Article

COVID-19 Model with High- and Low-Risk Susceptible

Population Incorporating the Effect of Vaccines
Alhassan Ibrahim 1,2 , Usa Wannasingha Humphries 1, *, Amir Khan 3 , Saminu Iliyasu Bala 2 ,
Isa Abdullahi Baba 1,2 and Fathalla A. Rihan 4,5

1 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology,

Thonburi (KMUTT), 126 Pracha Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand;
alhassan.i@mail.kmutt.ac.th (A.I.); iababa.mth@buk.edu.ng (I.A.B.)
2 Department of Mathematical Sciences, Bayero University, Kano Kano 700006, Nigeria; sibala.mth@buk.edu.ng
3 Department of Mathematics and Statistics, University of Swat, Khyber 01923, Pakistan;
amirkhan@uswat.edu.pk
4 Department of Mathematical Sciences, College of Science, UAE University,
Al Ain 15551, United Arab Emirates; frihan@uaeu.ac.ae
5 Department of Mathematics, Faculty of Science, Helwan University, Cairo 11795, Egypt
* Correspondence: usa.wan@kmutt.ac.th

Abstract: It is a known fact that there are a particular set of people who are at higher risk of
getting COVID-19 infection. Typically, these high-risk individuals are recommended to take more
preventive measures. The use of non-pharmaceutical interventions (NPIs) and the vaccine are playing
a major role in the dynamics of the transmission of COVID-19. We propose a COVID-19 model
with high-risk and low-risk susceptible individuals and their respective intervention strategies.
We find two equilibrium solutions and we investigate the basic reproduction number. We also
carry out the stability analysis of the equilibria. Further, this model is extended by considering the
vaccination of some non-vaccinated individuals in the high-risk population. Sensitivity analyses
and numerical simulations are carried out. From the results, we are able to obtain disease-free and
Citation: Ibrahim, A.; Humphries, endemic equilibrium solutions by solving the system of equations in the model and show their global
U.W.; Khan, A.; Iliyasu Bala, S.; Baba, stabilities using the Lyapunov function technique. The results obtained from the sensitivity analysis
I.A.; Rihan, F.A. COVID-19 Model shows that reducing the hospitals’ imperfect efficacy can have a positive impact on the control of
with High- and Low-Risk Susceptible COVID-19. Finally, simulations of the extended model demonstrate that vaccination could adequately
Population Incorporating the Effect of control or eliminate COVID-19.
Vaccines. Vaccines 2022, 1, 0.
https://doi.org/
Keywords: equilibrium solutions; stability analysis; COVID-19; global stability; sensitivity; vaccine
Academic Editor: Giuseppe La Torre

Received: 21 November 2022

Accepted: 15 December 2022
1. Introduction
Published: 19 December 2022
The COVID-19 pandemic is still among the most devastating infectious diseases in
Publisher’s Note: MDPI stays neutral
the world. It is caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2)
with regard to jurisdictional claims in
and is transmitted within individuals through respiratory droplets created when someone
published maps and institutional affil-
that is infected coughs or sneezes or via direct contact with saliva, nasal discharge, and
iations.
sputum of an infected individual [1,2]. The COVID-19 virus has been responsible for over
6 million deaths around the world by the end of March 2022, as a result, it has become the
most significant global health disaster since the 1918 influenza pandemic [3,4].
Copyright: © 2022 by the authors.
The fight against COVID-19 made significant strides in early 2022 when the reported
Licensee MDPI, Basel, Switzerland. daily number of cases plunged, and hospitalizations were down by almost 27.9%. These
This article is an open access article changes are attributed to the wide implementation of COVID-19 control strategies on a large
distributed under the terms and scale, which includes keeping in touch with the authorities, getting vaccinated, properly
conditions of the Creative Commons wearing a recommended facemask, avoiding crowded areas, or maintaining distance
Attribution (CC BY) license (https:// between the self and others, yet an infection is still detected [5,6]. Therefore, it is reasonable
creativecommons.org/licenses/by/ to speculate that transmission cannot be ruled out completely, and given the potential
4.0/). importance of such transmission, urgent research on this subject is imperative [7,8].

Vaccines 2022, 1, 0. https://doi.org/10.3390/vaccines1010000 https://www.mdpi.com/journal/vaccines

Vaccines 2022, 1, 0 2 of 21

Although COVID-19 vaccines are readily available, there is no antiviral treatment that
has been recommended at this time for any of the COVID-19 variants [9]. Several clinical
trials are being conducted to evaluate the effectiveness of potential treatments. Despite
this, there are treatments available that can help relieve symptoms and help shorten the
length of the illness as well [10,11]. There are a few treatments that are aimed at relieving
symptoms, as well as supporting the respiratory system. Hospitalization may be necessary
for some people, especially if they have an underlying medical condition that requires
treatment [12].
There are more than 30 COVID-19 vaccines approved by National regulatory authori-
ties around the world in which most of them are proven to be effective [13]. Additionally,
more than half of the world’s population received at least one dose of the vaccine [14].
Many people refused to take the optimal dosage of the vaccine due to various reasons,
including due to fake information related to its safety and long-term effects. Whatever
the reason, there are significant consequences when individuals opt out of vaccines [15].
Outbreaks of COVID-19 are more likely to occur in communities where vaccination rates
are low. This not only puts unvaccinated individuals at risk but also increases the chances
that these diseases will spread to other people, including low-risk individuals.
The COVID-19 vaccines are the subject of extensive debate on the ideal dosage that
would offer the greatest degree of protection. It has been claimed by several researchers
that taking the booster dose is crucial for managing the condition and preventing major
illness or even death induced by this disease. As an interim measure, the most effective
way to curb the spread of this disease is through vaccination on a regular basis and in
accordance with the recommended schedule [16].
There has been numerous studies conducted to better understand how the virus
spreads from one person to another. The use of mathematical models to predict disease
transmission is beneficial for analysis and prediction, especially when tested against acces-
sible disease data. The initial models generally consist of deterministic ordinary differential
equations of the SEIR form which are rigorously analyzed by different authors taking differ-
ent stages into account. See, for instance, Swati et al. in [17] analyze the COVID-19 model
considering hospitalized individuals, and home-quarantined, or home-treated individuals.
According to Iboi [18], a mathematical model was developed for evaluating the influence of
the NPIs on the transmission pattern of the COVID-19 pandemic, as well as investigating
the impact of the early removal of social distancing and community lockdown measures.
Their work was extended by Riyapan et al. in [19], in which he added a compartment.
Tylicki et al. in [20] shows that the susceptible class is divided into two: the high
risk and the low risk. They indicated that the elderly, males, hypertensive, and patients
with comorbidities, constitute the most well-characterized high-risk group for severe
manifestations of COVID-19. None of the research mentioned above considered this
division in detail mathematically.
Understanding the dynamics of the interactions between those at low risk and those
at high risk is crucial to the control and prevention of COVID-19. This requires some
mathematical models, which have the potential to enhance our understanding of the
parameters associated with the increase in infection rates. Hence, our aim in this paper is
to examine the dynamics of COVID-19 transmission in two susceptible populations setting
(low- and high-risk individuals) through mathematical modeling, where we also use the
non-linear expression of the incidence rate. This incidence rate expression is proved to be
more realistic than the corresponding bilinear incidence rate [21].
Here, we use the COVID-19 features to develop a model and utilized some data that
are either real, estimated, or from the literature. In order to identify the most sensitive
parameter associated with the model basic reproduction number R0 , a sensitivity analysis
was conducted.
We organize this paper as follows: in Section 1, we give the introduction; Section 2 is
the model formulation; in Section 3, we analyse the model qualitatively, which includes
computing the reproduction number using a technique called the next generation ma-
Vaccines 2022, 1, 0 3 of 21

trix, calculating disease-free equilibrium points, identifying endemic equilibrium points,

analysing the stability of these equilibrium points on a local and global scale. In Section 4,
we perform sensitivity analysis and numerical simulations of the model; in Section 5,
we extend the model by considering movement from the high-risk to low-risk suscepti-
ble population which is caused due to vaccination of some unvaccinated individuals in
the high-risk population. We provide discussion and conclusions regarding our results
in Section 6.

2. Model Formulation
In this part, we modify an existing COVID-19 model proposed by Pal Bajiya et al.
in [22]. The modification procedure is presented below. We define N (t) as the total human
population at time t divided into six sub-populations: high-risk susceptible S1 , low-risk
susceptible S2 , exposed E, infected I, hospitalized H, and recovered R individuals:

N ( t ) = S1 ( t ) + S2 ( t ) + E ( t ) + I ( t ) + H ( t ) + R ( t ) . (1)

Our model also accounts for certain demographic impacts by assuming that all sub-
populations have a natural death rate (µ > 0) and a net inflow of humans to the two
susceptible population at a rate Λ. Next, we define the incident fraction as follows:

β( I + ϵH )
η= , (2)
N
where β is the effective contact rate, and ϵ is a modification factor that measures hospital in-
efficacy.
The high-risk susceptible population S1 (t) are those that are unvaccinated, human
with underlying medical conditions demonstrated to have a higher risk of death due to
COVID-19 [23], as well as the elderly [24]. Undocumented migrants associated with limited
access to healthcare because of legal, administrative, social barriers, etc., are also part
of this population. The high-risk susceptible population S1 (t) is reduced by the natural
death rate µ and the force of infection η. The parameter σ1 incorporates the impact of
the non-pharmaceutical intervention (wearing masks, physical or social distancing, and
washing hands regularly) by individuals in S1 on the number of contacts,

dS1
= (1 − ρ)Λ − ((1 − σ1 )η + µ)S1 .
dt
Similarly, the low-risk susceptible population S2 (t) includes those that are not in S1 (t),
and are recruited either by birth, or screened of COVID-19 by the authorities at the point of
entry through any of the borders. S2 (t) also reduced by natural death rate µ and force of
infection η, following effective contacts with an infected individual. Here, σ2 represents
the same interventions with S1 . High-risk individuals must take additional precautions in
addition to the above-mentioned actions to reduce transmission, this implies 0 ≤ σ2 < σ1 ,

dS2
= ρΛ − ((1 − σ2 )η + µ)S2 .
dt
Exposed individuals E(t) stands for the number of people who were exposed to the
virus. They are generated by the population of susceptible (high or low risk) humans that
become exposed or contact with the infected persons, and decreases as a result of infection
at a rate α and natural death at the rate µ,

dE
= η ((1 − σ1 )S1 + (1 − σ2 )S2 ) − (α + µ) E.
dt
Vaccines 2022, 1, 0 4 of 21

Infected individuals I (t) stands for the number of people who are infected but have
not been detected. It is reduced by either hospitalization at the rate of ν2 , recovery without
being hospitalized at a rate ν1 , the natural death µ, and COVID-19 induced death at a rate δ,

dI
= αE − (ν1 + ν2 + µ + δ) I.
dt
Hospitalized individuals H (t) are those who have been diagnosed with COVID-19
and have been isolated by the authorities. They are reduced due to recovery at the rate τ,
the natural death µ, and COVID-19 induced death at a rate δ,

dH
= ν2 I − (τ + µ + δ) H.
dt
Recovered individuals R(t) represents the number of recovered, undiagnosed individ-
uals, who are not being officially identified and those that recovered due to the impact of
isolation. The only decline in this population is due to the natural death rate µ,

dR
= ν1 I + τH − µR.
dt
Table 1 contains the model’s state variables, whereas Table 2 lists the model parameters
and their descriptions. Based on the assumptions stated above, Figure 1, below, describes
the flow transmission of the infection from one compartment to another. The interactions is
represented by the system of non-linear ordinary differential equations below:

dS1
= (1 − ρ)Λ − ((1 − σ1 )η + µ)S1 ,
dt
dS2
= ρΛ − ((1 − σ2 )η + µ)S2 ,
dt
dE
= η ((1 − σ1 )S1 + (1 − σ2 )S2 ) − (α + µ) E,
dt (3)
dI
= αE − (ν1 + ν2 + µ + δ) I,
dt
dH
= ν2 I − (τ + µ + δ) H,
dt
dR
= ν1 I + τH − µR.
dt

Table 1. Description of the variables

Variables Description
S1 Number of high-risk susceptible population
S2 Number of low-risk susceptible population
E Number of exposed population
I Number of infected population
H Number of hospitalize population
R Number of recovered population
N Total human population
Vaccines 2022, 1, 0 5 of 21

Table 2. Description of the parameters.

Parameter Description
Λ Recruitment rate
β Effective contact rate
ρ Fraction of newly recruited individuals moving to S2
σ1 Rate of reduction in infectiousness in S1
σ2 Rate of reduction in infectiousness in S2
µ The natural death rate
α The rate of progression from exposed population to infected population
ϵ The rate of hospital inefficacy
ν1 The recovery rate of infected in
ν2 The hospitalization rate of infected individuals
τ The recovery rate of the hospitalized individuals
δ The COVID-19 induced mortality rate

Figure 1. Schematic diagram of the COVID-19 model.

The basic dynamic characteristics of the model will now be discussed. Since the model
tracks populations of humans. In the same way as [25], the following theorem shows that
all the state variables are non-negative for all time t > 0.

Theorem 1. Consider the model (3) with initial conditions S1 (0) > 0, S2 (0) ≥ 0, E(0) ≥ 0,
I (0) ≥ 0, H (0) ≥ 0, and R(0) ≥ 0 then the solution is positive in R6+ .

Proof. Using the first equation of the model (3) given by

dS1
= (1 − ρ)Λ − ((1 − σ1 )η + µ)S1 ,
dt
we have
dS1
≥ −((1 − σ1 )η + µ)S1 .
dt
Solving for S1 R
S1 ( t ) ≥ S1 ( 0 ) e − ((1−σ1 )η +µ)dt
,
which implies that
S1 (t) ≥ 0, since ((1 − σ1 )η + µ) > 0.
Similarly, S2 (t), E(t), I (t), H (t), R(t) can be demonstrated to be positive. Therefore,
the solution of the model (3) is a positive quantity in R6+ for all t ≥ 0.
Vaccines 2022, 1, 0 6 of 21

Theorem 2. The model system’s (3) solution is bounded in

Λ
D = {(S1 , S2 , E, I, H, R) ∈ R6+ |0 : N ≤ }.
µ

Proof. By combining all equations in (3) we obtain

dN
= Λ − µN − δ( I + H ) ≤ Λ − µN,
dt
dN
+ µN ≤ Λ,
dt
therefore
Λ
N (t) ≤ N (0)e−µt + (1 − e−µt ),
µ
where N (0) is the initial values, i.e., N (t) = N (0) at t = 0.
Λ
Following from [26], it can be observed that N (t) −→ µ as t −→ ∞. Hence, N (t) is
Λ
bounded as 0 ≤ N (t) ≤ µ. So, all solutions in R6+ of the model (3) eventually enter in
D.
The proposed model is well presented epidemiologically and mathematically from
above in the region D . As a result, analysing the qualitative dynamics of model in D
is adequate.

3. Model Analysis
The model system (3) admits two equilibria: disease free equilibrium (DFE) and
endemic equilibrium (EE).

3.1. Disease Free Equilibrium

If there is no disease, the DFE of the model system (3) is calculated as follows:
Setting E = 0, I = 0, H = 0, R = 0 and denoted as

(1 − ρ)Λ ρΛ
 
0 0 0 0 0 0
DFE = (S1 , S2 , E , I , H , R ) = , , 0, 0, 0, 0 .
µ µ

We computed the basic reproduction number in a similar manner as in [27]. Consider-

ing the infected block x = { E, I, H } we obtain the matrix F and V as follows:

 
0 β((1 − σ1 )(1 − ρ) + (1 − σ2 )ρ) βϵ((1 − σ1 )(1 − ρ) + (1 − σ2 )ρ)
F = 0 0 0 , (4)
0 0 0
 
α+µ 0 0
V =  −α ν1 + ν2 + µ + δ 0 . (5)
0 −ν2 τ+µ+δ
Therefore, the basic reproduction R0 is as follows

αβ(ϵν2 + τ + µ + δ)((σ1 − σ2 )ρ + (1 − σ1 ))
R0 = . (6)
(ν1 + ν2 + µ + δ)(α + µ)(τ + µ + δ)
The basic reproduction number R0 defined as the number of new infections in a
population induced by a single infected person within a given period of time. If the R0 < 1,
the DFE is said to be locally stable, otherwise is said to be unstable [17].
Vaccines 2022, 1, 0 7 of 21

Theorem 3. The disease free equilibrium point DFE of the model (3) is said to be locally asymptoti-
cally stable (LAS) if R0 < 1 and unstable if R0 > 1.

Proof. From the Jacobian matrix evaluated at the DFE defined as

−µ 0 0 − β(σ1 − 1)(ρ − 1) − βϵ(σ1 − 1)(ρ − 1) 0
 
 0 −µ 0 βρ(σ2 − 1) βϵρ(σ2 − 1) 0 
 
 0 0 −( α + µ) β((σ1 − σ2 )ρ + (1 − σ1 )) βϵ((σ1 − σ2 )ρ + (1 − σ1 )) 0 
JDFE = . (7)
 0
 0 α −(µ + δ + ν1 + ν2 ) 0 0 

 0 0 0 ν2 −(µ + τ + δ) 0 
0 0 0 ν1 τ −µ
λ = −µ is a negative root of multiplicity 3 (λ1 = λ2 = λ6 = −µ). A reduced Jacobian
matrix defined as JrDFE below has three (3) eigenvalues, which correspond to the remaining
eigenvalues of JDFE .
 
−(α + µ) β((σ1 − σ2 )ρ + (1 − σ1 )) βϵ((σ1 − σ2 )ρ + (1 − σ1 ))
JrDFE = α −(µ + δ + ν1 + ν2 ) 0 . (8)
0 ν2 −(µ + τ + δ)

Define k1 = α + µ, k2 = (σ1 − σ2 )ρ + (1 − σ1 ), k3 = ν1 + ν2 + µ + δ, and k4 = µ + τ + δ.

The characteristic polynomial of the matrix Jr DFE is determined as

λ3 + c1 λ2 + c2 λ + c3 = 0. (9)

where
c1 = k 4 + k 3 + k 1 ,
c2 = k1 k3 + k1 k4 + k3 k4 − αβk2 ,
c3 = k1 k3 k4 − αβk2 k4 − αβϵν2 k2 = k1 k3 k4 − αβk2 (k4 + ϵν2 )
= k1 k3 k4 (1 − αβk2k(1kk43+
k4
ϵν2 )
)
= k 1 k 3 k 4 (1 − R0 ).
According to Routh–Hurwitz criterion, the sufficient and necessary condition for
stability is
c1 > 0, c3 > 0, c1 c2 − c3 > 0. (10)
Because all of the model parameters are positive, the first inequality in (10) is automat-
ically satisfied, the second inequality is satisfied when R0 < 1. As a result c1 c2 − c3 > 0 as
long as c3 > 0 holds. Hence, the disease free equilibrium DFE of the model (3) is locally
asymptotically stable when R0 < 1 and unstable whenever R0 > 1.

For global stability of the DFE, we have the following theorem

Theorem 4. The disease-free equilibrium (DFE) of the model (3) is globally asymptotically stable
(GAS) if R0 < 1.

Proof. Considering the following Voltera-type Lyapunov function

L0 = (S1 − S10 ln(S1 )) + (S2 − S20 ln(S2 )) + f 1 E + f 2 I + f 3 H (11)

where f 1 , f 2 , and f 3 > 0 are the Lyapunov coefficient. The corresponding derivative of L0
( dL 0
dt ) is given by

µ ( S 1 − S 10 ) 2 µ ( S 2 − S 20 ) 2
L˙0 = − − + f 1 Ė + f 2 İ + f 3 Ḣ. (12)
S1 S2

µ ( S 1 − S 10 ) 2 µ ( S 2 − S 20 ) 2
L˙0 = − − + f 1 (η (1 − σ1 )S1 + η (1 − σ2 )S2 − k1 E)
S1 S2 (13)
+ f 2 (αE − k3 I ) + f 3 (ν2 I − k4 H ).
Vaccines 2022, 1, 0 8 of 21

At the DFE, we linearize the (14). We note that near the DFE,
(1− ρ ) Λ ρΛ S1 S2
S1 ≤ µ , S2 ≤ µ and therefore N ≤ (1 − ρ) and N ≤ ρ.
Using this relation, we have

µ ( S 1 − S 10 ) 2 µ ( S 2 − S 20 ) 2
L˙0 ≤ − − + f 1 ( β( I + ϵH )(1 − ρ)(1 − σ1 ) + β( I + ϵHρ(1 − σ2 ))
S1 S2 (14)
− f 1 k1 E + f 2 (αE − k3 I ) + f 3 (ν2 I − k4 H ).

µ ( S 1 − S 10 ) 2 µ ( S 2 − S 20 ) 2
L˙0 ≤ − − + f 1 ( β( I + ϵH )k2 − k1 E) + f 2 (αE − k3 I )
S1 S2 (15)
+ f 3 (ν2 I − k4 H ).
We choose
αβϵk2
f 1 = α, f 2 = k1 , and f 3 = k4
(15) becomes,

µ ( S 1 − S 10 ) 2 µ(S2 − S20 )2
L˙0 ≤ − − + α( β( I + ϵH )k2 − k1 E) + k1 (αE − k3 I )
S1 S2 (16)
ϵα
+ (ν2 I − k4 H ).
k4

µ ( S 1 − S 10 ) 2 µ ( S 2 − S 20 ) 2
 
αβ(ϵν2 + k4 )
=− − + Ik1 k3 −1 . (17)
S1 S2 k1 k3 k4
µ ( S 1 − S 10 ) 2 µ ( S 2 − S 20 ) 2
=− − + Ik1 k3 [ R0 − 1]. (18)
S1 S2
Clearly, dL
dt ≤ 0 whenever R0 ≤ 0. As a result, and according to the Lasalle invariance
0

principle [28], the DFE is said to be globally asymptotically stable.

3.2. Endemic Equilibrium

Finding the endemic equilibrium solutions S1∗ , S2∗ , E∗ , I ∗ , H ∗ , and R∗ defined as EE
helps to further analyze the model. In order to obtain this endemic equilibrium, we solve (3)
substituting k1 , k2 , k3 , and k4 as defined in Section 3.1 simultaneously in terms η (i.e., force
of infection) and obtained

Λ (1 − ρ )
S1∗ = ,
(1 − σ1 )η ∗ + µ
ρΛ
S2∗ = ,
(1 − σ2 )η ∗ + µ
η ∗ Λ((1 − σ2 )(1 − σ1 )η ∗ + k2 µ)
E∗ = ,
k1 ((1 − σ1 )η ∗ + µ)((1 − σ2 )η ∗ + µ)
(19)
αη ∗ Λ((1 − σ2 )(1 − σ1 )η ∗ + k2 µ)
I∗ = ,
k1 k3 ((1 − σ1 )η ∗ + µ)((1 − σ2 )η ∗ + µ)
αν2 η ∗ Λ((1 − σ2 )(1 − σ1 )η ∗ + k2 µ)
H∗ = ,
k1 k3 k4 ((1 − σ1 )η ∗ + µ)((1 − σ2 )η ∗ + µ)
αη ∗ Λ(k4 ν1 + τν2 )((1 − σ2 )(1 − σ1 )η ∗ + k2 µ)
R∗ = .
µk1 k3 k4 ((1 − σ1 )η ∗ + µ)((1 − σ2 )η ∗ + µ)

where
β( I ∗ + ϵH ∗ )
η∗ = , (20)
N∗
and N ∗ = S1∗ + S2∗ + E∗ + I ∗ + H ∗ + R∗
Vaccines 2022, 1, 0 9 of 21

From Equation (20) we obtained η ∗ = 0 as one of the solutions (which corresponds to

DFE) and the following equation which is quadratic:

aη ∗2 + bη ∗ + c = 0. (21)

where
a = k5 k6 (αµk4 + αµν2 + ατν2 + αk4 ν1 + µk3 k4 ),
b = µρk1 k3 k4 k5 + αµ2 k2 k4 + αµ2 k2 ν2 + αµτk2 ν2 + αµk2 k4 ν1 + µ2 k2 k3 k4 + µk1 k3 k4 k6 −
βαϵµk5 k6 ν2 − βαµk4 k5 k6 − µρk1 k3 k4 k6 ,
c = µ2 k1 k3 k4 − βαϵµ2 k2 ν2 − βαµ2 k2 k4 = µ2 k1 k3 k4 (1 − R0 ),
where k5 = 1 − σ1 and k6 = 1 − σ2 . √
2
From the Equation (21) we have η ∗ = −b± 2ab −4ac .
Clearly, a is greater than zero. When R0 > 1, we have the following:

c < 0,
√ √
−b + b2 − 4ac −b − b2 − 4ac
=⇒ > 0 and < 0.
2a 2a
This shows that there is a unique endemic equilibrium point.
When R0 < 1, we have

c > 0,
=⇒ ac > 0 therefore − 4ac < 0,
=⇒ if b2 > 4ac then b2 − 4ac > 0,
p
=⇒ −b ± b2 − 4ac < 0,
otherwise no real roots of (21).

Global Stability Analysis of the Endemic Equilibrium

Following [29–31], Theorem 6 below is established

Theorem 5. The unique endemic equilibrium of (3) is GAS in D if R0 > 1, provided that

Iη ∗
  
η
1− ∗ 1− ∗ ≥0 (22)
η I η
and
Hη ∗
  
η
1− ∗ 1− ∗ ≥0 (23)
η H η
are satisfied.

Proof. In this case, we use the approach in [29–31] to establish the proof. If we consider a
Lyapunov function:
     
∗ ∗ S1 ∗ ∗ S2 ∗ ∗ E
L1 (t) = ω1 S1 − S1 − S1 ln ∗ + ω2 S2 − S2 − S2 ln ∗ + ω3 E − E − E ln ∗
S1 S2 E
    (24)
∗ ∗ I ∗ ∗ H
+ω4 I − I − I ln ∗ + ω5 H − H − H ln ∗ .
I H
Vaccines 2022, 1, 0 10 of 21

where ωi > 0(i = 1, 2, 3, 4, 5) are constants to be determined. It is easy to see that

L1 ≥ 0 for all S1 , S2 , E, I, H > 0, and L1 = 0 ⇐⇒ (S1 , S2 , E, I, H ) = (S1∗ , S2∗ , E∗ , I ∗ , H ∗ ).
From (3), we have the solution below:

(1 − ρ)Λ − ((1 − σ1 )η + µ)S1 = 0,

ρΛ − ((1 − σ2 )η + µ)S2 = 0,
η ((1 − σ1 )S1 + (1 − σ2 )S2 ) − (α + µ) E = 0,
(25)
αE − (ν1 + ν2 + µ + δ) I = 0,
ν2 I − (τ + µ + δ) H = 0,
ν1 I + τH − µR = 0.

We can differentiate L1 along these solutions which is given by

S1∗ ˙ S2∗ ˙ E∗
     
˙
L1 (t) = ω1 1 − S1 + ω 2 1 − S2 + ω 3 1 − Ė
S1 S2 E
(26)
I∗ H∗
   
+ ω4 1 − İ + ω5 1 − Ḣ.
I H

By direct computation from (26) , we have

S1∗ ˙ S1∗
   
ω1 1 − S1 = ω 1 1 − ((1 − ρ)Λ − ((1 − σ1 )η + µ)S1 )
S1 S1
S∗
 
= ω1 1 − 1 ((1 − ρ)Λ − (1 − σ1 )ηS1 − µS1 )
S1
S1∗
 
= ω1 1 − ((1 − σ1 )η ∗ S1∗ + µS1∗ − (1 − σ1 )ηS1 − µS1 )
S1
S1∗
 
= ω1 1 − ((1 − σ1 )η ∗ S1∗ − (1 − σ1 )ηS1 − µS1 + µS1∗ )
S1
S1∗
   
ηS1
= ω1 1 − ((1 − σ1 )η S1 1 − ∗ ∗ − µ(S1 − S1∗ ))
∗ ∗
(27)
S1 η S1

S ∗   
S ∗ 
ηS
= ω1 (1 − σ1 )η ∗ S1∗ 1 − 1 1 − ∗ 1∗ − ω1 1 − 1 µ(S1 − S1∗ )
S1 η S1 S1

S ∗  
(S1 − S1∗ )2
ηS
= ω1 (1 − σ1 )η ∗ S1∗ 1 − 1 1 − ∗ 1∗ − ω1 µ
S1 η S1 S1

S ∗  
ηS
≤ ω1 (1 − σ1 )η ∗ S1∗ 1 − 1 1 − ∗ 1∗
S1 η S1

S ∗ 
ηS η
= ω1 (1 − σ1 )η ∗ S1∗ 1 − ∗ 1∗ − 1 + ∗ ,
η S1 S1 η
Vaccines 2022, 1, 0 11 of 21

and
S∗ S2∗
   
ω2 1 − 2 S˙2 = ω2 1 − (ρΛ − ((1 − σ2 )η + µ)S2 )
S2 S2
S2∗
 
= ω2 1 − (ρΛ − (1 − σ2 )ηS2 − µS2 )
S2
S2∗
 
= ω2 1 − ((1 − σ2 )η ∗ S2∗ + µS2∗ − (1 − σ2 )ηS2 − µS2 )
S2
S2∗
 
= ω2 1 − ((1 − σ2 )η ∗ S2∗ − (1 − σ2 )ηS2 − µS2 + µS2∗ )
S2
S2∗
   
ηS
= ω2 1 − ((1 − σ2 )η ∗ S2∗ 1 − ∗ 2∗ − µ(S1 − S2∗ )) (28)
S2 η S2
S∗ S∗
    
ηS2
= ω2 (1 − σ2 )η ∗ S2∗ 1 − 2 1 − ∗ ∗ − ω2 1 − 2 µ(S2 − S2∗ )
S2 η S2 S2

S ∗  
(S2 − S2∗ )2
ηS2
= ω2 (1 − σ2 )η ∗ S2∗ 1 − 2 1 − ∗ ∗ − ω2 µ
S2 η S2 S2

S ∗  
ηS2
≤ ω2 (1 − σ2 )η ∗ S2∗ 1 − 2 1− ∗ ∗
S2 η S2

S ∗ 
ηS 2 η
= ω2 (1 − σ2 )η ∗ S2∗ 1 − ∗ ∗ − 2 + ∗ ,
η S2 S2 η

and
E∗ E∗
   
ω3 1− Ė = ω3 1 − (η (1 − σ1 )S1 + η (1 − σ2 )S2 − k1 E)
E E
E∗
 
= ω3 1 − (η (1 − σ1 )S1 + η (1 − σ2 )S2
E
 
∗ ∗ E ∗ ∗ E
− η (1 − σ1 )S1 ∗ + η (1 − σ2 )S2 ∗ )
E E

E ∗  
ηS1 E

∗ ∗
= ω3 1 − (η (1 − σ1 )S1 ∗ ∗ − ∗
E η S1 E
 
ηS E
+ η ∗ (1 − σ2 )S2∗ ∗ 2∗ − ∗ ) (29)
η S2 E

E ∗  ηS E

1
= ω3 η ∗ (1 − σ1 )S1∗ 1 − −
E η ∗ S1∗ E∗
E∗
  
ηS2 E
+ ω3 η ∗ (1 − σ2 )S2∗ 1 − −
E η ∗ S2∗ E∗
ηS E∗
 
ηS E
= ω3 (1 − σ1 )η ∗ S1∗ ∗ 1∗ − ∗ − ∗ 1 ∗ + 1
η S1 E η S1 E
ηS2 E∗
 
∗ ∗ ηS2 E
+ ω3 (1 − σ2 )η S2 ∗ ∗ − ∗ − ∗ ∗ + 1 ,
η S2 E η S2 E
Vaccines 2022, 1, 0 12 of 21

and
I∗ I∗
   
ω4 1 − İ = ω4 1 − (αE − k3 I )
I I
I∗
  
I
= ω4 1 − αE − αE∗ ∗
I I
 ∗  E  (30)
I I
= ω4 αE∗ 1 − −
I E∗ I∗
I∗ E
 
E I
= ω4 αE∗ − − + 1 ,
E∗ I∗ IE∗

and
H∗ H∗
   
ω5 1− Ḣ = ω5 1 − (ν2 I − k4 H )
H H
H∗
  
∗ H
= ω5 1 − ν2 I − ν2 I
H H∗
 ∗   (31)
∗ H I H
= ω5 ν2 I 1 − − ∗
H I∗ H

I H ∗
H I

∗
= ω5 ν2 I − ∗− +1 .
I∗ H H I∗

η ∗ ((1−σ )S∗ +(1−σ )S∗ )

1 1 2 2 η ∗ (1−σ2 )S2∗
Substituting ω1 = ω2 = ω3 = 1, ω4 = αE∗ , and ω5 = ν2 I ∗ ,
and (27)–(31) into (26), we have

S1∗ ηS1 E∗
 
˙ ∗ ∗ E η
L1 (t) ≤ (1 − σ1 )η S1 2 − − ∗− ∗ ∗ + ∗
S1 E η S1 E η
∗
ηS E∗
 
S E η
+(1 − σ2 )η ∗ S2∗ 2 − 2 − ∗ − ∗ 2 ∗ + ∗
S2 E η S2 E η
I∗ E
 
E I
+(1 − σ1 )η ∗ S1∗ − − + 1 (32)
E∗ I∗ IE∗
I∗ E
 
E I
+(1 − σ2 )η ∗ S2∗ − − + 1
E∗ I∗ IE∗
H∗ I
 
I H
+(1 − σ2 )η ∗ S2∗ ∗ − ∗ − + 1 .
I H H I∗

Following the idea in [31,32], suppose we have a function defined as χ( x ) = 1 −

x + ln( x ), if x > 0, we have χ( x ) ≤ 0, and if x = 1, we have χ(1) = 0. Thus , x − 1 ≥
ln( x ) for x > 0 Using this relation we find that

S1∗ ηS1 E∗
 
E η
2− − ∗− ∗ ∗ + ∗ =
S1 E η S1 E η
∗ ∗ ∗ Iη ∗
  
η Iη S ηS E E I
−1 + ∗ 1− ∗ + 3 − 1 − ∗ 1∗ − ∗ − ∗ + ∗
η I η S1 η S E I η E I
 ∗  1 ∗
ηS1 E∗
  
S1 Iη E I
≤− −1 − ∗ ∗ −1 − ∗ −1 − ∗ + ∗ (33)
S1 η S1 E I η E I
 ∗
S ηS1 E∗ Iη ∗

E I
≤ − ln 1 ∗ ∗ ∗ − ∗+ ∗
S1 η S1 EI η E I
   
I I E E
= ∗ − ln ∗ + ln ∗ − ∗ .
I I E E
Vaccines 2022, 1, 0 13 of 21

Likewise,

S2∗ ηS2 E∗
 
E η
2− − ∗− ∗ ∗ + ∗ =
S2 E η S2 E η
∗ ∗ ∗ Hη ∗
  
η Hη S ηS E E H
−1 + ∗ 1− ∗ + 3 − 2 − ∗ 2∗ − ∗ − ∗ + ∗
η H η S2 η S2 E H η E H
 ∗
ηS2 E∗ Hη ∗
    
S2 E H
≤− −1 − ∗ ∗ −1 − ∗
−1 − ∗ + ∗ (34)
S2 η S2 E H η E H
 ∗
S2 ηS2 E∗ Hη ∗

E H
≤ − ln ∗ ∗ ∗
− ∗+ ∗
S2 η S2 EH η E H
   
H H E E
= ∗ − ln + ln ∗ − ∗ .
H H∗ E E

Meanwhile, one can still verify that

I∗ E
   ∗   ∗ 
E I I E E I I E E I
∗
− ∗ − ∗ +1 = − −1 + − ≤ − ln + −
E I IE IE∗ E∗ I∗ IE∗ E∗ I∗
 ∗       (35)
I E E I E E I I
≤ − ln − ln + − = − ln − + ln .
I E∗ E∗ I∗ E∗ E∗ I∗ I∗

Similarly, we have
H∗ I
   ∗   ∗ 
I H H I I H H I I H
− ∗− +1 = − −1 + − ≤ − ln + −
I∗ H H I∗ H I∗ I∗ H∗ H I∗ I∗ H∗
 ∗       (36)
H I I H I I H H
≤ − ln − ln + − = − ln − + ln .
H I∗ I∗ H∗ I∗ I∗ H∗ H∗

Substituting (33)–(36) in (32) we have,

     
˙ ∗ ∗ I I E E
L1 (t) ≤ (1 − σ1 )η S1 ∗ − ln ∗ + ln − ∗ +
I I E∗ E
     
∗ ∗ H H E E
(1 − σ2 )η S2 − ln + ln ∗ − ∗ +
H∗ H∗ E E
    
E E I I
(1 − σ1 )η ∗ S1∗ − ln − + ln + (37)
E∗ E∗ I∗ I∗
    
E E I I
(1 − σ2 )η ∗ S2∗ − ln − + ln +
E∗ E∗ I∗ I∗
    
I I H H
(1 − σ2 )η ∗ S2∗ − ln − + ln .
I∗ I∗ H∗ H∗

Hence, (27)–(37) ensure that dL dL1

dt ≤ 0. It is easy to see that dt = 0 holds only for
1

S1 = S1∗ , S2 = S2∗ , E = E∗ , I = I ∗ , H = H ∗ , and R = R∗ . In a similar manner in [28], every

solutions of our model system (3) with initial conditions in D approaches the stable EE as
t −→ ∞. Hence, EE is GAS equilibrium of (3) on D .

4. Numerical Simulations
In addition to the theoretical results from the model analysis, it is also crucial that
the model equations are simulated in order to gain an understanding of the model. This
simulations were conducted using MATLAB R2022b on a mid-range personal laptop that
features an i7 processor and 8GB of RAM on a Windows 11 operating system. In the
Supplementary Materials, a copy of the simulation code that was used can be found. While
it is a challenge to find suitable data for model simulation, the lack of an appropriate answer
to this question continues to be a major problem. We used parameters provided in the
Vaccines 2022, 1, 0 14 of 21

existing literature to simulate and perform a sensitivity analysis of our model. Occasionally,
we assumed the values which can be seen in Table 3.

Table 3. Parameter, values, and boundary

Parameters Values/Source Boundary/Source

β 0.6886, assumed [0, 1], estimated
ρ 0.7, assumed [0.5, 1], estimated
σ1 0.4521, [33] [0.213, 0.51], estimated
σ2 0.2757, [33] [0.2, 0.5], estimated
µ 0.0079, assumed [0.0074, 0.008], estimated
α 0.1667, [19] [0, 1], estimated
ϵ 0.0075, assumed [0.3, 0.4], estimated
ν1 0.1, [34] [0.1, 0.2], estimated
ν2 0.2, [34] [0.195, 0.39], [35]
τ 0.005, assumed [0.03, 0.08], estimated
δ 0.0015 assumed [0.0013, 0.0023], estimated

We begin the numerical simulation of our model by first letting β = 0.2693 so that
R0 = 0.9380 < 1 for various initial conditions. When R0 < 1 Figure 2a–d shows that
the system has a DFE that is asymptotically stable which supports the result stated in
Theorem 3. In Figure 3, we examine the scenario in which R0 = 2.3986 > 1 for various
initial conditions. When R0 > 1 Figure 3a–d shows that the system has a DFE and it is
unstable whenever R0 > 1.

(a) (b)

(c) (d)

Figure 2. Time series plot of the model (3) with different initial conditions (described by a color
scheme). The parameter values are given in Table 3 with β = 0.2693 so that R0 = 0.9380 < 1; (a) the
high-risk susceptible, (b) the low-risk susceptible, (c) exposed, and (d) infectious.
Vaccines 2022, 1, 0 15 of 21

(a) (b)

(c) (d)

Figure 3. Time series plot of the model (3) with different initial conditions (described by a color
scheme). The parameter values are given in Table 3 with β = 0.6886 so that R0 = 2.3986 > 1; (a) the
high-risk susceptible, (b) the low-risk susceptible, (c) exposed, and (d) infectious.

Sensitivity Analysis
In mathematical modeling, numerous parameter values are uncertain, this could be
due to incorrect parameter estimation and uncertainty regarding the accurate values of the
parameters. Therefore, it is reasonable to carry out sampling and sensitivity analysis to
identify the parameters that significantly affect the output of our model. We used a Sam-
pling and Sensitivity Analysis Tool (SaSAT) in [36] to recognise these parameters. This is an
efficient tool that enables us to analyze the sensitivity. We first have values and boundaries
assigned to our parameters as in Table 3. We then applied uniform probability distributions
to each of the parameters, in accordance with the suggestion of [36]. We evaluated Partial
Rank Correlation Coefficients (PRCC) with 1000 samples for each parameter per run, which
was resulted from the Latin hypercube sampling (LHS).
Figure 4 shows how varying the parameters of model (3) changes the behavior of R0 .
The five (5) most sensitive parameters affecting R0 are β, τ, σ2 , ν1 , and ϵ. Rising the value of
the recovery rate of the hospitalized individuals, rate of reduction in infectiousness of the
low-risk susceptible which is proportional to rising the value of σ1 might cause the value of
R0 to drop. Meanwhile, reducing the contact rate, hospital inefficacy might also cause the
value of R0 to drop significantly. This study demonstrate that isolating and hospitalizing
the infected persons reduces the spread of infection. Figure 5 shows the change in behavior
of R0 while varying the value of the most sensitive parameters.
Vaccines 2022, 1, 0 16 of 21

Figure 4. Tornado plot for the model parameter sensitivities affecting the basic reproduction num-
ber R0 .

(a) (b)

(c) (d)

Figure 5. Cont.
Vaccines 2022, 1, 0 17 of 21

(e) (f)

Figure 5. Response surface plot showing the change in behavior of R0 while varying the value
of the most sensitive parameters. (a) Plot of R0 vs. τ and β. (b) Plot of R0 vs. τ and µ. (c) Plot of
R0 vs. ν1 and µ. (d) Plot of R0 vs. σ2 and σ1 . (e) Plot of R0 vs. ϵ and β (f) Plot of R0 vs. ν2 and β.

5. Extended Model with Vaccination

Vaccine effectiveness has so far been the driving force in the dynamics of COVID-19
transmission. In this section, we carried out numerical simulations to test for various
situations in relation to the parameters under close examination. We propose two main
parameters ω, which indicates the level of vaccination for the high-risk population, and ν3 ,
which represents the effectiveness of the vaccine among the vaccinated individuals in the
low-risk population.
In summary, as a result of incorporating an imperfect COVID-19 vaccine we believe:
i. The number of high-risk individuals reduces.
ii. Infections can be prevented with some degree of efficacy.
Next, we see the effect of vaccination and its effectiveness on the dynamics of COVID-
19 by simulating the related vaccine parameters of the model (3) over time.

The Effect of Vaccine on the Disease Dynamics

Some high-risk individuals after being vaccinated at the rate ω move to the low-
risk class and further move to the recovered class depending on the vaccine efficacy,
ν3 = 0.01, ν3 = 0.05, ν3 = 0.1, ν3 = 0.15, ν3 = 0.2, and ν3 = 0.3.
In Figure 6, we can see that as vaccination coverage increases, there is a striking
decline in the number of infected individuals at each level of effectiveness of the vaccine.
In Figure 7, we can see that the hospitalized population decreases relatively quickly over
time at various levels of ν3 . Therefore, improving the vaccination rates for people at high
risk will result in a decrease in the transmission of COVID-19 in the future.
A look at Figure 8 illustrates, on the other hand, what the effects of vaccination may
be on the recovered class, it can be concluded that the population of the recovered class
increases as the value of ω increases from 0.01 to 0.3.
Vaccines 2022, 1, 0 18 of 21

(a) (b)

Figure 6. The effect of the vaccine on infected COVID-19 individuals for ν3 = 0.25 and ν3 = 0.75.
(a) ν3 = 0.25. (b) ν3 = 0.75.

(a) (b)

Figure 7. The effect of the vaccine on hospitalized COVID-19 individuals for ν3 = 0.25 and ν3 = 0.75.
(a) ν3 = 0.25. (b) ν3 = 0.75.

(a) (b)

Figure 8. The effect of the vaccine on recovered individuals for ν3 = 0.25 and ν3 = 0.75. (a) ν3 = 0.25.
(b) ν3 = 0.75.

6. Discussion and Conclusions

As it stands now, there is still no specific drug for COVID-19, so many treatment guide-
lines were developed in the literature based on patient treatment records for physicians
and healthcare personnel as a guide for decision-making in the treatment of patients with
Vaccines 2022, 1, 0 19 of 21

COVID-19. Understanding the influence of the NPIs on both low and high-risk individuals,
the impact of the vaccine on the high-risk population can help us inform public health
policy as it was discussed in [37]. It is equally vital to evaluate the impact of hospital
efficacy/inefficacy.
In this paper, we present a model which explains the transmission and spread of
COVID-19, incorporating a fraction ρ low-risk and the remaining (1 − ρ) high-risk of the
human population. The model also captures NPIs by both low- and high-risk individuals,
to reduce infection transmission and insisted that the high-risk individuals take more
intervention than the low-risk, similar to what is mentioned in [38]. The model was later
extended by incorporating the vaccination of some high-risk individuals and vaccine
efficacy. This study has been able to draw some important conclusions both mathematically
and biologically, which are summarized below.
Many models of the SEIR form in literature, see [21,22,39,40], did not take into con-
sideration the segregation in the susceptible class into low- and high-risk individuals.
However, considering this division is essential in providing more insight into the transmis-
sion dynamics of the disease, as well as guiding the relevant authorities in introducing and
providing plans to curtail the spread of the disease.
(i) A GAS DFE occurs in the model without vaccination every time the associated basic
reproduction number is less than 1 and there are multiple endemic equilibria, which
is a GAS in a particular case.
(ii) Based on numerical simulations it is indicated that if some high-risk individuals
were vaccinated and moved to low-risk, the disease could be reduced to a minimum.
However, this is dependent on the coverage of vaccinations.
(iii) The Latin Hypercube Sampling to create 1000 samples and the resulting Partial Rank
Correlation Coefficients (PRCCs) were used to perform a sensitivity study of the
model parameter values. A tornado plot is used to visually display the results. The
parameters τ, (hospitalized recovery rate), σ1,2 , (reduction in infectiousness of risk
individuals), according to sensitivity analysis, greatly lessen an epidemic if increased.
However, if the rates of person-to-person interaction and hospital inefficacy are
reduced, transmission also decreases.
In order to stop the epidemic, it is crucial to improve vaccination of the high-risk
individuals. Law enforcement must also be strengthened in order to restrict undocu-
mented immigration.

Future Work Directions

• Some new models with a different approach of incidence fraction can be proposed.
• Several dynamical features of COVID-19 were captured by our model, though other
population compartments might be added and furthermore implement optimal control
strategies when having access to more detailed and authentic COVID-19 data.

Author Contributions: All authors contributed equally to the manuscript and typed. All authors
have read and agreed to the published version of the manuscript.
Funding: This research was funded Petchra by “Pra Jom Klao Ph.D. Research Scholarship from King
Mongkut’s University of Technology”.
Data Availability Statement: There are no underlying data.
Acknowledgments: The authors are grateful to the editors and reviewers for their anticipated
thoughtful and insightful comment that will improved the manuscript.
Conflicts of Interest: The authors declare no conflicts of interest.
Vaccines 2022, 1, 0 20 of 21

Abbreviations
The following abbreviations are used in this manuscript:

COVID-19 Coronavirus disease 2019

NPIs Non-pharmaceutical interventions against COVID-19
DFE Disease-free equilibrium
EE Endemic equilibrium
LAS Locally asymptotically stable
GAS Globally asymptotically stable
SaSAT Sampling and sensitivity analysis tool
PRCC Partial rank correlation coefficient
LHS Latin hypercube sampling
ASQ Alternative state quarantine

References
1. Acter, T.; Uddin, N.; Das, J.; Akhter, A.; Choudhury, T.R.; Kim, S. Evolution of severe acute respiratory syndrome coronavirus
2 (SARS-CoV-2) as coronavirus disease 2019 (COVID-19) pandemic: A global health emergency. Sci. Total. Environ. 2020, 730,
138996.
2. Anderson, D.E.; Sivalingam, V.; Kang, A.E.Z.; Ananthanarayanan, A.; Arumugam, H.; Jenkins, T.M.; Hadjiat, Y.; Eggers, M.
Povidone-iodine demonstrates rapid in vitro virucidal activity against SARS-CoV-2, the virus causing COVID-19 disease. Infect.
Dis. Ther. 2020, 9, 669–675.
3. Elmancy, L.; Alkhatib, H.; Daou, A. SARS-CoV-2: An Analysis of the Vaccine Candidates Tested in Combatting and Eliminating
the COVID-19 Virus. Vaccines 2022, 10, 2086.
4. Beach, B.; Clay, K.; Saavedra, M. The 1918 influenza pandemic and its lessons for COVID-19. J. Econ. Lit. 2022, 60, 41–84.
5. Ilatova, E.; Abraham, Y.S.; Celik, B.G. Exploring the Early Impacts of the COVID-19 Pandemic on the Construction Industry in
New York State. Architecture 2022, 2, 457–475.
6. LaMattina, J.L. Pharma and Profits: Balancing Innovation, Medicine, and Drug Prices; John Wiley & Sons: Hoboken, NJ, USA, 2022.
7. Vivian, D.T. Media Literacy and Fake News: Evaluating the Roles and Responsibilities of Radio Stations in Combating Fake
News in the COVID-19 Era. Int. J. Inf. Manag. Sci. 2022, 6, 33–49.
8. Hughes, D. What is in the so-called COVID-19 “Vaccines”? Part 1: Evidence of a Global Crime Against Humanity. Int. J. Vaccine
Theory Pract. Res. 2022, 2, 455–586.
9. Kumari, M.; Lu, R.M.; Li, M.C.; Huang, J.L.; Hsu, F.F.; Ko, S.H.; Ke, F.-Y.; Su, S.-C.; Liang, K.-H.; Yuan, J.P.-Y. A critical overview of
current progress for COVID-19: Development of vaccines, antiviral drugs, and therapeutic antibodies. J. Biomed. Sci. 2022, 29,
1–36.
10. Niknam, Z.; Jafari, A.; Golchin, A.; Danesh Pouya, F.; Nemati, M.; Rezaei-Tavirani, M.; Rasmi, Y. Potential therapeutic options for
COVID-19: An update on current evidence. Eur. J. Med. Res. 2022, 27, 1–15.
11. Favilli, A.; Mattei Gentili, M.; Raspa, F.; Giardina, I.; Parazzini, F.; Vitagliano, A.; Borisova, A.V.; Gerli, S. Effectiveness and safety
of available treatments for COVID-19 during pregnancy: A critical review. J. Matern.-Fetal Neonatal Med. 2022, 35, 2174–2187.
12. Graham, E.L.; Koralnik, I.J.; Liotta, E.M. Therapeutic approaches to the neurologic manifestations of COVID-19. Neurotherapeutics
2022 19, 1435–1466.
13. Md Khairi, L.N.H.; Fahrni, M.L.; Lazzarino, A.I. The race for global equitable access to COVID-19 vaccines. Vaccines 2022, 10,
1306.
14. Lukaszuk, K.; Podolak, A.; Malinowska, P.; Lukaszuk, J.; Jakiel, G. Cross-Reactivity between Half Doses of Pfizer and AstraZeneca
Vaccines—A Preliminary Study. Vaccines 2022, 10, 521.
15. Liu, X.; Zhao, N.; Li, S.; Zheng, R. Opt-out policy and its improvements promote COVID-19 vaccinations. Soc. Sci. Med. 2022, 307,
115120.
16. Moore, S.; Hill, E.M.; Dyson, L.; Tildesley, M.J.; Keeling, M.J. Retrospectively modeling the effects of increased global vaccine
sharing on the COVID-19 pandemic. Nat. Med. 2022, 28, 2416–2423.
17. Tyagi, S.; Martha, S.C.; Abbas, S.; Debbouche, A. Mathematical modeling and analysis for controlling the spread of infectious
diseases. Chaos Solitons Fractals 2021, 144, 1107072.
18. Iboi, E.; Sharomi, O.O.; Ngonghala, C.; Gumel, A.B. Mathematical modeling and analysis of COVID-19 pandemic in Nigeria.
MedRxiv 2020. https://doi.org/10.1101/2020.05.22.20110387.
19. Riyapan, P.; Shuaib, S.E.; Intarasit, A. A mathematical model of COVID-19 pandemic: A case study of Bangkok, Thailand. Comput.
Math. Methods Med. 2021, 2021, 6664483.
20. Tylicki, L.; Puchalska-Reglińska, E.; Tylicki, P.; Och, A.; Polewska, K.; Biedunkiewicz, B.; Parczewska, A.; Szabat, K.; Wolf, J.;
D˛ebska-Ślizień, A. Predictors of mortality in hemodialyzed patients after SARS-CoV-2 infection. J. Clin. Med. 2022, 11, 285.
21. He, S.; Peng, Y.; Sun, K. SEIR modeling of the COVID-19 and its dynamics. Nonlinear Dyn. 2020, 101, 1667–1680.
22. Bajiya, V.P.; Bugalia, S.; Tripathi, J.P. Mathematical modeling of COVID-19: Impact of non-pharmaceutical interventions in India.
Chaos Interdiscip. J. Nonlinear Sci. 2020, 30, 113143.
Vaccines 2022, 1, 0 21 of 21

23. Chen, T.; Wu, D.; Chen, H.; Yan, W.; Yang, D.; Chen, G.; Ma, K.; Xu, D.; Yu, H.; Wang, H. Clinical characteristics of 113 deceased
patients with coronavirus disease 2019: Retrospective study. BMJ 2020, 368, m1091.
24. Wang, C.; Horby, P.W.; Hayden, F.G.; Gao, G.F. A novel coronavirus outbreak of global health concern. Lancet 2020, 395, 470–473.
25. Mtisi, E.; Rwezaura, H.; Tchuenche, J.M. A mathematical analysis of malaria and tuberculosis co-dynamics. Discret. Contin. Dyn.
Syst.-B 2009, 12, 827.
26. Prathumwan, D.; Trachoo, K.; Chaiya, I. Mathematical modeling for prediction dynamics of the coronavirus disease 2019
(COVID-19) pandemic, quarantine control measures. Symmetry 2020, 12, 1404.
27. Van den Driessche, P.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of
disease transmission. Math. Biosci. 2002, 180, 1–2.
28. La, S.; Joseph, P. The Stability of Dynamical Systems; Society for Industrial and Applied Mathematics: Philadelphia, PA, USA, 1976.
29. Garba, S.M.; Gumel, A.B.; Bakar, M. Abu. Backward bifurcations in dengue transmission dynamics. Math. Biosci. 2008, 215, 11–25.
30. Roop, O.P.; Chinviriyasit, W.; Chinviriyasit, S. The effect of incidence function in backward bifurcation for malaria model with
temporary immunity. Math. Biosci. 2015, 256, 47–64.
31. Yang, C.; Wang, X.; Gao, D.; Wang, J. Impact of awareness programs on cholera dynamics: Two modeling approaches Bull. Math.
Biol. 2017, 79, 2109–2131.
32. Salihu, S.M.; Shi, Z.; Chan, H.S.; Jin, Z.; He, D. A mathematical model to study the 2014–2015 large-scale dengue epidemics in
Kaohsiung and Tainan cities in Taiwan, China. Math. Biosci. Eng. 2019, 6, 3841–3863
33. Asempapa, R.; Oduro, B.; Apenteng, O.O.; Magagula, V.M. A COVID-19 mathematical model of at-risk populations with
non-pharmaceutical preventive measures: The case of Brazil and South Africa. Infect. Dis. Model. 2022, 7, 45–61.
34. Eikenberry, S.E.; Mancuso, M.; Iboi, E.; Phan, T.; Eikenberry, K.; Kuang, Y.; Kostelich, E.; Gumel, Abba B. To mask or not to mask:
Modeling the potential for face mask use by the general public to curtail the COVID-19 pandemic. Infect. Dis. Model. 2020, 5,
293–308.
35. Bala, S.I.; Gimba, B. Global sensitivity analysis to study the impacts of bed-nets, drug treatment, and their efficacies on a two-strain
malaria model. Math. Comput. Appl. 2019, 24, 32.
36. Hoare, A.; Regan, D.G.; Wilson, D.P. Sampling and sensitivity analyses tools (SaSAT) for computational modelling. Theor. Biol.
Med. Model. 2008, 5, 1–18.
37. Cianetti, S.; Pagano, S.; Nardone, M.; Lombardo, G. Model for taking care of patients with early childhood caries during the
SARS-Cov-2 pandemic. Int. J. Environ. Res. Public Health 2020, 17, 3751.
38. Carli, E.; Pasini, M.; Pardossi, F.; Capotosti, I.; Narzisi, A.; Lardani, L. Oral Health Preventive Program in Patients with Autism
Spectrum Disorder. Children 2022, 9, 535.
39. Baba, I.A.; Rihan, F.A. A fractional–order model with different strains of COVID-19. Phys. A Stat. Mech. Its Appl. 2022, 603,
127813.
40. Baba, I.A.; Sani, M.A.; Nasidi, B.A. Fractional dynamical model to assess the efficacy of facemask to the community transmission
of COVID-19. Comput. Methods Biomech. Biomed. Eng. 2022, 25, 1588–1598.