key: cord-0956547-8u1wfelz
authors: Sintunavarat, Wutiphol; Turab, Ali
title: Mathematical analysis of an extended SEIR model of COVID-19 using the ABC-Fractional operator
date: 2022-02-17
journal: Math Comput Simul
DOI: 10.1016/j.matcom.2022.02.009
sha: 35569b8ba69c50b6efdc03b0da095a3db2071936
doc_id: 956547
cord_uid: 8u1wfelz

This paper aims to suggest a time-fractional [Formula: see text] model of the COVID-19 pandemic disease in the sense of the Atangana-Baleanu-Caputo operator. The proposed model consists of six compartments: susceptible, exposed, infected (asymptomatic and symptomatic), hospitalized and recovered population. We prove the existence and uniqueness of solutions to the proposed model via fixed point theory. Furthermore, a stability analysis in the context of Ulam-Hyers and the generalized Ulam-Hyers criterion is also discussed. For the approximate solution of the suggested model, we use a well-known and efficient numerical technique, namely the Toufik-Atangana numerical scheme, which validates the importance of arbitrary order derivative [Formula: see text] and our obtained theoretical results. Finally, a concise analysis of the simulation is proposed to explain the spread of the infection in society.

Infectious diseases have been a constant threat to humanity. This threat has increased in recent decades due to the emergence and re-emergence of several lethal infectious diseases. Mathematical analysis and modeling enable officials J o u r n a l P r e -p r o o f

The literature includes several mathematical models that aim to explain the dynamics of COVID-19's development. Three phenomenological models are provided in [34] , which have been validated for outbreaks of other diseases other than COVID-19, seeking to produce and test short-term estimates of the total recorded events. Other research (see, e.g., [35] ) suggests minor modifications of 40 the SEIR form, including stochastic components.

COVID-19 is a disease triggered by a modern virus that creates a worldwide emergency and requires a model that considers its unique characteristics. In specific, creating a model that integrates the following would be appropriate:

• the impact of undetected infected individuals (see [36] ) indicates that 45 COVID-19 is based on the ratio of cases observed overestimated gross infected people;

• the impact of various sanitary and contagious factors on hospitalized individuals (differentiating those with moderate and extreme conditions to survive from others that would ultimately die); 50 • calculating bed demands in hospitals (one of the main issues confronting policy-makers tackling COVID- 19) .

The main objective of this research is to study an epidemic fractional-order (S P E P I A P I SP P H P R P ) model, which investigates the significance of COVID-19 spread in society. The paper is organized in the following sections. The deriva- The model described here is an update of the SEIR model having two additional compartments. We propose an (S P E P I A P I SP P H P R P ) constituent model, which contains the susceptible, exposed, infected (asymptomatic and symptomatic), hospitalized, and recovered population (see Figure 1 ). Here, we give the following fractional system:

where Π := n × N, N is the total number of individuals and n is the birth rate, 70 ABC D ϑ 0 denotes the ABC fractional derivative of order ϑ ∈ [0, 1) and all other parameters are described in Table 1 . It is important to note that the model's parameters are non-negative and having dimensions 1 time ϑ . The total population at time t is N which is further divided into S P (t), E P (t), I A P (t), I SP P (t), H P (t) and R P (t). The key assumptions are as follows: 75 1. natural mortality and birth rates are same; 2. the natural causes of death account for fatalities are presented in compartments S P and R P ; 3. the model's death population is made up of individuals who died as a result of exposure κ ϑ 1 E P (t), during infectious period (κ ϑ 2 I A P (t) and κ ϑ 3 I SP P (t)) and 80 hospitalization κ ϑ 4 H P (t); 4. the total number of dead population from each compartment can be cal-

J o u r n a l P r e -p r o o f Journal Pre-proof Incubation period of an exposed individual

Fraction of the exposed population that becomes asymptomatic after the incubation period and the remaining of the population are symptomatic, respectively

Infectious rates of an asymptomatic and a symptomatic individual, respectively ℓ ϑ Recovery rate through hospitalization

Mortality rates of the exposed, asymptomatic, symptomatic and hospitalized populations, respectively 5. other natural disasters have had a minimal effect on the population; therefore, we are neglecting them. 85 

The following concepts and proven results would be needed in the sequel. 

Based on the above definition, Atangana and Baleanu [24] gave the following definition of the Atangana-Baleanu fractional derivative in Caputo sense. 

where ω is the normalization function defined by ω(ϑ) = ϑ 2−ϑ , for all 0 ≤ ϑ < 1.

Also, E ϑ stands for the Mittag-Leffler function which is a generalization of the exponential function (see [6, 38, 39] ).

, then we get the so-called Caputo-Fabrizo differential operator. Further, it is to be noted that 24] ). The Laplace transform of (3.1) is defined as follows:

Definition 3.5 ( [24] ). Let Ψ ∈ H 1 (a, b), where a, b ∈ R with a < b, and ϑ ∈ [0, 1). The corresponding integral in ABC sense are defined as follows:

where Γ is a gamma function.

Lemma 3.6. The solution of the given problem for 0 ≤ ϑ < 1,

where Ψ : [0, T ] → R is an unknown function and Ψ 0 is a fixed constant, provided by For the qualitative analysis, we define the Banach space

T ] with 0 < T < ∞, under the norm given for

Throughout this paper, for v = (v 1 , v 2 , ..., v 6 ) ∈ R 6 , we use this notion |·| as the taxicab norm in R 6 .

We shall examine the existence and uniqueness of a solution to our main model (2.1) with the condition (2.2) in this section. Let us write the model

J o u r n a l P r e -p r o o f Journal Pre-proof

with the condition (2.2). By using (2.1), we transform (4.2) with (2.2) to the following system:

,

Using Lemma 3.6, the model (4.3) can be turned to the fractional integral equation in the sense of ABC fractional integral as follows:

Here we impose the following conditions: (∆ 1 ) there exists two constants p 1 > 0 and p 2 > 0 such that

J o u r n a l P r e -p r o o f Journal Pre-proof

for each Ψ,Ψ ∈ F and t ∈ [0, T ].

We start with the following result. Proof. First, we see that if Ψ ∈ F, then CΨ ∈ F. For each Ψ ∈ D Z := {Ψ ∈ F :

Hence, C is uniformly bounded on D Z .

Next, we have to prove that the operator C is equicontinuous. For this, let

Then

Due to the fact that t 1 → t 2 , the right hand side of the preceding inequality approaches to zero. As a result of the Arzelá-Ascoli theorem, C is compact and therefore completely continuous. Proof.

is clear that the operator C : S→ F as given in (4.9) is completely continuous.

. This implies that

As a result, S is bounded. Hence, the operator has at least one fixed point, which happens to be the solution of the COVID-19 model (2.1)-(2.2). 

J o u r n a l P r e -p r o o f Journal Pre-proof

for all t ∈ [0, T ]. It can be seen that the operator C is well defined and the unique solution of (4.9) is just the fixed point of C. Here, we consider D Z = {Ψ ∈ F : ∥Ψ∥ ≤ Z} as a closed and convex set with Z ≥Λ 2 1−Λ1 , wherẽ

This implies that 

Here, we discuss some critical aspects of the model (2.1) or, more precisely, the system (4.3), including the boundedness and positivity of the solutions for 130 t ≥ 0.

The boundary of solutions for the nonlinear system (2.1) with non-negative initial conditions is determined in this subsection. Our major goal is to prove that the feasible region produced in R 6 + is positively invariant with respect to 135 the fractional model (2.1).

Theorem 5.1. The feasible region of the proposed fractional model (2.1) is given by

The existence and uniqueness of the solution of model (2.1) have already been shown in the preceding section; all that remains is to demonstrate that the set Ω is positively invariant with respect to the condition (2.2). For the proof of Theorem 5.1, the following theorem will be utilized. Proof. To obtain the fractional derivative of total population, we add all the relations in system (2.1). So

J o u r n a l P r e -p r o o f

By applying the Laplace transform on both sides of above inequality, we obtain

and then

where N (0) represent the initial value of the total population and ω is the normalization function defined in Definition 3.2. This implies that

Now, by applying the inverse Laplace transform, we have

where ℘ = ϑw ω(ϑ)+(1−ϑ)ν , and E a,b is the Mittag-Leffler function with two parameters a > 0 and b > 0 may be defined by the following series

whose Laplace transform is

provided that s >| Λ | 1/a . For a, b > 0, the Mittag-Leffler function satisfies

and for the case a = ϑ, b = ϑ + 1 and ℏ = −℘t ϑ , we have

From [24] , it is clear that the Mittag-Leffler function is bounded for all t > 0 and possess an asymptotic behavior. Therefore, from (5.2) and (5.3), we can 

In this subsection, we will show that for all t ≥ 0, the state variables S P , E P , I A P , I SP P , H P and R P are positive. This characteristic is essential in order to demonstrate that our model is physically feasible.

Theorem 5.3. The solution space S P , E P , I A P , I SP P , H P , R P of the system (2.1) will remain positive forever with any non-negative initial data.

Proof. First equation of the model (2.1) is rearranged to give

As all the solutions are bounded, therefore, we let where q := ξ ϑ A ρ 1 + ξ ϑ SP ρ 2 + d ϑ is a constant. Applying the Laplace transform on both sides of (5.5), we obtain

. Now, by applying the inverse Laplace transform, we have

. Because both of the values on the right-hand side of (5.6) are positive. Therefore, for every t ≥ 0, the solution S P (t) is also remains positive. Likewise, for every t ≥ 0 corresponding to any non-negative initial data, we can simply argue that E P > 0, I A P > 0, I SP P > 0, H P > 0, R P > 0. As a result, the solutions in R 6 + will always be positive. 160 

Here, by solving the following system yields the equilibrium points of the suggested fractional model (2.1):

The proposed model (2.1) has a unique non-negative Corona free equilibrium at the point after simple computations given by

and a unique non-negative Corona present equilibrium at the point given by

in the epidemiological region Ω, where 

The maximum absolute eigenvalue of the positive matrix FG −1 is the value of R 0 for the model (2.1). That is, is that this number is evaluated using the next-generation matrix approach by assuming coronavirus-free equilibrium in the system (2.1) and calculating the second derivative of infectious classes. As a result, the transmission and transition matrices are denoted by F and G [40] , respectively, where:

Hence, the strength number A 0 is

A negative strength number indicates that the system (2.1) will have a single 175 magnitude, either a maximum with two infection sites suggesting a single wave or a quick drop from the coronavirus-free equilibrium. As a result, the infection will grow to a minimal point with the renewal process, then stabilize or halt as requested later. The definitions that follow are required in the upcoming results. Let ϵ > 0 and consider the following inequality:

where ϵ = max (ϵ j ) T , j = 1, 2, ..., 6.

Definition 6.1. The proposed problem (3.5), which is equivalent to model

2), is Ulam-Hyers stable if there exists Ξ > 0 such that, for every ϵ > 0 and for each solutionΨ ∈ F satisfying inequality (6.1), there exist a solution Ψ ∈ F of problem (4.3) with the initial conditionΨ(0) = Ψ(0) such that such that

where φ = max (φ j ) T , j = 1, 2, ..., 6. (ii) for t ∈ [0, T ], we have the following model

where h(t) = (h 1 (t), h 2 (t), ..., h 6 (t)) T . satisfies the given relation

whereΨ h (t) is a solution of (6.2),Ψ(t) satisfies (6.1) and Λ :

Proof. In view of (ii) of Remark 6.3 and Lemma 3.6, the solution of (6.2) is

given bỹ

J o u r n a l P r e -p r o o f Journal Pre-proof Furthermore, we havẽ

Using (i) of Remark 6.3, we get

Hence, we get the desired result. 

Thus, by (∆ 1 ) and Lemma 6.4, we obtain

Hence,

As, Λp 3 < 1, we obtain

Remark 6.6. If we set φ(ϵ) = Ξϵ such that φ(0) = 0, then from the above 205 theorem, we conclude that the proposed model (2.1) with (2.2) is generalized

Ulam-Hyers stable.

In this section, we need to examine the estimated solutions of the ABC fractional-order model (2.1) with (2.2). The theoretical models are then ob-210 tained by employing the current scheme. The ABC fractional derivative is utilized to develop the computational procedure for the simulation of our model (2.1).

In this subsection, we offer a numerical technique for analyzing and predicting the numerical stability of a Coronavirus fractional model (2.1) with (2.2) based on a recently developed Toufik-Atangana criterion [45] . We first outline the concept briefly before applying it to the fractional model (2.1) with (2.2) to get an iterative approach. The system (2.1) may be represented using the fundamental theorem of fractional calculus. 

The function Ψ(ς, z(ς)) can be approximated over [t j , t j+1 ], using the interpolation polynomial

By substituting this in (7.1), we obtain

Finally, we get the approximate solution as:

Hence, we obtain the following recursive formula for the model equations:

J o u r n a l P r e -p r o o f It means the susceptible individuals will expand over time, while the exposed 230 population will decline. It is apparent from Figures 4 and 5 that, with time, the affected community (asymptomatic I A P (t) and symptomatic I SP P (t)) hits their stable points. As COVID-19 infection cases initially rose, the amount of hospitalized patients H P (t) was higher (as seen in Figure 6 ), although it would decrease with time. Also, the number of COVID-19 contaminated classes 235 decreases over time; therefore, the recovered class R P (t) eventually decreases (see Figure 7 ). 

In this work, we developed a fractional-order (S P E P I A P I SP P H P R P ) model Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

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This project is funded by National Research Council of Thailand (NRCT) N41A640092.

The authors declare no conflict of interest.

All authors contributed equally to the manuscript and typed, read, and approved the final manuscript.