Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme

Rahila Naz; Aasim Ullah Jan; Attaullah; Salah Boulaaras; Rafik Guefaifia

doi:10.1515/nleng-2024-0028

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Computational analysis of the Covid-19 model using the continuous Galerkin–Petrov scheme

Rahila Naz , Aasim Ullah Jan , Attaullah , Salah Boulaaras und Rafik Guefaifia

Veröffentlicht/Copyright: 15. Oktober 2024

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Aus der Zeitschrift Nonlinear Engineering Band 13 Heft 1

Abstract

Epidemiological models feature reliable and valuable insights into the prevention and transmission of life-threatening illnesses. In this study, a novel SIR mathematical model for COVID-19 is formulated and examined. The newly developed model has been thoroughly explored through theoretical analysis and computational methods, specifically the continuous Galerkin–Petrov (cGP) scheme. The next-generation matrix approach was used to calculate the reproduction number ( R 0 ) . Both disease-free equilibrium (DFE) ( E 0 ) and the endemic equilibrium ( E ⁎ ) points are derived for the proposed model. The stability analysis of the equilibrium points reveals that ( E 0 ) is locally asymptotically stable when R 0 < 1 , while E ⁎ is globally asymptotically stable when R 0 > 1 . We have examined the model’s local stability (LS) and global stability (GS) for endemic equilibrium and DFE based on the number ( R 0 ) . To ascertain the dominance of the parameters, we examined the sensitivity of the number ( R 0 ) to parameters and computed sensitivity indices. Additionally, using the fourth-order Runge–Kutta (RK4) and Runge–Kutta–Fehlberg (RK45) techniques implemented in MATLAB, we determined the numerical solutions. Furthermore, the model was solved using the continuous cGP time discretization technique. We implemented a variety of schemes like cGP(2), RK4, and RK45 for the COVID-19 model and presented the numerical and graphical solutions of the model. Furthermore, we compared the results obtained using the above-mentioned schemes and observed that all results overlap with each other. The significant properties of several physical parameters under consideration were discussed. In the end, the computational analysis shows a clear image of the rise and fall in the spread of this disease over time in a specific location.

Keywords: epidemiological model; basic reproduction number; stability analysis; sensitivity analysis; numerical comparison; nonlinear equations

1 Introduction

Infections caused by viruses have consistently posed a significant danger to humanity throughout our collective history. Several recent epidemics have caused significant loss of life and widespread devastation worldwide. For example, the Spanish Flu pandemic, for instance, which tragically devastated the lives of millions of people across the globe. Numerous diseases, such as HIV/AIDS, have a devastating impact on countless lives each year as they rapidly spread among populations. Recently, instances of coronavirus pandemics have been documented [1,2,3,4,5]. A respiratory disorder was apparently identified in 2019 in the Hubei, province of China as an infectious illness brought on by a novel strain of the COVID-19 viral infection, also known as Respiratory Syndrome COVID-19, previously known as 2019-nCoV [6,7,8]. SARS, a coronavirus outbreak that destroyed more than 800 lives and caused more than 8,000 positive cases. Middle East Respiratory Syndrome (MERS) is thought to have spread from its original location in the Kingdom of Saudi Arabia to other nearby and far-off nations, particularly those along the Persian Gulf. In a few situations, MERS is already a contributing cause [9]. World Health Organization (WHO) initially received the report on December 31, 2019, regarding the outbreak of a contagious disease. On January 30, 2020, WHO formally identified it as COVID-19, a pandemic that exists and adversely affects the whole planet [10,11]. According to reports, the illness first affected animals before moving on to people. It was asserted by the media that, in the initial stage of the pandemic, the disease is transmitted by bats [12]. When the entire region was sealed in Wuhan in January and February 2020, the pandemic expanded rapidly. Later, reports of positive cases also came from the USA and other countries on both sides of the Pacific and Atlantic. Then, it was discovered that the illness was contagious and could spread through physical contact between individuals. In the middle of March 2020, the WHO proclaimed the virus to be a pandemic [13]. CoV-2 can remain on a surface for hours or days, depending on how sunshine, climatic change, and the surface material affect it. By engaging with a virus-contaminated object or material and then touching one’s lips, nose, or eyes, COVID-19 could be transmitted. Sadly, there are other ways for the virus to spread as well. Social isolation outside of the home decreases the chances of coming into contact with infectious objects or contagious people [14]. The government implemented a series of control measures, including strict lockdowns, social distancing, limits on gatherings, and the mandatory use of face masks to effectively reduce the transmission of COVID-19. Many countries implemented rigorous measures to verify the interactions of individuals who were known to be infected with COVID-19. This was done in order to effectively control the spread of the virus, and any incidents were promptly isolated for medical attention [15].

On March 7, 2021, sources stated that the pandemic has spread to numerous nations, with 116.17 million and above confirmed cases and 2.58 million losses [16]. Forty-one registered admitted positive events were confirmed as COVID-19 positive in China on January 2, 2020. More than 25 different Chinese provinces confirmed 571 COVID-19 incidents on January 22, 2020 [7,17]. On January 30, 2020, China had 7,734 COVID-19 occurrences that tested positive, as well as 90 overseas cases that were exported to roughly 13 other nations, including India, France, Germany, Canada, the UAE, and the USA. Over 4,667,780 positive cases were discovered worldwide by October 31, 2020 (Asia: 13,461,293 COVID patients, Africa: 1,776,595 COVID patients, Europe: 9,840,736 cases, America: 20,546,580 cases, Oceania: 41,880 cases, and others: 696 cases), resulting in 1,189,499 casualties (Asia: 239,675 losses, Africa: 42,688 losses, Europe: 265,565 losses, America: 640,513 losses, and others: 7 deaths) [18,19].

For better comprehension of how infectious and contagious diseases spread, mathematical models are useful. It is an effective instrument for assessing processes and phenomena in the actual world [20–24]. In 1760, Bernoulli became the first mathematician to suggest a novel mathematical model that depicts how infectious and contagious diseases spread. Later, the subject caught the interest of numerous scientists and researchers. These models make it easier to comprehend a wide range of physical and biological events and how they work. Models in this subject now range from the straightforward to the complex and intricate. Using mathematical models, many infectious and non-infectious diseases have been studied (see, for example, previous studies [21,22]). Computational equations are used by researchers to comprehend and examine the dynamics of a disease; see previous studies [25,26] for more information.

The local stability (LS) and global stability (GS) of the endemic and negative pool equilibria (disease-free equilibria [DFE]) were evaluated by the researchers using nonlinear numerical analytic approaches. Similar to this, researchers have carried out a thorough analysis of the innovative COVID-19 mathematical models that cover several perspectives of the disease. Global and local dynamics are studied along with stability theory and numerical simulations. In this area, we have been introduced to several outstanding studies [27–33]. Epidemiological models were used to obtain accurate and valuable information regarding the prevention and spread of illness. Consequently, our objective is to investigate the COVID-19 problem. In order to describe the COVID-19 epidemic, this work seeks to develop a SEQIRP model originating from the SIR model. Based on the reproduction number (R ₀), the model’s analysis will provide an understanding of the stable and unstable states [34–36]. Additionally, sensitivity analysis identifies the parameter that the system relies on the most, and numerical simulation illustrates how parameters change over time to forecast the COVID-19 outbreak. Ben Makhlouf et al. [37] investigated the existence and uniqueness of Hadamard Itô–Doob Stochastic delay fractional integral equations (HIDSDFIE) under non-Lipschitz conditions and by using the successive approximation. Rhaima et al. [38] discussed the existence and uniqueness properties pertaining to a class of fractional Hadamard Itô–Doob stochastic integral equations (FHIDSIE). Our study centers around the utilization of the Picard iteration technique (PIT), which not only establishes these fundamental properties but also unveils the remarkable averaging principle within FHIDSIE. Makhlouf et al. [39] developed a systematic discussion of the existence and uniqueness of the solution of a family of proportional Liouville–Caputo fractional stochastic differential equations by applying the Banach fixed point technique. Makhlouf et al. [40] investigated the existence and Ulam–Hyers stability (UHS) results in the context of mixed Hadamard and Riemann–Liouville fractional stochastic differential equations (HRFSDEs).

2 Model formulation

The overall human population N(t) is divided into six categories: susceptible ( S ), exposed ( E ), under quarantine ( Q ), infected ( I ), and death (P). The part of the population that gets healthy is denoted by R . The variables S(t) stands for the susceptible group, E ( t ) for diseased but not contagious individuals, and Q ( t ) for quarantine, which has the potential to produce either uninfected, infected outcomes, or fatal outcomes depending on the circumstances, I ( t ) stands for the number of infected individuals who are cured or they would die, and R ( t ) stands for those who have recovered and been cured of the illness. The variable P ( t ) depicts the disease that results in death. The population as a whole is represented by N ( t ) . The flowchart of the SEQIRP mathematical model is represented in Figure 1.

Figure 1

The graphical representation of the SEQIRP model.

All of the categories that make up the population ( N ) are Sus ( S ), Exp ( E ), Qua ( Q ), Inf ( I ), Rec ( R ), and death ( P ). Mathematically,

N ( t ) = S ( t ) + E ( t ) + Q ( t ) + I ( t ) + R ( t ) + P ( t ) .

The modified framework comprises a system of differential equations, which is illustrated as follows:

(1) d S d t = − ( β 1 + β 2 ) SI N + β 1 SE + ϵ Q ,

(2) d E d t = β 1 SI N − ( γ + μ + η + σ ) E ,

(3) d Q d t = β 2 SI N + γ E − ( μ + ν + τ ) Q ,

(4) d I d t = η E + ν Q − ( δ + μ + r ) I ,

(5) d R d t = r I − μ R ,

(6) d P d t = μ ( E + Q + S ) + δ I .

Figure 1 shows the pictorial diagram of the model.

Table 1 defines the parameters used in the model. Knowing that S ( t ) , E(t), Q(t), I(t), R(t), and P(t) are portions of all the population, we may state the following:

s ( t ) + e ( t ) + q ( t ) + i ( t ) + r ( t ) + p ( t ) = 1 ,

where s(t) = S ( t ) N , e(t) = E ( t ) N , q(t) = Q ( t ) N , i(t) = I ( t ) N , r(t) = R ( t ) N , p(t) = P ( t ) N .

Table 1

Description of variables used in the model

Parameters	Physical interpretation	Values
τ	Death rates for classes that were exposed to death	0.0002
β 1	Rate of contact between exposed and susceptible classes	0.0517901
β 2	Rate of contact between susceptible and quarantine classes	0.2
δ	Mortality rate due to corona viruses in infected individual class	1.6728 × 10⁻⁵
γ	Transfer rate of exposed people to quarantine	2.0138 × 10⁻⁴
η	Rate of people shifting from the Exp class to the Inf class	0.4478
μ	Rate of natural death	0.0106
ν	Transfer ratio of quarantined people to the class of infected people	3.2084 × 10⁻⁴
σ	Quarantine class to death class mortality ratio	0.01
r	Recovering rate of those who are infected	5.7341 × 10⁻⁵
ϵ	Transfer ratio from the quarantine class to the susceptible class	0.6

Now, we can assume the system of Eqs. (1)–(6) as follows:

(7) d s ( t ) d t = − ( β 1 + β 2 ) s ( t ) i ( t ) + β 1 e ( t ) + ϵ q ( t ) ,

(8) d e ( t ) d t = β 1 s ( t ) i ( t ) − ( γ + μ + η + σ ) e ( t ) ,

(9) d q ( t ) d t = β 2 s ( t ) i ( t ) + γ e ( t ) − ( μ + ν + τ ) q ( t ) ,

(10) d i ( t ) d t = η e ( t ) + ν q ( t ) − ( δ + μ + r ) i ( t ) ,

(11) d r ( t ) d t = r i ( t ) − μ r ( t ) ,

(12) d p ( t ) d t = μ ( e ( t ) + q ( t ) + s ( t ) ) + δ i ( t ) .

3 Numerical analysis of the model

Here, we perform the computational analysis of the system consisting of Eqs. (7)–(12).

3.1 Equilibria

At equilibrium, the left-hand side (LHS) of the system of Eqs. (7)–(12) will be zero, i.e.,

d s ( t ) d t = 0 , d e ( t ) d t = 0 , d q ( t ) d t = 0 , d i ( t ) d t = 0 , d r ( t ) d t = 0 .

DFEs ( E 0 ) are equilibrium points without an infection. Consequently, the system of Eqs. (7)–(12) always maintains an equilibrium at E 0 = ( S 0 , 0, 0, 0, 0) = (1, 0, 0, 0, 0). The equilibrium points with an infection called endemic equilibrium ( E ⁎ ) . Therefore, point ( E ⁎ ) from the system of Eqs. (7)–(12) is given by

s = β 1 e + ϵ q ( β 1 + β 2 ) i = S ⁎ ,

e = β 1 s i ( γ + μ + η + σ ) = E ⁎ ,

q = β 2 s i + γ e ( μ + ν + τ ) = Q ⁎ ,

i = η e + ν q ( δ + μ + r ) = I ⁎ ,

r = r μ i = R ⁎ .

Thus, E ⁎ = ( S ⁎ , E ⁎ , Q ⁎ , I ⁎ , R ⁎ ) is the unique endemic equilibrium of the system of Eqs. (7)–(12).

3.2 Basic reproduction number ( R 0 )

The basic transmission rate, or the basic reproduction number R 0 , is an essential indicator for analyzing disease transmission patterns. It provides insights into the dynamics of disease spread and the effectiveness of control measures. If R 0 < 1 , then the DFE is stable, and the disease ceases to exist in the community. If R 0 > 1 , the endemic equilibrium exists because the disease spreads throughout the community. The number R 0 is obtained by the method known as the next-generation matrix.

Let X = ( E , Q , I ) , then it follows from the system of Eqs. (7)–(12) that

d X d t = ℱ − V ,

where ℱ = β 1 s i β 2 s i 0 and V = ( γ + μ + η + σ ) e − γ e + ( μ + ν + τ ) q − η e − ν q + ( δ + μ + r ) i .

The Jacobian of the matrices ℱ and V , which are symbolized by F and V, are given as follows:

F = 0 0 β 1 0 0 β 2 0 0 0 , V = γ + μ + η + σ 0 0 − γ μ + ν + τ 0 − η − ν δ + μ + r .

The multiplicative inverse of the matrix V is

V − 1 = 1 γ + μ + η + σ 0 0 γ ( γ + μ + η + σ ) ( μ + ν + τ ) 1 μ + ν + τ 0 γ ν + η ( μ + ν + τ ) ( γ + μ + η + σ ) ( μ + ν + τ ) ( δ + μ + r ) ν ( μ + ν + τ ) ( δ + μ + r ) 1 δ + μ + r .

The next-generation matrix for the system of Eqs. (7)–(12) is

F V − 1 = β 1 ( γ ν + η ( μ + ν + τ ) ) ( γ + μ + η + σ ) ( μ + ν + τ ) ( δ + μ + r ) β 1 ν ( μ + ν + τ ) ( δ + μ + r ) β 1 δ + μ + r β 2 ( γ ν + η ( μ + ν + τ ) ) ( γ + μ + η + σ ) ( μ + ν + τ ) ( δ + μ + r ) β 2 ν ( μ + ν + τ ) ( δ + μ + r ) β 2 δ + μ + r 0 0 0 .

The eigenvalues of the matrix F V − 1 is λ 1 = λ 2 = 0 and

λ 3 = 1 ( μ + ν + τ ) ( δ + μ + r ) β 1 ( γ ν + η ( μ + ν + τ ) ) ( γ + μ + η + σ ) + β 2 ν . Therefore, R 0 is the highest (dominant) among the two eigenvalues of F V − 1 . As a result, we have

(13) R 0 = 1 ( μ + ν + τ ) ( δ + μ + r ) β 1 ( γ ν + η ( μ + ν + τ ) ) ( γ + μ + η + σ ) + β 2 ν .

This is the system’s necessary reproductive number R 0 .

3.3 Stability analysis

The stability of the equilibrium points E 0 and E ⁎ on a local and global basis are discussed in this section.

Theorem 1

When R 0 < 1 , the DFE E 0 is locally asymptotically stable. If R 0 = 1 , E 0 is locally stable. When R 0 > 1 , then point E 0 is an unstable saddle point.

Proof

The Jacobian matrix at point E 0 is given by

(14) J ( E 0 ) = 0 β 1 ϵ − ( β 1 + β 2 ) 0 0 − ( γ + μ + η + σ ) 0 β 1 0 0 γ − ( μ + ν + τ ) β 2 0 0 η ν − ( δ + μ + r ) 0 0 0 0 r − μ ,

E 0 is locally asymptotically stable [34] if all the eigenvalues of J ( E 0 ) have a negative real part, which means R 0 < 1 . Again, E 0 is unstable if at least one of the eigenvalues of J ( E 0 ) has a positive real part, which means R 0 > 1 . Now, from J ( E 0 ) , we obtain the two eigenvalues: λ 1 = 0 , λ 2 = − μ ; the other eigenvalues of J ( E 0 ) are determined by the following equation:

(15) λ 3 + C 3 λ 2 + C 4 λ + C 5 = 0 ,

where A = ( γ + μ + η + σ ) , B = ( μ + ν + τ ) , C = ( δ + μ + r ) ,

C 3 = ( A + B + C ) > 0 ,

C 4 = ( A B + A C + B C − β 1 η − β 2 ν ) ,

C 5 = A B C − B β 1 η − A β 2 ν − β 1 γ ν ,

C 4 = A B + A C ( 1 − R 0 ) + B C ( 1 − R 0 ) + B A β 1 η > 0 if R 0 < 1 ,

C 5 = A B C ( 1 − R 0 ) > 0 if R 0 < 1 ,

C 3 C 4 − C 5 = C C 4 + C ( A 2 + B 2 ) ( 1 − R 0 ) + B 2 C β 1 η + A 2 C β 2 ν + A B 2 + A 2 B + A B C > 0 if R 0 < 1 .

The Routh–Hurwitz criterion is satisfied as C 3 > 0 , C 4 > 0 , C 5 > 0 , and C 3 C 4 − C 5 > 0 if R 0 < 1 .

Therefore, all eigenvalues of Eq. (15) have a negative real part. On the other hand, one eigenvalue is zero, which means the reproductive number R 0 = 1 ( μ + ν + τ ) ( δ + μ + r ) β 1 ( γ ν + η ( μ + ν + τ ) ) ( γ + μ + η + σ ) + β 2 ν leads us to a value 1, which is the precise information of system (14) based on the study of Shahrear et al. [34]. The remaining eigenvalues of system (14) have a negative real part. This concludes the analysis of the DFE of system (14), which means E 0 is locally stable [34].□

Theorem 2

When R 0 > 1 , E ⁎ (endemic equilibrium) is locally asymptotically stable.

Proof

The Jacobian matrix of the model (7)–(12) at E ⁎ = ( S ⁎ , E ⁎ , Q ⁎ , I ⁎ , R ⁎ ) is

J ( E ⁎ ) = − ( β 1 + β 2 ) I ⁎ β 1 ϵ − ( β 1 + β 2 ) S ⁎ 0 β 1 I ⁎ − ( γ + μ + η + σ ) 0 β 1 S ⁎ 0 β 2 I ⁎ γ − ( μ + ν + τ ) β 2 S ⁎ 0 0 η ν − ( δ + μ + r ) 0 0 0 0 r − μ .

Trace

J ( E ⁎ ) = − ( β 1 + β 2 ) I ⁎ − ( γ + μ + η + σ ) − ( μ + ν + τ ) − ( δ + μ + r ) − μ < 0 .

Now, det

J ( E ⁎ ) = − μ ( β 1 + β 2 ) ( γ + μ + η + σ ) ( μ + ν + τ ) ( δ + μ + r ) I ⁎ + β 1 2 μ ( μ + ν + τ ) ( δ + μ + r ) I ⁎ + ϵ μ β 1 γ ( δ + μ + r ) I ⁎ + ϵ μ β 2 ( γ + μ + η + σ ) ( δ + μ + r ) I ⁎ .

From the study of Shahrear et al. [34], the equilibrium E ⁎ of the system of Eqs. (7)–(12) has a negative real part. Thus, we conclude that E ⁎ of the system is locally asymptotically stable, so R 0 > 1. This finalizes the proof.□

Theorem 3

The system of Eqs. (7)–(12) has no periodic orbits.

Proof

We apply Dulac’s criterion to prove this. Now, let X = ( S , E , Q , I , R ) . Pursing the Dulac’s function

G = 1 S E ,

in the case,

G d S ⁎ d t = − ( β 1 + β 2 ) I ⁎ E ⁎ + β 1 S ⁎ + ϵ Q ⁎ S ⁎ E ⁎ ,

G d E ⁎ d t = β 1 I ⁎ E ⁎ − ( γ + μ + η + σ ) S ⁎ ,

G d Q ⁎ d t = β 2 I ⁎ E ⁎ + γ S ⁎ − ( μ + ν + τ ) Q ⁎ S ⁎ E ⁎ ,

G d I ⁎ d t = η S ⁎ + ν Q ⁎ S ⁎ E ⁎ − ( δ + μ + r ) I ⁎ S ⁎ E ⁎ ,

G d R ⁎ d t = r I ⁎ S ⁎ E ⁎ − μ R ⁎ S ⁎ E ⁎ .

Thus,

d G X d t = ∂ ∂ S ⁎ G d S ⁎ d t + ∂ ∂ E ⁎ G d E ⁎ d t + ∂ ∂ Q ⁎ G d Q ⁎ d t + ∂ ∂ I ⁎ G d I ⁎ d t + ∂ ∂ R ⁎ G d R ⁎ d t ,

d G X d t = ∂ ∂ S ⁎ − ( β 1 + β 2 ) I ⁎ E ⁎ + β 1 S ⁎ + ϵ Q ⁎ S ⁎ E ⁎ + ∂ ∂ E ⁎ β 1 I ⁎ E ⁎ − ( γ + μ + η + σ ) S ⁎ + ∂ ∂ Q ⁎ β 2 I ⁎ E ⁎ + γ S ⁎ − ( μ + ν + τ ) Q ⁎ S ⁎ E ⁎ + ∂ ∂ I ⁎ η S ⁎ + ν Q ⁎ S ⁎ E ⁎ − ( δ + μ + r ) I ⁎ S ⁎ E ⁎ + ∂ ∂ R ⁎ r I ⁎ S ⁎ E ⁎ − μ R ⁎ S ⁎ E ⁎ ,

d G X d t = − β 1 S ⁎2 − ϵ Q ⁎ S ⁎2 E ⁎ − β 1 I ⁎ E ⁎2 − ( μ + ν + τ ) S ⁎ E ⁎ − ( δ + μ + r ) S ⁎ E ⁎ − μ S ⁎ E ⁎ ,

d G X d t = − β 1 S ⁎2 + ϵ Q ⁎ S ⁎2 E ⁎ + β 1 I ⁎ E ⁎2 + ( δ + 3 μ + ν + τ + r ) S ⁎ E ⁎ < 0 .

There is no periodic orbit in the system of Eqs. (7)–(12) as a result. The proof is now complete.□

Theorem 4

The endemic equilibrium E ⁎ for the system of Eqs. (7)–(12) is globally asymptotically stable whenever R 0 > 1.

3.4 Sensitivity analysis

The reproductive number R 0 clearly identifies the start of disease transmission. To identify which model parameters have an important impact on the fundamental reproduction number ( R 0 ) and, subsequently, the spread of disease, we calculate the sensitivity indices of R 0 to those values. Assuming parameter P i , the normalized sensitivity indices of R 0 are given by

I P i R 0 = ∂ R 0 ∂ P i P i R 0 .

In this study, we examine the sensitivity analysis of R 0 to the model parameters. We use the parameter values from Table 1 in this case. We obtained R 0 from Eq. (13), which contains an explicit formula for

R 0 = 1 ( μ + ν + τ ) ( δ + μ + r ) β 1 ( γ ν + η ( μ + ν + τ ) ) ( γ + μ + η + σ ) + β 2 ν .

The parameters’ sensitivity is determined as follows:

I β 1 R 0 = ∂ R 0 ∂ β 1 β 1 R 0 = γ ν + η ( μ + ν + τ ) ( γ + μ + η + σ ) ( μ + ν + τ ) ( δ + μ + r ) β 1 R 0 ,

when the parameter values from Table 1 were used, we obtained 0.44.

I β 2 R 0 = ∂ R 0 ∂ β 2 β 2 R 0 = ν ( μ + ν + τ ) ( δ + μ + r ) β 2 R 0 ,

after using the parameter values from Table 1, we obtained a value of −0.80.

I δ R 0 = ∂ R 0 ∂ δ δ R 0 = − 1 ( μ + ν + τ ) ( δ + μ + r ) 2 β 1 ( γ ν + η ( μ + ν + τ ) ) ( γ + μ + η + σ ) + β 2 ν δ R 0 ,

after using the parameter values from Table 1, we obtained a value of −0.16.

We may argue that if β 1 increases by 10%, then R 0 increases by 10% |0.44| = 0.044, which can cause an outbreak. Likewise, if β 1 drops by 10%, then R 0 falls by 10% |0.44| = 0.044, indicating that the value of β 1 significantly affects the value of R 0 . For controlling the sickness, it is necessary to lower the rate of β 1 . The WHO and the government encouraged quarantine and other preventative measures to deal with the pandemic. Additionally, the number R 0 can be affected by changing the values of β 2 and δ .

3.5 Numerical schemes

3.5.1 Continuous Galerkin–Petrov (cGP) technique

The Galerkin technique is a powerful tool for numerically exploring significant difficulties in various real-world problems. This approach is often used for complex problems and can handle nonlinear systems and complicated issues (refer previous studies [41–52] for more information). This section discusses the numerical schemes used to solve the aforesaid model to assess the dynamic behavior of the model. The system of ODEs for the considered model can be written as follows:

Find V ˜ : [ 0 , t max ] → V = ℝ d ; then d t V ˜ ( t ) = F ( t , V ˜ ( t ) ) ∀ t ∈ R ,

(16) V ˜ ( 0 ) = V ˜ 0 ,

where d t is the time derivative of V ˜ ( t ) , R = [ 0 , T ] is the whole interval, V ˜ ( t ) = ( V ˜ 1 ( 0 ) , V ˜ 2 ( 0 ) , V ˜ 3 ( 0 ) ) ∈ V ⇒ V ˜ ( 0 ) = ( V ˜ 1 ( 0 ) , V ˜ 2 ( 0 ) , V ˜ 3 ( 0 ) ) ∈ V are the initial values of V ˜ ( t ) at t = 0 .

We further assume that ( V ˜ 1 ( 0 ) , V ˜ 2 ( 0 ) , V ˜ 3 ( 0 ) ) = ( T ( t ) , R ( t ) , V ( t ) ) , which implies that ( V ˜ 1 ( 0 ) , V ˜ 2 ( 0 ) , V ˜ 3 ( 0 ) ) = ( T ( 0 ) , R ( 0 ) , V ( 0 ) ) . The function F = ( f 1 , f 2 , f 3 ) is nonlinear and is defined as F : R × S → S .

The weak formulation (see previous studies [10,11,12,13,14,15,17] for explanation) of problem (16) is as follows: Find V ˜ ∈ X such that V ˜ ( 0 ) = V ˜ 0 and

(17) ∫ I 〈 d t V ˜ ( t ) , v ( t ) 〉 d t = ∫ I 〈 F ( t , V ˜ ( t ) , v ( t ) ) 〉 d t for all v ϵ Y ,

where the test space and solution, respectively, are represented by X and Y in order to describing the time discretization of a variation kind problem (16).

Identify the function t → V ˜ ( t ) ; then , we describe the space C ( R , S ) = C 0 ( R , S ) . It is the continuous function space V ˜ : R → S possessing the following standard norm:

∥ V ˜ ∥ C ( R , S ) = ∥ t ∈ I Sup V ˜ ∥ K .

We shall utilize the space L 2 ( R , S ) as the space of the functions that are discontinuous and defined as

L 2 ( R , S ) = V ˜ : R → S : ∥ V ˜ ∥ L 2 ( R , S ) = ∫ R ∥ V ˜ ∥ S 2 d t < ∞ 1 / 2 .

During time discretization, the intervals R are divided into N subintervals R τ = [ t τ − 1 , t τ ] , where τ = 1 , … , N and 0 = t 0 < t 1 < t 2 < ⋯ < t N − 1 < t N = T . The parameter j indicates the time discretization with the largest time step size j = max j τ 1 ≤ τ ≤ N , where j τ = t τ − t τ − 1 , the length of the nth time interval R τ . The time step is calculated from the following collection of time intervals M j = { R 1 , … , R N } . We perceive the solution V ˜ : R → S on all time intervals R τ through a function V ˜ j : R → S described by

X j l = { V ˜ ∈ C ( R , S ) : V ˜ I τ ∈ ℙ l ( R τ , S ) for all I τ ∈ M j } ,

where

ℙ l ( R τ , S ) = V ˜ : R τ → S : V ˜ ( t ) = ∑ s = 0 l U s t s , for all t ∈ R τ , U s ∈ S , ∀ s .

The discrete test space for V ˜ j is Y j l ⊂ Y and is defined by

(18) Y j k = { v ∈ L 2 ( R , S ) : v | R τ ∈ ℙ k − 1 ( R τ , S ) ∀ R τ ∈ M j } ,

which is made up of l − 1 piecewise polynomials (see previous studies [41–52] for details) and at time step ending nodes, it is discontinuous. Multiplying Eq. (16) with test functions v j ∈ Y j k and then integrating in excess of the interval R, we obtain the discrete-time case. Find V ˜ ∈ X j k , so that V ˜ j ( 0 ) = 0 and

(19) ∫ R 〈 V ˜ j ' ( t ) , v j ( t ) 〉 d t = ∫ R 〈 F ( t , V ˜ j ( t ) , v j ( t ) ) 〉 d t ∀ v j ϵ Y j l .

Using the test functions v j ( t ) = ν ψ ( t ) with ν ∈ S and a scalar function ψ : R → R , which is zero on R | R τ and a polynomial of order less than or equal to l − 1 on the time interval R τ = [ t τ − 1 , t τ ] . Now, we find V ˜ j | I τ ∈ ℙ k − 1 ( R τ , S ) such that

(20) ∫ R τ 〈 d t V ˜ j ( t ) , v j ( t ) φ ( t ) 〉 d t = ∫ R τ 〈 F ( t , V ˜ j ( t ) , v ) 〉 φ ( t ) d t ∀ v ϵ S ∀ φ ∈ ℙ l − 1 ( R τ ) ,

with the initial conditions V ˜ j | R τ ( t τ − 1 ) = V ˜ j | R τ − 1 ( t τ − 1 ) for τ ≥ 2 and V ˜ j | R τ ( t τ − 1 ) = V ˜ 0 for τ = 1 . To determine cGP(l), we will find V ˜ j | R τ − 1 ∈ ℙ l ( R τ , S ) such that V ˜ j ( t τ − 1 ) = V ˜ τ − 1 . This leads us to

(21) ∑ s = 0 l w ˆ s d t V ˜ j ( t τ , s ) φ ( t τ , s ) = ∑ s = 0 k w ˆ s F ( t τ , s , V ˜ j ( t τ , s ) ) φ ( t τ , s ) ∀ φ ∈ ℙ l − 1 ( R τ ) ,

where the weights are denoted by w ˆ s . These weights are represented by t ˆ ∈ [ − 1 , 1 ] , s = 0 , 1 , 2 , 3 , … , l . We utilize a polynomial ansatz to determine V ˜ j for every time interval R τ as follows:

(22) V ˜ j ( t ) = ∑ s = 0 k U τ s ϕ τ , s ( t ) ∀ t ∈ R τ ,

where the coefficients U τ s are the components of S and the functions ϕ τ , s ∈ ℙ l ( R τ ) are the Lagrange basis functions (see previous studies [41–52] for more details) w.r.t. l + 1 appropriate nodal points t τ , s ∈ R τ satisfying the conditions specified below:

(23) ϕ τ , s ( t τ , r ) = δ r , s , r , s = 0 , 1 , 2 , … , l .

where δ r , s denotes the Kronecker delta and is defined as follows:

δ r , s = 1 , if r = s , 0 , if r ≠ s .

Here, t τ , s are described as the quadrature points [46] of “ ( l + 1 ) - point Gauß-Lobatto” formula (for more information, see previous studies [41–52]) on the R τ interval. Choosing initial conditions is important, we can therefore set t τ , 0 = t τ − 1 which indicates the initial conditions for Eq. (20):

U τ 0 = V ˜ j | R τ − 1 if τ ≥ 2 ,

for τ = 1 ⇒ U τ 0 = V ˜ 0 .

We define the basis functions ϕ τ , s ∈ ℙ l ( R τ ) via the affine reference transformation (see previous studies [41–52] for detailed explanation) T ¯ : R ˆ → R τ where R ˆ = [ − 1 , 1 ] and

t = T ¯ ( t ˆ ) = t τ − t τ − 1 2 + j τ 2 t ˆ ∈ R τ ∀ t ˆ ∈ R τ , τ = 1 , 2 , 3 … , N .

Let ϕ ˆ s ∈ ℙ l ( R ˆ ) , s = 0 , 1 , … , l , illustrate the basis functions that satisfy

ϕ ˆ s ( t ˆ r ) = δ r , s r , s = 0 , 1 , 2 , … , l ,

in which t ˆ 0 = − 1 and t ˆ r , r = 1 , 2 , … , l , are the quadrature points based on the interval R ˆ . The basis functions on the specified interval R τ are defined by the mapping

ϕ τ , s ( t ) = ϕ ˆ s ( t ˆ r ) with t ˆ = T ¯ τ − 1 ( t ) = 2 j τ t + t τ − 1 − t τ 2 ∈ R ˆ .

Similarly, the appropriate reference basis functions φ ˆ ∈ ℙ l − 1 ( R ˆ ) also define the test basis functions, which are denoted by φ τ , r , i.e.,

φ τ , r ( t ) = φ ˆ r ( T ¯ τ − 1 ( t ) ) ∀ t ∈ R τ , r = 1 , 2 , … , l .

According to the illustration (22), we learn that d t V ˜ j

(24) d t V ˜ j ( t ) = ∑ s = 0 k U τ s ϕ τ , s ' ( t ) ∀ t ∈ R τ .

By substituting Eq. (24) into Eq. (20), we obtain

∫ R τ 〈 d t V ˜ j ( t ) , v 〉 φ ( t ) d t = ∫ R τ ∑ s = 0 k 〈 U τ s 〉 ϕ τ ′ ( t ) φ ( t ) d t .

Next, the integral is converted into the referral interval R ˆ and calculated using the “ ( l + 1 ) -point Gauß–Lobatto quadrature” formula, which results in all test basis functions having the value ϕ ∈ ℙ l − 1 and for every ν ∈ S,

∫ R τ ∑ s = 0 l 〈 U τ s , v 〉 ϕ ˆ s ′ ( t ˆ ) φ ˆ ( t ˆ ) d t ˆ = ∫ R τ 〈 F w τ ( t ˆ ) , ∑ s = 0 l U τ s ( t ˆ ) , v 〉 φ ˆ ( t ˆ ) d t ˆ ∀ v ∈ S ,

⇒ ∑ μ = 0 l w ˆ μ ∑ μ = 0 l 〈 U τ s , v 〉 ϕ ˆ s ' ( t ˆ μ ) φ ˆ ( t ˆ μ ) = ∑ μ = 0 l w ˆ μ 〈 F w τ ( t ˆ ) , ∑ s = 0 l U τ s ( t ˆ ) , v 〉 φ ˆ ( t ˆ μ ) ,

where t ˆ μ ∈ [ − 1 , 1 ] is the integration point with t ˆ 0 = − 1 and t ˆ l = 1 and w ˆ μ represents the weights. If we pick the test functions φ τ , i ∈ ℙ l − 1 ( R τ ) such that

φ ˆ ( t ˆ μ ) = ( w ˆ ) − 1 δ r , μ r , μ = 1 , 2 , … , l .

The undetermined coefficient U τ s ∈ S for s = 1 … l , are found by using

∑ s = 0 l α r , s U τ s = j τ 2 { F ( t τ , r , U τ s ) + β r F ( t τ , 0 , U τ s ) } ∀ i = 1 , 2 , … , l ,

where U τ s = U τ − 1 s for τ > 1 and U 1 0 = V ˜ 0 for τ = 1 and

α r , s = ϕ ˆ s ′ ( t ˆ r ) + β r ϕ ˆ s ′ ( t ˆ 0 ) , β r = w ˆ 0 φ ˆ r ( t ˆ 0 ) .

We will discuss the cGP(k) approach for two cases k = 1 and k = 2 .

The cGP(1) method

The “two-point Gauß–Lobatto” formula was employed with t τ , 0 = t τ − 1 , t τ , 1 = t τ , and weights w ˆ 0 = w ˆ 1 = 1 that provides the famous Trapezoidal rule. We obtained α 1 , 0 = − 1 , α 1 , 1 = 1 , and β 1 = 1 . For just one coefficient U τ 1 = V ˜ j ( t τ ) ∈ S , the issue generates the block equation as follows:

α 1 , 1 U τ 1 − α τ , 0 U τ 0 = j τ 2 { F ( t τ , U τ 1 ) + F ( t τ − 1 , U τ 0 ) } .

The cGP(2) method

The weighted quadratic basis functions w ˆ 0 = w ˆ 2 = 1 / 3 , w ˆ 1 = 4 / 3 , and t ˆ 0 = − 1 , t ˆ 1 = 0 , t ˆ 2 = 1 are defined using the three-point Gauß–Lobatto formula (Simpson rule). Then, we obtain

α r , s = − 5 4 1 1 4 2 − 4 2 , β r = 1 2 − 1 , r = 1 , 2 , s = 0 , 1 , 2 .

Therefore, the system that needs to be fixed for U τ 1 , U τ 2 ∈ K from the well-known U τ 0 = U τ − 1 2 is as follows:

α 1 , 1 U τ 1 + α 1 , 2 U τ 2 = − α 1 , 0 U τ 0 + j τ 2 { F ( t τ , 1 , U τ 1 ) + β 1 F ( t τ , 0 , U τ 0 ) } ,

α 2 , 1 U τ 1 + α 2 , 2 U τ 2 = − α 2 , 0 U τ 0 + j τ 2 { F ( t τ , 2 , U τ 2 ) + β 2 F ( t τ , 0 , U τ 0 ) } ,

where U τ 0 indicates the initial conditions at the current time interval.

4 Numerical comparison and discussion

This section focuses on the numerical solution of the novel model of COVID-19 using the Galerkin scheme. The parameters listed in Table 1 are used to construct the model. In order to gain insight into the behavior of specific parameters in the suggested model, we explore different values for some of the parameters while keeping all other parameters constant. The model illustrates the geometric representation of the interactions and correlation between all parameters. It is clear from the figures that changing the values of the parameters will result in different effects on the dynamic behavior of the system. We only planned for the situation where one quantity (parameter) was varied. We explored the variations of ( η ) and ( μ ), as shown in Figures 2–13, across the six compartments S ( t ) , E ( t ) , Q ( t ) , I ( t ) , R ( t ) , and P ( t ) , respectively. Figures 2, 4–9 show an increasing behavior in the sensitive population over time as the value of η is increased, while they show the decreasing behavior for population dynamics of exposed class E ( t ) , as shown in Figure 3. The effects of various μ values on the exposed population are analyzed in Figures 8 and 13. Figures 8 and 12 illustrate the decreasing behavior for the population dynamics of S , E , Q , I , and R by increasing the values of μ . By increasing the values of μ , the concentration of P increases, as displayed in Figure 13. Figures 14–19 show the comparison graphs for the Galerkin method, RK methods, and Runge–Kutta–Fehlberg method solutions, respectively. Additionally, the Galerkin, fourth-order Runge–Kutta (RK4) and Runge–Kutta–Fehlberg methods are used for numerical computations in a certain manner and throughout a range of time intervals t ∈ [ 0 , 1 ] , as shown by the arithmetic values for various step sizes in Figures 14–19.

$Figure 2 Influence of η \eta on S ( t ) S(t) .$

Figure 2

Influence of η on S ( t ) .

$Figure 3 Influence of η \eta on E ( t ) E(t) .$

Figure 3

Influence of η on E ( t ) .

$Figure 4 Impact of η \eta on Q ( t ) Q(t) .$

Figure 4

Impact of η on Q ( t ) .

$Figure 5 Impact of η \eta on I ( t ) I(t) .$

Figure 5

Impact of η on I ( t ) .

$Figure 6 Effect of η \eta on R ( t ) R(t) .$

Figure 6

Effect of η on R ( t ) .

$Figure 7 Effect of η \eta on P ( t ) P(t) .$

Figure 7

Effect of η on P ( t ) .

$Figure 8 Influence of μ \mu on S ( t ) S(t) .$

Figure 8

Influence of μ on S ( t ) .

$Figure 9 Influence of μ \mu on E ( t ) E(t) .$

Figure 9

Influence of μ on E ( t ) .

$Figure 10 Impact of μ \mu on Q ( t ) Q(t) .$

Figure 10

Impact of μ on Q ( t ) .

$Figure 11 Impact of μ \mu on I ( t ) I(t) .$

Figure 11

Impact of μ on I ( t ) .

$Figure 12 Effect of μ \mu on R ( t ) R(t) .$

Figure 12

Effect of μ on R ( t ) .

$Figure 13 Effect of μ \mu on P ( t ) P(t) .$

Figure 13

Effect of μ on P ( t ) .

Figure 14

Comparison results for S ( t ) .

Figure 15

Comparison results for E ( t ) .

Figure 16

Comparison results for I ( t ) .

Figure 17

Comparison results for Q ( t ) .

Figure 18

Comparison results for R ( t ) .

Figure 19

Comparison results for P ( t ) .

Here, we graphically depict and match the outcomes. Figures 14–19 compare the results of the Galerkin and RK techniques, which are substantially more similar. In Figures 20–22, the mesh grid graphs are presented. Ultimately, we conclude that the numerical method presented in this study may be relied upon to provide solutions that are quite adaptable and precise when applied to situations of a similar nature.

Figure 20

The mesh grid graph of the Galerkin scheme.

Figure 21

The mesh grid graph of the RK4 scheme.

Figure 22

The mesh grid graph of the RK45 scheme.

5 Conclusion

In the present study, an innovative mathematical model was developed to illustrate the dynamics of different population groups, including susceptible, exposed, infected, quarantined, recovered, and deceased individuals. This model leads to a system of differential equations. The LS and GS of the DFE ( E 0 ) and endemic equilibrium ( E ⁎ ) are examined for the proposed model. The number ( R 0 ) is evaluated using the next-generation matrix procedure. It is observed that when R 0 < 1 and R 0 > 1 , the DFE ( E 0 ) is found to be locally asymptotically stable and unstable, respectively. Additionally, the endemic equilibrium E ⁎ for the system is globally asymptotically stable when R 0 > 1 . The model is solved numerically using the Galerkin–Petrov scheme and discusses the influence of different clinical parameters on the variation of state variables. The process of determining the sensitivity indices involves partial derivative analysis, which computes the partial derivatives of the model output ( R 0 ) with respect to the model parameters, aims to quantify the importance of each parameter in determining the behavior of the system based on the updated SIR model that has been demonstrated. Additionally, in order to assess the accuracy and dependability of the scheme, we numerically solved the model using the well-known traditional RK4 method and Runge–Kutta–Fehlberg (RK45) method. This simulation illustrates how the rates of infection, recovery, and death can change over time. Graphical illustrations of each result are displayed. It can be noted that the results obtained through all methods coincide with each other. Our future plans are focused on applying the proposed Galerkin scheme to a variety of mathematical models in population biology and epidemiology and other complex real-world problems consisting of non-linear differential equations.

Acknowledgment

The authors are thankful to the editor and anonymous reviewers for meticulously reading the manuscript, and giving us valuable comments and suggestions, which helped us to improve the quality of the manuscript.

Funding information: There is no funding source available for this research.
Author contributions: All the authors contributed equally.
Conflict of interest: All the authors declare that they have no conflicts of interest.
Data availability statement: No new data were created or analyzed in this study. Data sharing is not applicable to this article.

References

[1] Al-Tawfiq JA, Hinedi K, Ghandour J, Khairalla H, Musleh S, Ujayli A, et al. Middle East respiratory syndrome coronavirus: A case-control study of hospitalized patients. Clin Infect Dis. 2014 Jul;59(2):160–5.10.1093/cid/ciu226Suche in Google Scholar PubMed PubMed Central

[2] Azhar EI, El-Kafrawy SA, Farraj SA, Hassan AM, Al-Saeed MS, Hashem AM, et al. Evidence for camel-to-human transmission of MERS coronavirus. N Engl J Med. 2014 Jun;370(26):2499–505.10.1056/NEJMoa1401505Suche in Google Scholar PubMed

[3] Hwan KY, Mi LS, Shin CC, Yun CS, Me HS, Seo SY. The characteristics of Middle Eastern respiratory syndrome coronavirus transmission dynamics in South Korea. Osong Public Health Res Perspect. 2016;7(1):49–55. 10.1016/j.phrp.2016.01.001Suche in Google Scholar PubMed PubMed Central

[4] Abdel-Aty AH, Khater MM, Dutta H, Bouslimi J, Omri M. Computational solutions of the HIV-1 infection of CD4+ T-cells fractional mathematical model that causes acquired immunodeficiency syndrome (AIDS) with the effect of antiviral drug therapy. Chaos Solitons Fract. 2020 Oct;139:110092.10.1016/j.chaos.2020.110092Suche in Google Scholar PubMed PubMed Central

[5] Alnaser WE, Abdel-Aty M, Al-Ubaydli O. Mathematical prospective of coronavirus infections in Bahrain, Saudi Arabia and Egypt. Inf Sci Lett. 2020;9(1):51–64.10.18576/isl/090201Suche in Google Scholar

[6] Rothan HA, Byrareddy SN. The epidemiology and pathogenesis of coronavirus disease (COVID-19) outbreak. J Autoimmun. 2020 May 1;109:102433.Suche in Google Scholar

[7] Lu H. Drug treatment options for the 2019-new coronavirus (2019-nCoV). Biosci Trends. 2020 Feb;14(1):69–71.10.5582/bst.2020.01020Suche in Google Scholar PubMed

[8] Bassetti M, Vena A, Giacobbe DR. The novel Chinese coronavirus (2019‐nCoV) infections: Challenges for fighting the storm. Eur J Clin Invest. 2020;50(3):e13209.Suche in Google Scholar

[9] Chen Z, Zhang W, Lu Y, Guo C, Guo Z, Liao C, et al. From SARS-CoV to Wuhan 2019-nCoV outbreak: Similarity of early epidemic and prediction of future trends. 2020 Jan. bioRxiv preprint. 10.1101/2020.01.24.919241.Suche in Google Scholar

[10] Coronavirus E. 13,968 cases and 223 deaths https://www.worldometers.info/coronavirus/country/ethiopia. Accessed on 2020 Jul 27.Suche in Google Scholar

[11] Wrapp D, Wang N, Corbett KS, Goldsmith JA, Hsieh CL, Abiona O, et al. Cryo-EM structure of the 2019-nCoV spike in the prefusion conformation. Science. 2020 Mar;367(6483):1260–3.10.1126/science.abb2507Suche in Google Scholar PubMed PubMed Central

[12] Bozkurt F, Yousef A, Baleanu D, Alzabut J. A mathematical model of the evolution and spread of pathogenic coronaviruses from natural host to human host. Chaos Solitons Fract. 2020 Sep;138:109931.10.1016/j.chaos.2020.109931Suche in Google Scholar PubMed PubMed Central

[13] World Health Organization. Air quality guidelines for Europe. World Health Organization. Eorope: Regional Office for Europe. 2000.Suche in Google Scholar

[14] Zhu N, Zhang D, Wang W, Li X, Yang B, Song J, et al. A novel coronavirus from patients with pneumonia in China, 2019. N Engl J Med. 2020 Feb;382(8):727–33.10.1056/NEJMoa2001017Suche in Google Scholar PubMed PubMed Central

[15] Odukoya OO, Adejimi AA, Isikekpei B, Jim CS, Osibogun A, Ogunsola FT. Epidemiological trends of coronavirus disease 2019 in Nigeria: from 1 to 10,000. Niger Postgrad Med J. 2020 Oct;27(4):271–9.10.4103/npmj.npmj_233_20Suche in Google Scholar PubMed

[16] Steffens I. A hundred days into the coronavirus disease (COVID-19) pandemic. Eurosurveillance. 2020 Apr;25(14):2000550.10.2807/1560-7917.ES.2020.25.14.2000550Suche in Google Scholar PubMed PubMed Central

[17] Rothan HA, Byrareddy SN. The epidemiology and pathogenesis of coronavirus disease (COVID-19) outbreak. J Autoimmun. 2020 May;109:102433.10.1016/j.jaut.2020.102433Suche in Google Scholar PubMed PubMed Central

[18] Bassetti M, Vena A, Giacobbe DR. The novel Chinese coronavirus (2019‐nCoV) infections: Challenges for fighting the storm. Eur J Clin Invest. 2020 Mar;50(3):e13209.10.1111/eci.13209Suche in Google Scholar PubMed PubMed Central

[19] Chughtai AA, Seale H, Islam MS, Owais M, Macintyre CR. Policies on the use of respiratory protection for hospital health workers to protect from coronavirus disease (COVID-19). Int J Nurs Stud. 2020 May;105:103567.10.1016/j.ijnurstu.2020.103567Suche in Google Scholar PubMed PubMed Central

[20] Ranjan R, Prasad HS. A fitted finite difference scheme for solving singularly perturbed two point boundary value problems. Inf Sci Lett. 2020;9(2):65–73.10.18576/isl/090202Suche in Google Scholar

[21] Brauer F. Mathematical epidemiology: Past, present, and future. Infect Dis Model. 2017 May;2(2):113–27.10.1016/j.idm.2017.02.001Suche in Google Scholar PubMed PubMed Central

[22] EnKo PD. On the course of epidemics of some infectious diseases. Int J Epidemiol. 1989;18(4):749–55.10.1093/ije/18.4.749Suche in Google Scholar PubMed

[23] Hadhoud AR. Quintic non-polynomial spline method for solving the time fractional biharmonic equation. Appl Math Inf Sci. 2019;13:507–13.10.18576/amis/130323Suche in Google Scholar

[24] Ereú J, Giménez J, Pérez L. On solutions of nonlinear integral equations in the space of functions of Shiba-bounded variation. Appl Math Inf Sci. 2020;14(2020):393–404.10.18576/amis/140305Suche in Google Scholar

[25] Castillo-Chavez C, Blower S, van den Driessche P, Kirschner D, Yakubu AA, editors. Mathematical approaches for emerging and reemerging infectious diseases: Models, methods, and theory. Berlin, Germany: Springer Science & Business Media; 2002 May.10.1007/978-1-4613-0065-6Suche in Google Scholar

[26] Kumar D, Singh J, Al Qurashi M, Baleanu D. A new fractional SIRS-SI malaria disease model with application of vaccines, antimalarial drugs, and spraying. Adv Differ Equ. 2019 Dec;2019(1):1–9.10.1186/s13662-019-2199-9Suche in Google Scholar

[27] Shaikh AS, Shaikh IN, Nisar KS. A mathematical model of COVID-19 using fractional derivative: outbreak in India with dynamics of transmission and control. Adv Differ Equ. 2020 Jul;2020(1):373.10.1186/s13662-020-02834-3Suche in Google Scholar PubMed PubMed Central

[28] Khan MA, Atangana A. Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alex Eng J. 2020 Aug;59(4):2379–89.10.1016/j.aej.2020.02.033Suche in Google Scholar

[29] Ndaïrou F, Area I, Nieto JJ, Torres DF. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fract. 2020 Jun;135:109846.10.1016/j.chaos.2020.109846Suche in Google Scholar PubMed PubMed Central

[30] Lin Q, Zhao S, Gao D, Lou Y, Yang S, Musa SS, et al. A conceptual model for the coronavirus disease 2019 (COVID-19) outbreak in Wuhan, China with individual reaction and governmental action. Int J Infect Dis. 2020 Apr;93:211–6.10.1016/j.ijid.2020.02.058Suche in Google Scholar PubMed PubMed Central

[31] Abdo MS, Shah K, Wahash HA, Panchal SK. On a comprehensive model of the novel coronavirus (COVID-19) under Mittag-Leffler derivative. Chaos Solitons Fract. 2020 Jun;135:109867.10.1016/j.chaos.2020.109867Suche in Google Scholar PubMed PubMed Central

[32] Yousaf M, Zahir S, Riaz M, Hussain SM, Shah K. Statistical analysis of forecasting COVID-19 for upcoming month in Pakistan. Chaos Solitons Fract. 2020 Sep;138:109926.10.1016/j.chaos.2020.109926Suche in Google Scholar PubMed PubMed Central

[33] Atangana A. Modelling the spread of COVID-19 with new fractal-fractional operators: can the lockdown save mankind before vaccination? Chaos Solitons Fract. 2020 Jul;136:109860.10.1016/j.chaos.2020.109860Suche in Google Scholar PubMed PubMed Central

[34] Shahrear P, Rahman SS, Nahid MM. Prediction and mathematical analysis of the outbreak of coronavirus (COVID-19) in Bangladesh. Results Appl Math. 2021 May;10:100145.10.1016/j.rinam.2021.100145Suche in Google Scholar PubMed PubMed Central

[35] Ahmed I, Modu GU, Yusuf A, Kumam P, Yusuf I. A mathematical model of Coronavirus Disease (COVID-19) containing asymptomatic and symptomatic classes. Results Phys. 2021 Feb;21:103776.10.1016/j.rinp.2020.103776Suche in Google Scholar PubMed PubMed Central

[36] Baleanu D, Abadi MH, Jajarmi A, Vahid KZ, Nieto JJ. A new comparative study on the general fractional model of COVID-19 with isolation and quarantine effects. Alex Eng J. 2022 Jun;61(6):4779–91.10.1016/j.aej.2021.10.030Suche in Google Scholar

[37] Ben Makhlouf A, Mchiri L, Mtiri F. Existence, uniqueness, and averaging principle for Hadamard Itô–Doob stochastic delay fractional integral equations. Math Methods Appl Sci. 2023 Sep;46(14):14814–27.10.1002/mma.9346Suche in Google Scholar

[38] Rhaima M, Mchiri L, Ben Makhlouf A. The existence and averaging principle for a class of fractional Hadamard Itô–Doob stochastic integral equations. Symmetry. 2023 Oct;15(10):1910.10.3390/sym15101910Suche in Google Scholar

[39] Makhlouf AB, Mchiri L, Srivastava HM. Some existence and uniqueness results for a class of proportional Liouville-Caputo fractional stochastic differential equations. Bull Sci Math. 2023 Dec;189:103349.10.1016/j.bulsci.2023.103349Suche in Google Scholar

[40] Rhaima M, Mchiri L, Makhlouf AB, Ahmed H. Ulam type stability for mixed Hadamard and Riemann–Liouville fractional stochastic differential equations. Chaos Solitons Fract. 2024 Jan 1;178:114356.10.1016/j.chaos.2023.114356Suche in Google Scholar

[41] Attaullah, Zeb K, Khan I, Ahmad R, Eldin SM. Transmission dynamics of a novel HIV/AIDS model through a higher-order Galerkin time discretization scheme. Sci Rep. 2023 May;13(1):7421.10.1038/s41598-023-34696-6Suche in Google Scholar PubMed PubMed Central

[42] Attaullah RD, Weera W. Galerkin time discretization scheme for the transmission dynamics of HIV infection with non-linear supply rate. J AIMS Math. 2022;6:11292–310.10.3934/math.2022630Suche in Google Scholar

[43] Alyobi S, Yassen MF. A study on the transmission and dynamical behavior of an HIV/AIDS epidemic model with a cure rate. AIMS Math. 2022;7(9):17507–28.10.3934/math.2022965Suche in Google Scholar

[44] Attaullah YM, Alyobi S, Al-Duais FS, Weera W. On the comparative performance of fourth order Runge–Kutta and the Galerkin–Petrov time discretization methods for solving nonlinear ordinary differential equations with application to some mathematical models in epidemiology. AIMS Math. 2023;8(2):3699–729.10.3934/math.2023185Suche in Google Scholar

[45] Attaullah JM, Alyobi S, Yassen MF, Weera W. A higher order Galerkin time discretization scheme for the novel mathematical model of COVID-19. AIMS Math. 2023 Jan;8(2):3763–90.10.3934/math.2023188Suche in Google Scholar

[46] Attaullah, Yüzbaşı Ş, Alyobi S, Yassen MF, Weera W. A higher‐order Galerkin time discretization and numerical comparisons for two models of HIV infection. Comput Math Methods Med. 2022;2022(1):3599827.10.1155/2022/3599827Suche in Google Scholar PubMed PubMed Central

[47] Khan MT, Alyobi S, Yassen MF, Prathumwan D. A computational approach to a model for HIV and the Immune System Interaction. Axioms. 2022;11(10):578.10.3390/axioms11100578Suche in Google Scholar

[48] Khurshaid A, Alyobi S, Yassen MF, Prathumwan D. Computational framework of the SVIR epidemic model with a non-linear saturation incidence rate. Axioms. 2022 Nov;11(11):651.10.3390/axioms11110651Suche in Google Scholar

[49] Attaullah ZebK, Mohamed A. The influence of saturated and bilinear incidence functions on the dynamical behavior of HIV model using Galerkin scheme having a polynomial of order two. CMES-Comput Model Eng Sci. 2023;136(2):1661–85.10.32604/cmes.2023.023059Suche in Google Scholar

[50] Jan R, Yüzbaşı Ş. Dynamical behaviour of HIV infection with the influence of variable source term through Galerkin method. Chaos Solitons Fract. 2021 Nov;152:111429.10.1016/j.chaos.2021.111429Suche in Google Scholar

[51] Attaullah, Sohaib M. Mathematical modeling and numerical simulation of HIV infection model. Results Appl Math. 2020 Aug;7:100118.10.1016/j.rinam.2020.100118Suche in Google Scholar

[52] Attaullah RJ, Jabeen A. Solution of the HIV infection model with full logistic proliferation and variable source term using Galerkin scheme. Matrix Sci Math. 2020;4(2):37–43.Suche in Google Scholar

Received: 2024-05-26

Revised: 2024-07-30

Accepted: 2024-04-12

Published Online: 2024-10-15

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2024-0028

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epidemiological model; basic reproduction number; stability analysis; sensitivity analysis; numerical comparison; nonlinear equations

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