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Stability analysis and numerical simulations of the fractional COVID-19 pandemic model

  • Ahmad Alalyani ORCID logo EMAIL logo and Sayed Saber
Published/Copyright: May 30, 2022
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International Journal of Nonlinear Sciences and Numerical Simulation
From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 3

Abstract

The purpose of this article is to formulate a simplified nonlinear fractional mathematical model to illustrate the dynamics of the new coronavirus (COVID-19). Based on the infectious characteristics of COVID-19, the population is divided into five compartments: susceptible S(t), asymptomatic infection I(t), unreported symptomatic infection U(t), reported symptomatic infections W(T) and recovered R(t), collectively referred to as (SIUWR). The existence, uniqueness, boundedness, and non-negativeness of the proposed model solution are established. In addition, the basic reproduction number R 0 is calculated. All possible equilibrium points of the model are examined and their local and global stability under specific conditions is discussed. The disease-free equilibrium point is locally asymptotically stable for R 0 leq1 and unstable for R 0 > 1. In addition, the endemic equilibrium point is locally asymptotically stable with respect to R 0 > 1. Perform numerical simulations using the Adams–Bashforth–Moulton-type fractional predictor–corrector PECE method to validate the analysis results and understand the effect of parameter variation on the spread of COVID-19. For numerical simulations, the behavior of the approximate solution is displayed in the form of graphs of various fractional orders. Finally, a brief conclusion about simulation on how to model transmission dynamics in social work.

Keywords: COVID-19; mathematical modeling; the predictor–corrector PECE method of Adams–Bashforth–Moulton type
MSC (2010): 92D30; 92D25; 92C42; 34C60

1 Introduction

The COVID-19 pandemic has spread around the world, and people are aware of the disease and are taking pandemic precautions. Infection COVID-19 is direct and indirect contact and may be caused by droplet infection from person to person [1, 2]. But despite all, COVID-19 is spreading very rapidly. In some countries such as Spain, Australia, Serbia, China, and India, the fourth wave of COVID-19 has begun. In the absence of effective vaccines and specific antivirals, mathematical modeling plays an important role in better understanding the dynamics of the disease and developing strategies for managing the rapid spread of infectious diseases increase. Vaccines are needed to prevent the spread of the disease. But without vaccines, people have to maintain social distance. To maintain social distance, you need to follow modeling rules. Predictions have important implications for the fight against the outbreak of COVID-19 with medical plans during this period. Another effect is the media effect, the use of masks, and non-drug intervention, have changed by a combination of media recognition, such as hand sanitizer (see [3], [4], [5], [6], [7], [8], [9]).

The history of coronavirus dates back to the 1930s when poultry is infected with infectious bronchitis virus (IBV). It was first described in humans in the 1960s [10]. The epidemic of Covid19, which began in China’s Wuhan City in early December 2019, the spread is, the UK, Italy, Spain, and France, has spread to temporarily some of the countries of Europe. The spread of new coronavirus (COVID-19) is a very frightening global problem. At the time of the 2021 August 15, confirmed cases of 207,758,858 in the world, a vaccine dose of 28,168,759, death of 372,205, there is a healing case of 186,235,389 [11].

Decision-makers need a mathematical model to predict a pandemic in order to ease the blockade, reopen the economy, and maintain people’s mental and psychological well-being to balance health care and the economy. It is clear that this virus is an infectious disease that easily infects individuals and can be modeled as a system of a nonlinear differential equation. There is much research in the literature on analyzing the effects of COVID-19 through mathematical modeling. See [2, 12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36].

Basic reproduction number R 0 is one of the most important quantities of infectious diseases. This is because R 0 measures the spread of the disease. At R 0 ≤ 1, the spread of the disease stops, but at R 0 = 1, an infected person can infect one person on average. The epidemic of the disease has stabilized. If R 0 must be greater than 1, the disease can spread and become epidemic. In this context, see [8]. The value of R 0 in China was about 2.5 [13] in the early stages of coronavirus outbreaks.

Liu et al. [24], Khoshnaw et al. [25], and Erturka et al. [36] studied a system of five ordinary differential equations of the first order, two of which have quadratic nonlinearity, with respect to five unknowns, which are the numbers of certain classes of individuals (susceptible, asymptomatic infected, unreported symptomatic infectious, reported symptomatic infectious, recovered), collectively (SIUWR). The main objective of this work is to describe the fractional mathematical modeling and dynamics of a novel corona virus (COVID-19) SIUWR model using the algorithm of a Caputo fractional derivative. Existence, uniqueness, non-negativity and boundedness of the solutions are presented. In addition, the simple reproduction number R 0 has been determined as R 0 = 0.2330. Furthermore, by constructing appropriate Lyapunov functions, the local and global asymptotic stability of the free-equilibrium point are studied.

2 Models’ description

2.1 The integer-order SIUWR model

Liu et al. [24] used the following mathematical model to introduce the COVID-19 outbreak in China:

(2.1) d S d t = − β S ( I + U ) ,   S ( 0 ) = S 0 > 0 , d I d t = β S ( I + U ) − ( γ + δ ) I ,   I ( 0 ) = I 0 > 0 , d U d t = δ I − η U ,   U ( 0 ) = U 0 > 0 , d W d t = γ I − η W ,   W ( 0 ) = W 0 > 0 .

Khoshnaw et al. [25] extended model (2.1) including recovered individuals (R), unreported symptomatic mortality (α 1 U) and mortality symptomatic reported (α 2 W). The model considered includes five compartments with susceptible individuals S(t), asymptomatic infections I(t), unreported symptomatic infections U(t), symptomatic infection W(t) and recovered R(t). The proposed model (SIUWR) of asymptomatic transmission is sensitive to unreported infections that have been recovered, defined as

(2.2) d S d t = − β S ( I + U ) ,   S ( 0 ) = S 0 > 0 , d I d t = β S ( I + U ) − ( γ + δ ) I ,   I ( 0 ) = I 0 > 0 , d U d t = δ I − ( η + α 1 ) U ,   U ( 0 ) = U 0 > 0 , d W d t = γ I − ( η + α 2 ) W ,   W ( 0 ) = W 0 > 0 , d R d t = η W + η U ,   R ( 0 ) = R 0 > 0 .

The parameters for the novel (COVID-19) SIUWR model (2.2) are given in Table 1.

Table 1:

The WHO situation report (National Health Commission of the Republic of China), provided the values for parameters and beginning populations used in this study [24, 25, 37].

Symbols Biological definitions Estimated values
S(0) The rate of natural birth 11.081 × 106
I(0) Infected people who were initially asymptomatic 3.62
U(0) Initially, symptomatic infected individuals went unreported 0.2
W(0) Symptomatic infected people were first reported 4.13
R(0) Individuals who have already recovered 0
β The rate of transmission between susceptible individuals and asymptomatic infected individuals 4.44 × 10−8
γ Asymptomatic infected to reported symptomatic infected transition rate 0.1142
δ Asymptomatic infected to unreported symptomatic infected transition rate 0.6
η Average time symptomatic has infectious symptoms 1/7
α 1 The unreported symptomatic death rate 1.5 × 10−4
α 2 The reported symptomatic death rate 1.7826 × 10−5

2.2 The fractional-order SIUWR model

The Euler gamma function is Γ ( z ) = ∫ 0 ∞ e − t t z − 1 d t . The function f ( t ) : R + → R , t > 0, which has the fractional integral operator of order ν ∈ R + is defined as

I ν f ( t ) = 1 Γ ( ν ) ∫ 0 t ( t − s ) ν − 1 f ( s ) d s .

The Caputo fractional derivative of order ν > 0, n − 1 < ν < n, n ∈ N, according to [38], is defined as

D ν f ( t ) = 1 Γ ( n − ν ) ∫ 0 t f ( n ) ( s ) ( t − s ) ν + 1 − n d s , n − 1 < ν < n , d n d t n f ( t ) , ν = n .

The SIUWR model’s fractional-order form for 0 < ν < 1, given by Eq. (2.2) is described as being

(2.3) D ν S = − β S ( I + U ) ,   S ( 0 ) = S 0 > 0 , D ν I = β S ( I + U ) − ( γ + δ ) I ,   I ( 0 ) = I 0 > 0 , D ν U = δ I − ( η + α 1 ) U ,   U ( 0 ) = U 0 > 0 , D ν W = γ I − ( η + α 2 ) W ,   W ( 0 ) = W 0 > 0 , D ν R = η W + η U ,   R ( 0 ) = R 0 > 0 .

Obviously, since the fractional parameter has higher arbitrary real-valued orders, the SIUWR model given by Eq. (2.3) has greater degrees of freedom. The two SIUWR models defined by Eqs. (2.2) and (2.3) have the same steady states. The model includes six constant coefficients selected that explain the strength of different processes. Such coefficients are basically model parameters.

2.3 Solutions’ properties

Let X(t) = (S(t),I(t),U(t),W(t),R(t)) T , then it is possible to reformulate system (2.3) as follows:

D ν X ( t ) = F ( X ( t ) ) ,

where

F ( X ( t ) ) = F 1 ( X ) F 2 ( X ) F 3 ( X ) F 4 ( X ) F 5 ( X ) = − β S ( I + U ) β S ( I + U ) − ( γ + δ ) I δ I − ( η + α 1 ) U γ I − ( η + α 2 ) W η W + η U .

We are now looking into the existence and uniqueness of fractional system solutions (2.3) in Ω × (0, T] with

Ω = { ( S , I , U , W , R ) ∈ R 5 : S ≥ 0 , I ≥ 0 , U ≥ 0 , W ≥ 0 , R ≥ 0 , max ( | S | , | I | , | U | , | W | , | R | ) ≤ ζ } .

In order to demonstrate the global existence of system solutions (2.3), the following lemma is necessary.

Lemma 1

([39], Lemma 3.1). Assume that the vector function F : R 5 → R 5 satisfies the following conditions:

  1. F(X) and ∂ F ∂ X are continuous.

  2. ∥ F(X) ∥ ≤ κ + λ ∥ X ∥, for all X ∈ R 5. Here κ and λ are two positive constants. The system (2.3) then has a unique solution.

Next, we will prove the non-negativity of this solution.

Since

D ν S | S = 0 = 0 , D ν I | I = 0 = β S U > 0 , D ν U | U = 0 = δ I > 0 , D ν W | W = 0 = γ I > 0 , D ν R | R = 0 = η W + η U > 0 .

The solution’s non-negativity will then be proved. Using Lemmas 5 and 6 in [40], the fractional-order model (2.3) solutions are non-negative. Finally, we establish the boundedness of solution. Assume that

N = S + I + U + W + R

is the total population at time t. In the fractional-order model (2.3), one obtains a direct sum of Eq. (2.3).

D ν N + λ N ≤ 3 ζ ,

where λ = min{1, α 1, α 2}. Following to [41]; Lemma 9, one obtains

0 ≤ N ≤ N ( 0 ) E ν ( − γ ν ) + r ν E ν , ν + 1 ( − γ ν ) ,

where E ν is the Mittag–Leffler function. According to [40]; Lemma 5 and Corollary 6, one obtains:

0 ≤ N ≤ 3 ζ λ , t → ∞ .

As a result, starting in Ω, the solutions of the fractional-order model (2.3) are uniformly bounded in the region

Λ = ( S , I , U , W , R ) ∈ Ω + : 0 ≤ S + I + U + W + R ≤ 3 ζ λ .

3 Stability analysis

For I = 0, we can easily obtain the infection-free equilibrium E 0 = S * , 0,0,0 , R * . Let x = ( S , W , R ) T and y = ( I , U ) T , then we have

D ν y = B ( y ) − W ( y ) ,

where

B ( y ) = β S ( I + U ) 0 , W ( y ) = ( γ + δ ) I − δ I + ( η + α 1 ) U .

The Jacobian matrices of the two matrices B ( y ) and W ( y ) , at the infection-free equilibrium point E 0, are, respectively,

H = β S * β S * 0 0 , K = γ + δ 0 − δ η + α 1 .

Thus

K − 1 = 1 γ + δ 0 δ ( η + α 1 ) ( γ + δ ) 1 η + α 1 .

The matrix H.K −1 has a spectral radius and is given by

ρ ( H . K − 1 ) = δ β S * + β S * ( η + α 1 ) ( η + α 1 ) ( γ + δ ) .

Then, the basic reproduction number of the fractional-order model (2.3) is

R 0 = δ β S * ( η + α 1 ) ( − β S * + γ + δ ) .

Lemma 2

If R 0 < 1 , the infection-free equilibrium E 0 = S * , 0,0,0 , R * is local asymptotic stable in Ω and unstable if R 0 > 1 .

Proof

For the fractional-order model (2.3) at E 0 = S * , 0,0,0 , R * , the Jacobian matrix J(E 0) of it is given by

J ( E 0 ) = 0 − β S * − β S * 0 0 0 β S * − ( γ + δ ) β S * 0 0 0 δ − ( η + α 1 ) 0 0 0 γ 0 − ( η + α 2 ) 0 0 0 η η 0 ,

and its characteristic equation is given by

(3.1) λ 2 [ λ + η + α 2 ] [ λ 2 + η + α 1 − β S * + γ + δ λ + ( η + α 1 ) ( − β S * + γ + δ ) − δ β S * ] = 0 .

From Eq. (3.1), the eigenvalues are obtained by: λ 1 = λ 2 = 0, λ 3 = −(η + α 2), and the other values can obtains from the equation

(3.2) λ 2 + η + α 1 − β S * + γ + δ λ + ( η + α 1 ) ( − β S * + γ + δ ) − δ β S * = 0 ,

The infection-free equilibrium point E 0 is stable according to the Routh–Hurwits [42] criterion if and only if all the eigenvalues are < 0 . Obviously, the eigenvalues λ 1, λ 2, λ 3 are all negatives. Thus the stability of Eq. (3.2) depends on whether it is λ 4 < 0, λ 5 < 0, or not. If R 0 < 1 , λ 4, λ 5, is computed directly, the sufficient stability condition is given by

R 0 = δ β S * ( η + α 1 ) ( − β S * + γ + δ ) .

That is, the required and adequate condition is R 0 < 1 that system (2.3) is locally asymptotically stable at E 0. Otherwise when R 0 > 1 , system (2.3) is not stable. So the facts follows. ■

Lemma 3

Infection-free equilibrium point E 0 in Ω is globally asymptotically stable if R 0 < 1 and unstable if R 0 > 1 .

Proof

We need to construct an effective Lyapunov function to prove this outcome, as follows:

L = δ I + − β S * + γ + δ U .

The fractional derivative of L, with respect to t, gives

D ν L = δ D ν I + − β S * + γ + δ D ν U = δ β S * − γ − δ − β S * + γ + δ I + δ β S * − ( η + α 1 ) ( − β S * + γ + δ ) U ≤ δ β S * − ( η + α 1 ) ( − β S * + γ + δ ) U

Therefore, if R 0 < 1 , then D ν L ≤ 0. In addition, {(S, I, U, W, R) ∈ Ω : D ν L = 0} is the largest invariant set. The one-tone {E 0}. Thus, E 0 is asymptotically stable globally by using the invariance principle of LaSalle [43]. ■

Now, we will compute the stability analysis of the epidemic point. Let

D ν S ( t ) = 0 ,   D ν I ( t ) = 0 ,   D ν U ( t ) = 0 ,   D ν W ( t ) = 0 ,   D ν R ( t ) = 0 .

For I* > 0, the unique endemic-equilibrium point is: E* = (S*, I*, U*, W*, R*), where

S * = ( γ + δ ) ( η + α 1 ) β ( η + α 1 + δ ) , I * = I * , U * = δ η + α 1 I * , W * = γ η + α 2 I * , R * = R * .

Now we examine the local stability of the endemic equilibrium point E*.

Lemma 4

If R 0 > 1 , E* is locally asymptotically stable in Ω.

Proof

At the endemic equilibrium point E* = (S*, I*, U*, W*, R*), the fractional order system (2.3) has the Jacobian matrix J(E*), which is given by

J ( E * ) = − β ( I * + U * ) − β S * − β S * 0 0 β ( I * + U * ) β S * − ( γ + δ ) β S * 0 0 0 δ − ( η + α 1 ) 0 0 0 γ 0 − ( η + α 2 ) 0 0 0 η η 0 ,

and its characteristic equation is given by

(3.3) λ ( λ + ( η + α 2 ) ) a 1 λ 3 + a 2 λ 2 + a 3 λ + a 4 = 0 ,

where

a 1 = 1 , a 2 = γ + δ + η + σ + α 1 + β ( I * + U * − S * ) , a 3 = − δ β S * + β ( γ + δ ) ( I * + U * ) − β S * ( η + α 1 ) + ( η + α 1 ) ( γ + δ ) + β ( η + α 1 ) ( I * + U * ) , a 4 = β ( η + α 1 ) ( γ + δ ) ( I * + U * ) .

From Eq. (3.3), we conclude that the characteristic values are given by: λ 1 = 0, λ 2 = −(η + α 2), and the other values can be obtained from the equation

a 1 λ 3 + a 2 λ 2 + a 3 λ + a 4 .

Based on Routh–Hurwitz conditions [42], the Routh–Hurwitz Array for the Jacobian matrix J(E*) is as follows:

D 1 ( E * ) = − a 1 a 2 a 2 a 4 ( a 2 a 3 − a 4 a 1 ) a 2 0 a 4 0 .

If one verifies that ( a 2 a 3 − a 4 a 1 ) a 2 has the same sign as a 3, then the three eigenvalues have negative real parts. Since a 1 > 0, a 2 > 0, a 3 > 0, a 4 > 0, ( a 2 a 3 − a 4 a 1 ) a 2 > 0 and a 2 a 3 > a 4 a 1 are held. As a result, the Routh–Hurwitz stability conditions are satisfied and the endemic-equilibrium point is locally asymptotically stable. ■

4 Numerical simulations

To perform numerical simulation of the fractional model of COVID-19 epidemic (2.3), we use parameter value from [24, 25], which are summarized in Table 1. The values of parameters and initial populations S(0) = 11.081 × 106, I(0) = 3.62, U(0) = 0.2, W(0) = 4.13, R(0) = 0 are obtained from [24, 25]. When calculating, we get R 0 = 0.2330. As a result, in system (2.3), there is a disease-free equilibrium point E* which is asymptotically globally stable, according to Lemma 3 (see Figure 1).

Figure 1: Dynamics of (a) susceptible individuals S(t), (b) asymptomatic infected individuals I(t), (c) unreported symptomatic infected individuals U(t), (d) reported symptomatic infected individuals W(t), (e) recovered individuals R(t), and (f) Lyapunov function L(t) for different values of ν at R0 = 0.23301.
Figure 1:

Dynamics of (a) susceptible individuals S(t), (b) asymptomatic infected individuals I(t), (c) unreported symptomatic infected individuals U(t), (d) reported symptomatic infected individuals W(t), (e) recovered individuals R(t), and (f) Lyapunov function L(t) for different values of ν at R0 = 0.23301.

4.1 Dynamics of (S − I − U − W − R) for different values of ν

For S(t), I(t), U(t), W(t), R(t), and L(t) of the infectious equilibrium point using various initial conditions as in Table 1 (such that R 0 = 0.23301), Figure 1 depicts the behavior of the attained outcomes by the projected solution process. The solutions of (2.3) converge to a single disease-free equilibrium E 0, as expected, which is globally asymptotically stable. There are distinctive effects of fractional orders; the solution curves for 0 < ν < 1 show delay in the epidemic peak and flatten faster, see Figure 1(b) and (d). The effects of nu are considerably more evident for smaller orders; for example, compare nu = 0.95 and nu = 0.75 in Figure 1(c) and (d). The number of infected individuals decreases significantly for smaller fractional orders, but the number of susceptible individuals increases, as seen in Figure 1(a). After 60 days, the number of susceptible populations decreases gradually and becomes stable, while the dynamics of recovered individuals increase and then return to flat. The number of infected individuals completely changes between 40 and 80 days in both asymptomatic and symptomatic groups. We looked at the relationship between asymptomatic infected people and (a) unreported symptomatic infected individuals, (b) reported symptomatic infected individuals, (c) susceptible individuals, (d) recovered individuals, and (e) Lyapunov function for different fractional order ν in Figure 2. The dynamical relations for reported and unreported symptomatic states are the same, however the model dynamics for susceptible and recovered groups are slightly different.

Figure 2: For different fractional orders ν, the relationship between asymptomatic infected individuals and (a) unreported symptomatic infected, (b) reported symptomatic infected, (c) susceptible, (d) recovered, and (e) Lyapunov function.
Figure 2:

For different fractional orders ν, the relationship between asymptomatic infected individuals and (a) unreported symptomatic infected, (b) reported symptomatic infected, (c) susceptible, (d) recovered, and (e) Lyapunov function.

4.2 Transition rate δ

With δ = 0.0285, 0.15, 0.6, we investigated the effects of the transition rate between asymptomatically infected and unreported symptomatically infected individuals on (a) asymptomatic infected individuals, (b) unreported symptomatic infected individuals, and (c) reported symptomatic infected individuals. The number of unreported symptomatic infected patients increases dramatically when the amount of δ is increased from 0.0285 to 0.6, as seen in Figure 3(b). If the transition rate values δ decrease, there will be more asymptomatic infected and reported symptomatic infected cases, as shown in Figure 3(a), (b) and (c).

Figure 3: Transition rate δ = 0.0285, 0.15, 0.6 for (a) asymptomatic infected individuals, (b) unreported symptomatic infected individuals, and (c) reported symptomatic infected individuals in computer simulations using MATLAB parameters using δ impact.
Figure 3:

Transition rate δ = 0.0285, 0.15, 0.6 for (a) asymptomatic infected individuals, (b) unreported symptomatic infected individuals, and (c) reported symptomatic infected individuals in computer simulations using MATLAB parameters using δ impact.

4.3 Transition rate γ

Using γ = 0.1142, 0.26, 0.32, we investigated the effects of the transition rate between asymptomatic infected and reported symptomatic infected on (a) asymptomatic infection, (b) unreported symptomatic infection, and (c) reported symptomatic infection. It can be seen that the dynamics of the model for such states becomes flatter as the value of γ increases. It is an important factor in disease control (Figure 4).

Figure 4: The behavior of the transition rate between asymptomatically infected and reported symptomatically infected on individuals in (a) asymptomatically infected individuals, (b) unreported symptomatic infected individuals, and (c) reported symptomatic infected individuals using γ = 0.1142, 0.26, 0.32 at ν = 0.85.
Figure 4:

The behavior of the transition rate between asymptomatically infected and reported symptomatically infected on individuals in (a) asymptomatically infected individuals, (b) unreported symptomatic infected individuals, and (c) reported symptomatic infected individuals using γ = 0.1142, 0.26, 0.32 at ν = 0.85.

4.4 Average time η

Using η = 0.002, 0.005, 0.1424 at ν = 0.85, we investigated the impacts of parameter (average time infected people have symptoms) on (a) unreported symptomatic infected people, (b) reported symptomatic infected people, and (c) recovered people in Figure 5. Figure 5(a) and (b) indicate that as the value of η decreases, the number of unreported and reported infected people increases significantly (b). In contrast, as the value of η increases more and larger, the dynamics of recovered people increase rapidly and become stable, as shown in Figure 5(c).

Figure 5: Using η = 0.002, 0.005, 0.1424 at ν = 0.85, the behavior of the average time symptomatic infectious have symptoms on (a) unreported symptomatic infected people, (b) reported symptomatic infected people, and (c) recovered people.
Figure 5:

Using η = 0.002, 0.005, 0.1424 at ν = 0.85, the behavior of the average time symptomatic infectious have symptoms on (a) unreported symptomatic infected people, (b) reported symptomatic infected people, and (c) recovered people.

We can see from the numbers that the given model has a high level of flexibility and is extremely dependent on the order. Furthermore, the fractional technique produces more interesting findings than the integer-order model and allows for better analysis of the outcomes. In comparison to the real results, the graphical simulations between ν = 0.75 and ν = 1 are more accurate. The presented classes illustrate a suitable nature between these values. The parameter has a significant impact on calculations. More modifications in the graphical calculations can be seen in future models with different values of η.

5 Discussions

The WHO situation report (the National Health Commission of the Republic of China) presented in [24, 25] was used to obtain the values of parameters and initial populations used in this study.

The results interestingly provide us with more understanding and identification of the main critical parameters of the model. It can be deduced that the transmission parameters between asymptomatic infected, both reported and unreported symptomatic infected, and the average time that infected people who have symptoms are extremely sensitive parameters regarding the variables of the model in computational simulations.

Unreported symptomatic death rate reported symptomatic death rate, and the transition rate between susceptible and asymptomatic infected people, on the other hand, are less sensitive parameters to the model dynamics. Therefore, using numerical simulations to identify the critical model parameters in this study is an effective way to further study the model both practically and theoretically, as well as provide some suggestions for future improvements to novel coronavirus vaccination programs, interventions, and disease control. On the basis of the suggested approaches for the updated COVID-19 model, some consequences and computational results are provided and summarized in the following points:

  1. The predictor–corrector PECE method of Adams–Bashforth–Moulton type was used to compute the model dynamics of susceptible, asymptomatic infected, both reported and unreported symptomatic individuals, and recovered individuals. The dynamical relationships are the same for reported and unreported symptomatic situations; however, the model dynamics for susceptible and recovered groups are slightly different.

  2. It is discussed how the transition rate affects asymptomatic infected, reported symptomatic infected, and unreported symptomatic infected people. In the dynamics of I, U, and W, this parameter plays an effective role.

  3. Asymptomatic infected people, reported symptomatic infected people, and unreported symptomatic infected people have also been affected by the transition rate. It is clear to observe that when the value of γ is increased, the model dynamics for such states become more flat. This is an important factor in order to control the disease.

  4. When the value of η decreases, the number of both unreported and reported infected people significantly increases, while when the value of η increases, the dynamics of recovered people significantly increase and become stable very quickly.

6 Conclusions

Results based on numerical simulations show that for various main model parameters, the model dynamics are significantly modified. We interestingly acknowledge that transfer rates between asymptomatic infected people, reported symptomatic infected people, and unreported symptomatic infected people are extremely critical parameters for the model variables in the transmission process of this disease. This supports international efforts to diminish the number of people affected by COVID-19 and prevent this disease from spreading more widely. This paper has another novelty, which is the identification of the critical model parameters that makes it easy to be used and improved theoretically and practically by biologists with less knowledge of mathematical modeling.

In this paper, with the Caputo fractional derivative and the basic functional response, we introduced and analyzed a new fractional order SIUWR epidemic model covering different types of incidence rates that exist in the literature. We have identified the nature and boundaries of nonnegative solutions. We have demonstrated the local and global stability of the disease-free equilibrium when R 0 < 1, indicating the disease’s extinction, after measuring the balance of our model. However, the disease-free balance becomes unstable when R 0 > 1, and that system (2.3) has an endemic equilibrium that is asymptotically stable globally. In this situation, the disease continues to remain present in the population.

In this approach, for numerical simulations, we stated different graphical results for the solution of the epidemic model at different values of ν. This study illustrates the implementations of the proposed method and the fractional operator considered, thus evaluating and forecasting the subsequent effects of actual word problems and comprehension.

Therefore, more suggestions to control COVID-19 disease can be made on the basis of the impact of each involved parameter on the model states. For any interventions and vaccine programs, that would be beneficial. Therefore, in order to manage this disease more efficiently, healthcare populations should pay more attention to quarantine sites. It can be strongly suggested that everyone should be isolated from others in quarantine locations and only use their distinct facilities, bedroom and toilet, in order to prevent the virus from spreading by touching shared surfaces. In addition, minimizing the interaction between asymptomatic–symptomatic people and susceptible people efficiently reduces the number of infected people. It appears appropriate to prepare a definite solution earlier rather than later to position the asymptomatic infectious individuals in quarantine sites. Future research on the identification of key critical elements could broaden the current COVID-19 explanations more broadly. It will be necessary that further suggested transmissions between the model groups are investigated by future research. For instance, by adding two transmission routes, the model can further develop, one of which is between unreported symptomatic infected and reported symptomatic infected and the other is between asymptomatic infected and recovered people.

The results of this research represent a significant step forward in forecasting model dynamics for the development of programs, interventions, and health care strategies in the future. From the cited figures, we can observe that more versatility depends greatly on the order and movements of the given model. In addition, the fractional model produces more interesting results than the integer-order model and allows for better analysis of the obtained results.

Data availability

The authors confirm that the data supporting the findings of this study are taken from [24, 25].

Corresponding author: Ahmad Alalyani, Department of Mathematics, Faculty of Science and Arts in Almandaq, Al-Baha University, Al-Baha, Saudi Arabia, E-mail:

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: The authors received no specific funding for this study.

  3. Conflict of interest statement: The authors declare that they have no conflicts of interest regarding the publication of this paper.

References

[1] S. Perlman and J. Netland, “Coronaviruses post-SARS: update on replication and pathogenesis,” Nat. Rev. Microbiol., vol. 7, pp. 439–450, 2009. https://doi.org/10.1038/nrmicro2147.Search in Google Scholar PubMed PubMed Central

[2] J. Cui, F. Li, and Z. L. Shi, “Origin and evolution of pathogenic coronaviruses,” Nat. Rev. Microbiol., vol. 17, pp. 181–192, 2019. https://doi.org/10.1038/s41579-018-0118-9.Search in Google Scholar PubMed PubMed Central

[3] K. Sarkar, S. Khajanchi, and J. J. Nieto, “Modeling and forecasting the COVID-19 pandemic in India,” Chaos, Solit. Fractals, vol. 139, p. 110049, 2020. https://doi.org/10.1016/j.chaos.2020.110049-DOI-PMC-PubMed.Search in Google Scholar

[4] S. Khajanchi and K. Sarkar, “Forecasting the daily and cumulative number of cases for the COVID-19 pandemic in India,” Chaos, vol. 30, no. 7, 2020, Art no. 071101. https://doi.org/10.1063/5.0016240.Search in Google Scholar PubMed PubMed Central

[5] P. Samui, J. Mondal, and S. Khajanchi, “A mathematical model for COVID-19 transmission dynamics with a case study of India,” Chaos, Solit. Fractals, vol. 140, p. 110173, 2020. https://doi.org/10.1016/j.chaos.2020.110173.Search in Google Scholar PubMed PubMed Central

[6] R. K. Rai, S. Khajanchi, P. K. Tiwari, et al.., “Impact of social media advertisements on the transmission dynamics of COVID-19 pandemic in India,” J. Appl. Math. Comput., vol. 68, pp. 19–44, 2021. https://doi.org/10.1007/s12190-021-01507-y.Search in Google Scholar PubMed PubMed Central

[7] S. Khajanchi, K. Sarkar, and J. Mondal, Dynamics of the COVID-19 Pandemic in India, 2021, arXiv:2005.06286v2.10.21203/rs.3.rs-27112/v1Search in Google Scholar

[8] S. Khajanchi, S. Bera, and T. K. Roy, “Mathematical analysis of the global dynamics of a HTLV-I infection model, considering the role of cytotoxic T-lymphocytes,” Math. Comput. Simulat., vol. 180, pp. 354–378, 2021.10.1016/j.matcom.2020.09.009Search in Google Scholar

[9] S. Khajanchi, K. Sarkar, J. Mondal, K. Sooppy Nisar, and S. F. Abdelwahab, “Mathematical modeling of the COVID-19 pandemic with intervention strategies,” Results Phys., vol. 25, p. 104285, 2021. https://doi.org/10.1016/j.rinp.2021.104285.Search in Google Scholar PubMed PubMed Central

[10] J. S. Kahn and K. McIntosh, “History and recent advances in coronavirus discovery,” Pediatr. Infect. Dis. J., vol. 24, no. 11 Suppl, pp. S223–S227, 2005. https://doi.org/10.1097/01.inf.0000188166.17324.60.Search in Google Scholar PubMed

[11] World Health Organization, Coronavirus, World Health Organization, 2020, cited January 19. Available from: https://www.who.int/health-topics/coronavirus.Search in Google Scholar

[12] A. A. Algaissi, N. K. Alharbi, M. Hassanain, and A. M. Hashem, “Preparedness and response to COVID-19 in Saudi Arabia: building on MERS experience,” J. Infect. Public Health, vol. 13, pp. 834–838, 2020. https://doi.org/10.1016/j.jiph.2020.04.016.Search in Google Scholar PubMed PubMed Central

[13] R. M. Anderson, H. Heesterbeek, D. Klinkenberg, and T. D. Hollingsworth, “How will country-based mitigation measures influence the course of the COVID-19 epidemic?” Lancet (London, England), vol. 395, no. 10228, pp. 931–934, 2020. https://doi.org/10.1016/s0140-6736(20)30567-5.Search in Google Scholar

[14] W. Aletreby, A. Alharthy, F. Faqihi, et al.., “Dynamics of SARS-CoV-2 outbreak in the Kingdom of Saudi Arabia: a predictive model,” Saudi Crit. Care J., vol. 4, pp. 79–83, 2020. https://doi.org/10.4103/sccj.sccj_19_20.Search in Google Scholar

[15] M. Nouri-Vaskeh and L. Alizadeh, “Fecal transmission in COVID-19: a potential shedding route,” J. Med. Virol., vol. 92, pp. 1731–1732, 2020. https://doi.org/10.1002/jmv.25816.Search in Google Scholar PubMed PubMed Central

[16] K. Linka, M. Peirlinck, F. Sahli Costabal, and E. Kuhl, “Outbreak dynamics of COVID-19 in Europe and the effect of travel restrictions,” Comput. Methods Biomech. Biomed. Eng., vol. 11, pp. 710–717, 2020. https://doi.org/10.1080/10255842.2020.1759560.Search in Google Scholar PubMed PubMed Central

[17] M. Dur-e-Ahmad and M. Imran, “Transmission dynamics model of coronavirus COVID-19 for the outbreak in most affected countries of the world,” Int. J. Interact. Multimed. Artif. Intell., vol. 6, pp. 7–10, 2020. https://doi.org/10.9781/ijimai.2020.04.001.Search in Google Scholar

[18] Y. G. Sanchez, Z. Sabir, and J. L. G. Guirao, “Design of a nonlinear SITR fractal model based on the dynamics of a novel coronavirus (COVID-19),” Fractals, vol. 28, no. 8, p. 2040026, 2020. https://doi.org/10.1142/s0218348x20400265.Search in Google Scholar

[19] Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, R. Ren, K. Leung, E. Lau, J. Y. Wong, X. Xing, N. Xiang, Y. Wu, C. Li, Q. Chen, D. Li, T. Liu, J. Zhao, M. Li, and Z. Feng, “Early transmission dynamics in Wuhan, China, of novel coronavirus-infected Pneumonia,” N. Engl. J. Med., vol. 382, no. 13, pp. 1199–1207, 2020. https://doi.org/10.1056/nejmoa2001316.Search in Google Scholar

[20] C. Huang, Y. Wang, X. Li, L. Ren, J. Zhao, Y. Hu, et al.., “Clinical features of patients infected with 2019 novel coronavirus in Wuhan, China,” Lancet, vol. 395, no. 10223, pp. 497–506, 2020. https://doi.org/10.1016/s0140-6736(20)30183-5.Search in Google Scholar

[21] J. F. Chan, S. Yuan, K. H. Kok, K. K. To, H. Chu, J. Yang, et al.., “A familial cluster of pneumonia associated with the 2019 novel coronavirus indicating personto- person transmission: a study of a family cluster,” Lancet, vol. 395, no. 10223, pp. 514–523, 2020. https://doi.org/10.1016/s0140-6736(20)30154-9.Search in Google Scholar PubMed PubMed Central

[22] N. Zhu, D. Zhang, W. Wang, X. Li, B. Yang, J. Song, et al.., “A novel coronavirus from patients with pneumonia in China, 2019,” N. Engl. J. Med., vol. 382, no. 8, pp. 727–733, 2020. https://doi.org/10.1056/nejmoa2001017.Search in Google Scholar PubMed PubMed Central

[23] S. Zhao, S. S. Musa, Q. Lin, J. Ran, G. Yang, W. Wang, et al.., “Estimating the unreported number of novel coronavirus (2019-nCoV) cases in China in the first half of January 2020: a data-driven Modelling analysis of the early outbreak,” J. Clin. Med., vol. 9, no. 2, p. 388, 2020. https://doi.org/10.3390/jcm9020388.Search in Google Scholar PubMed PubMed Central

[24] Z. Liu, P. Magal, O. Seydi, and G. Webb, “Understanding unreported cases in the COVID-19 epidemic outbreak in Wuhan, China, and the importance of major public health interventions,” Biology, vol. 9, no. 3, p. 50, 2020. https://doi.org/10.3390/biology9030050.Search in Google Scholar PubMed PubMed Central

[25] S. H. Khoshnaw, M. Shahzad, M. Ali, and F. Sultan, “A quantitative and qualitative analysis of the COVID-19 pandemic model,” Chaos, Solit. Fractals, vol. 138, p. 109932, 2020. https://doi.org/10.1016/j.chaos.2020.109932.Search in Google Scholar PubMed PubMed Central

[26] W. Gao, P. Veeresha, H. M. Baskonus, D. Prakasha, and P. Kumar, “A new study of unreported cases of 2019-nCoV epidemic outbreaks,” Chaos, Solit. Fractals, vol. 138, p. 109929, 2020. https://doi.org/10.1016/j.chaos.2020.109929.Search in Google Scholar PubMed PubMed Central

[27] T. Kuniya, “Prediction of the epidemic peak of coronavirus disease in Japan,” J. Clin. Med., vol. 9, no. 3, p. 789, 2020 https://doi.org/10.3390/jcm9030789.Search in Google Scholar PubMed PubMed Central

[28] De Falco, I., Della Cioppa, A., Scafuri, U., Tarantino, E., “Coronavirus Covid-19 spreading in Italy: optimizing an epidemiological model with dynamic social distancing through differential evolution,” 2020 arXiv:2004.00553v3. https://doi.org/10.48550/arXiv.2004.00553.Search in Google Scholar

[29] M. A. Khan and A. Atangana, “Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative,” Alex. Eng. J., vol. 59, no. 4, pp. 2379–2389, 2020. https://doi.org/10.1016/j.aej.2020.02.033.Search in Google Scholar

[30] BogochII, A. Watts, A. Thomas-Bachli, C. Huber, M. U. G. Kraemer, and K. Khan, “Pneumonia of unknown aetiology in Wuhan, China: potential for international spread via commercial air travel,” J. Trav. Med., vol. 27, no. 2, p. taaa008, 2020. https://doi.org/10.1093/jtm/taaa008 [PMID: 31943059; PMCID: PMC7107534].Search in Google Scholar PubMed PubMed Central

[31] J. T. Wu, K. Leung, and G. M. Leung, “Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study,” Lancet, vol. 395, no. 10225, pp. 689–697, 2020. https://doi.org/10.1016/S0140-6736(20)30260-9 [Epub 2020 Jan 31. Erratum in: Lancet. PMID: 32014114; PMCID: PMC7159271].Search in Google Scholar PubMed PubMed Central

[32] T. M. Chen, J. Rui, Q. P. Wang, et al.., “A mathematical model for simulating the phase-based transmissibility of a novel coronavirus,” Infect. Dis. Poverty, vol. 9, p. 24, 2020. https://doi.org/10.1186/s40249-020-00640-3.Search in Google Scholar PubMed PubMed Central

[33] S. Zhao, Q. Lin, J. Ran, S. S. Musa, G. Yang, W. Wang, Y. Lou, D. Gao, L. Yang, D. He, and M. H. Wang, “Preliminary estimation of the basic reproduction number of novel coronavirus (2019-nCoV) in China, from 2019 to 2020: a data-driven analysis in the early phase of the outbreak,” Int. J. Infect. Dis., vol. 92, pp. 214–217, 2020. https://doi.org/10.1016/j.ijid.2020.01.050.Search in Google Scholar PubMed PubMed Central

[34] K. Liang, “Mathematical model of infection kinetics and its analysis for COVID-19, SARS and MERS,” Infect. Genet. Evol., vol. 82, p. 104306, 2020. https://doi.org/10.1016/j.meegid.2020.104306.Search in Google Scholar PubMed PubMed Central

[35] M. H. Alshehri, F. Z. Duraihem, A. Ahmad, and S. Saber, “A Caputo (discretization) fractional-order model of glucose-insulin interaction: numerical solution and comparisons with experimental data,” J. Taibah Univ. Sci., vol. 15, no. 1, pp. 26–36, 2021. https://doi.org/10.1080/16583655.2021.1872197.Search in Google Scholar

[36] V. S. Erturka and P. Kumar, “Solution of a COVID-19 model via new generalized Caputo-type fractional derivatives,” Chaos, Solit. Fractals, vol. 139, p. 110280, 2020. https://doi.org/10.1016/j.chaos.2020.110280.Search in Google Scholar PubMed PubMed Central

[37] Wikipedia website, Wikipedia, 2020, [Google Scholar]. Available from: https://en.wikipedia.org/wiki/coronavirus [Accessed 04 07, 2020].Search in Google Scholar

[38] I. Podlubny, Fractional Differential Equations, New York, NY, USA, Academic Press, 1999.Search in Google Scholar

[39] W. Lin, “Global existence theory and chaos control of fractional differential equations,” J. Math. Anal. Appl., vol. 332, pp. 709–726, 2007. https://doi.org/10.1016/j.jmaa.2006.10.040.Search in Google Scholar

[40] A. Boukhouima, K. Hattaf, and N. Yousfi, “Dynamics of a fractional order HIV infection model with specific functional response and cure rate,” Int. J. Differ. Equ., vol. 2017, Art no. 8372140. https://doi.org/10.1155/2017/8372140.Search in Google Scholar

[41] S. K. Choi, B. Kang, and N. Koo, “Stability for Caputo fractional differential systems,” Abstr. Appl. Anal., vol. 2014, 2014, Art no. 631419. https://doi.org/10.1155/2014/631419 .Search in Google Scholar

[42] A. Hurwitz, “On the conditions under which an equation has only roots with negative real parts,” in Rpt. in Selected Papers on Mathematical Trends in Control Theory, R. T. Ballman, Ed., New York, Dover, 1964.Search in Google Scholar

[43] J. P. LaSalle, The Stability of Dynamics Systems, Philadelphia: PA, SIAM, 1976.Search in Google Scholar
Received: 2021-01-30
Revised: 2021-09-16
Accepted: 2022-04-07
Published Online: 2022-05-30

© 2022 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
  4. Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
  5. Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
  6. A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
  7. Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
  8. Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
  9. Elastic trend filtering
  10. A new approach to representations of homothetic motions in Lorentz space
  11. Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
  12. Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
  13. Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
  14. N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
  15. Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
  16. Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
  17. Singularity analysis and analytic solutions for the Benney–Gjevik equations
  18. Trajectory controllability of nonlinear fractional Langevin systems
  19. Similarity transformations for modified shallow water equations with density dependence on the average temperature
  20. Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
  21. Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
  22. Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
  23. Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
  24. A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
https://doi.org/10.1515/ijnsns-2021-0042
Keywords for this article
COVID-19; mathematical modeling; the predictor–corrector PECE method of Adams–Bashforth–Moulton type

Articles in the same Issue

  1. Frontmatter
  2. Original Research Articles
  3. One-dimensional optimal system and similarity transformations for the 3 + 1 Kudryashov–Sinelshchikov equation
  4. Hopf bifurcations in a network of FitzHugh–Nagumo biological neurons
  5. Gravity modulation effect on weakly nonlinear thermal convection in a fluid layer bounded by rigid boundaries
  6. A new numerical formulation for the generalized time-fractional Benjamin Bona Mohany Burgers’ equation
  7. Chebyshev spectral method for solving a class of local and nonlocal elliptic boundary value problems
  8. Adaptive extragradient methods for solving variational inequalities in real Hilbert spaces
  9. Elastic trend filtering
  10. A new approach to representations of homothetic motions in Lorentz space
  11. Numerical simulation of variable-order fractional differential equation of nonlinear Lane–Emden type appearing in astrophysics
  12. Stability analysis and numerical simulations of the fractional COVID-19 pandemic model
  13. Numerical solutions of the Bagley–Torvik equation by using generalized functions with fractional powers of Laguerre polynomials
  14. N-fold Darboux transformation and exact solutions for the nonlocal Fokas–Lenells equation on the vanishing and plane wave backgrounds
  15. Closed form soliton solutions to the space-time fractional foam drainage equation and coupled mKdV evolution equations
  16. Master-slave synchronization in the Duffing-van der Pol and Φ6 Duffing oscillators
  17. Singularity analysis and analytic solutions for the Benney–Gjevik equations
  18. Trajectory controllability of nonlinear fractional Langevin systems
  19. Similarity transformations for modified shallow water equations with density dependence on the average temperature
  20. Non-equidistant partition predictor–corrector method for fractional differential equations with exponential memory
  21. Dynamical analysis of fractional-order of IVGTT glucose–insulin interaction
  22. Dynamic response of Mathieu–Duffing oscillator with Caputo derivative
  23. Non-Newtonian fluid flow having fluid–particle interaction through a porous zone in a channel with permeable walls
  24. A class of computationally efficient Newton-like methods with frozen inverse operator for nonlinear systems
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