented with a time delay. The effect on the population’s susceptible
, infected
, and recovered
cases can be seen; whereas the proportion of susceptible individuals
declines as the fractional order
increases, while the proportion of recovered individuals
declines with the fixed vaccination rate
From the analysis of
Figure 5,
Figure 6b and
Figure 7, it is evident that increased vaccination doses can minimize and delay the peak of infection to a greater extent. Thus, improving vaccination efficacy can help to prevent the spread of COVID-19 more effectively. However, in practice, the effectiveness of vaccines cannot be improved quickly, and COVID-19 cannot be quickly controlled by simply increasing vaccination rates. Therefore, in addition to vaccinations, several non-pharmaceutical measures must be used.
Figure 5a and
Figure 6b compare the trends in the number of affected individuals for the vaccination rate
and fractional order
, with a rate
within 100 days. The peak of the infected population
in the case of a vaccination rate of
occurs around day 15 (
Figure 5a) and day 2 (
Figure 6b), with roughly 135 and 130 asymptomatic individuals, respectively.
Daily COVID-19 cases in the UAE were considered for the time period from 25 March to 30 July 2020 for the population without vaccination, from 15 July 2021 to 15 December 2021 for a 50% vaccinated population, and from 15 January 2022 to 20 May 2022 for a 100% vaccinated population. The daily COVID-19 updates were obtained from [
55]. The graphical data in
Figure 4 and
Figure 8 illustrate the evolution of diagnosed infected and recovered cases with and without delays, demonstrating the biological impact of delay parameters. We fit model (
2) to daily new infected cases of COVID-19 in the UAE from 25 March to 30 July 2020 in
Figure 4a,d and fit model (
2) to daily new recovered cases of COVID-19 for the UAE in
Figure 8a,d.
Figure 4b,e, show the fitting of model (
2) to the cumulative daily COVID-19 vaccination data in the UAE from 15 July 2021 to 15 December 2021 for a 50% vaccinated population, and in
Figure 8b,e, we fit model (
2) to the recovered cases after 50% of the UAE population was vaccinated. In
Figure 4c,f, the model is fitted to the cumulative daily COVID-19 vaccination data in the UAE from 15 January 2022 to 20 May 2022 for 100%, and in
Figure 8c,f, we fit the model to the recovered cases after 100% of the UAE population was vaccinated. The number of diagnosed infected and recovered cases is highly impacted by delays, as seen in
Figure 4 and
Figure 8. As a result, the plot of model (
2) without delays is different from that of the clinical data. Thus, we can conclude that delays are crucial to understanding the dynamic behavior of COVID-19 around the world, particularly in the UAE. The two figures,
Figure 4 and
Figure 8, individually demonstrate how well our model fits the three data sets from [
55]. Therefore, our vaccine efficacy models are efficient in describing the spread of COVID-19 in the UAE.
Figure 3 and
Figure 5 demonstrate that vaccination (
) is effective in lowering infection rates and that early treatment and management of COVID-19 have a positive impact. It is also clear that despite the high efficiency of COVID-19 vaccines, the outbreak was difficult to control. One of the reasons for this is the occurrence of bifurcation, where
converges to a non-zero constant. In this instance, COVID-19 coexists with humans for a considerable amount of time before becoming an endemic disease. If
converges to zero, the rate of convergence is quite slow since the basic reproduction number is too large and close to one. We need to perform a calculation to determine the bifurcating parameter
that causes this. It is not difficult to obtain the bifurcating parameter
and the critical frequency
(see
Table 3 and
Figure 2b). Moreover, the bifurcating parameter varies according to model parameters such as the transmission rate
, recovery rate
(see
Table 2), vaccination rate
(see
Table 4), and efficacy rate
(see
Table 5). Since
, bifurcation appears in the vaccination model (
2), as shown in
Figure 2a (the curve corresponding to the fractional-order
) and
Figure 9. We compare the dynamics of the number of infected cases without time delay with different transmission rates
within 300 days (see
Figure 10) for with and without vaccination (see
Figure 11). It can be seen that the level of infectiousness decreases as the vaccination rate
if fixed. This demonstrates that COVID-19 can be effectively contained by increasing the vaccination rate
.
Recent research has focused on investigating the long-term effects of vaccination on controlling COVID-19 incidence, utilizing mathematical modeling as seen in [
56,
57]. In biological systems with memory, both time delays and fractional orders play a significant role, which provides the model with greater flexibility. To investigate the impact of vaccination coverage on disease incidence, we simulate the COVID-19 fractional-order time-delayed vaccine model (
2).
Figure 5 and
Figure 7 present comparison plots for the Caputo fractional-order model, considering various values of the fractional-order parameter
; the graphical results are presented for comparison. The number of infected individuals decreases as vaccination efficacy rates increase. As can be seen from these visual findings, vaccination rates play a crucial role in controlling infections. The infected population decreases as the fractional order increases. Furthermore, simulations were conducted to demonstrate the dynamics of daily COVID-19 cases in the UAE when vaccination coverage increased by 50% and 100%.
Figure 6 and
Figure 9 illustrate the resulting graphical interpretation. The graphs in
Figure 6 show that a higher vaccination rate significantly reduces the peaks of infected curves. In particular, a 100% increase in vaccination coverage can significantly reduce infection peaks and even eradicate the disease. Therefore, these results demonstrate that if the vaccination rate is high enough and the vaccines are used effectively, pandemics can be eliminated not only in the chosen region but also globally.
Remark 4. A mathematical model serves as a theoretical basis for formulating and simulating epidemic prevention measures, as well as a tool for predicting and analyzing epidemic spread. The fractional model has been found to be more effective than the integer-order model [36,58]. The purpose of this study is to investigate possible issues arising from vaccination for COVID-19. In [5], the authors examined whether multiple vaccination strategies could affect COVID-19 dynamics in a population using the Atangana-Baleanu derivative. Here, we present the dynamics of the time-delayed COVID-19 disease model using the Caputo fractional derivative.