key: cord-0965868-gi40crwr
authors: Pitchaimani, M.; Saranya Devi, A.
title: Fractional dynamical probes in COVID-19 model with control interventions: a comparative assessment of eight most affected countries
date: 2022-03-19
journal: Eur Phys J Plus
DOI: 10.1140/epjp/s13360-022-02556-3
sha: cb942642698d58473e6a939a5ae278a6d93562c7
doc_id: 965868
cord_uid: gi40crwr

The ultimate aim of the article is to predict COVID-19 virus inter-cellular behavioral dynamics using an infection model with a quarantine compartment. Internal viral dynamics and stability attributes are thoroughly investigated around stable equilibrium states to probe possible ways in reducing rapid spread by incorporating fractional-order components into epidemic systems. Furthermore, a fractional optimal problem was built and studied with three control measures to restrict the widespread of COVID-19 infections and exhibit perfect protection. It is found that by following [Formula: see text] of control strategies can eradicate the infectives. Furthermore, the time frame of sixteen months has been divided into four short periods to grasp the pandemic, as the pandemic’s parameters change over time. Finally, using real data, we estimated the parameters of the model system and the expression of the basic reproduction number [Formula: see text] for the most affected countries, China, USA, UK, Italy, France, Germany, Spain, and Iran.

Coronavirus is the most lethal of all viruses because it has the most destructive effect on humans. Individuals infected with the coronavirus are quarantined, leaving the infectives vulnerable and worried. The virus was so contagious that the government had to implement lockdown over the country during the pandemic, resulting in widespread financial distress. The infection causes us to lose both our health and our country's wealth, which has never happened before in history. The lack of proper therapy for the virus adds to the severity of the situation for humanity. In addition, the virus's successive waves cause more and more severe harm. Due to the reasons above, it is very reasonable to study the dynamics of the corona virus through mathematical equations [1] [2] [3] [4] [5] [6] [7] [8] [9] .

Real-life problems can also be modeled through ordinary and partial differential equations that do not depend on past history. However, the model investigated under classical derivatives and integrals suffers by the restriction for the use of various degrees of freedom. After noticing some limitations imposed by models with local classical derivatives, many authors converted to fractional calculus, a comparatively new and widely used field of mathematical analysis in which nonlocal differential operators possessing memory effects are used to model natural and physical phenomena showing anomalous behavior and nonlocal dynamics [10] [11] [12] [13] [14] [15] [16] . The use of fractional derivatives in the COVID-19 model under study is considered since memory effects significantly impact the evolution of an epidemiological process related to humans, and memory effects play a significant role in disease transmission. Furthermore, memory effects are appropriate to include in epidemiological investigations of real dynamical processes since such systems rely on memory strength, governed by order of fractional derivative [17, 18] .

In the literature, different types of fractional operators are available for understanding the model dynamics in a better way. Such operators are Riemann-Liouville, Caputo, Weyl, Hadamard, Riesz, general fractional derivative and Erdelyi-Kober, and many others [17, 19] , were each with its own set of benefits and drawbacks. We know that we require fractional-order initial conditions for solving the mathematical models in the sense of the Riemann-Liouville fractional derivative, which makes them difficult to work. On the other hand, the Caputo fractional operator removes this restriction and allows the use of initial conditions with integer-order derivatives that have obvious physical meaning. Due to the above-mentioned reason, the Caputo fractional operator is considered in the present study to model the COVID-19 dynamics. The Caputo fractional operator has subsequently been used to describe a variety of infectious diseases, and related application problems [17] . In addition, such nonlocal fractional operators are not only effectively used for modeling infectious disease, but also they have proven helpful to improve the performance of various physical and engineering systems [17, 18] .

Mathematical modeling plays a vital role in converting natural phenomena into mathematical equations. This allows real-time phenomena to be tested quickly without having to wait for a real-world situation. As a result, it has piqued the interest of many researchers. It acts as better tool for them to model and experiment with the problem based on their imagination. a e-mail: mathsaran430@gmail.com (corresponding author) (2) • People at asymptomatic compartment A(t) are COVID-19 infected by symptoms that are yet to develop. Asymptomatic compartment gain population from the exposed compartment. Asymptomatic population decrease when it becomes symptomatic, recovered at the rate σ 2 , r 1 , respectively. In addition, E(t) also decrease by natural death rate μ. Thus,

• COVID-19 infectives with symptoms make up the population in the symptomatic compartment I (t). They get increased when asymptomatic become symptomatic with the transmission rate σ 2 . Symptomatic population decrease when they are divided into hospitalized and recovered compartments with the transmission rates h 1 and r 2 . They also decrease by disease-induced death rate d 1 and die naturally at the rate μ. Thus, D α t I (t) = σ 2 A(t) − (h 1 + r 2 + d 1 + μ)I (t). • People at quarantine compartment Q(t) are population divided from exposed class E(t) with the transmission rate δ 1 . Quarantined population decrease when they are divided to hospitalized, recovered compartment with the transmission rates h 2 , r 3 and die naturally at the rate μ. Hence, the quarantine population Q(t) is expressed as follows:

• H (t) denotes a hospitalized compartment. People in the hospitalized compartment H (t) are separated into two groups: symptomatic class I (t) and quarantined class Q(t), with transmission rates of h 1 and h 2 , respectively. Hospitalized population decrease with the recovery rate r 4 , disease-induced death rate d 2 and die naturally at the rate μ. Thus,

and hospitalized H (t) classes with the transmission rates r 1 , r 2 , r 3 and r 4 , respectively. Recovered population decrease only with natural death rate μ. Thus,

The proposed COVID-19 model (2) involves the assumptions as follows:

i. S(t) is composed of uninfected individuals, who may be infected through disease transmission rates β 1 , β 2 and β 3 only from exposed, asymptomatic and symptomatic classes, respectively. But not from other infectives at quarantine and hospitalized classes. ii. The population is homogeneously mixed and age-structure are ignored [28, 30] .

iii. The disease-induced death rate is considered only for the symptomatic and hospitalized class of infectives at a rate d 1 and d 2 since other infective classes are at less risk of death due to disease. iv. The asymptomatic, symptomatic, hospitalized and quarantine individuals get into recovered class at a rate r 1 , r 2 , r 3 and r 4 , respectively. v. All compartments are considered to have the same natural death rate symbolized by μ [31] . The proposed COVID-19 model (1) is governed by a system of nonlinear fractional-order differential equations as follows:

(1)

It is observed that the first six equations in the system (1) do not depend on the seventh equation, and so this equation can be omitted without loss of generality. This allows us to attack the system (1) by studying the subsystem (2) , which is governed by the system of nonlinear fractional-order differential equation as follows,

(2)

In system (2), the total population N (t) is divided into six compartments, such that N (t) = S(t)+ E(t)+ A(t)+ I (t)+ Q(t)+ H (t) because all six classes are mutually disjoint. The fractional derivative of model (2) is in the sense of Caputo. Here α ∈ (0, 1] is the order of the fractional derivative and D α t denotes d α dt α . The classical version of the proposed system (2) is retained when α = 1. There are two main advantages for using the Caputo fractional operator over the Riemann-Liouville operator that is given as follows: (i) The definition of Riemann-Liouville fractional derivative do not satisfy the property that the derivative of a constant is zero. (ii) The Caputo fractional derivative allows the initial condition similar to the one in an ordinary differential case, but this is not allowed in the case of Riemann-Liouville. The reasons described above suggest a preference for the Caputo fractional derivative in modeling natural phenomena, particularly the epidemic models [29] . The description of parameters used in system (2) is provided in Table 1 . The system (2), with the initial conditions 

Furthermore, we assume that

The upcoming sections deal with the analysis of the proposed COVID-19 model (2).

The nonlinear fractional-order COVID-19 model (2) is studied in this section for its analytical properties.

In this section, we recall some basic definitions of fractional-order derivatives. Consider the system

where D α is the Caputo fractional derivative which is given in the following definition.

Definition 1 [13] The Caputo fractional derivative of order α of a function

where t ≥ t 1 , (.) is the Gamma function, and n is the positive integer such that α ∈ (n − 1, n).

When α ∈ (0, 1), one has

Lemma 1 [13] Let x(t) be a continuous function on [t 1 , +∞) and satisfy

where α ∈ (0, 1], λ, μ ∈ R, and λ = 0, t 1 ≤ 0 is the initial time. Then

where E α (·) is the Mittag-Leffler function that is defined as

where α > 0, β > 0 and z ∈ C. When β = 1, one has E α (z)

Theorem 3.1.1 [10] . Consider the following commensurated fractional-order system 

We need to locate the eigenvalues to examine the local stability criteria of the equilibrium point. Hence, the following results are required.

Definition 2 [32] The discriminate D( f ) of a polynomial f (x) = x n + a 1 x n−1 + a 2 x n−2 + · · · + a n is given by D n ( f ) = (−1)

The discriminate of a polynomial plays an important role to define the nature of the roots of f (x) = 0. If

then

Theorem 3.1.2 [33] If the system (5) has a characteristic polynomial at x * defined by (8) , then according to the sign of the discriminate D 3 (P) given by (9) , condition (7) is satisfied in the following cases:

then its discriminate is

Consider the characteristic equation of the form

The following theorem provides important results in determining the stability criteria of the system (5).

Theorem 3.1.3 [34, 35] If the system (5) has a characteristic polynomial at x * defined by (12) , then according to the sign of the discriminate D 4 (P) given by (11) , condition (7) is satisfied in the following cases:

Consider the determinants i , i = 1 : 3 defined by, 1 = a 1 , 2 = a 1 1 a 3 a 2 and 3 = a 1 1 0 a 3 a 2 a 1 0 a 4 a 3 , 1. The equilibrium point x * is locally asymptotically stable for α = 1 if and only if a 4 > 0 and i > 0, i = 1 : 3.

2. If D 4 (P) > 0, a 1 > 0 and a 2 < 0, then the equilibrium point x * is unstable for α > 2 3 .

3. D 4 (P) < 0 and a i > 0, i = 1 : 4, then the equilibrium point x * is locally asymptotically stable for α < 1 3 . However, if D 4 (P) < 0, a 1 < 0, a 3 < 0 and a 2 > 0, a 4 > 0 then the equilibrium point x * is unstable.

, then the equilibrium point x * is locally asymptotically stable for all α ∈ (0, 1). 5. The equilibrium point x * is locally asymptotically stable only if a 4 > 0.

This section describes the non-negativity and boundedness of the proposed nonlinear fractional-order COVID-19 model (2).

We focus only on non-negative and bounded solutions since the proposed COVID-19 model (2) under consideration is biologically significant. Denote 

is positively invariant with respect to the system (2) .

Proof To prove the non-negativity of the system (2), we add all the compartments of the system (2), such that

We know that all the parameters are positive, thus we can obtain

, By applying Lemma 1, we get

Therefore, the positive invariance of the system (2) is given by

It can be observed that S(t), E(t), A(t), I (t), Q(t) and H (t) are bounded in an invariant set . This completes the proof.

The upcoming section deals with the computation of basic reproduction number (R 0 ) and the equilibrium points of the fractionalorder COVID-19 model (2).

The basic reproduction number (R 0 ) of the proposed nonlinear fractional-order COVID-19 model (2) is computed by adopting the next-generation matrix approach [36] is expressed as,

where

The proposed nonlinear fractional-order COVID-19 model (2) admits two equilibria as follows: i) The disease-free equilibrium point isÊ

ii) The endemic equilibrium point isÊ where

Next, we are going to analyze some of the basic properties that are local and global behavior of the fractional-order nonlinear system (2) at each of its equilibrium points.

This section discusses the criteria for the proposed COVID-19 model to be locally asymptotically stable at its positive equilibrium points. In the history of infectious disease modeling, the basic reproduction number R 0 plays a vital role in the disease dynamics. In the present, the dynamics of the system (2) depends on the corresponding basic reproduction number R 0 .

The disease-free equilibriumÊ 0 of the system (2) is locally asymptotically stable if R 0 < 1.

Proof The Jacobian matrix of system (2) at the disease-free equilibrium pointÊ 0 is given by

The characteristic equation of the above matrix is obtained as

where

Clearly, Eq. (14) has three negative real roots,

The remaining roots λ 4 , λ 5 and λ 6 are calculated by the following equation,

Let D 3 (P) be the discriminant of the characteristic polynomialP(λ) [32] . Thus,

We have the following result, by using the construction of fractional Routh-Hurwitz conditions provided in [32] that is given below.

The positive equilibrium pointÊ 0 of the system (2) is asymptotically stable for R 0 < 1 , if one of the following conditions holds for polynomialP(λ) and coefficients P 1 , P 2 , P 3 which are given by (15) .

1. If D 3 (P) > 0, then the necessary and sufficient condition for the equilibrium point to be locally asymptotically stable is

, then the equilibrium point is locally asymptotically stable. Also if D 3 (P) < 0,

, then all roots of theP(λ) = 0 satisfy the condition |arg(λ j )| < απ 2 , j = 1, 2, 3.

3. If D 3 (P) < 0, P 1 > 0, P 2 > 0, P 1 P 2 = P 3 , then the equilibrium point is locally asymptotically stable for all α ∈ [0, 1).

We strive to provide numerical support for our above arguments by using parameter values as = 2274, β 1 = 0.00003, β 2 = 0.000111, β 3 = 0.0001197, σ 1 = 0.094, σ 2 = 0.09, δ 1 = 0.0986, r 1 = 0.9999, r 2 = 0.16, r 3 = 0.2553, r 4 = 0.4449, h 1 = 0.10001, h 2 = 0.4129, d 1 = 0.002, d 2 = 0.001, μ = 0.3349 and α = 1. By using this parameters the value of R 0 is calculated as R 0 = 0.4958 < 1. We get the characteristic polynomial of the Jacobian matrix J (Ê 0 ) as, λ 6 + 4.464297402 * λ 5 + 7.784525049 * λ 4 + 6.765681308 * λ 3 +3.074701298 * λ 2 + 0.6885212084 * λ + 0.5933570162 = 0. It can be found that λ j < 0, j = 1, 2 · · · 6.

The disease-free equilibrium has simulated by setting all the infected compartments to zero. In the case of disease-free equilibrium, the basic reproduction number R 0 is limited to less than one. In this scenario, the community is assumed to be disease free and safe from epidemics. The threshold R 0 is biologically very much crucial as it gives the idea for the range of R 0 required to reduce the disease. Using this, we can maintain R 0 and relevant parameters to control the disease spread.

The endemic equilibriumÊ 1 of the system (2) is locally asymptotically stable if R 0 > 1.

Proof As the standard techniques in the theory of stability analysis suggests, the Jacobian matrix of the system (2) at the endemic equilibrium pointÊ 1 to identify the characteristic equation as follows:

The characteristic polynomial of the above matrix J (Ê 1 ) can be written as

where

,

Clearly, Eq.(18) has two negative real roots, λ 1 = −Q 3 , λ 2 = −H 3 . The remaining roots λ 3 , λ 4 , λ 5 and λ 6 are calculated by the following equation,

Let D 4 (B) be the discriminant of the characteristic polynomial B(λ), which was discussed in [35] . Thus,

We have the following result, by using the construction of fractional Routh-Hurwitz conditions provided in [35] that is discussed below.

The positive equilibrium pointÊ 1 of the system (2) is asymptotically stable for R 0 > 1 , if one of the following conditions holds for polynomial B(λ) and coefficients B 1 , B 2 , B 3 , B 4 which are given by (19) .

Consider the determinants i , i = 1 : 3 defined by

3. D 4 (B) < 0 and B i > 0, i = 1 : 4, then the equilibrium pointÊ 1 is locally asymptotically stable for α < 1 3 . However, if

, then the equilibrium pointÊ 1 is locally asymptotically stable for all α ∈ (0, 1). 5. The equilibrium pointÊ 1 is locally asymptotically stable only if B 4 > 0.

We strive to provide numerical support for our above arguments by using parameter values as

.001, μ = 0.3349 and α = 1. By using this parameters the value of R 0 is calculated as R 0 = 2.9295 > 1.

We get the characteristic polynomial of the Jacobian matrix J (Ê 1 ) as,

The roots of above equation are as follows,

It can be found that λ j < 0, j = 1, 2 . . . 6. The local stability of the endemic equilibrium of the fractional-order COVID-19 model (2) biologically represents the surveillance of the infected population in the community. In case of endemic equilibrium population at infected compartments, i.e, E(t), A(t), I (t), Q(t), H (t) tends to constant. The above phenomena happen as the basic reproduction number R 0 > 1.

The upcoming subsection deals with the global stability analysis of equilibria of the proposed COVID-19 model (2).

To establish global stability, we construct suitable Lyapunov functionals and use LaSalle's invariance principle theory.

Lemma 2 [37] let y(t) ∈ R + be derivable and continuous function. Then, for any time t ≥ t 0 ,

∀α ∈ (0, 1), y * ∈ R + . Note that for α = 1, the inequalities in (22) becomes equalities. Let us denote

in upcoming results.

be any positive solution of the system (2) with the initial condition (3), define a Lyapunov functional W 1 (t) as follows,

where

Differentiating W 1 (t) along the solution of system (2), we obtain

Using μ = S 0 in (24), we obtain

Using (13) in (25), we obtain

It follows from Eq.

By the LaSalle invariance principle, the disease-free equilibriumÊ 0 of the model (2) is globally asymptotically stable.

be any positive solution of the system (2) with the initial condition (3), define a Lyapunov functional W 2 (t) as follows,

where,

Differentiating W 2 (t) along the solution of system (2), we obtain

Note that,

Using Eq. (29) in Eq. (28), we obtain

Using Eq. (13) in Eq. (30), we obtain

It follows from Eq. 1 and H (t) = H 1 . By the LaSalle invariance principle, the endemic equilibriumÊ 1 of the model (2) is globally asymptotically stable.

The whole theory of stability analysis of the epidemiological model goes behind the threshold R 0 , as it depicts the scenario. Here, the basic reproduction number acts as the threshold parameter. The community suffers from an epidemic outbreak when R 0 > 1 and enjoys the safer community when R 0 < 1. The main objective of Theorems (3.5.1)-(3.5.2) is to analyze the global stability around equilibrium points of the nonlinear fractional-order COVID-19 model (2) . That is to find conditions for global stability around equilibria and work out the relations among these stability conditions. Theorems (3.5.1)-(3.5.2) reveal that the nonlinear fractional-order COVID-19 model (2) always return to its corresponding equilibrium points with time, meaning thereby, the solution trajectories of the system will be attracted toward the equilibrium point with time and establishing the global stability of the system at equilibrium points.

The following section discusses the sensitivity analysis of the basic reproduction number R 0 .

The sensitivity of the basic reproduction number R 0 of the nonlinear fractional-order system (2) is discussed in this section. The analysis for the sensitivity of R 0 is given attention because R 0 is the indicator of an epidemic's magnitude, and the entire dynamics of the system (2) depends on the threshold R 0 .

Sensitivity analysis (SA) naturally follows uncertainty analysis (UA) as it measures the statistical distribution of differences in model outputs to various input sources. For most biological models, input variables are parameters, and it is not always established with a reasonable degree of certainty due to natural variability, and measurement error [38, 39] . In this subsection, we illustrate the most familiar sampling technique: the Latin hypercube sampling (LHS) to accomplish UA. There are twelve parameters involved in R 0 , and the uncertainty analysis has performed for eight out of twelve parameters. The eight parameters are β 1 , β 2 , β 3 , σ 1 , σ 2 , δ 1 , h 1 and d 1 . Each parameter is assumed to be a random variable with a corresponding probability density function. The other four parameters ( , μ, r 1 , r 2 ) chosen with the fixed values given in Table 1 have not been considered for sensitivity analysis. The probability density functions are based on biological information of the natural history of influenza [38] . The eight parameters follow the following probability distributions:

• The transmission rate by infectives at exposed class β 1 follows normal distribution with mean and standard deviation 0.00005 and 0.0000000004, respectively. • The transmission rate by infectives at asymptomatic class β 2 follows normal distribution with mean and standard deviation 0.00251 and 0.0000000002, respectively. • The transmission rate by infectives at symptomatic class β 3 follows normal distribution with mean and standard deviation 0.001197 and 0.0000000001, respectively. • The progression rate by infectives from exposed to symptomatic class σ 1 follows triangular distribution with minimum, mode and maximum as 0.0065, 0.007 and 0.0075, respectively. • The progression rate by infectives from asymptomatic to symptomatic class σ 2 follows triangular distribution with minimum, mode and maximum as 0.0085, 0.009 and 0.0095, respectively. • Rate at which the exposed individuals are diminished by quarantine δ 1 follows gamma distribution with mean and standard deviation 0.0986 and 0.0004, respectively. • Rate at which symptomatic infectives are hospitalized h 1 follows gamma distribution with mean and standard deviation 0.1001 and 0.0001, respectively. • Diseases induced rate form symptomatic infectives d 1 , follows gamma distribution with mean and standard deviation 0.002 and 0.00000001, respectively.

Sensitivity analysis is performed in this section to determine the main parameter contributing to the variability in the outcome of the basic reproduction number depending on its estimation uncertainty. Between the values of R 0 and each of the eight parameters produced from the uncertainty analysis [38, 39] , the partial rank correlation coefficient (PRCCs) is estimated. Scatter plots have been plotted to compare R 0 against each of eight parameters: β 1 , β 2 , β 3 , σ 1 , σ 2 , δ 1 , h 1 and d 1 as shown in Fig.  2 from LHS with sample size 1000. These scatter plots indicate the linear relationships (monotonicity) between the outcome of R 0 and input parameters. The PRCCs value for R 0 and each of eight parameters enlisted in Table 2 and graphically represented in Fig.  3 .

The parameter with positive PRCCs is directly proportional to R 0 , i.e., β 1 , β 2 , β 3 , σ 1 and σ 2 , whereas the parameter with negative PRCCs is inversely proportional to R 0 , i.e., δ 1 , h 1 and d 1 . After using the sample from LHS, we observe that the transmission rate by infectives at asymptomatic phase β 2 and the transmission rate by infectives at exposed class β 1 are highly correlated Table 2 . Moderate correlation exists between the transmission rate by infectives at symptomatic phase β 3 , the rate at which the asymptomatic becomes symptomatic σ 2 with R 0 corresponding value is 0.1150 and 0.1150. Weak correlation has been observed between the rate at which the exposed become asymptomatic σ 1 , rate at which the exposed individuals are diminished by quarantine δ 1 , rate at which symptomatic infectives are hospitalized h 1 and diseases induced rate form symptomatic infectives d 1 with R 0 and corresponding values are 0.0442, −0.0541, −0.008 and −0.0089, respectively. Hence, we can conclude that β 1 and β 2 are the most important parameters in determining the R 0 .

An optimal control approach has been used in the system of fractional differential equations to reduce the number of infected people and to abate the outbreak of the epidemic [37, [40] [41] [42] [43] . We are developing a COVID-19 model (2) by incorporating specific control measures to prevent the spread of COVID-19, which results in the formation of a fractional optimal problem. The first control function v 1 (t) represents a transmission control rate that reduces the number of exposed by a factor (1 − v 1 (t) ). Control of v 1 (t) is the proportion of the susceptible people who follow proper non-pharmaceutical interventions i.e., lock-down, who use proper face mask, introducing social distancing, using proper sanitation change their behavior per unit time. The second control function v 2 (t) is a rapid test among asymptomatic population. The third control function v 3 (t) is a treatment among symptomatic population. The proposed model (2) is being modified as a result of these control measures, as follows:

with the non-negative initial conditions

When v i (t) = 1, the control measure is fully effective, and when v i (t) = 0, the control measure does not work, with i = 1, 2, 3, i.e., 0 ≤ v i (t) < 1. Our aim is to reduce the number of people exposed while minimizing the cost of control measures, which can be achieved by considering the following optimal control problem to minimize the objective functional given by

relied on the state system provided by (32) in accordance with non-negative initial conditions (33) . In Eq. 

subjected to the state system given in (32) , where the control set is defined as

The Lagrangian L and Hamiltonian H for the fractional optimal problem (32)- (36) are, respectively, given by [40, 42, 43 ]

and

Eur. Phys. J. Plus (2022) 137:370

The above equation can be written as

where S , E , A , I , Q , and H are the adjoint variables. Now we have to prove the necessary conditions for the optimality of the fractional system (32) . For the optimal control v(t), that minimizes the performance index

subjected to the dynamical constraints

with initial conditions

where π(t) and v(t) are the state and control variables, respectively, L and ω are differentiable functions, and α ∈ (0, 1]. We have the following theorem.

is a minimizer of (40) under the dynamic constraint (41) and the boundary condition (42) , then there exists a function such that the triplet (π, v, ) satisfies

for the Hamiltonian H(t, π, v, ) = L(t, π, v) + T (t, π, v).

Proof For the proof of theorem (5.0.1), viewers are recommended to see [37, 40, 41] , in which the authors present evidence in detail. This ends the proof of the theorem (5.0.1).

Theorem 5.0.2 Let S 1 , E 1 , A 1 , I 1 , Q 1 and H 1 be optimal state solutions with associated optimal control variables v * 1 , v * 2 , v * 3 for the optimal control problems (32) and (34) . Then there exist adjoint variables S , E , A , I , Q and H satisfy the following:

with transversality conditions or boundary conditions S = 0, E = 0, A = 0, I = 0, Q = 0 and H = 0.

Furthermore, the control functions v * 1 , v * 2 and v * 3 are given by v * 1 = min 1, max 0,

Proof The adjoint system (44), i.e., S , E , A , I , Q and H are obtained from the Hamiltonian H as 

The numerical methods used for solving ordinary differential equations cannot be used directly to solve fractional differential equations because of nonlocal nature of the fractional differential operator. A modification in Adams-Bashforth-Moulton predictorcorrector algorithm is proposed by Diethelm et al. in [44, 45] to solve fractional differential equations. Consider the initial value problem

where f is in general a nonlinear function of its arguments. The initial value problem (46) is equivalent to the Volterra integral equation

Consider the uniform grid {t n = nh/n = 0, 1, . . . , N } for some integer N and h := T /N . Let y h (t n ) denote the approximation to y(t n ). Assume that we have already calculated approximations y h (t j ), j = 1, 2, . . . , n and want to obtain y h (t n+1 ) by means of the equation [44, 45] 

where 

The preliminary approximation y P h (t n+1 ) is called predictor and is given by

Error in this method is

where p = min(2, 1 + α). Numerical analysis has been carried out using MATLAB(R2015a), to represent the system (2) graphically. A nonlinear fractionalorder COVID-19 model (2) has been solved numerically by adopting predictor-corrector algorithm [44] [45] [46] , as discussed above. Here, the figures are plotted with the initial conditions as S(t) = 600, E(t) = 500, A(t) = 400, I (t) = 300, Q(t) = 200, and H (t) = 100, with the values of parameters described in Table 1 .

In this subsection, we deal with numerical analysis for the fractional-order COVID model (2). Figure 4 has been plotted using the parameters listed in Table 1 for α = 0.95, where R 0 = 0.4958 < 1. It can be observed from Fig. 4 that the susceptible population survive, and all the infected population tend to zero. This scenario is due to the value of basic reproduction number R 0 being less than unity, which biologically implies that there is no infected population to spread the disease among the susceptible. As in this article, we are dealing with infectious diseases, and it is essential to discuss the infected population, which is highly credible for disease spread. The disease spread is ascertained by the number of infected people and the disease transmission rate. Figure 5 has been plotted using the parameters listed in Table 1 (2) for α = 0.95, where R 0 = 2.9285 > 1 that the susceptible population is less than the exposed population. This biologically denotes the endemic outbreak of the coronavirus in the society, and this scenario is due to R 0 > 1. The infectives at the asymptomatic phase are higher than the infectives at the symptomatic phase. It is due to the reason ratio of symptomatic individuals who are hospitalized, quarantined and recovered. According to the sensitivity analysis, the highly sensitive parameter is β 2 , which is the disease transmission rate from the infected population at asymptomatic phase A(t) to the susceptible population S(t). It biologically communicates that the transmission rate of exposed individuals β 1 and the transmission rate of infected individuals at symptomatic stage β 3 are less sensitive than the transmission rate of infected individuals at asymptomatic stage β 2 . This is since exposed individuals are only exposed to the disease and are not infected. Symptomatic individuals are infected and aware of the disease, so they are either hospitalized or quarantined. As a result, the symptomatic infectives has a lower chance of spreading. Infected individuals in the asymptomatic stage are unaware that they are infected because they have no symptoms. As a result, they have a high risk of spreading the disease. This biologically demonstrates that β 2 has a higher sensitivity than β 1 and β 3 .

Naturally, information about disease behavior in the past helps people protect themselves from the spread of the disease. The role of being aware of the past dynamics of the solution trajectories. The control of disease spread has a significant influence in knowing their history, which helps people decide what preventive measures to take. If people know the past about disease in their area, they can use various preventative measures, such as vaccination. On the other hand, fractional derivative plays a vital role in interpreting memory effects in dynamic systems. As α approaches 1, the memory effects are decreased. Figure 6 has been plotted using the same parameters used for Fig. 4 for different values of α, where R 0 = 0.4958 < 1. Figure  7 has been plotted using the same parameters used for Fig. 5 for different values of α, where R 0 = 2.9285 > 1. It can be seen from Figs. 6 and 7, that fractional-order solution is the trace of its integer order. The findings indicate that the order of the fractional derivative has a significant impact on the dynamic process. In addition, the results show that the memory effect is zero for α = 1. In case of fractional-order system memory effect is indirectly proportional to the value of α.

As transmission rate plays a crucial role in disease spread, this subsection deals with its impact over its dynamics. Figures 8, 9 and 10 has been plotted with the same parameters and initial condition used to plot Fig. 5 with α = 0.98. Let us have a brief on transmission rates β 1 , β 2 and β 3 .

• The parameter β 1 denotes the disease transmission rate between susceptible population S(t) and exposed population E(t). The solution trajectory of the symptomatic population I (t) varies with β 1 , as shown in Fig. 8 . It is observed that the symptomatic population I (t) increases as β 1 increases. It is also witnessed from Fig. 3 that the transmission rate β 1 has the second-highest sensitivity value while β 2 is the first highest sensitivity value and thirdly, the β 3 . The partial rank correlation coefficient value 6 Denotes graph trajectories of system (2) for different values of α, where R 0 = 0.4958 < 1 of β 1 is found to be 0.1354 has been enlisted in Table 2 . This scenario biologically implies that the transmission rate β 1 has the eligibility of higher disease spread than β 3 . • The parameter β 2 represents the disease transmission rate between susceptible population S(t) and asymptomatic population A(t). The solution trajectory of the symptomatic population I (t) varies with β 2 , as shown in Fig. 9 . It is observed that the symptomatic population I (t) increases as β 2 increases. It is also witnessed from Fig. 3 that the transmission rate β 2 has the highest sensitivity value compared to all the parameters. The partial rank correlation coefficient value of β 2 is found to be 0.1981 has been enlisted in Table 2 . This scenario biologically implies that the transmission rate β 2 has a crucial role in disease spread than other transmission rates β 1 and β 3 . • The parameter β 3 denotes the disease transmission rate between susceptible population S(t) and symptomatic population I (t).

The solution trajectory of the symptomatic population I (t) varies with β 3 , as shown in Fig. 10 . It is witnessed that the symptomatic population I (t) increases as β 3 increases. This scenario biologically implies that the symptomatic population depends on β 3 , i.e., the symptomatic population is directly proportional to the value of β 3 . So, this Fig. 10 conveys to us that the symptomatic population can be controlled once the transmission rate β 3 is controlled. This scenario results in explaining to us that COVID-19 can only be controlled with proper control measures so that the transmission rate β 3 can be reduced, automatically the symptomatic population.

According to the above analysis of disease transmission rates β 1 , β 2 , and β 3 , the virus living in infected individuals is both asymptomatic and symptomatic in the case of COVID-19. Therefore, unknowingly, the susceptible population becomes a victim of Page In strategy 1, we set the control measure v 1 = 0 (non-pharmaceutical interventions), v 2 = 0 (rapid test to infectives at asymptomatic stage) and v 3 = 0 (treatment to infectives at symptomatic stage). From Fig. 11 , it can be observed that implementing the control strategies v 1 and v 2 to the proposed COVID-19 model (2) helps us to decrease the infected population. Figure 11d portrays that the infected individuals can be vanished within 50 days. This scenario emphasis implementing the two control measures v 1 and v 2 and also aids in the elimination of the exposed, asymptomatic, and quarantined population after some In strategy 2, the control measures are v 1 = 0 (non-pharmaceutical interventions), v 2 = 0 (rapid test for infectives at an asymptomatic stage), and v 3 = 0 (treatment to infectives at symptomatic stage). It can be seen in Fig. 12 that applying the control strategies v 2 and v 3 to the proposed COVID-19 model (2) helps us to reduce infectives, but it does not enable us to eliminate them. This scenario demonstrates the significance of disease transmission rates. The factors β 1 , β 2 , and β 3 significantly influence COVID-19 spread than the other parameters. Controlling disease transmission among susceptible is more important than doing rapid tests for asymptomatic patients and initiating treatment for symptomatic infectives after infection. In strategy 3, we adopted the control measures v 1 = 0 (non-pharmaceutical interventions), v 2 = 0 (rapid test to infectives at asymptomatic stage) and v 3 = 0 (treatment to infectives at symptomatic stage). It can be seen in Fig. 13 that implementing the control strategies v 1 and v 3 to the proposed COVID-19 model (2) helps us to reduce infectives, but it does not enable us to eliminate them. Even if the transmission rate is reduced by implementing (v 1 = 0) and the infectives are treated properly (v 3 = 0), the infectives cannot be eradicated since the infectives at the asymptomatic stage are more hazardous than those at the symptomatic stage. When asymptomatic people do not take quick tests, they are unaware of the virus, which spreads to the rest of the community. As a result, the infectives cannot be eradicated. In strategy 4, we set the control measure v 1 = 0 (non-pharmaceutical interventions), v 2 = 0 (rapid test to infectives at asymptomatic stage) and v 3 = 0 (treatment to infectives at symptomatic stage). From Fig. 14 , it can be observed that implementing the control strategies v 1 , v 2 and v 3 to the proposed COVID-19 model (2) helps us to wipe out the infected population. It can be observed from Fig. 14d that the infectives can be eliminated after 20 days. When the control measure increases, the infected population tends to zero and susceptible increase gradually.

The first and fourth strategies are the strongest since they help eliminate the infectious agents. Although strategies 1 and 4 behave similarly, infectives can be eradicated more quickly with strategy 4 than with strategy 1. Figures 11 and 14 show that the infected population can only be removed if at least 60% of the control measures are performed, but not less than that. Even after implementing three procedures, the infectives are lowered but not removed below 60%. As a result, all three control methods should be established and made to utilize at least 60% for the infectives to vanish after a while. The discussion above proposes the ideal approach for researchers, policymakers, and the government to comprehend the impact of COVID-19 infection control strategies. Table 3 .

The data were obtained from datahub (https://datahub.io/core/covid-19) 

In this section, we fit the infected cases of COVID-19 at symptomatic phase I (t) of system (2) to data to determine the trend of newly infected COVID-19 cases. Curve fitting be mathematically expressed as,

where θ i = (Y i −Ŷ ) and n refers to the data points and RSS refers to the sum of square error estimate which is assumed to follow a normal distribution. This article breaks down the 16 month of study into four phases. Phase 1 covers the months of May 2020 to August 2020 for 4 months. Phase 2 spans the months of September 2020 to December 2020 and represents a 4 months. Then, from January 2021 to April 2021, Phase 3 represents four months. Finally, Phase 4 spans the months of May 2021 to August 2021, for 4 months. It is important to observe that the cases of COVID-19 increase day by day. The results show a rise in COVID-19 cases between May 2020 and Aug 2021. Initial condition assumed to plot Figs. 15, 16, 17 and 18 are given in Table 3 . Figures 15, 16, 17 and 18 depicts that our model (2) well fits with the real data of China, USA, UK, Italy, France, Germany, Spain and Iran for phase 1, phase 2, phase 3 and phase 4, respectively. Estimated parameter values are for all eight countries for phase 1, phase 2, phase 3 and phase 4 are provided in Table 4 . Estimated value of R 0 using the estimated parameters is provided in Table 5 . modeled to study its dynamics, bearing all of this in mind. The study wraps up by applying adequate control methods, such as non-pharmaceutical interventions to susceptible, rapid testing to asymptomatic corona population and administering treatment to symptomatic corona population at appropriate time intervals. In 2020, Anwarud Din et al. [48] have analyzed the covid model with the case study of china. After this, Utkucan Sahin and Tezcan Sahin [49] have forecasted the cumulative number of confirmed cases of COVID-19 in the USA, UK, and Italy. Following this, Duccio Fanelli and Francesco Piazza [50] have corned a problem of COVID-19 in China, Italy and France. Next, Thomas Gotz and Peter Heidrich [51] have performed parameter estimation of the COVID-19 model for Germany. After this, Antonio Guirao [52] has presented the Covid-19 outbreak in Spain with control response. Finally, Jin Zhao et al. [53] studied the modeling of the COVID-19 Pandemic Dynamics in Iran. All the above work have modeled and analyzed the COVID-19 model for a few months for at most three countries. But this article deals with the fractional-order COVID-19 model with control strategies. Also, data calibration have been performed for the most affected eight countries China, UK, USA, Italy, France, Germany, Spain and Iran. In addition, for data calibration, the sixteen months taken understudy has been divided into four phases, with each phase having four months. This segmentation gives a deeper insight into COVID-19 dynamics for the different time intervals for different countries, which has never been addressed in the literature to the best of the author's knowledge. Moreover, this article contributes parameter estimation and basic reproduction number estimation for four phases of eight countries.

The novelty of our study lies in analyzing the COVID-19 model through Caputo fractional derivative along with quarantine and hospitalized compartments. This article evaluates the proposed model for basic reproduction number, equilibrium points, sensitivity analysis and (local and global) stability of its equilibria. Sensitivity analysis for the parameters of the basic reproduction number R 0 has been calculated. The proposed model (2) is developed by implementing control strategies into it, which gives us the fractional optimal problem. The impact of control strategies has been discussed both theoretically and graphically. It is evidenced from Fig.  11 , 12, 13 and 14 that implementing all three control strategies at the same time with 60% can help us to wipe out the infectives. The work provides a theoretical and pictorial representation of the dynamics of the COVID-19 model (2) via Caputo fractional derivative. Furthermore, the study emphasis the analytical properties of the proposed COVID-19 model (2) , which is used to capture its dynamics.

This section discusses the limitations of the current research. Such limitations will pave the way for future research in this field. Some of the significant constraints of the study are listed as follows:

i. The discussed COVID-19 model could be modified with comorbidities and vaccinated compartments with a case study. ii. The discussed COVID-19 model could be extended to a stochastic case and solve a stochastic control problem. iii. Discrete-time delays can be incorporated into the discussed COVID-19 model for further investigation. iv. Comparative study of frequency-dependent and density-dependent can be adopted in the proposed COVID-19 model. v. The concept of short memory was developed in the numerical approaches of fractional differential equations. Predictor-corrector algorithm is one of the most common method used to derive numerical solution of fractional-order systems with long memory effect. A predictor-corrector algorithm with short memory effect was examined by some researchers in the literature [54] [55] [56] , which is notable for its low computational cost. In the case of the short memory principle, memory length is fixed and it describes the recent past instead of the whole history. Therefore, it is worth checking the stability properties of fractional-order systems with fixed memory length. Furthermore, the fractional derivative with short memory degrades into the normal one if the memory length is high enough. On the other hand, the nonlocal characteristic of a predetermined fractional derivative becomes a local characteristic when the memory length is small enough. In general, the short memory principle in the theory of fractional calculus [55] is promising and applicable to a vast class of fractional ordinary differential equations. The concept of short memory is also used in the study of fractional calculus, modeling of memristors, and neural networks [54] . Furthermore, the introduced short memory model can be examined for its positivity and boundedness, and stability properties can be discussed. We can now propose a new fractional-order COVID-19 model with a short term memory principle to study its stability properties. vi. As already discussed in Sect. 1, there are many fractional derivatives available in the literature such as, Riemann-Liouville, Caputo, Weyl, Hadamard, Riesz, general fractional derivative and Erdelyi-Kober [19] . The topic of the general fractional derivative has recently piqued the interest of fractional calculus scholars. One of the most advantages of using general fractional derivative is that, by analyzing the stability properties of general fractional derivative, one can trace the stability properties of all other derivatives, which are the special cases of general fractional derivative. In the article [57] , the author investigated an extended fractional differentiation, which generalizes the Riemann-Liouville and the Hadamard fractional derivatives into a single form, which when a parameter is fixed at different values or by taking limits produces the above derivatives as special cases. The primary problem with fractional operators and their generalized counterparts is accurately defining them in the appropriate function space. In the article [58] [59] [60] , the authors have discussed the generalized fractional derivative. Using some specific function in the introduced general fractional derivative, we can get the standard Caputo fractional derivative, Hadamard, Katugampola, and exponential-type fractional derivatives. However, the stability properties of the generalized fractional-order system are left to future work. We can now propose a new COVID-19 model with a generalized fractional derivative to study its stability properties.

As a result of the preceding discussion, the study suggests numerous avenues for future research. The current study allows us to see the impact of memory in COVID-19 modeling.

A SIR model assumption for the spread of COVID-19 in different communities

Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative

Risk estimation and prediction of the transmission of coronavirus disease-2019 (COVID-19) in the mainland of China excluding Hubei province

A mathematical model for simulating the phase-based transmissibility of a novel corona virus

Mathematical model of infection kinetics and its analysis for COVID-19

Stability analysis of COVID-19 model with fractional order derivative and a delay in implementing the quarantine strategy

A multi-region discrete time mathematical modeling of the dynamics of Covid-19 virus propagation using optimal control

Impact of social media advertisements on the transmission dynamics of COVID-19 pandemic in India

Mathematical modeling of COVID-19 spreading with asymptomatic infected and interacting peoples

Analysis of a fractional order eco-epidemiological model with prey infection and type 2 functional response

Threshold dynamics of an HIV-TB co-infection model with multiple time delays

An Investigation on analytical properties of delayed fractional order HIV model: a case study

Dynamic analysis of a delayed fractional order SIR model with saturated incidence and treatment functions

An introduction to the fractional calculus and fractional differential equations

Fractional differential equations

Application of fractions calculus

The memory effect on fractional calculus: an application in the spread of COVID-19

Fractional q-deformed chaotic maps: a weight function approach

A note on fractional order derivatives and table of fractional derivatives of some special functions

The analysis of a time delay fractional COVID-19 model via Caputo type fractional derivative

A numerical simulation of fractional order mathematical modeling of COVID-19 disease in case of Wuhan China

Analysis and forecast of COVID-19 spreading in China, Italy and France

Analysis of a COVID-19 model: optimal control, stability and simulations

Mathematical model for coronavirus disease 2019 (COVID-19) containing isolation class

Modeling and forecasting the COVID-19 pandemic in India

Stochastic probical strategies in a delay virus infection model to combat COVID-19

Analysis of a COVID-19 model: optimal control, stability and simulations

Short-term predictions and prevention strategies for COVID-19: a model based study

Properties of Caputo operator and its applications to linear fractional differential equations

Impact of social media advertisements on the transmission dynamics of COVID-19 pandemic in India

Mathematical assessment of the role of pre-exposure prophylaxis on HIV transmission dynamics

On some Routh-Hurwitz conditions for fractional order differential equations and their Applications in Lorenz, Rossler, Chua and Chen systems

Stability results for fractional differential equations with applications to control processing

Dynamical analysis and circuit simulation of a new fractional order hyperchaotic system and its discretization

Stability conditions, hyperchaos and control in a novel fractional order hyperchaotic system

On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

Global dynamics of a fractional order model for the transmission of HIV epidemic with optimal control

Uncertainty and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza

A methodology for performing global uncertainty and sensitivity analysis in systems biology

Stability analysis of a fractional order model for the HIV/AIDS epidemic in a patchy environment

Fractional optimal control of an HIV/AIDS epidemic model with random testing and contact tracing

A general formulation and solution scheme for fractional optimal control problems

Effect of partial immunity on transmission dynamics of dengue disease with optimal control

A predictor-corrector approach for the numerical solution of fractional differential equations

An algorithm for the numerical solution of differential equations of fractional order

A predictor-corrector scheme for solving nonlinear delay differential equations of fractional order

Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China

Forecasting the cumulative number of confirmed cases of COVID-19 in Italy, UK and USA using fractional nonlinear grey Bernoulli model

Analysis and forecast of COVID-19 spreading in China, Italy and France

Early stage COVID-19 disease dynamics in Germany: models and parameter identification

The Covid-19 outbreak in Spain. A simple dynamics model, some lessons, and a theoretical framework for control response

Modeling the COVID-19 pandemic dynamics in Iran and China

Short memory fractional differential equations for new memristor and neural network design. Nonlinear Dyn

Short memory principle and a predictor-corrector approach for fractional differential equations

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The author wishes to thank the editor and anonymous referees for their helpful comments and suggestions for the significant improvement of the manuscript.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

The world has never witnessed a population loss as devastating as the coronavirus has produced over decades. To safeguard the uninfected population, all countries threatened by the epidemic were forced to follow control tactics like using face masks, limiting large gatherings, treating infected people, etc. As control strategies differ in each country, the infected population in each country changes simultaneously. While a few countries could control a pandemic, others experienced a rise in new cases. To witness this scenario for multiple countries simultaneously, we divided the 16 (MAY 2020 TO AUG 2021) months into four phases, each with four months. Let's also take a look at WHO's latest situation reports. We can see that coronavirus transmission and expansion criteria cannot be the same in different countries. We tried to understand the COVID-19 transmission dynamics in this work by looking at multiple countries. We looked at China, the USA, the UK, Italy, France, Germany, Spain, and Iran to see how COVID-19 spreads in various nations. Figures 19 and 20 show a graphical depiction of the real data for these eight countries used in the study.The behavior of the viral infection is unknown to the scientist, and as before predicted by the scientist, the infection spread rapidly over the world. Because the virus kills the individual and spreads quickly, doctors face a colossal task. Furthermore, people all around the globe are suffering due to a lack of medical resources. The unique corona virus dynamics have been mathematically