key: cord-0731456-k0lw1r3z
authors: Akindeinde, S. O.; Okyere, Eric; Adewumi, A. O.; Lebelo, R. S.; Fabelurin, Olanrewaju. O.; Moore, Stephen. E.
title: Caputo Fractional-order SEIRP model for COVID-19 epidemic
date: 2021-05-04
journal: nan
DOI: 10.1016/j.aej.2021.04.097
sha: b9ef834f2b63937a2368a09265ef0732eba34fcd
doc_id: 731456
cord_uid: k0lw1r3z

We propose a Caputo-based fractional compartmental model for the dynamics of the novel COVID-19 transmission dynamics. The newly proposed nonlinear fractional order differential equation epidemic model is an extension a recently formulated integer-order COVID-19 mathematical model. Using basic concepts such as continuity and Banach fixed-point theorem, the existence and uniqueness of the solution to the proposed model were shown. Furthermore, we analyze the stability of the model in the context of Ulam-Hyers and generalized Ulam-Hyers stability criteria. The concept of next-generation matrices was used to compute the basic reproduction number R 0 , a number that determines the spread or otherwise of the disease into the general population. We also investigated the local asymptotic stability for the derived disease-free equilibrium point. Numerical simulation of the constructed epidemic model was carried out using the fractional Adam-Bashforth-Moulton method to validate the obtained theoretical results.

In the first quarter of 2020, the World Health Organization (WHO) declared COVID-19 as a pandemic that had affected several countries in all continents, see [1] . The novel coronavirus has several mutations that have occurred in many countries with varied symptoms. Some of the most common symptoms are fever, dry cough, and tiredness whiles other less common symptoms include body aches and pains, sore throat, diarrhea, loss of taste or smell or a rash on the skin [1, 2] . It is well-known that COVID-19 is transmitted by means of either direct or indirect contact, droplet spray such as sneezing in short-range transmission and airborne transmission such as aerosol in long-range transmission [3] .

Epidemiological modeling of infectious disease using differential equations with integer-order to examine and investigate the transmission dynamics of epidemics is widely studied in literature [4] [5] [6] [7] [8] . Mathematical modeling of epidemics in literature reveals that nonlinear dynamical equations can give some important insight into the transmission dynamics or dynamical behaviors of disease spread. The recent COVID-19 outbreaks around the world have attracted a lot of interest in the mathematical modeling of this highly contagious disease by constructing realistic nonlinear compartmental mathematical models that are driven by data to better understand the transmission dynamics of the epidemics [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] . A new eight-compartmental deterministic model for the COVID-19 epidemic that captures awareness campaign programs and hospitalization control strategies for both severe and mild cases of infections in Nigeria is proposed in a recent study by Musa et al. [22] .

They fitted the proposed nonlinear dynamical model to the infected cumulative cases in Nigeria from 29 March to 12 June 2020. Their results reveal that, if awareness campaigns are not properly and effectively implemented, there could be an upward trend of the infections in the population.

Memon et al. in [23] formulated and analyzed a SEQIJR epidemic model that assesses the role of quarantine and isolation as control measures in preventing or minimizing the spread of COVID-19 outbreaks in Pakistan.

Fractional order differential equations are very useful and powerful mathematical modeling tools to study biological and engineering systems. This is because differential operators in these types of equations or models are related to systems with memory dynamics which naturally exists in most biological and engineering systems, see [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] . In [38] , a new efficient and novel numerical method based on hybrid Chelyshkov functions is constructed to solve some fractional order models that are driven by time-dependent control functions. Rezapour and his co-authors in [39] utilized the Caputo-Fabrizio derivative operator in fractional calculus and extended the nonlinear integer-order anthrax disease model developed and analyzed by Githire et al [40] . Using the Picard-Lindelof method, they provided an existence criterion of solutions for their proposed fractional order anthrax disease epidemic model. Differential equations that are characterized by Caputo fractional order derivative have been used to study and analysed several infectious diseases transmission dynamics such as HIV/AIDS [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] , TB [52] [53] [54] , Malaria [55] [56] [57] , Dengue fever [58] [59] [60] [61] , Zika [62, 63] , Ebola [64] [65] [66] and Hepatitis B [67] [68] [69] [70] [71] [72] . In [73, 74] , backward bifurcation dynamics is explored in a basic and realistic vaccination epidemic model. Therein, the author considered rigorous qualitative analysis of the formulated mathematical model and also presented numerical simulations to confirm the obtained theoretical results. Javidi et al. [75] extended and studied the cholera epidemic mathematical model formulated by the author in [76] to capture Caputo fractional order derivative. The author in [77] used Lyapunov functions that are of Volterra-type and investigated uniform asymptotic stability of some basic epidemic models (SIS, SIR, SIRS) and the well-known Ross vector-borne diseases in Caputo sense. A 3-compartmental deterministic modeling structure (normal weight-overweightobese) is constructed to explore obesity epidemic dynamics in a variable population using Caputo fractional order nonlinear system [78] . The basic SEIR mathematical model driven by Caputo fractional derivative based on the assumption of variable population dynamics is studied in [79] . The authors gave a detailed qualitative stability analysis for their new and realistic deterministic model.

In a recent epidemiological model formulation for Zika virus infection, the authors in [80] utilized a nonlinear system of differential equations in the sense of Caputo fractional order derivative operator of which the total human and mosquitoes populations were each grouped into two compartmental classes (susceptible people, infected people; susceptible mosquitoes, infected mosquitoes). In [81] , the authors developed and studied SEIR-type epidemic model using both classical and Caputo fractional order differential operators to describe the transmission dynamics of Rubella epidemic in Pakistan.

Due to the powerful nature of fractional order differential equations and their derivative operators used in constructing realistic mathematical representations of real-world problems in science, finance, and engineering [82] [83] [84] [85] [86] [87] [88] , some recent studies have considered the mathematical modeling of this deadly COVID-19 pandemic using some of these useful derivative operators. Naik et al. [89] provided a detailed qualitative analysis as well as parameter estimation and numerical simulations for a nonlinear COVID-19 epidemiological model constructed with Caputo and Atangana-Baleanu fractional derivative operators. In another study [90] , the authors formulated an autonomous nonlinear Atangana-Baleanu fractional order differential equation model to study the COVID-19 epidemic in Nigeria. Baleanu et al. [91] in their new study considered an extended version of the integer-order epidemic model proposed and analyzed by Chen et al. [92] to include Caputo-Fabrizio derivative.

With the use of fixed point theory, they proved that the nonlinear Caputo-Fabrizio fractional order COVID-19 model has a unique solution. They also applied the homotopy analysis transform method and generated the approximate solution for the model problem in convergent series. A Ca-puto fractional order deterministic epidemic model for COVID-19 infection is developed and studied in [93] . They used the well-known Banach contraction mapping principle to establish the existence and uniqueness of the solution for the mathematical model. In a new mathematical modeling study [94] , the authors proposed a SEIQRDP fractional order deterministic model characterized by Caputo derivative to examine the novel coronavirus epidemic. A SEIPAHRF Caputo fractional order compartmental model is proposed to analyze COVID-19 transmission dynamics in Wuhan [95] .

They numerically solved their proposed model and compared it with the initial 67 days reported data on confirmed infected and death cases in Wuhan city. In recent studies, the authors in [96] proposed a Caputo-type nonlinear COVID-19 epidemiological model to explore the significance of lockdown dynamics in controlling the spread of the infectious disease. They further presented the uniqueness and existence of solutions for the mathematical model under lockdown using Banach and Schauder fixed theorems. In [97] , a susceptible-exposed-infected-isolated-recovered compartmental modeling framework with constant total population dynamics is used to construct a new SEIQR Caputo fractional order COVID-19 mathematical model. Owusu-Mensah and co-authors [98] , have also studied a new COVID-19 epidemic model using Caputo derivative in fractional calculus. The well-known and powerful generalized form of Adams Bashforth-Moulton iterative scheme was applied to numerically solve their formulated nonlinear fractional order differential equation model.

In another study, the authors in [99] used an extended version of the classical SEIR epidemic model to construct a data-driven Caputo-based COVID-19 fractional order mathematical model.

Our motivation for this present study is based on the aforementioned literature on the application of fractional order differential equations in modelling nonlinear systems related to real-world problems and more especially infectious diseases' transmission dynamics. Recent research works and studies in literature extensively demonstrate or reveals that mathematical modelling of nonlinear systems with fractional order differential operators gives more realistic results than the classical integer-order based models, see, e.g, [81, 89, [100] [101] [102] . We therefore apply the well-known and reliable Caputo derivative operator in fractional calculus to extend an existing COVID-19 epidemic model [103] characterized by the classical integer order derivative. The goal of the present paper is two folds, first, we want to establish both the mathematical and epidemiological well-posedness of the integer-order model proposed in [103] and employ an approximate analytical technique to obtain long-term dynamics of the disease. Second, we modify and extend the existing epidemic model using dimensionally consistent Caputo derivative operator which has been extensively demonstrated in the literature to be one of the useful and powerful derivative operators to describe more efficiently memory effect dynamics that exist in real-world phenomena. It is important to mention that, our study is further motivated by the recent research works conducted by the authors in [104] [105] [106] .

The paper is organized as follows. Following the essential preliminaries on fractional calculus in Section 2, we examine in Section 3 both mathematical and epidemiological well-posedness of the integer-order model in [103] , and obtain disease dynamics using the approximate analytical technique proposed in [107, 108] . Section 4 of the paper is devoted to the formulation, well-posedness, local and global stability analysis of the fractional order model. Therein, using basic concepts such as continuity and Banach fixed-point theorem, the existence and uniqueness of the solution to the proposed model were shown. Furthermore, we discussed stability analysis of the model in the context of Ulam-Hyers and generalized Ulam-Hyers stability criteria. Also, the concept of next-generation matrices was used to compute the basic reproduction number R 0 , a number that determines the spread or otherwise of the disease into the general population. We conclude the paper with numerical simulation of the fractional order disease model using fractional Adam-Bashforth-Moulton; and discussion of results.

In this section, we introduce well-known definitions and Lemma in fractional calculus that are relevant to the current article. The interested reader can see the monograph [109] and the article [110] for the proofs and further references.

The fractional order integral of the function g ∈ L 1 ([0, b], R + ) of order α ∈ R + is defined the sense of Riemann-Liouville as 

where n = v + 1 and v denotes the smallest integer that is less or equal to v. 

In particular, if 0 < v ≤ 1, then

Lemma 2.4. [110] (Generalized Mean Value Theorem) For 0 < v ≤ 1, let g(t) ∈ C([a, b]) and

D v g(t) ∈ (a, b]. Then it holds

It is well known that the ravaging Covid-19 virus is transmitted through human-to-human interactions and transmission through the environment. When an infected person sneezes or coughs, the virus is released into the immediate environment which, experts believe, stays viable for as long as five days. We, therefore, consider two interacting populations of humans and pathogens denoted by N (t) and P (t) respectively. At any time t, the total population N (t) is assumed to comprise the susceptible population S(t), the exposed E(t), the asymptomatic infected I A (t), the symptomatic infected I S (t) and the recovered population R(t). Thus, N (t) = S(t) + I A (t) + I S (t) + R(t). These interactions are depicted in Figure 1 below as reported in [103] . In the figure, b denotes the rate at which the human population is born into susceptible class S(t). The terms β 1 SP 1+α 1 P and β 2 S(I A +I S )

represent rate at which the susceptible get infected by pathogens and through interactions with infectious asymptomatic I A (t) and infectious symptomatic I S (t). The denominators in the above terms factor in the adherence to recent experts' advice on social distancing and the use of face masks which minimizes contact with infectious individuals, and transmission through the environment.

Based on the above description, the authors in [103] proposed the following integer-order differential equation to model the dynamics of the transmission of COVID-19 taking into account environment and social distancing, namely properties allows us to extend the study in [103] to fractional order model, which accommodate the memory effect typically associated with epidemic disease models. In addition, we shall obtain approximate analytical solutions of the model via a multistage method proposed in [107] , [108] .

For the fact that (2) models human populations, all the model parameters are assumed nonnegative, see Table 1 for values and description of the model parameters. It then remains to show that unique solution exists for the model and that the state variables remain bounded and nonnegative for all time t > 0. 7   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 Proof. By defining y(t) = (S(t), E(t), I A (t), I S (t), R(t), P (t)) T , then (2) can be written as

By assumption the initial condi-

Thus computing and examining the entries of the Jacobian J(F (y)), e.g.

reveal that both the right hand side of (2) namely F and its Jacobian are continuous for t > 0. 8   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 Thus, F satisfies a Lipschitz condition on R 6 + . The existence and uniqueness of solution for some time interval (0, t f ) follows from Picard-Lindelof Theorem. (2) with non-negative initial data remains non-negative for all t > 0.

Proof. Let S(0) = S 0 > 0. It follows from the first equation of (2) that

and upon integration, one obtains

dτ.

and as a consequence, non-negativity of the remaining state variables are obtained. 

Proof. Using the fact that 2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 Thus considering the initial valued problem N (t) = b − µN, N (0) = N 0 , and invoking comparison theorem [111] , it follows that

For the pathogen population, using the fact that I A + I S ≤ N ≤ b/µ, the last equation of (2) yields

Here, we shall adopt the multistage technique proposed in [107] , [108] to compute approximate analytical solution to the integer-order model (2) . For that purpose, let us define U = 1 1+α 1 P and W = 1 1+α 2 (I A +I S ) so that (2) is transformed to an equivalent polynomial system

which is now amenable to the proposed technique. By writing S(t) = ∞ n=0 S n t n , it follows from the above polynomial system that the coefficients S n are obtained recursively through 3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 In a similar manner, approximate analytical solutions of the remaining compartments are computed.

Such series solution often have small interval of convergence. Therefore to improve convergence, we choose a safe step length h, (see [107] and [108] ) and compute a piecewise continuous approximate solution which is convergent in the entire integration interval. The dynamics of the disease progression for each sub-population are displayed in Figures 2a-3b and Figures 4a-4b below, confirming the results in [103] . 3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64 

Fractional derivatives are generally believed to model disease epidemics more realistically because of their capability to capture the memory effect often associated with the human body's 12   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 response to diseases. Here, we propose a fractional order variant of (2) given by

In the above, operator D α denotes Caputo fractional derivative of order 0 < α ≤

Noting that all the model parameters except α 1 , α 2 and δ have dimensions 1/t α , we have raised these parameters to power of α for dimensional consistency emphasized by [59] .

In this section, we examine the mathematical and biological well-posedness of the fractional order model. In essence, we prove that solution of the fractional model is bounded and remains positive as long as a positive initial condition is given. Furthermore, we prove the existence and uniqueness of the solution to the modified model.

Let X(t) = (S, E, I A , I S , R, P ) T and K(t, X(t)) = (φ i ) T , i = 1, 2, . . . 6 where

Then the dynamical system (5) can be written as

13 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 In the above, the condition X(0) ≥ 0 is to be interpreted component-wise. Problem (6) , which is equivalent to fractional differential equation (5), in turn has integral representation X(t) = X 0 + J α 0 + K(t, X(t)),

Next, we shall analyse model (5) 

. Furthermore, we define the operator P : E → E by

Note that operator P is well-defined due to the obvious continuity of K. Then

showing non-negative invariance of the axes. 14 1 2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 Now, since the solution to the model (5) is positive in the E − I A − I S − R − P plane, let t * > 0 such that S(t * ) = 0, E(t * ) > 0, I A (t * ) > 0, I S (t * ) > 0, R(t * ) > 0, P (t * ) > 0 and S(t) < S(t * ). On this plane,

By Caputo fractional mean value theorem (see Lemma 2.4), it holds

. Therefore, using (9), we obtain S(t) > S(t * ), contradicting our earlier assumption for t * . Thus, any solution S(t) is non-negative for all t ≥ 0. The remaining variables can be treated similarly. Hence, solution X(t) remains positive for all t ≥ 0.

Finally for boundedness, proceeding as in the integer order case (see the proof of Theorem 3.3),

Here, we establish existence, uniqueness and uniform stability of solutions to (4) through (5).

The following preliminary result is in order. Proof. From the first component of K, we observe that 3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 However,

Similarly, we obtain,

Altogether, we have

In a similar manner, one obtains

{β α 1 f 1 (t) + β α 1 f 2 (t) + β α 2 g 1 (t) + β α 2 g 2 (t) + β α 1 g 3 (t)}. 16   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 For the remaining components of K, it holds 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64

It then follows that operator P X ⊆ B κ and P is indeed a self-map. It remains at this point to prove that P is a contraction. Let X andX ∈ E satisfy (6). Then again using the result of Lemma

Thus if ΩL K < 1 then P is a contraction mapping, and consequently by the Banach contraction mapping principle, P has a unique fixed point on [0, b] which is solution of (5 

By setting the left hand side of (5) to zero, one obtains the equilbra points. Disease-free equilibrium points are those where I A = I S = 0 and P = 0. These immediately imply E = 0 and 18   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63 64 R = 0. Thus, we obtain b α − µ α S * = 0 or S * = b α /µ α . Consequently, the disease-free equilibrium point of the fractional order model is (S * , E * , I * A , I * S , R * , P * ) = ( b α µ α , 0, 0, 0, 0, 0). The dynamics and stability of a disease model are usually governed by the basic reproduction number R 0 which simply put, measure the average number of secondary infections resulting from introducing one infected individual into an entirely susceptible population. In other words, it determines the spread or otherwise of the infection into the entire population. We follow the recipes described in [115] to compute the basic reproduction number for the fractional order model. The approach is based on next-generation matrices.

The infected subsystem of (5) comprises the second, third, fourth and sixth equations. Therefore, letting x = (E, I A , I S , P ) T , then the infected subsystem can be written in the form

The matrices F and V represent the rate of production of new infections and transition of new infections respectively.

Linearizing the infected subsystem about the disease-free equilibrium point, we obtain the corresponding rate of infection and infection transition rate near the equilibrium, respectively, as

The basic reproduction number is the spectral radius of matrix K = F V −1 , the total production of new infections over the course of outbreak. Using symbolic computation Maple17 software, we 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 As suggested by [115] , the above matrix can be further reduced to a 2 × 2 matrix through a certain transformation matrix E, while still preserving the dominant eigenvalue. It is immediate from

Let us now define the reduced matrix

Consequently, it follows that ρ(K) = ρ(K S ) = 1 2 trace(K S ) + trace(K S ) 2 − 4 det(K S ) . Hence,

Here, we establish local stability of the disease-free equilibrium point E 0 of (5) by examining the nature of the eigenvalues of the linearization matrix of (5), namely the Jacobian J(E 0 ). Proof. The Jacobian of (5) evaluated at the disease-free equilibrum E 0 is given by

The characteristic equation of J(E 0 ) is computed directly as 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 where

with the previously defined parameters c α 1 = ψ α +µ α +ω α , c α 2 = µ α +σ α +γ α S , and c 3 = µ α +σ α +γ α A . Clearly −µ α is a negative eigenvalue with multiplicity two. The remaining eigenvalues are the roots of the equation

Routh-Hurwitz conditions are known to be necessary and sufficient for the roots of this equation to satisfy |arg(λ i )| > απ 2 . However, to simplify notation in the sequel, let us define the discriminant (or resultant) of P (λ) as

Using the results in [116] , we have the following i. If

, then for α = 1, the necessary and sufficient condition for the equilibrium point E 0 to be locally asymptotically stable are ∆ 1 > 0, ∆ 2 > 0, ∆ 3 = 0, B 4 > 0, and the above conditions are sufficient for E 0 to be locally asymptotically stable for all α ∈ [0, 1). 21   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 ii. If D(P ) > 0, B 1 > 0, B 2 < 0 and α > 2 3 , then the equilibrium point E 0 is unstable.

iii. If D(P ) < 0, B 1 > 0, B 2 > 0, B 3 > 0, B 4 > 0 and α < 1 3 , then the equilibrium point E 0 is locally asymptotically stable. Furthermore, if D(P ) < 0,

, then the equilibrium point E 0 is locally asymptotically stable, for all α ∈ (0, 1).

v. B 4 > 0 is the necessary condition for the equilibrium point E 0 to be locally asymptotically stable.

We establish the global stability of the fractional model (6) in the sense of Ulam-Hyers [117] .

The Ulam-Hyers stability has been an active research area since it was first introduced by Ulam For clarity of the discussion that follows, let us introduce the inequality given by

We say a functionX ∈ E is a solution of (10) if and only if there exists h ∈ E satisfying i. |h(t)| ≤ .

ii. D α

It is important to observe that by invoking (7) and property ii. above, simple simplification yields the fact that any functionX ∈ E satisfying (10) also satisfies the integral inequality

Definition 4.5. The fractional order model (6) (and equivalently (5)) is Ulam-Hyers stable if there exists C K > 0 such that for every > 0, and for each solutionX ∈ E satisfying (10), there exists a solution X ∈ E of (6) with 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 Definition 4.6. The fractional order model (6) (and equivalently (5)) is said to be generalized Ulam-Hyers stable if there exists a continuous function φ K : R + → R + with φ K (0) = 0, such that, for each solutionX ∈ E of (10), there exists a solution X ∈ E of (6) such that

We now present our result on the stability of the fractional order model. Proof. Let X be a unique solution of (6) guaranteed by Theorem 4.3;X satisfies (10) . Then recalling the expressions (7), (11) 

This section provides some illustrative numerical simulations to explain the dynamical behavior of the Caputo fractional order deterministic nonlinear COVID-19 mathematical model. The numerical solution of a nonlinear mathematical model using the appropriate iterative scheme is very important in mathematical modeling. For this purpose, we have used the Adams-type predictorcorrector iterative scheme constructed in [118, 119] to solve fractional order differential equations.

The method relies on the equivalent integral formulation of our fractional model (5) written in the form (6) . Consider a uniform discretization of [0, b] given by t n = nh, n = 0, 1, 2, . . . , N where 23   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 0 < h = b/n denote the step size. Now, given any approximation X h (t i ) ≈ X(t i ), we obtain the next approximation X h (t i+1 ) using the Adams-type predictor-corrector iterative scheme as follows;

Predictor :

Corrector :

For varying values of the derivative order α, the dynamics of the fractional order model are presented in Figures 5 below. We have used the model parameters in Table 1 as reported in [103] and initial data S Recall that the constants α 1 and α 2 represent the proportion of interaction with the infectious environment and infectious individuals respectively. Consider a scenario of a contaminated environ-ment α 1 = 0.05 where there is higher chances of contracting COVID-19 through the environment than through infectious individuals I A or I S , (α 1 = 0.1). In this situation, a rapid decline in the susceptible population is noticeable in figure 7a during the first ten days. This is expected as more people get exposed through the virus-laden environment, explaining the rapid rise in the exposed population and the infected population within the same time period, figure 7b, 7c and figure 7d.

In comparison with other scenarios, namely α 2 = 0.05, α 1 = 0.1 where there is a higher risk of contracting COVID-19 virus through contacts with infectious individuals than through the environment; and α 2 = 0.1, α 1 = 0.1 depicting generally low infection rate from both sources, the number of exposed, asymptomatic infectious and symptomatic infectious populations peaked in the former. 25   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   0   10  20  30  40  50  60  70  80  90  100 Time t (days) 3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   0   10  20  30  40  50  60  70  80  90  100 Time t (days) 27   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65   0   10  20  30  40  50  60  70  80  90  100 Time t (days) 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62 63 64

We have proposed a fractional order epidemic model for COVID-19 disease dynamics based on the integer-order model of [103] . This studies presented:

• Qualitative analysis of both integer order and fractional order models. For the integer order, we used the multistage technique to present the analytic solution via a polynomial system.

• Also, we established the well-posedness of the fractional model via the Banach fixed point theorem. We proved the local asymptotic stability of the fractional model by using the concept of next-generation matrices for computing basic reproduction number R 0 .

• Furthermore, we proved the global stability of the model in the sense of Ulam-Hyer stability criteria.

• Finally, numerical simulations confirmed the established properties of the proposed model, and more importantly elucidated the need for compliance with regulations on basic preventive measures such as social distancing, quarantining of infectious and infected individuals, and frequent hand washing to rid the population of the deadly virus.

Indeed, the analysis of the proposed model is far from being complete. Therefore, future research efforts in this direction will consider the following improvements of the present work:

• extension of the model capture stochastic dynamics;

• parameter estimation of model parameters, including the order of the fractional derivative, based on the vast available data on COVID-19; and

• effectiveness of various available COVID-19 vaccines using optimal control formulation.

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article. 29   1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64 1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65  1  2  3  4  5  6  7  8  9  10  11  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  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  60  61  62  63  64  65 

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