Mathematical analysis of the transmission dynamics of viral infection with effective control policies via fractional derivative

Rashid Jan; Normy Norfiza Abdul Razak; Salah Boulaaras; Ziad Ur Rehman; Salma Bahramand

doi:10.1515/nleng-2022-0342

Article Open Access

Mathematical analysis of the transmission dynamics of viral infection with effective control policies via fractional derivative

Rashid Jan , Normy Norfiza Abdul Razak , Salah Boulaaras , Ziad Ur Rehman and Salma Bahramand

Published/Copyright: October 27, 2023

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

It is well known that viral infections have a high impact on public health in multiple ways, including disease burden, outbreaks and pandemic, economic consequences, emergency response, strain on healthcare systems, psychological and social effects, and the importance of vaccination. Mathematical models of viral infections help policymakers and researchers to understand how diseases can spread, predict the potential impact of interventions, and make informed decisions to control and manage outbreaks. In this work, we formulate a mathematical model for the transmission dynamics of COVID-19 in the framework of a fractional derivative. For the analysis of the recommended model, the fundamental concepts and results are presented. For the validity of the model, we have proven that the solutions of the recommended model are positive and bounded. The qualitative and quantitative analyses of the proposed dynamics have been carried out in this research work. To ensure the existence and uniqueness of the proposed COVID-19 dynamics, we employ fixed-point theorems such as Schaefer and Banach. In addition to this, we establish stability results for the system of COVID-19 infection through mathematical skills. To assess the influence of input parameters on the proposed dynamics of the infection, we analyzed the solution pathways using the Laplace Adomian decomposition approach. Moreover, we performed different simulations to conceptualize the role of input parameters on the dynamics of the infection. These simulations provide visualizations of key factors and aid public health officials in implementing effective measures to control the spread of the virus.

Keywords: viral dynamics; differential equations; mathematical operators; stability analysis; dynamical behavior; public health policies

1 Introduction

Concerted global efforts have recently been made to establish a worldwide surveillance network to combat the emergence and re-emergence of infectious illnesses. Scientists and researchers from diverse fields are actively engaged in rapidly assessing potentially urgent situations. Mathematical modeling plays a crucial role in predicting, assessing, and developing control strategies for potential outbreaks [1–3]. To improve our understanding and modeling of contagious dynamics, including diseases like COVID-19, SARS, and Nipah virus infection, it is important to examine the influence of different factors, including interaction terms and prevailing social, ecological, and demographic factors. These infectious diseases have caused numerous illnesses and deaths worldwide, profoundly impacting the affected nations’ economic and social structures. Hence, it is evident that these contagious illnesses pose a significant risk to society. The study of infectious diseases has garnered considerable attention, enabling researchers to better comprehend their dynamics [4–6]. Mathematical models are primarily focused on aspects such as transmission mode, susceptibility, and infectious period. However, as the field of epidemiology has expanded, it has become crucial to consider additional factors, including geographic, economic, and demographic conditions, which have a substantial effect on the transmission of the infection. Therefore, enhancing the realism of mathematical models by incorporating a broader range of variables and parameters is essential.

It is acknowledged that mathematics is a fundamental tool in many fields of science and engineering [7,8]. It is used as an essential tool in physics, biology, economics, and many other areas of research [9,10]. In epidemiology, numerous mathematical models have been formulated to conceptualize the dynamics of infections with different assumptions [11–13]. Mathematical modeling is an important tool in infectious disease epidemiology, utilized to link the biological process of transmission and the emergent dynamics of infections [14]. Mathematical models can be used to analyze the emergent dynamics of observed epidemics, predict the future course of an outbreak, and evaluate interventions to prevent an epidemic [15]. These models have been widely used to assess the effectiveness of vaccination policies, determine the best vaccination ages and target groups, and evaluate the impact of different interventions [16]. In recent years, these models have been widely used to study the transmission dynamics of COVID-19 and evaluate the impact of different interventions [17,18]. A mathematical model of COVID-19 containing asymptomatic and symptomatic classes was proposed to describe the outbreak of the disease. The model included parameters such as the transmission rate, the incubation period, and the proportion of asymptomatic cases [19]. Haq et al. [20] proposed a mathematical model for COVID-19 infection with vaccination and quarantine to study the transmission mechanism of the disease. Their model included parameters such as the vaccination rate, the quarantine rate, and the transmission rate.

Forecasting and mathematical modeling are used by the centers for disease control and prevention (CDC) to advise public health decision-makers about the potential impact of COVID-19. Forecasts are developed by CDC partners in the COVID-19 Forecast Hub using mathematical models, and they are intended to influence public health choices about pandemic planning, resource allocation, social distancing measures, and other interventions. Ndarou et al. [21] presented a compartmental model for the spread of COVID-19, with a special focus on the transmissibility of super-spreaders. The model included parameters such as the transmission rate, the incubation period, and the proportion of super-spreaders. To examine the disease processes and anticipate the presumed abundance of infected people in Italy, a five-dimensional mathematical model of COVID-19 was developed. The transmission rate, incubation period, and fraction of asymptomatic cases were all factors in the model [22]. Zhang et al. [23] conducted an analysis of a virus model considering delayed transmission and treatment, examining two distinct methods of transmission. Kifle and LemechaObsu [24] delved into assessing how optimal control functions could minimize the transmission of the disease. Keno and Etana [25] introduced a mathematical model for COVID-19 that incorporated optimal control with cost-effective strategies. In addition, the parameters of the model were estimated using real data of confirmed cases in Ethiopia from 1 October 2022 to 30 October 2022. In this work, we will structure a mathematical model for COVID-19 with home quaranine and hospitalization in a fractional framework to enhance our understanding of the transmission dynamics of this viral infection, optimize control strategies, and contribute valuable information for informed decision-making in public health.

Fractional derivatives are important for epidemic models because they provide a powerful instrument for incorporating memory and hereditary properties of the systems, which are important in modeling the spread of infectious diseases [26,27]. These operators allow for more efficient modeling and can capture long-term memory effects, the heterogeneity of the population, and the impact of nonpharmaceutical interventions. The use of fractional derivatives in epidemic models is a novel approach that can provide new insights into the dynamics of infectious diseases and help in the development of more effective strategies for controlling their spread. This article is organized as follows: Section 2 provides an overview of the essential ideas and significant discoveries of fractional theory. In Section 3, we construct an epidemic model to offer a more realistic understanding of COVID-19 with home quarantine and hospitalization. The evaluation of the proposed model’s uniqueness and existence is presented in Section 4, while Section 5 discusses the stability requirements based on the Ulam–Hyers framework. We introduce a numerical approach to solve the model and conduct a numerical analysis of COVID-19 by varying the input factors of the system in Section 6. At the end, the conclusion and final remarks of the article are provided in the last section.

2 Fundamental concepts

This section provides an exposition of the rudimentary definitions and results of fractional calculus, which will be applied in analyzing the proposed model. Fractional models are often more efficient than integer-order models because the option of derivative order gives more degree of freedom to explore the dynamics of the system. As a result, researchers have directed their attention toward studying fractional systems due to their diverse practical applications. The basic definitions and results are summarized as follows:

Definition 2.1

[28] Consider q ( t ) be a function such that q ( t ) ∈ L 1 ( [ g , h ] , R ) and let ℏ be the fractional order such that

(1) I g + g ℏ q ( t ) = 1 Γ ( ℏ ) ∫ 0 t ( t − f ) ℏ − 1 q ( f ) d f ,

which denotes the fractional integral and ℏ ∈ ( 0 , 1 ] .

Definition 2.2

[28] Assume q ( t ) be a function with q ( t ) ∈ C j [ g , h ] , then

(2) D 0 + ℏ L C q ( t ) = 1 Γ ( j − ℏ ) ∫ 0 t ( t − f ) j − ℏ − 1 h j ( f ) d f ,

which denotes the renowned derivative of Caputo.

Lemma 2.1

[29] Let q ( h ) be a function, then take the following system:

(3) D 0 + ℏ L C q ( t ) = w ( t ) , t ∈ [ 0 , χ ] , j − 1 < ℏ < j , q ( 0 ) = w 0 ,

where w ( t ) ∈ C ( [ 0 , χ ] ) , and then

(4) q ( t ) = ∑ i = 0 j − 1 d i t i f o r i = 0 , 1 , … , j − 1 a n d d i ∈ R .

Definition 2.3

The Laplace transform for the Caputo derivative is

(5) £ [ D 0 + ℏ L C q ( t ) ] = f ℏ q ( f ) − ∑ k = 0 j − 1 f ℏ − k − 1 q k ( 0 ) ,

with j − 1 < ℏ < j . Moreover, take the norm on Z as

(6) ‖ q ‖ = max t ∈ [ 0 , χ ] { ∣ q ∣ , for all q ∈ Z } .

Theorem 2.1

[30] Consider Z to be a Banach space in a way that G : Z → Z is compact and continuous. Then, there exists a fixed point of G if

(7) X = { q ∈ Z : q = ς G b , ς ∈ ( 0 , 1 ) }

is bounded.

3 Dynamics of the infection

In the construction of the COVID-19 model, the host population is denoted as Q and is divided into seven subclasses. The subclasses are designated as follows: K for susceptible individuals, A for exposed individuals, I 1 for asymptomatic individuals in the early stage, I 2 for symptomatic individuals in the later stage, C for hospitalized individuals, D for home quarantined individuals, and F for recovered individuals.

The susceptible class experiences growth at a rate denoted by β , which represents the recruitment rate. However, the class decreases at a rate determined by η K and ω K ( I 1 + I 2 ) , where η represents the natural death rate and ω is the transmission coefficient of the infection. Consequently, the rate of change in the susceptible class can be expressed as follows:

d K d t = β − η K − ω K ( I 1 + I 2 ) .

The term ω K ( I 1 + I 2 ) increases the population of the exposed individuals, while the term ( τ + η ) A decreases the population of the class, where τ represents the acquiring infection rate. Therefore, the rate of change in the exposed class is stated as follows:

d A d t = ω K ( I 1 + I 2 ) − ( τ + η ) A .

The class of asymptomatic individuals at an early stage grows at the rate τ A , while it lowers at the rate ( φ 1 + ρ + η ) I 1 , where φ 1 represents the rate at which the asymptomatic individuals receive antiviral treatment and ρ represents the rate of transition of individuals from the asymptomatic to the symptomatic class. Then, the dynamics of the asymptomatic class at an early stage is mathematically given as follows:

d I 1 d t = τ A − ( φ 1 + ρ + η ) I 1 .

The population of symptomatic individuals at later stage increases at the rate ρ I 1 , while it decreases at the rate ( φ 2 + σ + ν 1 + η + ξ ) I 2 , where φ 2 represents the rate at which the symptomatic population receive antiviral treatment, σ represents the rate of hospitalization, ν 1 represents the recovery rate of later-stage symptomatic infectious population, and ξ is the death rate due to the infection. Thus, the dynamics of the symptomatic class at a later stage is given as follows:

d I 2 d t = ρ I 1 − ( φ 2 + σ + ν 1 + η + ξ ) I 2 .

The term σ I 2 increases the population of hospitalized individuals, while the terms ( ν 3 + η + α ξ ) C lower the population, where ν 3 represents the recovery rate of hospitalized individuals and 0 < α < 1 as the hospitalized individuals in class C have a lower mortality rate compared to infectious individuals in class I 2 . Therefore, the dynamics of the hospitalized class is

d C d t = σ I 2 − ( ν 3 + η + α ξ ) C .

The terms φ 1 I 1 and φ 2 I 2 increase the population of home quarantined individuals, while the terms ( ν 2 + η ) D decrease the population, where ν 2 is the recovery rate of quarantined individuals. Thus, the dynamics of the home quarantined class is

d D d t = φ 1 I 1 + φ 2 I 2 − ( ν 2 + η ) D .

The class of recovered individuals grows at the rates ν 1 I 2 , ν 3 C , and ν 2 D , while its population lowers at the rate η F . Thus, the dynamics of the recovered class is stated as follows:

d F d t = ν 1 I 2 + ν 3 C + ν 2 D − η F .

From the above, we have the below dynamics for the transmission of the infection:

(8) d K d t = β − η K − ω K ( I 1 + I 2 ) , d A d t = ω K ( I 1 + I 2 ) − ( τ + η ) A , d I 1 d t = τ A − ( φ 1 + ρ + η ) I 1 , d I 2 d t = ρ I 1 − ( φ 2 + σ + ν 1 + η + ξ ) I 2 , d C d t = σ I 2 − ( ν 3 + η + α ξ ) C , d D d t = φ 1 I 1 + φ 2 I 2 − ( ν 2 + η ) D , d F d t = ν 1 I 2 + ν 3 C + ν 2 D − η F ,

where

K ( 0 ) ≥ 0 , A ( 0 ) ≥ 0 , I 1 ( 0 ) ≥ 0 , I 2 ( 0 ) ≥ 0 , C ( 0 ) ≥ 0 , D ( 0 ) ≥ 0 , F ( 0 ) ≥ 0 .

Moreover, the total population is given as follows:

Q = K + A + I 1 + I 2 + C + D + F .

Due to the nonlocal nature of biological processes, fractional systems offer increased reliability and accuracy. Furthermore, fractional systems possess inherent characteristics related to inheritance and the ability to retain information about their past and present states, which can be utilized for future predictions. The Liouville-Caputos derivative, a recently developed technique, offers a more precise explanation of the nonlocal behavior exhibited by biological processes. Consequently, the Liouville-Caputo (LC) fractional operator is employed to represent the COVID-19 infection as follows:

(9) D t i 0 L C K = β − η K − ω K ( I 1 + I 2 ) , D t i 0 L C A = ω K ( I 1 + I 2 ) − ( τ + η ) A , D t i 0 L C I 1 = τ A − ( φ 1 + ρ + η ) I 1 , D t i 0 L C I 2 = ρ I 1 − ( φ 2 + σ + ν 1 + η + ξ ) I 2 , D t i 0 L C C = σ I 2 − ( ν 3 + η + α ξ ) C , D t i 0 L C D = φ 1 I 1 + φ 2 I 2 − ( ν 2 + η ) D , D t i 0 L C F = ν 1 I 2 + ν 3 C + ν 2 D − η F ,

where the Liouville-Caputo’s is symbolized by 0 L C D t i and the memory index is represented by the symbol i .

Theorem 3.1

The solutions ( K , A , I 1 , I 2 , C , D , F ) of fractional model (9) of COVID-19 infection are positivity and boundedness.

Proof

To achieve the result, we will proceed in the following manner:

(10) D t i 0 L C K ∣ K = 0 = β ≥ 0 , D t i 0 L C A ∣ A = 0 = ω K ( I 1 + I 2 ) ≥ 0 , D t i 0 L C I 1 ∣ I 1 = 0 = τ A ≥ 0 , D t i 0 L C I 2 ∣ I 2 = 0 = ρ I 1 ≥ 0 , D t i 0 L C C ∣ C = 0 = σ I 2 ≥ 0 , D t i 0 L C D ∣ D = 0 = φ 1 I 1 + φ 2 I 2 ≥ 0 , D t i 0 L C F ∣ F = 0 = ν 1 I 2 + ν 3 C + ν 2 D ≥ 0 .

Hence, the solutions of our fractional model (9) are positive. To show that the solutions of themodel are bounded. We add all the equations of the model (8) and get

d Q d t = β − η Q − ξ I 2 − α ξ C ,

where K ( t ) , A ( t ) , I 1 ( t ) , I 2 ( t ) , C ( t ) , D ( t ) , F ( t ) ≥ 0 , which shows that d Q d t is bounded by β − η Q . By utilizing standard comparison theorem [31], we achieve that

0 < Q ( t ) ≤ β η ( 1 − e − η t ) + Q ( 0 ) .

Therefore,

lim t → + ∞ sup Q ( t ) ≤ β η .

For C = 0 , I 2 = 0 , if Q ( t ) > β η , then d Q d t < 0 . Then, we obtain 0 < Q ( t ) ≤ β η . In addition Q ( t ) ≤ β η if Q ( 0 ) ≤ β η . As a result, the region Γ = { ( K , A , I 1 , I 2 , C , D , F ) : K + A + I 1 + I 2 + C + D + F ≤ β η } is invariant. Thus, the solution of system (9) is positive and bounded.

The disease-free steady-state of our proposed fractional model (9) for COVID-19 is denoted as ℰ 0 ( K , A , I 1 , I 2 , C , D , F ) and is given by ℰ 0 = β η , 0 , 0 , 0 , 0 , 0 , 0 . In Section 4, we will investigate the solution of the recommended model of COVID-19 infection in detail.

4 Theory of existence

Here, we will examine the qualitative aspects of the proposed fractional system (9) of COVID-19 with the help of mathematical skills. To achieve this, we will proceed as follows:

(11) ℋ 1 ( t , K , A , I 1 , I 2 , C , D , F ) = β − η K − ω K ( I 1 + I 2 ) , ℋ 2 ( t , K , A , I 1 , I 2 , C , D , F ) = ω K ( I 1 + I 2 ) − ( τ + η ) A , ℋ 3 ( t , K , A , I 1 , I 2 , C , D , F ) = τ A − ( φ 1 + ρ + η ) I 1 , ℋ 4 ( t , K , A , I 1 , I 2 , C , D , F ) = ρ I 1 − ( φ 2 + σ + ν 1 + η + ξ ) I 2 , ℋ 5 ( t , K , A , I 1 , I 2 , C , D , F ) = σ I 2 − ( ν 3 + η + α ξ ) C , ℋ 6 ( t , K , A , I 1 , I 2 , C , D , F ) = φ 1 I 1 + φ 2 I 2 − ( ν 2 + η ) D , ℋ 7 ( t , K , A , I 1 , I 2 , C , D , F ) = ν 1 I 2 + ν 3 C + ν 2 D − η F ,

and this can be written as follows:

(12) D 0 + i L C ℋ ( t ) = F ( t , ℋ ( t ) ) , t ∈ [ 0 , χ ] , ℋ ( 0 ) = ℋ 0 , 0 < i ≤ 1 ,

where

(13) ℋ ( t ) = K ( t ) , A ( t ) , I 1 ( t ) , I 2 ( t ) , C ( t ) , D ( t ) , F ( t ) . ℋ 0 ( t ) = K 0 , A 0 , I 10 , I 20 , C 0 , D 0 , F 0 . F ( t , ℋ ( t ) ) = ℋ 1 ( t , K , A , I 1 , I 2 , C , D , F ) . ℋ 2 ( t , K , A , I 1 , I 2 , C , D , F ) . ℋ 3 ( t , K , A , I 1 , I 2 , C , D , F ) . ℋ 4 ( t , K , A , I 1 , I 2 , C , D , F ) . ℋ 5 ( t , K , A , I 1 , I 2 , C , D , F ) . ℋ 6 ( t , K , A , I 1 , I 2 , C , D , F ) . ℋ 7 ( t , K , A , I 1 , I 2 , C , D , F ) .

By utilizing Lemma (2.1), we can write the system (12) as follows:

(14) ℋ ( t ) = ℋ 0 ( t ) + 1 Γ ( i ) ∫ 0 t ( t − f ) i − 1 F ( f , ℋ ( f ) ) d f .

In order to investigate our proposed system, we utilized the below stated Lipschitz criteria:

(C1) For l ∈ [ 0 , 1 ) , there exist N F and G F such that

(15) ∣ F ( t , ℋ ( t ) ) ∣ ≤ N ℋ ∣ ℋ ∣ l + G F .

(C2) We have K F > 0 and ℋ , ℋ ¯ ∈ Z such that

(16) ∣ F ( t , ℋ ) − F ( t , ℋ ¯ ) ∣ ≤ K F [ ∣ ℋ − ℋ ¯ ∣ ] .

Now consider a map ℬ on Z such that

(17) ℬ ℋ ( t ) = ℋ 0 ( t ) + 1 Γ ( i ) ∫ 0 t ( t − f ) i − 1 F ( f , ℋ ( f ) ) d f .

If criteria C 1 and C 2 are satisfied, then there exists at least one solution to Eq. (12). For further investigation of the solution of COVID-19 model, we will proceed in the following manner:

Theorem 4.1

At least one solution exists for the system (9) of COVID-19 if assumptions C1 and C2 are satisfied.

Proof

To establish the validity of the result, we will utilize Schaefer’s fixed-point theorem. The theorem will be presented in the following four steps:

P1: In the initial stage, we will establish the continuity of the operator ℬ . We assume that ℋ i is continuous for i = 1 , 2 , … , 9 , which ensures the continuity of F ( t , ℋ ( t ) ) . Then, consider ℋ v , ℋ ∈ Z such that ℋ v → ℋ , and we have ℬ ℋ v → ℬ ℋ . Moreover, consider

(18) ‖ ℬ ℋ v − ℬ ℋ ‖ = max t ∈ [ 0 , χ ] 1 Γ ( i ) ∫ 0 t ( t − f ) i − 1 Q v ( f , ℋ v ( f ) ) d f − 1 Γ ( i ) ∫ 0 t ( t − f ) i − 1 F ( f , ℋ ( f ) ) d s ≤ max t ∈ [ 0 , χ ] ∫ 0 t ( t − f ) i − 1 Γ ( i ) × ∣ F v ( f , ℋ v ( f ) ) − F ( f , ℋ ( f ) ) ∣ d f ≤ χ i K F Γ ( i + 1 ) ‖ ℋ v − ℋ ‖ → 0 as v → ∞ .

We obtain that ℬ ℋ v → ℬ ℋ is continuous as F is continuous, which guarantees the continuity of ℬ .

P2: In this step, we will establish the boundedness of ℬ . Let ℋ ∈ X , then the operator ℬ satisfies the following:

(19) ‖ ℬ ℋ ‖ = max t ∈ [ 0 , χ ] ℋ o ( t ) + 1 Γ ( i ) ∫ 0 t ( t − f ) i − 1 F ( f , ℋ ( f ) ) d f ≤ ∣ ℋ 0 ∣ max t ∈ [ 0 , χ ] 1 Γ ( i ) ∫ 0 t ∣ ( t − f ) i − 1 ∣ ∣ F ( f , ℋ ( f ) ) ∣ d f ≤ ∣ ℋ 0 ∣ + χ i Γ ( i + 1 ) { U Z ∣ ∣ ℋ ∣ ∣ l + A F } .

Next, we will show the boundedness of ℬ ( B ) for a bounded subset B of Z . Consider ℋ ∈ B , as S is bounded, then there is U ≥ 0 such that

(20) ‖ ℋ ‖ ≤ U ∀ ℋ ∈ B .

Thus, for any ℋ ∈ B , we obtain that

(21) ‖ ℬ W ‖ ≤ ∣ ℋ 0 ∣ + χ i Γ ( i + 1 ) [ U F ‖ ℋ ‖ l + A F ] ≤ ∣ ℋ 0 ∣ + χ i Γ ( i + 1 ) [ U F U l + A F ] .

As a result, we obtain that the operator ℬ ( B ) is bounded.

P3: For equicontinuity, consider c 1 , c 2 ∈ [ 0 , χ ] in a manner that c 1 ≥ c 2 , then we have

(22) ∣ ℬ ℋ ( c 1 ) − ℬ ℋ ( c 1 ) ∣ = 1 Γ ( i ) ∫ 0 c 1 ∣ ( c 1 − f ) i − 1 ∣ ∣ F ( f , ℋ ( f ) ) ∣ d f − 1 Γ ( i ) ∫ 0 c 2 ∣ ( c 2 − f ) i − 1 ∣ ∣ F ( f , ℋ ( f ) ) ∣ d f ≤ 1 Γ ( i ) ∫ 0 c 1 ∣ ( c 1 − f ) i − 1 ∣ − 1 Γ ( i ) ∫ 0 c 2 ∣ ( c 2 − f ) i − 1 ∣ ∣ F ( f , ℋ ( f ) ) ∣ d f ≤ χ i Γ ( i + 1 ) [ U F ∣ ∣ ℋ ∣ ∣ l + A F ] [ c 1 i − c 2 i ] ,

which approaches to zero as c 1 approaches to c 2 . This illustrates the relative compactness of ℬ ( B ) with the help of the Arzela-Ascoli theorem.

P4: Assume a set X as

(23) X = { ℋ ∈ Z : ℋ = ς ℬ ℋ , ς ∈ ( 0 , 1 ) } .

To establish the boundedness of set X , consider ℋ ∈ X , then for any t ∈ [ 0 , χ ] , the following holds true:

(24) ‖ ℋ ‖ = ς ‖ ℬ ℋ ‖ ≤ ς ∣ ℋ 0 ∣ χ i Γ ( i + 1 ) [ U F ∣ ∣ ℋ ∣ ∣ l + A F ] .

This indicates the bounded nature of set X , confirming that the operator ℬ possesses a fixed point as a consequence of Schaefer’s theorem. Hence, we can deduce that there exists at least one solution of our recommended system (12) of the infection.

Remark 4.1

If C 1 holds true for l = 1 , then Theorem 4.1 can be established for χ i U Z Γ ( i + 1 ) < 1 .

Theorem 4.2

System (12) of COVID-19 has a unique solution, if χ i U Z Γ ( i + 1 ) < 1 holds true.

Proof

In order to achieve the result, we will utilize the Banach’s contraction theorem by assuming that ℋ , ℋ ¯ ∈ Z as

(25) ‖ ℬ ℋ − ℬ ℋ ¯ ‖ ≤ max t ∈ [ 0 , χ ] 1 Γ ( i ) ∫ 0 t ∣ ( t − f ) i − 1 ∣ ∣ F ( f , ℋ ( f ) ) − F ( f , ℋ ¯ ( f ) ) ∣ d f ≤ χ i U F Γ ( i + 1 ) ∣ ∣ ℋ − ℋ ¯ ∣ ∣ .

As, there exists a unique fixed point of ℬ , therefore model (12) of COVID-19 has a unique solution.

5 Ulam–Hyers stability

Our primary objective in this section is to examine the proposed model of COVID-19 for Ulam–Hyers stability (UHS). This stability idea was initially given by Ulam in 1940 and subsequently further developed by Hyers [32,33]. The UHS concept has been widely applied by several researchers in various fields of study [34–36]. The underlying theory is as follows:

Assume that T : Z → Z such that

(26) T G = G for G ∈ Z .

Definition 5.1

Eq. (26) will be UHS if, for every solution ℋ ∈ Z and ℜ > 0 , we have

(27) ∣ ∣ G − T G ∣ ∣ ≤ ℜ for t ∈ [ 0 , χ ] .

Moreover, a unique solution ℋ ¯ exists for Eq. (26) such that 0 < C l and the following statement holds true:

(28) ‖ G ¯ − G ‖ ≤ C l ℜ , t ∈ [ 0 , χ ] .

Definition 5.2

Let ℋ and ℋ ¯ be solution to Eq. (26), then the system (26) will be generalized UHS if

(29) ‖ G ¯ − G ‖ ≤ F ( ℜ ) ,

where the image of 0 is 0 and F ∈ C ( R , R ) .

Remark 5.1

If the solution ℋ ¯ ∈ Z satisfies Eq. (28) and ∀ t ∈ [ 0 , χ ] , then the following statements hold true:

∣ ζ ( t ) ∣ ≤ ℜ , where ζ ∈ C ( [ 0 , χ ] ; R )
T G ¯ ( B ) = G ¯ + ζ ( B ) ,

After small changes, Eq. (12) can be written as follows:

(30) D 0 + i C G ( t ) = ℋ ( t , G ( t ) ) + ζ ( t ) , G ( 0 ) = G 0 .

Lemma 5.1

System (30) also satisfies

(31) ∣ G ( t ) − B G ( t ) ∣ ≤ a ℜ , w h e r e a = χ i Γ ( i + 1 ) .

This theorem can be proved by utilizing Remark 5.1 and Lemma 2.1.

Theorem 5.1

The solution to Eq. (12) will be UHS and generalized UHS on Lemma 5.1 if the condition χ i L ℋ Γ ( i + 1 ) < 1 satisfies.

Proof

Consider that G ∈ X and G ¯ ∈ X are the solutions of the system (12), then

(32) ∣ G ( t ) − G ¯ ( t ) ∣ = ∣ G ( t ) − G ¯ ( t ) ∣ ≤ ∣ G ( t ) − B G ¯ ( t ) ∣ ≤ ∣ G ( t ) − B G ¯ ( t ) ∣ ≤ a ℜ + χ ϰ L N Γ ( ϰ + 1 ) ∣ G ( t ) − G ¯ ( t ) ∣ ≤ a ℜ 1 − χ ϰ L N Γ ( ϰ + 1 ) ,

which shows that the system (12) of COVID-19 is UHS and generalized Ulam-Hyers stable.

Definition 5.3

The solution to Eq. (26) will be Ulam–Hyers–Rassias stable (UHRS) if for any G ∈ Z , there exists

(33) ‖ G − T G ‖ ≤ ϒ ( t ) ℜ , for t ∈ [ 0 , χ ] ,

where ℜ > 0 and ϒ ∈ C [ [ 0 , χ ] , R ] . If C l > 0 , then there is a unique solution G ¯ of the system (26) satisfying

(34) ‖ G ¯ − G ‖ ≤ C l ϒ ( t ) ℜ

∀ p t ∈ [ 0 , χ ] .

Definition 5.4

Assume that ℋ ¯ is a unique solution and ℋ be any other solution to Eq. (26) in such a way that

(35) ‖ G ¯ − G ‖ ≤ C l , ϒ ϒ ( t ) ℜ ,

where t ∈ [ 0 , χ ] and ϒ ∈ D [ [ 0 , χ ] , R ] in a manner that C l , ϒ and ℜ > 0 , which shows that the solution to Eq. (26) is generalized UHRS.

Remark 5.2

Consider G ¯ ∈ X , this solution will satisfy Eq. (28) if ∀ t ∈ [ 0 , χ ] , we have

∣ ζ ( t ) ∣ ≤ ℜ ϒ ( t ) , where ζ ( t ) ∈ C ( [ 0 , χ ] ; R )
T G ¯ ( t ) = G ¯ + ζ ( t ) .

Lemma 5.2

The perturb system 5.1 holds the condition

(36) ∣ G ( t ) − B G ( B ) ∣ ≤ a ϒ ( t ) ℜ , w h e r e a = χ i Γ ( i + 1 ) .

Using Lemma 5.2 and Remark 2.1, we can prove this result.

Theorem 5.2

If χ i L N Γ ( i + 1 ) < 1 , then the solution of system (12)will be UHRS and generalized UHRS on Lemma 5.2.

Proof

Let G ¯ ∈ Z is a unique solution and G ∈ Z be any other solution of system (12), then we have

(37) ∣ G ( t ) − G ¯ ( t ) ∣ = ∣ G ( t ) − G ¯ ( t ) ∣ ≤ ∣ G ( t ) − B G ¯ ( t ) ∣ ≤ ∣ G ( t ) − B G ¯ ( t ) ∣ ≤ a ϒ ( t ) ℜ + χ i L ℋ Γ ( i + 1 ) ∣ G ( t ) − G ¯ ( t ) ∣ ≤ a ϒ ( t ) ℜ 1 − χ i L ℋ Γ ( i + 1 ) .

Therefore, UHRS and generalized UHRS are the solutions of Eq. (12).

6 Numerical scheme for the system

In this study, we will examine the dynamic behavior of the COVID-19 system. To accomplish this, we will utilize the Laplace transform to formulate a framework for the system (9). The following are the steps involved in this method:

(38) D t i 0 L C K = β − η K − ω K ( I 1 + I 2 ) , D t i 0 L C A = ω K ( I 1 + I 2 ) − ( τ + η ) A , D t i 0 L C I 1 = τ A − ( φ 1 + ρ + η ) I 1 , D t i 0 L C I 2 = ρ I 1 − ( φ 2 + σ + ν 1 + η + ξ ) I 2 , D t i 0 L C C = σ I 2 − ( ν 3 + η + α ξ ) C , D t i 0 L C D = φ 1 I 1 + φ 2 I 2 − ( ν 2 + η ) D , D t i 0 L C F = ν 1 I 2 + ν 3 C + ν 2 D − η F ,

(39) ℒ [ K ( t ) ] = K 0 s + 1 s i ℒ [ β − η K − ω K ( I 1 + I 2 ) ] , ℒ [ A ( t ) ] = A 0 s + 1 s i ℒ [ ω K ( I 1 + I 2 ) − ( τ + η ) A ] , ℒ [ I 1 ( t ) ] = I 10 s + 1 s i ℒ [ τ A − ( φ 1 + ρ + η ) I 1 ] , ℒ [ I 2 ( t ) ] = I 20 s + 1 s i ℒ [ ρ I 1 − ( φ 2 + σ + ν 1 + η + ξ ) I 2 ] , ℒ [ C ( t ) ] = C 0 s + 1 s i ℒ [ σ I 2 − ( ν 3 + η + α ξ ) C ] , ℒ [ D ( t ) ] = D 0 s + 1 s i ℒ [ φ 1 I 1 + φ 2 I 2 − ( ν 2 + η ) D ] , ℒ [ F ( t ) ] = F 0 s + 1 s i ℒ [ ν 1 I 2 + ν 3 C + ν 2 D − η F ] ,

where

(40) K ( t ) = ∑ v = 0 ∞ K v ( t ) , A ( t ) = ∑ v = 0 ∞ A v ( t ) , I 1 ( t ) = ∑ v = 0 ∞ I 1 v ( t ) , I 2 ( t ) = ∑ v = 0 ∞ I 2 v ( t ) , C ( t ) = ∑ v = 0 ∞ C v ( t ) , D ( t ) = ∑ v = 0 ∞ D v ( t ) , F ( t ) = ∑ v = 0 ∞ F v ( t ) .

Utilizing Adomian polynomials to decompose the nonlinear terms as

K ( t ) ( I 1 ( t ) + I 2 ( t ) ) = ∑ v = 0 ∞ F v ( t ) , with F v ( t ) = 1 v ! d v d z v ∑ k = 0 v z k S k ( t ) ∑ k = 0 v z k ( I 1 k ( t ) + I 2 k ) z = 0 ,

we obtain

(41) ℒ ∑ v = 0 ∞ K v ( t ) = K 0 s + 1 s i ℒ × β − η ∑ v = 0 ∞ K v ( t ) − ω ∑ v = 0 ∞ F v ( t ) , ℒ ∑ v = 0 ∞ A v ( t ) = A 0 s + 1 s i ℒ × ω ∑ v = 0 ∞ F v ( t ) − ( τ + η ) ∑ v = 0 ∞ A v ( t ) , ℒ ∑ v = 0 ∞ I 1 v ( t ) = I 10 s + 1 s i ℒ × τ ∑ v = 0 ∞ A v ( t ) − ( φ 1 + ρ + η ) ∑ v = 0 ∞ I 1 v ( t ) , ℒ ∑ v = 0 ∞ I 2 v ( t ) = I 20 s + 1 s i ℒ × ρ ∑ v = 0 ∞ I 1 v ( t ) − ( φ 2 + σ + ν 1 + η + ξ ) ∑ v = 0 ∞ I 2 v ( t ) , ℒ ∑ v = 0 ∞ C v ( t ) = C 0 s + 1 s i ℒ × σ ∑ v = 0 ∞ I 2 v ( t ) − ( ν 3 + η + α ξ ) ∑ v = 0 ∞ C v ( t ) , ℒ ∑ v = 0 ∞ D v ( t ) = D 0 s + 1 s i ℒ × φ 1 ∑ v = 0 ∞ I 1 v ( t ) + φ 2 ∑ v = 0 ∞ I 2 v ( t ) − ( ν 2 + η ) ∑ v = 0 ∞ D v ( t ) , ℒ ∑ v = 0 ∞ F v ( t ) = F 0 s + 1 s i ℒ × ν 1 ∑ v = 0 ∞ I 2 v ( t ) + ν 3 ∑ v = 0 ∞ C v ( t ) + ν 2 ∑ v = 0 ∞ D v ( t ) − η ∑ v = 0 ∞ F v ( t ) .

(42) ℒ [ K 0 ( t ) ] = K 0 s , ℒ [ A 0 ( t ) ] = A 0 s , ℒ [ I 10 ( t ) ] = I 10 s , ℒ [ I 20 ( t ) ] = I 20 s , ℒ [ C 0 ( t ) ] = C 0 s , ℒ [ D 0 ( t ) ] = D 0 s , ℒ [ F 0 ( t ) ] = F 0 s .

Then, we have

(43) ℒ [ K 1 ( t ) ] = 1 s i ℒ [ β − η K 0 ( t ) − ω F 0 ( t ) ] , ℒ [ A 1 ( t ) ] = 1 s i ℒ [ ω F 0 ( t ) − ( τ + η ) A 0 ( t ) ] , ℒ [ I 11 ( t ) ] = 1 s i ℒ [ τ A 0 ( t ) − ( φ 1 + ρ + η ) I 10 ] , ℒ [ I 21 ( t ) ] = 1 s i ℒ [ ρ I 10 − ( φ 2 + σ + ν 1 + η + ξ ) I 20 ] , ℒ [ C 1 ( t ) ] = 1 s i ℒ [ σ I 20 − ( ν 3 + η + α ξ ) C 0 ( t ) ] , ℒ [ D 1 ( t ) ] = 1 s i ℒ [ φ 1 I 10 + φ 2 I 20 − ( ν 2 + η ) D 0 ( t ) ] , ℒ [ F 1 ( t ) ] = 1 s i ℒ [ ν 1 I 20 + ν 3 C 0 ( t ) + ν 2 D 0 ( t ) − η F 0 ( t ) ] ,

and

(44) ℒ [ K 2 ( t ) ] = 1 s i ℒ [ β − η K 1 ( t ) − ω F 1 ( t ) ] , ℒ [ A 2 ( t ) ] = 1 s i ℒ [ ω F 1 ( t ) − ( τ + η ) A 1 ( t ) ] , ℒ [ I 12 ( t ) ] = 1 s i ℒ [ τ A 1 ( t ) − ( φ 1 + ρ + η ) I 11 ] , ℒ [ I 22 ( t ) ] = 1 s i ℒ [ ρ I 11 − ( φ 2 + σ + ν 1 + η + ξ ) I 21 ] , ℒ [ C 2 ( t ) ] = 1 s i ℒ [ σ I 21 − ( ν 3 + η + α ξ ) C 1 ( t ) ] , ℒ [ D 2 ( t ) ] = 1 s i ℒ [ φ 1 I 11 + φ 2 I 21 − ( ν 2 + η ) D 1 ( t ) ] , ℒ [ F 2 ( t ) ] = 1 s i ℒ [ ν 1 I 21 + ν 3 C 1 ( t ) + ν 2 D 1 ( t ) − η F 1 ( t ) ] .

Moreover, we obtain

(45) ℒ [ K ( v + 1 ) ( t ) ] = 1 s i ℒ [ β − η K v ( t ) − ω F v ( t ) ] , ℒ [ A ( v + 1 ) ( t ) ] = 1 s i ℒ [ ω F v ( t ) − ( τ + η ) A v ( t ) ] , ℒ [ I 1 ( v + 1 ) ( t ) ] = 1 s i ℒ [ τ A v ( t ) − ( φ 1 + ρ + η ) I 1 v ] , ℒ [ I 2 ( v + 1 ) ( t ) ] = 1 s i ℒ [ ρ I 1 v − ( φ 2 + σ + ν 1 + η + ξ ) I 2 v ] , ℒ [ C ( v + 1 ) ( t ) ] = 1 s i ℒ [ σ I 2 v − ( ν 3 + η + α ξ ) C v ( t ) ] , ℒ [ D ( v + 1 ) ( t ) ] = 1 s i ℒ [ φ 1 I 1 v + φ 2 I 2 v − ( ν 2 + η ) D v ( t ) ] , ℒ [ F ( v + 1 ) ( t ) ] = 1 s i ℒ [ ν 1 I 2 v + ν 3 C v ( t ) + ν 2 D v ( t ) − η F v ( t ) ] ,

with the following initial conditions

(46) K 0 ( t ) = K 0 , A 0 ( t ) = A 0 , I 10 ( t ) = I 10 , I 20 ( t ) = I 20 , C 0 ( t ) = C 0 , D 0 ( t ) = D 0 , F 0 ( t ) = F 0 .

For further simplification, we will proceed in the following manner:

(47) K 1 ( t ) = ℒ − 1 1 s i ℒ [ β − η K 0 ( t ) − ω F 0 ( t ) ] , A 1 ( t ) = ℒ − 1 1 s i ℒ [ ω F 0 ( t ) − ( τ + η ) A 0 ( t ) ] , I 11 ( t ) = ℒ − 1 1 s i ℒ [ τ A 0 ( t ) − ( φ 1 + ρ + η ) I 10 ] , I 21 ( t ) = ℒ − 1 1 s i ℒ [ ρ I 10 − ( φ 2 + σ + ν 1 + η + ξ ) I 20 ] , C 1 ( t ) = ℒ − 1 1 s i ℒ [ σ I 20 − ( ν 3 + η + α ξ ) C 0 ( t ) ] , D 1 ( t ) = ℒ − 1 1 s i ℒ [ φ 1 I 10 + φ 2 I 20 − ( ν 2 + η ) D 0 ( t ) ] , F 1 ( t ) = ℒ − 1 1 s i ℒ [ ν 1 I 20 + ν 3 C 0 ( t ) + ν 2 D 0 ( t ) − η F 0 ( t ) ] ,

and

(48) K 2 ( t ) = ℒ − 1 1 s i ℒ [ β − η K 1 ( t ) − ω F 1 ( t ) ] , A 2 ( t ) = ℒ − 1 1 s i ℒ [ ω F 1 ( t ) − ( τ + η ) A 1 ( t ) ] , I 12 ( t ) = ℒ − 1 1 s i ℒ [ τ A 1 ( t ) − ( φ 1 + ρ + η ) I 11 ] , I 22 ( t ) = ℒ − 1 1 s i ℒ [ ρ I 11 − ( φ 2 + σ + ν 1 + η + ξ ) I 21 ] , C 2 ( t ) = ℒ − 1 1 s i ℒ [ σ I 21 − ( ν 3 + η + α ξ ) C 1 ( t ) ] , D 2 ( t ) = ℒ − 1 1 s i ℒ [ φ 1 I 11 + φ 2 I 21 − ( ν 2 + η ) D 1 ( t ) ] , F 2 ( t ) = ℒ − 1 1 s i ℒ [ ν 2 I 21 + ν 3 C 1 ( t ) + ν 2 D 1 ( t ) − η F 1 ( t ) ] .

Furthermore, we achieve that

(49) K ( v + 1 ) ( t ) = ℒ − 1 1 s i ℒ [ β − η K v ( t ) − ω F v ( t ) ] , A ( v + 1 ) ( t ) = ℒ − 1 1 s i ℒ [ ω F v ( t ) − ( τ + η ) A v ( t ) ] , I 1 ( v + 1 ) ( t ) = ℒ − 1 1 s i ℒ [ τ A v ( t ) − ( φ 1 + ρ + η ) I 1 v ] , I 2 ( v + 1 ) ( t ) = ℒ − 1 1 s i ℒ [ ρ I 1 v − ( φ 2 + σ + ν 1 + η + ξ ) I 2 v ] , C ( v + 1 ) ( t ) = ℒ − 1 1 s i ℒ [ σ I 2 v − ( ν 3 + η + α ξ ) C v ( t ) ] , D ( v + 1 ) ( t ) = ℒ − 1 1 s i ℒ [ φ 1 I 1 v + φ 2 I 2 v − ( ν 2 + η ) D v ( t ) ] , F ( v + 1 ) ( t ) = ℒ − 1 1 s i ℒ [ ν 1 I 2 v + ν 3 C v ( t ) + ν 2 D v ( t ) − η F v ( t ) ] .

Therefore, we obtain the following answer in the form of series:

(50) K ( t ) = K 0 ( t ) + K 1 ( t ) + K 2 ( t ) + K 3 ( t ) + … , A ( t ) = A 0 ( t ) + A 1 ( t ) + A 2 ( t ) + A 3 ( t ) + … , I 1 ( t ) = I 10 ( t ) + I 11 ( t ) + I 12 ( t ) + I 13 ( t ) + … , I 2 ( t ) = I 20 ( t ) + I 21 ( t ) + I 22 ( t ) + I 23 ( t ) + … , C ( t ) = C 0 ( t ) + C 1 ( t ) + C 2 ( t ) + C 3 ( t ) + … , D ( t ) = D 0 ( t ) + D 1 ( t ) + D 2 ( t ) + D 3 ( t ) + … , F ( t ) = F 0 ( t ) + F 1 ( t ) + F 2 ( t ) + F 3 ( t ) + … .

In this section of the research, we present the numerical findings that demonstrate the solution pathways of the system when the input parameters undergo fluctuations. The purpose of this section is to analyze how changes in the input parameters affect the system’s behavior and explore different scenarios to gain insights into the system’s response under varying conditions. By examining these numerical results, we aim to highlight the robustness and sensitivity of the system to different input variations. This section plays a crucial role in demonstrating the system’s behavior under different conditions, enhancing the understanding of its response, and providing valuable insights for engineering or scientific applications. For numerical purposes, we have made assumptions regarding the initial values of state variables and system parameters (Table 1).

Table 1

Input parameters along with their corresponding descriptions used in numerical simulation

Input factors	Interpretations
β	Susceptible individuals recruitment rate
η	Natural death rate
ω	The coefficient of disease transmission
τ	Acquiring infection rate
ρ	The rate of transition of individuals from asymptomatic to symptomatic class
φ 1	The rate at which the asymptomatic population receive antiviral treatment
φ 2	The rate at which the symptomatic population receive antiviral treatment
σ	The rate of hospitalization
ν 1	Recovery rate of later-stage symptomatic infectious population
ν 2	Recovery rate of quarantined population
ν 2	Recovery rate of hospitalized population
ξ	Disease-related death

In the first simulation, as depicted in Figures 1 and 2, we investigated the influence of the fractional parameter i on the dynamics of COVID-19 infection. By considering various values of the input parameter i , we examined the characteristic solution pathways of the system. The findings from these simulations clearly indicate that the fractional parameter significantly impacts the infection dynamics. Moreover, it emerges as a potential tool for controlling the spread of the infection within the society. Therefore, we strongly recommend further exploration and analysis of this fractional parameter by policymakers to better understand its potential in mitigating the infection’s impact on public health. Figure 3 demonstrates the influence of the input parameter ω on the dynamics of COVID-19 infection. Our observations reveal the critical nature of this parameter, as it exhibits a direct correlation with an increased risk of infection.

$Figure 1 Graphical view analysis of the solution pathways of (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with various assumptions of the parameter ı \imath , i.e., ı = 0.85 , 0.90 , 0.95 , 1.00 \imath =0.85,0.90,0.95,1.00 .$

Figure 1

Graphical view analysis of the solution pathways of (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with various assumptions of the parameter ı , i.e., ı = 0.85 , 0.90 , 0.95 , 1.00 .

$Figure 2 Plotting the solution pathways of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with different values of fractional order ı \imath , i.e., ı = 0.60 , 0.63 , 0.66 , 0.69 \imath =0.60,0.63,0.66,0.69 .$

Figure 2

Plotting the solution pathways of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with different values of fractional order ı , i.e., ı = 0.60 , 0.63 , 0.66 , 0.69 .

$Figure 3 Time series analysis of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with different values of the input parameter ω \omega , i.e., ω = 0.35 , 0.40 , 0.45 , 0.50 \omega =0.35,0.40,0.45,0.50 .$

Figure 3

Time series analysis of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with different values of the input parameter ω , i.e., ω = 0.35 , 0.40 , 0.45 , 0.50 .

In Figures 4 and 5, we have elucidated the effects of the input parameters τ and ρ on the dynamics of the COVID-19 infection system. Specifically, we have examined how variations in these parameters influence the behavior of asymptomatic and infected individuals within the system. In the second last simulation, as depicted in Figure 6, we investigated the influence of the input parameter σ on various classes within the COVID-19 infection system. These classes include the exposed, asymptomatic, infected, hospitalized, and quarantined individuals. The implementation of treatment through hospitalization appears to play a pivotal role in reducing the level of infection and significantly controlling the spread of the disease. The findings from this simulation underscore the importance of timely and appropriate medical interventions, such as hospitalization, in mitigating the COVID-19 outbreak. It highlights the potential effectiveness of medical measures in controlling the infection and emphasizes the significance of implementing healthcare strategies to combat the pandemic. These insights can serve as valuable guidance for policymakers and health authorities in formulating effective measures to combat COVID-19 and similar infectious diseases. However, it is essential to recognize that these conclusions are based on the specific model and parameter settings used in the simulation, and further investigations and real-world validations are necessary to fully comprehend the efficacy and implications of these strategies in actual public health scenarios.

$Figure 4 Graphical view analsysis of the solution pathways of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with different values of input parameter τ \tau , i.e., τ = 0.32 , 0.36 , 0.40 , 0.44 \tau =0.32,0.36,0.40,0.44 .$

Figure 4

Graphical view analsysis of the solution pathways of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with different values of input parameter τ , i.e., τ = 0.32 , 0.36 , 0.40 , 0.44 .

$Figure 5 Time series analysis of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with various values of input parameter ρ \rho , i.e., ρ = 0.0295 , 0.0395 , 0.0495 , 0.059 \rho =0.0295,0.0395,0.0495,0.059 .$

Figure 5

Time series analysis of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with various values of input parameter ρ , i.e., ρ = 0.0295 , 0.0395 , 0.0495 , 0.059 .

$Figure 6 Dynamical behaviour of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with various values of input parameter σ \sigma , i.e., σ = 0.0578 , 0.1078 , 0.1578 , 0.2078 \sigma =0.0578,0.1078,0.1578,0.2078 .$

Figure 6

Dynamical behaviour of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with various values of input parameter σ , i.e., σ = 0.0578 , 0.1078 , 0.1578 , 0.2078 .

In the final simulation, as illustrated in Figure 7, we have explored the role of home quarantining as an intervention measure to control the spread of infection. The primary focus of this simulation was to examine how the implementation of home quarantines can impact the dynamics of the infection and whether it proves to be an effective strategy in mitigating the disease within the society. It became evident that the implementation of home quarantines plays a crucial role in controlling the infection level within the population. By isolating infected individuals and limiting their interactions with others, the spread of the virus is curtailed, leading to a decline in the overall infection rate. The findings from this simulation lend strong support to the recommendation for policymakers to consider home quarantining as a viable and effective strategy to combat infectious disease outbreaks, including COVID-19. This intervention can be integrated with other control measures, such as testing, contact tracing, and vaccination campaigns, to create a comprehensive approach in containing the infection and safeguarding public health.

$Figure 7 Time series analysis of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with various assumptions of φ 1 {\varphi }_{1} , i.e., φ 1 = 0.020 , 0.025 , 0.030 , 0.035 {\varphi }_{1}=0.020,0.025,0.030,0.035 .$

Figure 7

Time series analysis of the (a) exposed (b) asymptomatic (c) symptomatic (d) hospitalized and (e) quarantined individuals of the model with various assumptions of φ 1 , i.e., φ 1 = 0.020 , 0.025 , 0.030 , 0.035 .

7 Conclusion

The viral infection COVID-19 is a global challenge that requires a multidisciplinary approach to control and prevent the disease. Scientists, policymakers, and public health officials are actively searching for effective strategies to address these challenges, including mathematical modeling, non-pharmaceutical interventions, global response strategies, online training courses, social and behavioral science, and enhancing public trust in COVID-19 vaccination. In this research, we constructed a mathematical model for COVID-19 with quarantine and hospitalization in a fractional framework. It has been shown that the solutions of the proposed model are positive and bounded. We examined the existence and uniqueness of the solutions of the system with the help of the fixed-point theorem in the context of Schaefer’s and Banach’s theorems. In addition to this, we established UHS results for our system of viral infection COVID-19. To assess the impact of different variables on the dynamics, we introduced a numerical approach to visualize the dynamical behavior of the system. Our findings highlighted the most critical parameters of the system, which are recommended to policymakers for the control and management of the infection. In our future work, we will formulate mathematical models that can account for the impact of new variants, vaccination, behavioral factors, and different populations and geographic regions. These models will help researchers, policymakers, and public health officials make informed decisions and develop effective interventions to combat the spread of COVID-19.

Funding information: Not applicable here.
Author contributions: All the authors contributed significantly to this research work. Rashid Jan conceptualized, formulated, and validated the model; Normy Norfiza Abdul Razak supervised, conceptualized, and analyzed the work; Salah Boulaaras supervised, formulated the model, and revised the whole work; Ziad Ur Rehman wrote the first draft and determined the analytic results; Salma Bahramand formulated, analyzed, and reviewed the work. Finally, all the authors checked and approved the work.
Conflict of interest: There is no conflict of interest concerning this project.
Data availability statement: The datasets used and/or analyzed during the current study available from the corresponding author on reasonable request.

References

[1] Gao D, Lou Y, He D, Porco TC, Kuang Y, Chowell G, et al. Prevention and control of Zika as a mosquito-borne and sexually transmitted disease: a mathematical modeling analysis. Sci Rep. 2016;6(1):28070. 10.1038/srep28070Search in Google Scholar PubMed PubMed Central

[2] Gholami S, Korosec CS, Farhang-Sardroodi S, Dick DW, Craig M, Ghaemi MS, et al. A mathematical model of protein subunits COVID-19 vaccines. Math Biosci. 2023;358:108970. 10.1016/j.mbs.2023.108970Search in Google Scholar PubMed PubMed Central

[3] Ghosh SK, Ghosh S. A mathematical model for COVID-19 considering waning immunity, vaccination and control measures. Sci Rep. 2023;13(1):3610. 10.1038/s41598-023-30800-ySearch in Google Scholar PubMed PubMed Central

[4] Tang TQ, Shah Z, Jan R, Alzahrani E. Modeling the dynamics of tumor-immune cells interactions via fractional calculus. Eur Phys J Plus. 2022;137(3):367. 10.1140/epjp/s13360-022-02591-0Search in Google Scholar

[5] Boulaaras S, Rehman ZU, Abdullah FA, Jan R, Abdalla M, Jan A. Coronavirus dynamics, infections and preventive interventions using fractional-calculus analysis. MATH. 2023;8(4):8680–701. 10.3934/math.2023436Search in Google Scholar

[6] Namazi H, Kulish VV, Wong A. Mathematical modelling and prediction of the effect of chemotherapy on cancer cells. Sci Rep. 2015;5(1):13583. 10.1038/srep13583Search in Google Scholar PubMed PubMed Central

[7] Kim T, Kim DS. Some identities on degenerate r-Stirling numbers via Boson operators. Russian J Math Phys. 2022;29(4):508–17. 10.1134/S1061920822040094Search in Google Scholar

[8] Kim T, SanKim D Degenerate r-Whitney numbers and degenerate r-Dowling polynomials via boson operators. Adv Appl Math. 2022;140:102394. 10.1016/j.aam.2022.102394Search in Google Scholar

[9] Kim T, Kim DS, Kim HK. Generalized degenerate Stirling numbers arising from degenerate boson normal ordering. 2023. arXiv: http://arXiv.org/abs/arXiv:2305.04302. 10.1080/27690911.2023.2245540Search in Google Scholar

[10] Kim T, SanKim D, KyungKim H. Normal ordering of degenerate integral powers of number operator and its applications. Appl Math Sci Eng. 2022;30(1):440–7. 10.1080/27690911.2022.2083120Search in Google Scholar

[11] Jan R, Jan A. MSGDTM for solution of fractional order dengue disease model. Int J Sci Res. 2017;6(3):1140–4. Search in Google Scholar

[12] Hethcote HW. The mathematics of infectious diseases. SIAM Rev. 2000;42(4):599–653. 10.1137/S0036144500371907Search in Google Scholar

[13] Jan A, Jan R, Khan H, Zobaer MS, Shah R. Fractional-order dynamics of Rift Valley fever in ruminant host with vaccination. Commun Math Biol Neurosci. 2020;2020:79. Search in Google Scholar

[14] Grassly NC, Fraser C. Mathematical models of infectious disease transmission. Nature Rev Microbiol. 2008;6(6):477–87. 10.1038/nrmicro1845Search in Google Scholar PubMed PubMed Central

[15] Huppert A, Katriel G. Mathematical modelling and prediction in infectious disease epidemiology. Clin Microbiol Infect. 2013;19(11):999–1005. 10.1111/1469-0691.12308Search in Google Scholar PubMed

[16] Kretzschmar M, Wallinga J. Mathematical models in infectious disease epidemiology. In: Krämer A, Kretzschmar M, Krickeberg K, editors. Modern infectious disease epidemiology: Concepts, methods, mathematical models, and public health. New York (NY), USA: Springer-Verlag; 2010. p. 209–21. 10.1007/978-0-387-93835-6_12Search in Google Scholar

[17] Gao S, Binod P, Chukwu CW, Kwofie T, Safdar S, Newman L, et al. A mathematical model to assess the impact of testing and isolation compliance on the transmission of COVID-19. Infect Disease Model. 2023;8(2):427–44. 10.1016/j.idm.2023.04.005Search in Google Scholar PubMed PubMed Central

[18] Imai N, Rawson T, Knock ES, Sonabend R, Elmaci Y, Perez-Guzman PN, et al. Quantifying the effect of delaying the second COVID-19 vaccine dose in England: a mathematical modelling study. Lancet Public Health. 2023;8(3):e174–83. 10.1016/S2468-2667(22)00337-1Search in Google Scholar PubMed PubMed Central

[19] 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;21:103776. 10.1016/j.rinp.2020.103776Search in Google Scholar PubMed PubMed Central

[20] Haq IU, Ullah N, Ali N, Nisar KS. A new mathematical model of COVID-19 with quarantine and vaccination. Mathematics. 2022;11(1):142. 10.3390/math11010142Search in Google Scholar

[21] Ndaïrou F, Area I, Nieto JJ, Torres DF. Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan. Chaos Solitons Fractals. 2020;135:109846. 10.1016/j.chaos.2020.109846Search in Google Scholar PubMed PubMed Central

[22] Ghosh JK, Biswas SK, Sarkar S, Ghosh U. Mathematical modelling of COVID-19: a case study of Italy. Math Comput Simulat. 2022;194:1–18. 10.1016/j.matcom.2021.11.008Search in Google Scholar PubMed PubMed Central

[23] Zhang T, Wang J, Li Y, Jiang Z, Han X. Dynamics analysis of a delayed virus model with two different transmission methods and treatments. Adv Differ Equ. 2020;1:1–17. 10.1186/s13662-019-2438-0Search in Google Scholar PubMed PubMed Central

[24] Kifle ZS, LemechaObsu L. Optimal control analysis of a COVID-19 model. Appl Math Sci Eng. 2023;31(1):2173188. 10.1080/27690911.2023.2173188Search in Google Scholar

[25] Keno TD, Etana HT. Optimal control strategies of COVID-19 dynamics model. J Math. 2023;2023:2050684. 10.1155/2023/2050684Search in Google Scholar

[26] Jan R, Boulaaras S. Analysis of fractional-order dynamics of dengue infection with non-linear incidence functions. Trans Inst Meas Control. 2022;44(13):2630–41. 10.1177/01423312221085049Search in Google Scholar

[27] Jan R, Qureshi S, Boulaaras S, Pham VT, Hincal E, Guefaifia R. Optimization of the fractional-order parameter with the error analysis for human immunodeficiency virus under Caputo operator. Discrete Contin Dynam Syst-S. 2023;16:2118–40. 10.3934/dcdss.2023010Search in Google Scholar

[28] Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. 1st ed. Amsterdam, The Netherlands: Elsevier; 2006. Search in Google Scholar

[29] Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Amsterdam, The Netherlands: Elsevier; 1998. Search in Google Scholar

[30] Granas A, Dugundji J. Elementary fixed point theorems. In Fixed Point Theory. New York, NY: Springer; 2003. p. 9–84. 10.1007/978-0-387-21593-8_2Search in Google Scholar

[31] Lakshmikantham V, Leela S, Martynyuk AA. Stability analysis of nonlinear systems. New York: M. Dekker; 1989. p. 249–75. 10.1142/1192Search in Google Scholar

[32] Ullam SM. Problems in modern mathematics (Chapter VI). New York (NY), USA: Wiley; 1940. Search in Google Scholar

[33] Hyers DH. On the stability of the linear functional equation. Proc NAS. 1941;27(4):222. 10.1073/pnas.27.4.222Search in Google Scholar PubMed PubMed Central

[34] Rassias TM. On the stability of the linear mapping in Banach spaces. Proc Amer Math Soc. 1978;72(2):297–300. 10.1090/S0002-9939-1978-0507327-1Search in Google Scholar

[35] Ali Z, Zada A, Shah K. On Ulam’s stability for a coupled systems of nonlinear implicit fractional differential equations. Bullet Malaysian Math Sci Soc. 2019;42(5):2681–99. 10.1007/s40840-018-0625-xSearch in Google Scholar

[36] Benkerrouche A, Souid MS, Etemad S, Hakem A, Agarwal P, Rezapour S, et al. Qualitative study on solutions of a Hadamard variable order boundary problem via the Ulam–Hyers–Rassias stability. Fractal Fracti. 2021;5(3):108. 10.3390/fractalfract5030108Search in Google Scholar

Received: 2023-08-29

Revised: 2023-10-03

Accepted: 2023-10-03

Published Online: 2023-10-27

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2022-0342

Keywords for this article

viral dynamics; differential equations; mathematical operators; stability analysis; dynamical behavior; public health policies

Creative Commons

BY 4.0