Fractional dynamical probes in COVID-19 model with control interventions: a comparative assessment of eight most affected countries

Abstract

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 $60\%$ 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 $R_0$ for the most affected countries, China, USA, UK, Italy, France, Germany, Spain, and Iran.

Introduction

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 [1–9].

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 [10–16]. 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.

In recent years, the coronavirus has been the subject of numerous articles published. Let us have a look at the few. First, Pushpendra and Vedat [20] investigated a time delay, fractional COVID-19 model, using a Caputo-type fractional derivative in 2020. In the same year, Ram Prasad and Renu Verma [21] studied a detailed analysis of the fractional-order COVID-19 model with a case study of Wuhan, China and also Fanelli and Piazza [22] presented on COVID-19 spreading in three nations: China, Italy and France. Following that, Shah et al. [23] published the COVID-19 model’s optimal methods in 2020. Finally, Zeb et al. [24] discussed the dynamics of the COVID-19 model with isolation class toward the end of 2020. Sarkar et al. [25] predicted the COVID-19 pandemic in India in 2020. Pitchaimani and Brasanna Devi [26] then used the stochastic delay to model and discuss COVID-19 the following year, in 2021 and also Araz [27] implemented optimal control on the COVID-19 model in the same year and discussed its impact.

In this article, we have analyzed the dynamical properties of COVID-19 model. Furthermore, we have extended the model by including optimal control into the model. The rest of the paper is organized as follows.

• i.In Sect. 2, the description of proposed COVID-19 model (2) is presented.
• ii.Section 3 presents some preliminary definitions along with the positivity of solution, the basic reproduction number, equilibrium points and also the stability (local and global) of proposed COVID-19 model (2) is discussed.
• iii.In Sect. 4, sensitivity and uncertainty analysis of $R_0$ are presented.
• iv.In Sect. 5, the fractional optimal problem is constructed and analyzed.
• v.Section 6 explains the numerical scheme called Adams–Bashforth–Moulton predictor-corrector method to solve the system of nonlinear fractional differential equation. Furthermore, it suggests numerical simulations of system (2) (without control) and (32) (with control) to illustrate the theoretical results.
• vi.In Sect. 7, we fit our model to daily new cases of COVID-19 infection for China, France, Germany, Iran, Italy, Spain, UK and USA.
• vii.In Sect. 8, the paper finishes with a discussion and conclusion.
• viii.In Sect. 9, we discuss some future challenges.

The model’s novelty lies in viewing a single fractional model in two dimensions, i.e., with and without optimal control, by incorporating quarantined and hospitalized compartments. Also, we fit our model to daily new cases of COVID-19 infection for China, France, Germany, Iran, Italy, Spain, the UK and the USA. To the best of the author’s knowledge, this has never been addressed in the literature for such a complex model. The upcoming section deals with the description of the proposed COVID-19 model (2).

Model presentation

The current work was inspired by a publication by Shahid et al. [28], which discusses a deterministic model of COVID-19. However, this article does not address the hospitalized compartment and stability analysis for endemic equilibrium point. The proposed model is designed in such a way that it overcomes all of the constraints of the work presented by Shahid et al. [28]. After being motivated by the successful advantages and applications of the Caputo fractional derivative in the field of mathematical modeling and particularly in mathematical epidemiology discussed in [17, 29], a new nonlinear fractional-order system for the COVID-19 epidemic has been suggested. To derive our model equations, we divide the total high-risk human population (denoted by N(t)) into seven mutually exclusive epidemiological classes, namely: susceptible to the disease S(t), exposed population E(t), infected population in asymptomatic phase A(t), infected population in symptomatic phase I(t), quarantined population Q(t), hospitalized population H(t) and recovered population R(t) at time t. Infected individuals at the exposed, asymptomatic and symptomatic classes can spread the infection to uninfected individuals at susceptible classes with varying intensity. The quarantine class of people are COVID-19 infected but not infectious as they are separated from the general population. The proposed COVID-19 model includes a total inflow of susceptible individuals into the region at a rate of $\mathrm{\Lambda }$ per unit time. New births, immigration and emigration were included in this parameter. A flow diagram depicting the complete dynamics of the system (2) is shown in Fig. 1.

Fig. 1.

Flowchart diagram for model (2)

• The susceptible population S(t) is composed of people recruited to the disease’s transmission area. The susceptible population increase when there is a lot of recruitment, and they reduce when they are divided into various infected compartments. The susceptible population is reduced in the proposed model by becoming exposed when they meet exposed, asymptomatic, symptomatic at the disease transmission rate ${\beta }_1$, ${\beta }_2$, ${\beta }_3$, respectively. In addition, S(t) also decreases when they die naturally at the rate $\mu$. Thus,

  $$\begin{array}{c}\hfill D_t^{\alpha }S(t)=\mathrm{\Lambda }-({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-\mu S(t).\end{array}$$
• The exposed E(t) population is infected but not virulent enough to spread the virus. Exposed individuals are increased when they get more infected population from susceptible. Exposed individuals are decreased when they become asymptomatic, quarantined, die naturally at the rate ${\sigma }_1$, ${\delta }_1$, $\mu$, respectively. Thus,

  $$\begin{array}{c}\hfill D_t^{\alpha }E(t)=({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-({\sigma }_1+{\delta }_1+\mu )E(t).\end{array}$$
• 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 ${\sigma }_2$, $r_1$, respectively. In addition, E(t) also decrease by natural death rate $\mu$. Thus,

  $$\begin{array}{c}\hfill D_t^{\alpha }A(t)={\sigma }_{1}E(t)-({\sigma }_2+r_1+\mu )A(t).\end{array}$$
• 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 ${\sigma }_2$. 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 $\mu$. Thus,

  $$\begin{array}{c}\hfill D_t^{\alpha }I(t)={\sigma }_{2}A(t)-(h_1+r_2+d_1+\mu )I(t).\end{array}$$
• People at quarantine compartment Q(t) are population divided from exposed class E(t) with the transmission rate ${\delta }_1$. 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 $\mu$. Hence, the quarantine population Q(t) is expressed as follows:

  $$\begin{array}{c}\hfill D_t^{\alpha }Q(t)={\delta }_{1}E(t)-(h_2+r_3+\mu )Q(t).\end{array}$$
• 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 $\mu$. Thus,

  $$\begin{array}{c}\hfill D_t^{\alpha }H(t)=h_{1}I(t)+h_{2}Q(t)-(r_4+d_2+\mu )H(t).\end{array}$$
• People at recovered compartment R(t) are population divided from asymptomatic A(t), symptomatic I(t), quarantined Q(t) 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 $\mu$. Thus,

  $$\begin{array}{c}\hfill D_t^{\alpha }R(t)=r_{1}A(t)+r_{2}I(t)+r_{3}Q(t)+r_{4}H(t)-\mu R(t).\end{array}$$

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 ${\beta }_1$, ${\beta }_2$ and ${\beta }_3$ 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 $\mu$ [31].

The proposed COVID-19 model (1) is governed by a system of nonlinear fractional-order differential equations as follows:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_t^{\alpha }S(t)=& \mathrm{\Lambda }-({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-\mu S(t),\hfill \\ \hfill D_t^{\alpha }E(t)=& ({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-({\sigma }_1+{\delta }_1+\mu )E(t),\hfill \\ \hfill D_t^{\alpha }A(t)=& {\sigma }_{1}E(t)-({\sigma }_2+r_1+\mu )A(t),\hfill \\ \hfill D_t^{\alpha }I(t)=& {\sigma }_{2}A(t)-(h_1+r_2+d_1+\mu )I(t),\hfill \\ \hfill D_t^{\alpha }Q(t)=& {\delta }_{1}E(t)-(h_2+r_3+\mu )Q(t),\hfill \\ \hfill D_t^{\alpha }H(t)=& h_{1}I(t)+h_{2}Q(t)-(r_4+d_2+\mu )H(t),\hfill \\ \hfill D_t^{\alpha }R(t)=& r_{1}A(t)+r_{2}I(t)+r_{3}Q(t)+r_{4}H(t)-\mu R(t).\hfill \end{array}\end{array}$$ 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,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_t^{\alpha }S(t)=& \mathrm{\Lambda }-({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-\mu S(t),\hfill \\ \hfill D_t^{\alpha }E(t)=& ({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-({\sigma }_1+{\delta }_1+\mu )E(t),\hfill \\ \hfill D_t^{\alpha }A(t)=& {\sigma }_{1}E(t)-({\sigma }_2+r_1+\mu )A(t),\hfill \\ \hfill D_t^{\alpha }I(t)=& {\sigma }_{2}A(t)-(h_1+r_2+d_1+\mu )I(t),\hfill \\ \hfill D_t^{\alpha }Q(t)=& {\delta }_{1}E(t)-(h_2+r_3+\mu )Q(t),\hfill \\ \hfill D_t^{\alpha }H(t)=& h_{1}I(t)+h_{2}Q(t)-(r_4+d_2+\mu )H(t).\hfill \end{array}\end{array}$$ 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 $\alpha \in (0,1]$ is the order of the fractional derivative and $D_t^{\alpha }$ denotes ${\displaystyle \frac{{\mathrm{d}}^{\alpha }}{\mathrm{d}{t}^{\alpha }}}$. The classical version of the proposed system (2) is retained when $\alpha =1$. 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

Table 1.

Parameters description

Parameters	Description	Values	Units
		[28]	[Assumed]
$\mathrm{\Lambda }$	The recruitment rate	2274	day${}^{-1}$
${\beta }_1$	The transmission rate by infectives at exposed class	0.00003	day${}^{-1}$
${\beta }_2$	The transmission rate by infectives at asymptomatic phase	0.000111	day${}^{-1}$
${\beta }_3$	The transmission rate by infectives at symptomatic phase	0.0001197	day${}^{-1}$
${\sigma }_1$	The rate at which the exposed become asymptomatic	0.094	day${}^{-1}$
${\sigma }_2$	The rate at which the asymptomatic become symptomatic	0.09	day${}^{-1}$
${\delta }_1$	Rate at which the exposed individuals are diminished by quarantine	0.0986	day${}^{-1}$
$r_1$	Recovery rate from asymptomatic individuals	0.9999	day${}^{-1}$
$r_2$	Recovery rate from symptomatic individuals	0.16	day${}^{-1}$
$r_3$	Recovery rate from quarantined individuals	0.2553	day${}^{-1}$
$r_4$	Recovery rate from hospitalized individuals	0.4449	day${}^{-1}$
$h_1$	Rate at which symptomatic infectives are hospitalized	0.10001	day${}^{-1}$
$h_2$	Rate at which quarantined individuals are hospitalized	0.4129	day${}^{-1}$
$d_1$	Diseases induced rate form symptomatic infectives	0.002	day${}^{-1}$
$d_2$	Diseases induced rate form hospitalized individuals	0.001	day${}^{-1}$
$\mu$	The natural death rate	0.3349	day${}^{-1}$

$$\begin{array}{c}\hfill S(0)=S_2,E(0)=E_2,A(0)=A_2,I(0)=I_2,Q(0)=Q_2\text{and}H(0)=H_2.\end{array}$$ 3

Furthermore, we assume that

$$\begin{array}{c}\hfill S(t)>0,E(t)\ge 0,A(t)\ge 0,I(t)\ge 0,Q(t)\ge 0,H(t)\ge 0,\mathrm{for}\mathrm{all}t>0.\end{array}$$ 4

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

Model analysis

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

Preliminaries

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

$$\begin{array}{c}\hfill D^{\alpha }x(t)=f(x),\alpha \in (0,1],x\in R^n,\end{array}$$ 5

where $D^{\alpha }$ is the Caputo fractional derivative which is given in the following definition.

Definition 1

[13] The Caputo fractional derivative of order $\alpha$ of a function $f(t)\in C^n([t_1,\infty ),\mathbb{R})$ is defined as

$$\begin{array}{c}\hfill D_t^{\alpha }f(t)={\displaystyle \frac{1}{\mathrm{\Gamma }(n-\alpha )}}{\int }_{t_1}^t{\displaystyle \frac{f^{(n)}(\xi )}{{(t-\xi )}^{\alpha +1-n}}}\mathrm{d}\xi ,\end{array}$$

where $t\ge t_1$, $\mathrm{\Gamma }(.)$ is the Gamma function, and n is the positive integer such that $\alpha \in (n-1,n)$.

When $\alpha \in (0,1)$, one has

$$\begin{array}{c}\hfill D_t^{\alpha }f(t)={\displaystyle \frac{1}{\mathrm{\Gamma }(1-\alpha )}}{\int }_{t_1}^t{\displaystyle \frac{f^{{}^{\prime }}(\xi )}{{(t-\xi )}^{\alpha }}}\mathrm{d}\xi .\end{array}$$

Lemma 1

[13] Let x(t) be a continuous function on $[t_1,+\infty )$ and satisfy

$$\begin{array}{c}\hfill D_t^{\alpha }x(t)\le -\lambda x(t)+\mu ,\end{array}$$

$x(t_1)=x_{t_1}$, where $\alpha \in (0,1],\lambda ,\mu \in \mathbb{R}$, and $\lambda \ne 0$, $t_1\le 0$ is the initial time. Then

$$\begin{array}{c}\hfill x(t)\le (x_{t_1}-{\displaystyle \frac{\mu }{\lambda }})E_{\alpha }(-\lambda {(t-t_1)}^{\alpha })+{\displaystyle \frac{\mu }{\lambda }},\end{array}$$ 6

where $E_{\alpha }(·)$ is the Mittag–Leffler function that is defined as

$$\begin{array}{c}\hfill E_{\alpha ,\beta }(z)=\sum _{k=0}^{\infty }{\displaystyle \frac{z_k}{\mathrm{\Gamma }(k\alpha +\beta )}},\end{array}$$

where $\alpha >0,\beta >0$ and $z\in \mathbb{C}$. When $\beta =1$, one has $E_{\alpha }(z):=E_{\alpha ,1}(z)$. Furthermore, $E_{1,1}(z)=e^z.$

Theorem 3.1.1

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

$$\begin{array}{cc}\hfill D_t^{\alpha }x(t)=& f(x),\hfill \\ \hfill x(0)=& x_0,\hfill \end{array}$$

with $\alpha \in (0,1]$, $x\in {\mathbb{R}}^n$ and $f:{\mathbb{R}}^n\to {\mathbb{R}}^n$, i.e., $f={(f_1,f_2\cdots f_n)}^T.$ The equilibrium points of the above system are calculated by solving the equation $f(x)=0.$ These equilibrium points are locally asymptotically stable if all eigenvalues ${\lambda }_j$ of the Jacobian matrix ${\displaystyle J=\frac{\partial f}{\partial x}}$ evaluated at the equilibrium points satisfy

$$\begin{array}{c}\hfill |arg({\lambda }_j)|>{\displaystyle \frac{\alpha \pi }{2}}.\end{array}$$ 7

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 $f(x)=x^n+a_1{x}^{n-1}+a_2{x}^{n-2}+\cdots +a_n$ is given by $D_n(f)={(-1)}^{\frac{n(n-1)}{2}}R(f.f^{\prime })$ where $f^{\prime }$ is the derivative of f(x).

Let $g(x)=x^l+b_1{x}^{l-1}+b_2{x}^{l-2}+\cdots +b_l$, hence R(f, g) is an $(n+l)\otimes (n+l)$ determinant.

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

$$\begin{array}{c}\hfill P(\lambda )={\lambda }^3+a_{1}x{\lambda }^2+a_2\lambda +a_3,\end{array}$$ 8

then

$$\begin{array}{c}\hfill D_3(P)=a_1^2{a}_2^2+18a_1{a}_2{a}_3-14a_3{a}_1^2-4a_2^3-27a_3^2.\end{array}$$ 9

Theorem 3.1.2

[33] If the system (5) has a characteristic polynomial at $x^{\ast }$ 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:

1. If $D_3(P)>0$, $a_i>0,i=1:3,a_1{a}_2-a_3>0$ and $\alpha \in (0,1].$

2. If $D_3(P)<0$, $a_i\ge 0,i=1:3,$ and $\alpha \in (\frac{2}{3},1]$.

3. If $D_3(P)<0$, $a_i\ge 0,i=1,2,a_1{a}_2=a_3$ and $\alpha \in (0,1]$.

If

$$\begin{array}{c}\hfill P(\lambda )={\lambda }^4+a_1{\lambda }^3+a_{2}x{\lambda }^2+a_3\lambda +a_4,\end{array}$$ 10

then its discriminate is

$$\begin{array}{cc}\hfill D_4(P)=& 18a_1^3{a}_2{a}_3{a}_4-27a_1^4{a}_4^2-4a_1^3{a}_3^3-4a_1^2{a}_2^3{a}_4+a_1^2{a}_2^3{a}_3^2+144a_1^2{a}_2{a}_3^2\hfill \\ \hfill & -6a_1^2{a}_3^2{a}_4-80a_1{a}_2^2{a}_3{a}_4+18a_1{a}_2{a}_3^3-192a_1{a}_2{a}_4^2+16a_2^4{a}_4\hfill \\ \hfill & -4a_2^3{a}_3^2-128a_2^2{a}_4^2+144a_1{a}_2^2{a}_4-27a_3^4+256a_4^3.\hfill \end{array}$$ 11

Consider the characteristic equation of the form

$$\begin{array}{c}\hfill {\lambda }^4+a_1{\lambda }^3+a_{2}x{\lambda }^2+a_3\lambda +a_4=0.\end{array}$$ 12

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 $x^{\ast }$ 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 ${\mathrm{\Delta }}_i,i=1:3$ defined by, ${\mathrm{\Delta }}_1=\left|\begin{array}{c}{a}_1\end{array}\right|$, ${\mathrm{\Delta }}_2=\left|\begin{array}{cc}{a}_1& 1\\ a_3& a_2\end{array}\right|$ and ${\mathrm{\Delta }}_3=\left|\begin{array}{ccc}{a}_1& 1& 0\\ a_3& a_2& a_1\\ 0& a_4& a_3\end{array}\right|$,

1. The equilibrium point $x^{\ast }$ is locally asymptotically stable for $\alpha =1$ if and only if $a_4>0$ and ${\mathrm{\Delta }}_i>0,i=1:3$.

2. If $D_4(P)>0,a_1>0$ and $a_2<0$, then the equilibrium point $x^{\ast }$ is unstable for $\alpha >{\displaystyle \frac{2}{3}}$.

3. $D_4(P)<0$ and $a_i>0,i=1:4$, then the equilibrium point $x^{\ast }$ is locally asymptotically stable for $\alpha <{\displaystyle \frac{1}{3}}$. However, if $D_4(P)<0,a_1<0,a_3<0$ and $a_2>0,a_4>0$ then the equilibrium point $x^{\ast }$ is unstable.

4. If $D_4(P)<0$, $a_i>0,i=1:4$ and $a_2={\displaystyle \frac{a_1{a}_4}{a_3}}+{\displaystyle \frac{a_3}{a_1}}$, then the equilibrium point $x^{\ast }$ is locally asymptotically stable for all $\alpha \in (0,1)$.

5. The equilibrium point $x^{\ast }$ is locally asymptotically stable only if $a_4>0$.

Well-posedness

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

Non-negativity and boundedness

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

$$\begin{array}{c}\hfill {\mathbb{R}}_{+}^6=\{x=(x_i)|x_i\ge 0,1\le i\le 6\}.\end{array}$$

Theorem 3.2.2

By the assumption (3), the set

$$\begin{array}{c}\hfill {\displaystyle \mathrm{\Omega }=\{(S(t),E(t),A(t),I(t),Q(t),H(t))\in {\mathbb{R}}_{+}^6|N(t)\le \frac{\mathrm{\Lambda }}{\mu },t\ge 0\}}\end{array}$$

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

$$\begin{array}{cc}& D_t^{\alpha }S(t)+D_t^{\alpha }E(t)+D_t^{\alpha }A(t)+D_t^{\alpha }I(t)+D_t^{\alpha }Q(t)+D_t^{\alpha }H(t)\hfill \\ \hfill & \le \mathrm{\Lambda }-\mu (S(t)+E(t))-(r_1+\mu )A(t)-(r_2+d_1+\mu )I(t)\hfill \\ \hfill & -(r_3+\mu )I(t)-(r_4+d_2+\mu )H(t).\hfill \end{array}$$

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

$$\begin{array}{c}\hfill D_t^{\alpha }N(t)\le \mathrm{\Lambda }-\mu (N(t)),\end{array}$$

By applying Lemma 1, we get

$$\begin{array}{c}\hfill {\displaystyle N(t)\le (\frac{-\mathrm{\Lambda }}{\mu }+N(0))E_{\alpha }(-\mu t^{\alpha })+\frac{\mathrm{\Lambda }}{\mu }.}\end{array}$$

Since $E_{\alpha }(-\mu t^{\alpha })\ge 0$ when ${\displaystyle N(0)\le \frac{\mathrm{\Lambda }}{\mu }}$, we have

$$\begin{array}{c}\hfill {\displaystyle N(t)\le \frac{\mathrm{\Lambda }}{\mu }.}\end{array}$$

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

$$\begin{array}{c}\hfill \mathrm{\Omega }=\{(S(t),E(t),A(t),I(t),Q(t),H(t))|N(t)\le \frac{\mathrm{\Lambda }}{\mu }\}.\end{array}$$ 13

It can be observed that S(t), E(t), A(t), I(t), Q(t) and H(t) are bounded in an invariant set $\mathrm{\Omega }$. This completes the proof. $\square$

The upcoming section deals with the computation of basic reproduction number ($R_0$) and the equilibrium points of the fractional-order COVID-19 model (2).

The basic reproduction number and equilibrium points

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,

$$\begin{array}{c}\hfill R_0=R_{01}+R_{02}+R_{03}={\displaystyle \frac{\mathrm{\Lambda }}{\mu }}({\displaystyle \frac{{\beta }_1}{E_3}}+{\displaystyle \frac{{\beta }_2{\sigma }_1}{E_3{A}_3}}+{\displaystyle \frac{{\beta }_3{\sigma }_1{\sigma }_2}{E_3{A}_3{I}_3}}),\end{array}$$

where

$$\begin{array}{c}\hfill R_{01}={\displaystyle \frac{\mathrm{\Lambda }{\beta }_1}{\mu E_3}},R_{02}={\displaystyle \frac{\mathrm{\Lambda }{\beta }_2{\sigma }_1}{\mu E_3{A}_3}},R_{03}={\displaystyle \frac{\mathrm{\Lambda }{\beta }_3{\sigma }_1{\sigma }_2}{\mu E_3{A}_3{I}_3}}.\end{array}$$

The proposed nonlinear fractional-order COVID-19 model (2) admits two equilibria as follows:

i) The disease-free equilibrium point is

ii) The endemic equilibrium point is

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_1=& {\displaystyle \frac{\mathrm{\Lambda }}{\mu R_0}},E_1={\displaystyle \frac{\mathrm{\Lambda }(R_0-1)}{E_3{R}_0}},A_1={\displaystyle \frac{{\sigma }_1{E}_1}{A_3}},I_1={\displaystyle \frac{{\sigma }_1{\sigma }_2{E}_1}{I_3{A}_3}},\hfill \\ \hfill Q_1=& {\displaystyle \frac{{\delta }_1{E}_1}{Q_3}}\text{and}{H}_1={\displaystyle \frac{E_1}{H_3}}({\displaystyle \frac{{\sigma }_1{\sigma }_2{h}_1}{I_3{A}_3}}+{\displaystyle \frac{h_2{\delta }_1}{Q_3}}),\hfill \end{array}\end{array}$$

where

$$\begin{array}{cc}\hfill E_3=& {\sigma }_1+{\delta }_1+\mu ,A_3={\sigma }_2+r_1+\mu ,I_3=h_1+r_2+d_1+\mu ,\hfill \\ \hfill Q_3=& h_2+r_3+\mu ,H_3=r_4+d_2+\mu .\hfill \end{array}$$

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.

The local stability

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 $R_0$.

Theorem 3.4.1

The disease-free equilibrium of the system (2) is locally asymptotically stable if $R_0<1$.

Proof

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

The characteristic equation of the above matrix is obtained as

$$\begin{array}{c}\hfill f(\lambda )=(\lambda +\mu )(\lambda +Q_3)(\lambda +H_3)({\lambda }^3+P_1{\lambda }^2+P_2\lambda +P_3)=0,\end{array}$$ 14

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill P_1& =A_3+I_3+(1-R_{01})E_3,\hfill \\ \hfill P_2& =((1-R_{01})E_3)(A_3+I_3)+A_3{E}_3(1-R_{02}),\hfill \\ \hfill P_3& =(1-R_{03})E_3{A}_3{I}_3.\hfill \end{array}\end{array}$$ 15

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

$$\begin{array}{c}\hfill {\lambda }_1=-\mu ,{\lambda }_2=-Q_3,{\lambda }_3=-H_3.\end{array}$$

The remaining roots ${\lambda }_4,{\lambda }_5$ and ${\lambda }_6$ are calculated by the following equation,

16

Let be the discriminant of the characteristic polynomial [32]. Thus,

17

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

Corollary 1

The positive equilibrium point of the system (2) is asymptotically stable for $R_0<1$ , if one of the following conditions holds for polynomial and coefficients $P_1,P_2,P_3$ which are given by (15).

1. If , then the necessary and sufficient condition for the equilibrium point to be locally asymptotically stable is $P_1>0,P_3>0,P_1{P}_2-P_3>0$.

2. If , $P_1\ge 0,P_2\ge 0,P_3>0,\alpha <{\displaystyle \frac{2}{3}}$, then the equilibrium point is locally asymptotically stable. Also if , $P_1<0,P_2<0,\alpha >{\displaystyle \frac{2}{3}}$, then all roots of the satisfy the condition $|arg({\lambda }_j)|<{\displaystyle \frac{\alpha \pi }{2}},j=1,2,3$.

3. If , $P_1>0,P_2>0,P_1{P}_2=P_3$, then the equilibrium point is locally asymptotically stable for all $\alpha \in [0,1)$. $\square$

Remark 1

We strive to provide numerical support for our above arguments by using parameter values as $\mathrm{\Lambda }=2274,{\beta }_1=0.00003,{\beta }_2=0.000111,{\beta }_3=0.0001197,{\sigma }_1=0.094,{\sigma }_2=0.09,{\delta }_1=0.0986,$ $r_1=0.9999,r_2=0.16,r_3=0.2553,r_4=0.4449,h_1=0.10001,h_2=0.4129,d_1=0.002,d_2=0.001,\mu =0.3349\text{and}\alpha =1.$ 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 as,

$$\begin{array}{cc}& {\lambda }^6+4.464297402\ast {\lambda }^5+7.784525049\ast {\lambda }^4+6.765681308\ast {\lambda }^3\hfill \\ \hfill & +3.074701298\ast {\lambda }^2+0.6885212084\ast \lambda +0.5933570162=0.\hfill \end{array}$$

The roots of above equation are as follows,

$$\begin{array}{cc}\hfill {\lambda }_1=& -1.479366945,{\lambda }_2=-1.003100000,{\lambda }_3=-0.7808000000,{\lambda }_4=-0.6191665350,\hfill \\ \hfill {\lambda }_5=& -0.3349000000\text{and}{\lambda }_6=-0.2469639220.\hfill \end{array}$$

It can be found that ${\lambda }_j<0,j=1,2\cdots 6$.

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.

Theorem 3.4.2

The endemic equilibrium 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 point to identify the characteristic equation as follows:

The characteristic polynomial of the above matrix can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g(\lambda )& =(\lambda +Q_3)(\lambda +H_3)({\lambda }^4+B_1{\lambda }^3+B_2{\lambda }^2+B_3\lambda +B_4)=0,\hfill \end{array}\end{array}$$ 18

where

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill B_1& =A_1{\beta }_2+E_1{\beta }_1+I_1{\beta }_3+A_3+I_3+\mu +(1-{\displaystyle \frac{R_{01}}{R_0}}),\hfill \\ \hfill B_2& =(1-{\displaystyle \frac{R_{01}}{R_0}})E_3(\mu +A_3+I_3)+(1-{\displaystyle \frac{R_{02}}{R_0}})E_3{A}_3\hfill \\ \hfill & +(A_3+E_3+I_3)(A_1{\beta }_2+E_1{\beta }_1+I_1{\beta }_3)+\mu (A_3+I_3),\hfill \\ \hfill B_3& =(1-{\displaystyle \frac{R_{03}}{R_0}})I_3{A}_3{E}_3+(1-{\displaystyle \frac{R_{01}}{R_0}})(\mu (A_3+I_3)+I_3{A}_3)E_3\hfill \\ \hfill & +(1-{\displaystyle \frac{R_{02}}{R_0}})(I_3+\mu )E_3{A}_3+I_3(E_3+A_3)(E_1{\beta }_1+I_1{\beta }_3)+\mu A_3{I}_3\hfill \\ \hfill & +A_1{A}_3{\beta }_2(E_3+I_3)+A_3{E}_3(E_1{\beta }_1+I_1{\beta }_3)+I_3{A}_1{E}_3{\beta }_2,\hfill \\ \hfill B_4& =A_3{E}_3{I}_3(2\mu +A_1{\beta }_2+E_1{\beta }_1+I_1{\beta }_3).\hfill \\ \hfill \end{array}\end{array}$$ 19

Clearly, Eq.(18) has two negative real roots, ${\lambda }_1=-Q_3,{\lambda }_2=-H_3.$ The remaining roots ${\lambda }_3,{\lambda }_4,{\lambda }_5$ and ${\lambda }_6$ are calculated by the following equation,

$$\begin{array}{c}\hfill B(\lambda )={\lambda }^4+B_1{\lambda }^3+B_2{\lambda }^2+B_3\lambda +B_4=0.\end{array}$$ 20

Let $D_4(B)$ be the discriminant of the characteristic polynomial $B(\lambda )$, which was discussed in [35]. Thus,

$$\begin{array}{cc}& D_4(B)=\left|\begin{array}{ccccccc}1& B_1& B_2& B_3& B_4& 0& 0\\ 0& 1& B_1& B_2& B_3& B_4& 0\\ 0& 0& 1& B_1& B_2& B_3& B_4\\ 4& 3B_1& 2B_2& B_3& 0& 0& 0\\ 0& 4& 3B_1& 2B_2& B_3& 0& 0\\ 0& 0& 4& 3B_1& 2B_2& B_3& 0\\ 0& 0& 0& 4& 3B_1& 2B_2& B_3\end{array}\right|.\hfill \\ \hfill D_4(B)=& 18a_1^3{B}_2{B}_3{B}_4-27B_1^4{B}_4^2-4B_1^3{B}_3^3-4B_1^2{B}_2^3{B}_4+B_1^2{B}_2^3{B}_3^2\hfill \\ \hfill & +144B_1^2{B}_2{B}_3^2-6B_1^2{B}_3^2{B}_4-80B_1{B}_2^{2}3B_4+18B_1{B}_2{B}_3^3\hfill \\ \hfill & -192B_1{B}_2{B}_4^2+16B_2^4{B}_4-4B_2^3{B}_3^2-128B_2^2{B}_4^2\hfill \\ \hfill & +144B_1{B}_2^2{B}_4-27B_3^4+256B_4^3.\hfill \end{array}$$ 21

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

Corollary 2

The positive equilibrium point of the system (2) is asymptotically stable for $R_0>1$ , if one of the following conditions holds for polynomial $B(\lambda )$ and coefficients $B_1,B_2,B_3,B_4$ which are given by (19).

Consider the determinants ${\mathrm{\Delta }}_i,i=1:3$ defined by

$$\begin{array}{c}\hfill {\mathrm{\Delta }}_1=\left|\begin{array}{c}{B}_1\end{array}\right|,{\mathrm{\Delta }}_2=\left|\begin{array}{cc}{B}_1& 1\\ B_3& B_2\end{array}\right|,{\mathrm{\Delta }}_3=\left|\begin{array}{ccc}{B}_1& 1& 0\\ B_3& B_2& B_1\\ 0& B_4& B_3\end{array}\right|.\end{array}$$

1. The equilibrium point is locally asymptotically stable for $\alpha =1$ if and only if $B_4>0$ and ${\mathrm{\Delta }}_i>0,i=1:3$.

2. If $D_4(B)>0,B_1>0$ and $B_2<0$, then the equilibrium point is unstable for $\alpha >{\displaystyle \frac{2}{3}}$.

3. $D_4(B)<0$ and $B_i>0,i=1:4$, then the equilibrium point is locally asymptotically stable for $\alpha <{\displaystyle \frac{1}{3}}$. However, if $D_4(B)<0,B_1<0,B_3<0$ and $B_2>0,B_4>0$ then the equilibrium point is unstable.

4. If $D_4(B)<0$, $B_i>0,i=1:4$ and $B_2={\displaystyle \frac{B_1{B}_4}{B_3}}+{\displaystyle \frac{B_3}{B_1}}$, then the equilibrium point is locally asymptotically stable for all $\alpha \in (0,1)$.

5. The equilibrium point is locally asymptotically stable only if $B_4>0$.

$\square$

Remark 2

We strive to provide numerical support for our above arguments by using parameter values as $\mathrm{\Lambda }=2274,{\beta }_1=0.00005,{\beta }_2=0.00251,{\beta }_3=0.001197,{\sigma }_1=0.094,{\sigma }_2=0.09,{\delta }_1=0.0986,r_1=0.9999,r_2=0.16,r_3=0.2553,$ $r_4=0.4449,h_1=0.10001,h_2=0.4129,d_1=0.002,d_2=0.001,\mu =0.3349\text{and}\alpha =1.$ 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 as,

$$\begin{array}{c}\hfill {\lambda }^6+4.328495670\ast {\lambda }^5+5.691032712\ast {\lambda }^4+1.660796461\ast {\lambda }^3+1.766397253\ast {\lambda }^2+1.291907695\ast \lambda +0.2269273954=0.\end{array}$$

The roots of above equation are as follows,

$$\begin{array}{cc}\hfill {\lambda }_1=& -2.199802549,{\lambda }_2=-1.003100000,{\lambda }_3=-0.7808000000,{\lambda }_4=-0.6320880487,\hfill \\ \hfill {\lambda }_5=& -0.3349000000\text{and}{\lambda }_6=-0.6221949274.\hfill \end{array}$$

It can be found that ${\lambda }_j<0,j=1,2\dots 6.$

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).

Global behavior at equilibrium points

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

Lemma 2

[37] let $y(t)\in {\mathbb{R}}^{+}$ be derivable and continuous function. Then, for any time $t\ge t_0$,

$$\begin{array}{c}\hfill {}_{t_0}{D}^{\alpha }[y(t)-y^{\ast }-y^{\ast }ln\frac{y(t)}{y^{\ast }}]\le (1-\frac{y^{\ast }}{y(t)}){}_{t_0}{D}^{\alpha }y(t),\end{array}$$ 22

$\forall \alpha \in (0,1),y^{\ast }\in {\mathbb{R}}^{+}.$

Note that for $\alpha =1$, the inequalities in (22) becomes equalities.

Let us denote

$$\begin{array}{c}\hfill [y(t)-y^{\ast }-y^{\ast }ln\frac{y(t)}{y^{\ast }}]=g({\displaystyle \frac{y(t)}{y^{\ast }}})\end{array}$$

in upcoming results.

Theorem 3.5.1

If $R_0<1$, the disease-free equilibrium of the system (2) is globally asymptotically stable.

Proof

Let (S(t), E(t), A(t), I(t), Q(t), H(t)) be any positive solution of the system (2) with the initial condition (3), define a Lyapunov functional $W_1(t)$ as follows,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill W_1(t)& ={\eta }_{1}g({\displaystyle \frac{S(t)}{S_0}})+{\eta }_{1}E(t)+{\eta }_{2}A(t)+{\eta }_{3}I(t)+{\eta }_{4}Q(t),\hfill \end{array}\end{array}$$ 23

where ${\eta }_1=Q(t)$, ${\eta }_2={\displaystyle \frac{{\eta }_1{E}_3}{{\sigma }_1}}$, ${\eta }_3={\displaystyle \frac{{\eta }_1{E}_{3}E{A}_3}{{\sigma }_1{\sigma }_2}}$ and ${\eta }_4={\displaystyle \frac{E_3\lambda }{Q_3\mu }}(1+{\displaystyle \frac{A_3}{{\sigma }_1}})$.

Differentiating $W_1(t)$ along the solution of system (2), we obtain

$$\begin{array}{cc}\hfill D_t^{\alpha }{W}_1(t)& \le {\eta }_1(1-{\displaystyle \frac{S_0}{S(t)}})(\mathrm{\Lambda }-({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-\mu S(t))\hfill \\ \hfill & +{\eta }_1(({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-E_{3}E(t))\hfill \\ \hfill & +{\eta }_2({\sigma }_{1}E(t)-A_{3}A(t))+{\eta }_3({\sigma }_{2}A(t)-I_{3}I(t))+{\eta }_4({\delta }_{1}E(t)-Q_{3}Q(t)).\hfill \end{array}$$ 24

Using  ${\displaystyle \frac{\mathrm{\Lambda }}{\mu }=S_0}$ in (24),  we  obtain

$$\begin{array}{cc}\hfill D_t^{\alpha }{W}_1(t)\le & -\frac{{\eta }_1\mu }{S(t)}{(S(t)-S_0)}^2+{\displaystyle \frac{{\eta }_1\mathrm{\Lambda }}{\mu }}({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))-{\eta }_1{E}_{3}E(t)\hfill \\ \hfill & +{\eta }_2({\sigma }_{1}E(t)-A_{3}A(t))+{\eta }_3({\sigma }_{2}A(t)-I_{3}I(t))+{\eta }_4({\delta }_{1}E(t)-Q_{3}Q(t)).\hfill \end{array}$$ 25

Using  (13) in (25),  we  obtain

$$\begin{array}{cc}\hfill D_t^{\alpha }{W}_1(t)\le & -\frac{{\eta }_1\mu }{S(t)}{(S(t)-S_0)}^2+{\eta }_1{E}_{3}E(t)(R_{01}-1)\hfill \\ \hfill & +{\eta }_{1}A(t){\displaystyle \frac{E_3{A}_3}{{\sigma }_1}}(R_{02}-1)+{\eta }_{1}I(t){\displaystyle \frac{E_3{A}_3{I}_3}{{\sigma }_1{\sigma }_2}}(R_{03}-1).\hfill \end{array}$$ 26

It follows from Eq. (26) that $D_t^{\alpha }{W}_1(t)\le 0$ with equality holding $S(t)=S_0,E(t)=A(t)=I(t)=Q(t)=H(t)=0$. By the LaSalle invariance principle, the disease-free equilibrium of the model (2) is globally asymptotically stable. $\square$

Theorem 3.5.2

If $R_0>1$, the endemic equilibrium of the system (2) is globally asymptotically stable.

Proof

Let (S(t), E(t), A(t), I(t), Q(t), H(t)) be any positive solution of the system (2) with the initial condition (3), define a Lyapunov functional $W_2(t)$ as follows,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill W_2(t)& =g({\displaystyle \frac{S(t)}{S_1}})+g({\displaystyle \frac{E(t)}{E_1}})+{\eta }_{1}g({\displaystyle \frac{A(t)}{A_1}})+{\eta }_{2}g({\displaystyle \frac{I(t)}{I_1}})\hfill \\ \hfill & +{\eta }_{3}g({\displaystyle \frac{Q(t)}{Q_1}})+{\eta }_{4}g({\displaystyle \frac{H(t)}{H_1}}),\hfill \end{array}\end{array}$$ 27

where,

$$\begin{array}{cc}& {\eta }_1=\frac{E(t)E_3}{R_0{\sigma }_1(\frac{\mathrm{\Lambda }}{\mu }+E_1)},{\eta }_2=\frac{{\eta }_{1}A(t)A_3}{{\sigma }_2(\frac{\mathrm{\Lambda }}{\mu }+A_1)},\hfill \\ \hfill & {\eta }_3=\frac{{\eta }_{2}I(t)I_3}{{\delta }_1(\frac{\mathrm{\Lambda }}{\mu }+E_1)}\text{and}{\eta }_4=\frac{{\eta }_{3}Q(t)Q_3}{h_1(\frac{\mathrm{\Lambda }}{\mu }+I_1)+h_2(\frac{\mathrm{\Lambda }}{\mu }+Q_1)}.\hfill \end{array}$$

Differentiating $W_2(t)$ along the solution of system (2), we obtain

$$\begin{array}{cc}\hfill D_t^{\alpha }{W}_2(t)& \le (1-{\displaystyle \frac{S_1}{S(t)}})(\mathrm{\Lambda }-({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-\mu S(t))\hfill \\ \hfill & +(1-{\displaystyle \frac{E_1}{E(t)}})(({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-({\sigma }_1+{\delta }_1+\mu )E(t))\hfill \\ \hfill & +{\eta }_1(1-{\displaystyle \frac{A_1}{A(t)}})({\sigma }_{1}E(t)-({\sigma }_2+r_1+\mu )A(t))\hfill \\ \hfill & +{\eta }_2(1-{\displaystyle \frac{I_1}{I(t)}})({\sigma }_{2}A(t)-(h_1+r_2+d_1+\mu )I(t))\hfill \\ \hfill & +{\eta }_3(1-{\displaystyle \frac{Q_1}{Q(t)}})({\delta }_{1}E(t)-(h_2+r_3+\mu )Q(t))\hfill \\ \hfill & +{\eta }_4(1-{\displaystyle \frac{H_1}{H(t)}})(h_{1}I(t)+h_{2}Q(t)-(r_4+d_2+\mu )H(t)).\hfill \end{array}$$ 28

Note that,

$$\begin{array}{c}\hfill \begin{array}{c}\hfill \mathrm{\Lambda }-({\beta }_1{E}_1+{\beta }_2{A}_1+{\beta }_3{I}_1)S_1-\mu S_1=0,\\ \hfill ({\beta }_1{E}_1+{\beta }_2{A}_1+{\beta }_3{I}_1)S_1-({\sigma }_1+{\delta }_1+\mu )E_1=0,\\ \hfill {\sigma }_1{E}_1-({\sigma }_2+r_1+\mu )A_1=0,\\ \hfill {\sigma }_2{A}_1-(h_1+r_2+d_1+\mu )I_1=0,\\ \hfill {\delta }_1{E}_1-(h_2+r_3+\mu )Q_1=0,\\ \hfill h_1{I}_1+h_2{Q}_1-(r_4+d_2+\mu )H_1=0.\end{array}\end{array}$$ 29

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

$$\begin{array}{cc}\hfill D_t^{\alpha }{W}_2(t)& \le -\frac{\mu }{S(t)}{(S(t)-S_1)}^2+S_1({\beta }_1{E}_1+{\beta }_2{A}_1+{\beta }_3{I}_1)(2-\frac{S_1}{S(t)}-\frac{E(t)}{E_1}-\frac{S(t)E_1}{E(t)S_1})\hfill \\ \hfill & -\frac{{\eta }_1{E}_{1}S(t)}{E(t)}({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))-{\eta }_2{A}_1{\sigma }_2\frac{E(t)}{A(t)}-{\eta }_3{I}_1{\sigma }_2\frac{A(t)}{I(t)}\hfill \\ \hfill & -{\eta }_4{Q}_1{\delta }_1\frac{E(t)}{Q(t)}-{\eta }_5\frac{H_1}{H(t)}(h_{1}I(t)+h_{2}Q(t))-{\eta }_5{H}_{3}H(t)\hfill \\ \hfill & -{\eta }_1{E}_{3}E(t)+{\eta }_2{\sigma }_1(E(t)+E_1)-{\eta }_2{A}_{3}A(t)+{\eta }_3{\sigma }_2(A(t)+A_1)\hfill \\ \hfill & -{\eta }_3{I}_{3}I(t)+{\eta }_4{\delta }_1(E(t)+E_1)-{\eta }_4{Q}_{3}Q(t)+{\eta }_5(h_1(I(t)+I_1)+h_2(Q(t)+Q_1)).\hfill \end{array}$$ 30

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

$$\begin{array}{cc}\hfill D_t^{\alpha }{W}_2(t)& \le -\frac{\mu }{S(t)}{(S(t)-S_1)}^2-{\eta }_5{H}_{3}H(t)+{\eta }_1{E}_{3}E(t)(\frac{1}{R_0}-1)\hfill \\ \hfill & +S_1({\beta }_1{E}_1+{\beta }_2{A}_1+{\beta }_3{I}_1)(2-\frac{S_1}{S(t)}-\frac{E(t)}{E_1}-\frac{S(t)E_1}{E(t)S_1})\hfill \\ \hfill & -\frac{{\eta }_1{E}_{1}S(t)}{E(t)}({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))-{\eta }_2{A}_1{\sigma }_2\frac{E(t)}{A(t)}\hfill \\ \hfill & -{\eta }_3{I}_1{\sigma }_2\frac{A(t)}{I(t)}-{\eta }_4{Q}_1{\delta }_1\frac{E(t)}{Q(t)}-{\eta }_5\frac{H_1}{H(t)}(h_{1}I(t)+h_{2}Q(t)).\hfill \end{array}$$ 31

It follows from Eq. (31) that $D_t^{\alpha }{W}_2(t)\le 0$ with equality holding $S(t)=S_1,E(t)=E_1,A(t)=A_1,I(t)=I_1,Q(t)=Q_1$ and $H(t)=H_1$. By the LaSalle invariance principle, the endemic equilibrium of the model (2) is globally asymptotically stable. $\square$

Remark 3

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 $R_0.$

Sensitivity analysis of $R_0$

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 $R_0$.

Uncertainty analysis for $R_0$

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 ${\beta }_1,{\beta }_2,{\beta }_3,{\sigma }_1,{\sigma }_2,{\delta }_1,h_1$ and $d_1$. Each parameter is assumed to be a random variable with a corresponding probability density function. The other four parameters $(\mathrm{\Lambda },\mu ,r_1,r_2)$ 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 ${\beta }_1$ follows normal distribution with mean and standard deviation 0.00005 and 0.0000000004, respectively.

• The transmission rate by infectives at asymptomatic class ${\beta }_2$ follows normal distribution with mean and standard deviation 0.00251 and 0.0000000002, respectively.

• The transmission rate by infectives at symptomatic class ${\beta }_3$ follows normal distribution with mean and standard deviation 0.001197 and 0.0000000001, respectively.

• The progression rate by infectives from exposed to symptomatic class ${\sigma }_1$ 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 ${\sigma }_2$ 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 ${\delta }_1$ 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 indices of ${\beta }_1,{\beta }_2,{\beta }_3,{\sigma }_1,{\sigma }_2,{\delta }_1,h_1$ and $d_1$ with $R_0$

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: ${\beta }_1,{\beta }_2,{\beta }_3,{\sigma }_1,{\sigma }_2,{\delta }_1,h_1$ 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.

Fig. 2.

These scatter plots representations for the basic reproduction number and eight sampled input parameters values are the outputs derived using a sample size of 1000 by the method of Latin hypercube sampling

Table 2.

PRCCs for $R_0$ and eight input parameters

S. no.	Parameter	PRCCs
1	${\sigma }_1$	0.0442
2	${\beta }_1$	0.1354
3	${\beta }_2$	0.1981
4	${\beta }_3$	0.1150
5	${\sigma }_2$	0.1150
6	${\delta }_1$	$-0.0541$
7	$h_1$	$-0.008$
8	$d_1$	$-0.0089$

Fig. 3.

The PRCCs between input parameters and output $R_0$

The parameter with positive PRCCs is directly proportional to $R_0$, i.e., ${\beta }_1,{\beta }_2,{\beta }_3,{\sigma }_1$ and ${\sigma }_2$, whereas the parameter with negative PRCCs is inversely proportional to $R_0$, i.e., ${\delta }_1,h_1$ and $d_1$. After using the sample from LHS, we observe that the transmission rate by infectives at asymptomatic phase ${\beta }_2$ and the transmission rate by infectives at exposed class ${\beta }_1$ are highly correlated with $R_0$ with the corresponding PRCCs values 0.1981 and 0.1354, respectively, tabulated in Table 2. Moderate correlation exists between the transmission rate by infectives at symptomatic phase ${\beta }_3$, the rate at which the asymptomatic becomes symptomatic ${\sigma }_2$ 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 ${\sigma }_1$, rate at which the exposed individuals are diminished by quarantine ${\delta }_1$, 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 ${\beta }_1$ and ${\beta }_2$ are the most important parameters in determining the $R_0$.

Fractional optimal system

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, 40–43]. 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:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_t^{\alpha }S(t)=& \mathrm{\Lambda }-(1-v_1(t))({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-\mu S(t),\hfill \\ \hfill D_t^{\alpha }E(t)=& (1-v_1(t))({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-({\sigma }_1+{\delta }_1+\mu )E(t),\hfill \\ \hfill D_t^{\alpha }A(t)=& {\sigma }_{1}E(t)-({\sigma }_2+r_1+v_2(t)+\mu )A(t),\hfill \\ \hfill D_t^{\alpha }I(t)=& {\sigma }_{2}A(t)-(h_1+r_2+d_1+v_3(t)+\mu )I(t),\hfill \\ \hfill D_t^{\alpha }Q(t)=& {\delta }_{1}E(t)-(h_2+r_3+\mu )Q(t),\hfill \\ \hfill D_t^{\alpha }H(t)=& h_{1}I(t)+h_{2}Q(t)-(r_4+d_2+\mu )H(t)\hfill \end{array}\end{array}$$ 32

with the non-negative initial conditions

$$\begin{array}{c}\hfill S(0)=S_0,E(0)=E_0,A(0)=A_0,I(0)=I_0,Q(0)=Q_0\text{and}H(0)=H_0.\end{array}$$ 33

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., $0\le v_i(t)<1$. 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

$$\begin{array}{c}\hfill \gamma (v_1,v_2,v_3)={\int }_0^{\tau }(b_{1}I(t)+\frac{c_1}{2}{v}_1^2(t)+\frac{c_2}{2}{v}_2^2(t)+\frac{c_3}{2}{v}_3^2(t))\mathrm{d}t\end{array}$$ 34

relied on the state system provided by (32) in accordance with non-negative initial conditions (33). In Eq. (34), $b_1$ represent the positive weight constant of infected population, while $c_1$, $c_2$ and $c_3$ are positive weight constants for transmission rate control, rapid test and treatment, respectively. The phrases ${\displaystyle \frac{c_1}{2}{v}_1^2}$, ${\displaystyle \frac{c_2}{2}{v}_2^2}$ and ${\displaystyle \frac{c_3}{2}{v}_3^2}$ represent the cost linked with similar interventions. It is considered that the costs are proportional to the square of the relevant control function. Our goal for fractional optimal control problem is this to find optimal control functions $v_1^{\ast },v_2^{\ast },v_3^{\ast }$ such that

$$\begin{array}{c}\hfill Z(v_1^{\ast },v_2^{\ast },v_3^{\ast })=min\{Z(v_1,v_2,v_3),(v_1,v_2,v_3)\in V\},\end{array}$$ 35

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

$$\begin{array}{c}\hfill V=\{(v_1,v_2,v_3)|v_i(t)\text{is a Lebesgue measuerable on}[0,1],i=1,2,3\}.\end{array}$$ 36

The Lagrangian $\mathcal{L}$ and Hamiltonian $\mathcal{H}$ for the fractional optimal problem (32)–(36) are, respectively, given by [40, 42, 43]

$$\begin{array}{c}\hfill \mathcal{L}(I,v_1,v_2,v_3)=b_{1}I+\frac{c_1}{2}{v}_1^2+\frac{c_2}{2}{v}_2^2+\frac{c_3}{2}{v}_3^2\end{array}$$ 37

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}& =\mathcal{L}(I,v_1,v_2,v_3)+{\mathrm{\Theta }}_{S_0}{D}^{\alpha }S(t)+{\mathrm{\Theta }}_{E_0}{D}^{\alpha }E(t)+{\mathrm{\Theta }}_{A_0}{D}^{\alpha }A(t)\hfill \\ \hfill & +{\mathrm{\Theta }}_{I_0}{D}^{\alpha }I(t)+{\mathrm{\Theta }}_{Q_0}{D}^{\alpha }Q(t)+{\mathrm{\Theta }}_{H_0}{D}^{\alpha }H(t).\hfill \end{array}\end{array}$$ 38

The above equation can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathcal{H}& =b_{1}I+\frac{c_1}{2}{v}_1^2+\frac{c_2}{2}{v}_2^2+\frac{c_3}{2}{v}_3^2\hfill \\ \hfill & +{\mathrm{\Theta }}_S\{\mathrm{\Lambda }-(1-v_1(t))({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-\mu S(t)\}\hfill \\ \hfill & +{\mathrm{\Theta }}_E\{(1-v_1(t))({\beta }_{1}E(t)+{\beta }_{2}A(t)+{\beta }_{3}I(t))S(t)-({\sigma }_1+{\delta }_1+\mu )E(t)\}\hfill \\ \hfill & +{\mathrm{\Theta }}_A\{{\sigma }_{1}E(t)-({\sigma }_2+r_1+v_2(t)+\mu )A(t)\}\hfill \\ \hfill & +{\mathrm{\Theta }}_I\{{\sigma }_{2}A(t)-(h_1+r_2+d_1+v_3(t)+\mu )I(t)\}\hfill \\ \hfill & +{\mathrm{\Theta }}_Q\{{\delta }_{1}E(t)-(h_2+r_3+\mu )Q(t)\}\hfill \\ \hfill & +{\mathrm{\Theta }}_H\{h_{1}I(t)+h_{2}Q(t)-(r_4+d_2+\mu )H(t)\},\hfill \end{array}\end{array}$$ 39

where ${\mathrm{\Theta }}_S,{\mathrm{\Theta }}_E,{\mathrm{\Theta }}_A,{\mathrm{\Theta }}_I,{\mathrm{\Theta }}_Q,$ and ${\mathrm{\Theta }}_H$ 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

$$\begin{array}{c}\hfill Z(v)={\int }_0^{\tau }\mathcal{L}(t,\pi ,v)dt\end{array}$$ 40

subjected to the dynamical constraints

$$\begin{array}{c}\hfill D_t^{\alpha }\pi (t)=\omega (t,\pi ,v)\end{array}$$ 41

with initial conditions

$$\begin{array}{c}\hfill \pi (0)={\pi }_0,\end{array}$$ 42

where $\pi (t)$ and v(t) are the state and control variables, respectively, $\mathcal{L}$ and $\omega$ are differentiable functions, and $\alpha \in (0,1]$. We have the following theorem.

Theorem 5.0.1

If $(\pi ,v)$ is a minimizer of (40) under the dynamic constraint (41) and the boundary condition (42), then there exists a function $\mathrm{\Theta }$ such that the triplet $(\pi ,v,\mathrm{\Theta })$ satisfies

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_t^{\alpha }\pi (t)& =\frac{\partial \mathcal{H}}{\partial \mathrm{\Theta }}(t,\pi (t),v(t),\mathrm{\Theta }(t))\hfill \\ \hfill D_t^{\alpha }\mathrm{\Theta }(t)& =-\frac{\partial \mathcal{H}}{\partial \pi }(t,\pi (t),v(t),\mathrm{\Theta }(t))\hfill \\ \hfill 0& =\frac{\partial \mathcal{H}}{\partial v}(t,\pi (t),v(t),\mathrm{\Theta }(t))\hfill \\ \hfill \theta (\tau )& =0\hfill \end{array}\end{array}$$ 43

for the Hamiltonian $\mathcal{H}(t,\pi ,v,\mathrm{\Theta })=\mathcal{L}(t,\pi ,v)+{\mathrm{\Theta }}^T(t,\pi ,v).$

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). $\square$

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 $v_1^{\ast },v_2^{\ast },v_3^{\ast }$ for the optimal control problems (32) and (34). Then there exist adjoint variables ${\mathrm{\Theta }}_S,{\mathrm{\Theta }}_E,{\mathrm{\Theta }}_A,{\mathrm{\Theta }}_I,{\mathrm{\Theta }}_Q$ and ${\mathrm{\Theta }}_H$ satisfy the following:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathrm{\Theta }}_S^{\prime }& =({\mathrm{\Theta }}_S-{\mathrm{\Theta }}_E)(1-v_1)({\beta }_{1}E+{\beta }_{2}A)+{\beta }_{3}I+\mu {\mathrm{\Theta }}_S\hfill \\ \hfill {\mathrm{\Theta }}_E^{\prime }& =(1-v_1){\beta }_{1}S+{\sigma }_1({\mathrm{\Theta }}_E-{\mathrm{\Theta }}_A)+{\delta }_1({\mathrm{\Theta }}_E-{\mathrm{\Theta }}_Q)+\mu {\mathrm{\Theta }}_E\hfill \\ \hfill {\mathrm{\Theta }}_A^{\prime }& =(1-v_1){\beta }_{2}S+{\sigma }_2({\mathrm{\Theta }}_A-{\mathrm{\Theta }}_I)+(v_2+r_1+\mu ){\mathrm{\Theta }}_A\hfill \\ \hfill {\mathrm{\Theta }}_I^{\prime }& =-b_1+(1-v_1){\beta }_{3}S+h_1({\mathrm{\Theta }}_I-{\mathrm{\Theta }}_H)+(v_3+r_2+d_1+\mu ){\mathrm{\Theta }}_I\hfill \\ \hfill {\mathrm{\Theta }}_Q^{\prime }& =h_2({\mathrm{\Theta }}_Q-{\mathrm{\Theta }}_H)+(r_3+\mu ){\mathrm{\Theta }}_Q\hfill \\ \hfill {\mathrm{\Theta }}_H^{\prime }& =(r_4+d_2-\mu ){\mathrm{\Theta }}_H,\hfill \end{array}\end{array}$$ 44

with transversality conditions or boundary conditions

$$\begin{array}{c}\hfill {\mathrm{\Theta }}_S=0,{\mathrm{\Theta }}_E=0,{\mathrm{\Theta }}_A=0,{\mathrm{\Theta }}_I=0,{\mathrm{\Theta }}_Q=0\mathrm{and}{\mathrm{\Theta }}_H=0.\end{array}$$

Furthermore, the control functions $v_1^{\ast },v_2^{\ast }$ and $v_3^{\ast }$ are given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill v_1^{\ast }& {\displaystyle =min(1,max(0,\frac{({\mathrm{\Theta }}_S-{\mathrm{\Theta }}_E)(1-v_1)({\beta }_1{E}_1+{\beta }_2{A}_1+{\beta }_3{I}_1)S(t)}{c_1})),}\hfill \\ \hfill v_2^{\ast }& {\displaystyle =min(1,max(0,\frac{{\mathrm{\Theta }}_{A}A(t)}{c_2})),}\hfill \\ \hfill v_3^{\ast }& {\displaystyle =min(1,max(0,\frac{{\mathrm{\Theta }}_{I}I(t)}{c_3})).}\hfill \end{array}\end{array}$$ 45

Proof

The adjoint system (44), i.e.,${\mathrm{\Theta }}_S^{\prime },{\mathrm{\Theta }}_E^{\prime },{\mathrm{\Theta }}_A^{\prime },{\mathrm{\Theta }}_I^{\prime },{\mathrm{\Theta }}_Q^{\prime }$ and ${\mathrm{\Theta }}_H^{\prime }$ are obtained from the Hamiltonian $\mathcal{H}$ as

$$\begin{array}{cc}& {\displaystyle -\frac{\mathrm{d}{\mathrm{\Theta }}_S}{\mathrm{d}t}=\frac{\partial \mathcal{H}}{\partial S},-\frac{\mathrm{d}{\mathrm{\Theta }}_E}{\mathrm{d}t}=\frac{\partial \mathcal{H}}{\partial E},}\hfill \\ \hfill & {\displaystyle -\frac{\mathrm{d}{\mathrm{\Theta }}_A}{\mathrm{d}t}=\frac{\partial \mathcal{H}}{\partial A},-\frac{\mathrm{d}{\mathrm{\Theta }}_I}{\mathrm{d}t}=\frac{\partial \mathcal{H}}{\partial I},}\hfill \\ \hfill & {\displaystyle -\frac{\mathrm{d}{\mathrm{\Theta }}_Q}{\mathrm{d}t}=\frac{\partial \mathcal{H}}{\partial Q},-\frac{\mathrm{d}{\mathrm{\Theta }}_H}{\mathrm{d}t}=\frac{\partial \mathcal{H}}{\partial H},}\hfill \end{array}$$

with zero final time conditions (transversality) conditions

$$\begin{array}{c}\hfill {\mathrm{\Theta }}_S(\tau )=0,{\mathrm{\Theta }}_E(\tau )=0,{\mathrm{\Theta }}_A(\tau )=0,{\mathrm{\Theta }}_I(\tau )=0,{\mathrm{\Theta }}_Q(\tau )=0\text{and}{\mathrm{\Theta }}_H(\tau )=0\end{array}$$

and the characterization of the fractional optimal control given by (45) is obtained by solving the equations

$$\begin{array}{c}\hfill {\displaystyle \frac{\partial \mathcal{H}}{\partial v_1}=0,\frac{\partial \mathcal{H}}{\partial v_2}=0\text{and}\frac{\partial \mathcal{H}}{\partial v_3}=0}\end{array}$$

on the interior of the control set and using the property of the control space V. This completes the proof of Theorem (5.0.2). $\square$

Numerical inspection

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 predictor-corrector algorithm is proposed by Diethelm et al. in [44, 45] to solve fractional differential equations.

Consider the initial value problem

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill D_t^{\alpha }y(t)& =f(t,y(t)),0\le t\le T,\hfill \\ \hfill y^{(k)}(0)& =y_0^{(k)},k=0,1,\dots ,m-1,\alpha \in (m-1,m],\hfill \end{array}\end{array}$$ 46

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

$$\begin{array}{c}\hfill \begin{array}{c}\hfill y(t)=\sum _{k=0}^{m-1}{y}_0^{(k)}{\displaystyle \frac{t^k}{{\displaystyle k!}}}+{\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}{\int }_0^t{(t-\tau )}^{\alpha -1}f(\tau ,y(\tau ))d\tau .\end{array}\end{array}$$ 47

Consider the uniform grid $\{t_n=nh/n=0,1,\dots ,N\}$ 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 $y_h(t_j),j=1,2,\dots ,n$ and want to obtain $y_h(t_{n+1})$ by means of the equation [44, 45]

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill y_h(t_{n+1})& =\sum _{k=0}^{m-1}{\displaystyle \frac{t_{n+1}^k}{{\displaystyle k!}}}{y}_0^{(k)}+{\displaystyle \frac{h^{\alpha }}{\mathrm{\Gamma }(\alpha +2)}}f(t_{n+1},y_h^P(t_{n+1}))\hfill \\ \hfill & +{\displaystyle \frac{h^{\alpha }}{\mathrm{\Gamma }(\alpha +2)}}\sum _{j=0}^n{a}_{j,n+1}f(t_j,y_n(t_j)),\hfill \end{array}\end{array}$$ 48

where

$$\begin{array}{c}\hfill a_{j,n+1}=\left\{\begin{array}{cc}{n}^{\alpha +1}-(n-\alpha ){(n+1)}^{\alpha },\hfill & \text{if}j=0,\hfill \\ {(n-j+2)}^{\alpha +1}+{(n-j)}^{\alpha +1}-2{(n-j+1)}^{\alpha +1},\hfill & \text{if}1\le j\le n,\hfill \\ 1,\hfill & \text{if}j=n+1.\hfill \end{array}\right)\end{array}$$ 49

The preliminary approximation $y_h^P(t_{n+1})$ is called predictor and is given by

$$\begin{array}{c}\hfill \begin{array}{c}\hfill y_h^P(t_{n+1})=\sum _{k=0}^{m-1}{\displaystyle \frac{t_{n+1}^k}{{\displaystyle k!}}}{y}_0^{(k)}+{\displaystyle \frac{1}{\mathrm{\Gamma }(\alpha )}}\sum _{j=0}^n{b}_{j,n+1}f(t_j,y_n(t_j)),\end{array}\end{array}$$ 50

where

$$\begin{array}{c}\hfill \begin{array}{c}\hfill b_{j,n+1}={\displaystyle \frac{h^{\alpha }}{\alpha }}({(n+1-j)}^{\alpha }-{(n-j)}^{\alpha }).\end{array}\end{array}$$ 51

Error in this method is

$$\begin{array}{c}\hfill \begin{array}{c}\hfill ma{x}_{j=0,1,\dots ,N}|y(t_j)-y_h(t_j)|=O(h^p),\end{array}\end{array}$$ 52

where $p=min(2,1+\alpha ).$

Numerical analysis has been carried out using MATLAB(R2015a), to represent the system (2) graphically. A nonlinear fractional-order COVID-19 model (2) has been solved numerically by adopting predictor-corrector algorithm [44–46], 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.

Dynamics of the system (2) without control strategies

In this subsection, we deal with numerical analysis for the fractional-order COVID model (2).

Biological interpretation when ${\mathit{R}}_{\mathbf{0}}\mathbf{<}\mathbf{1}$

Figure 4 has been plotted using the parameters listed in Table 1 for $\alpha =0.95$, 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.

Fig. 4.

Denotes graph trajectories of S(t), E(t), A(t), I(t), Q(t) and H(t) versus time t of system (2) for $\alpha =0.95$, where $R_0=0.4958<1$

Biological interpretation when ${\mathit{R}}_{\mathbf{0}}\mathbf{>}\mathbf{1}$

Figure 5 has been plotted using the parameters listed in Table 1 expect for ${\beta }_1=0.00005,{\beta }_2=0.00251,{\beta }_3=0.001197$ with $\alpha =0.95$, where $R_0=2.9285>1$. It can be observed from Fig. 5 that population of all the compartments survives. It is evidenced 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.

Fig. 5.

Denotes graph trajectories of S(t), E(t), A(t), I(t), Q(t) and H(t) versus time t of system (2) for $\alpha =0.95$, where $R_0=2.9285>1$

According to the sensitivity analysis, the highly sensitive parameter is ${\beta }_2$, 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 ${\beta }_1$ and the transmission rate of infected individuals at symptomatic stage ${\beta }_3$ are less sensitive than the transmission rate of infected individuals at asymptomatic stage ${\beta }_2$. 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 ${\beta }_2$ has a higher sensitivity than ${\beta }_1$ and ${\beta }_3$.

Impact of fractional-order

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 $\alpha$ approaches 1, the memory effects are decreased.

Figure 6 has been plotted using the same parameters used for Fig. 4 for different values of $\alpha$, where $R_0=0.4958<1$. Figure 7 has been plotted using the same parameters used for Fig. 5 for different values of $\alpha$, 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 $\alpha =1$. In case of fractional-order system memory effect is indirectly proportional to the value of $\alpha$.

Fig. 6.

Denotes graph trajectories of system (2) for different values of $\alpha$, where $R_0=0.4958<1$

Fig. 7.

Denotes graph trajectories of system (2) for different values of $\alpha$, where $R_0=2.9285>1$

Impact of disease transmission rates ${\mathit{\beta }}_{\mathbf{1}}\mathbf{,}{\mathit{\beta }}_{\mathbf{2}}\text{and}{\mathit{\beta }}_{\mathbf{3}}$

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 $\alpha =0.98$. Let us have a brief on transmission rates ${\beta }_1,{\beta }_2$ and ${\beta }_3$.

• The parameter ${\beta }_1$ 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 ${\beta }_1$, as shown in Fig. 8. It is observed that the symptomatic population I(t) increases as ${\beta }_1$ increases. It is also witnessed from Fig. 3 that the transmission rate ${\beta }_1$ has the second-highest sensitivity value while ${\beta }_2$ is the first highest sensitivity value and thirdly, the ${\beta }_3$. The partial rank correlation coefficient value of ${\beta }_1$ is found to be 0.1354 has been enlisted in Table 2. This scenario biologically implies that the transmission rate ${\beta }_1$ has the eligibility of higher disease spread than ${\beta }_3$.

• The parameter ${\beta }_2$ 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 ${\beta }_2$, as shown in Fig. 9. It is observed that the symptomatic population I(t) increases as ${\beta }_2$ increases. It is also witnessed from Fig. 3 that the transmission rate ${\beta }_2$ has the highest sensitivity value compared to all the parameters. The partial rank correlation coefficient value of ${\beta }_2$ is found to be 0.1981 has been enlisted in Table 2. This scenario biologically implies that the transmission rate ${\beta }_2$ has a crucial role in disease spread than other transmission rates ${\beta }_1$ and ${\beta }_3$.

• The parameter ${\beta }_3$ 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 ${\beta }_3$, as shown in Fig. 10. It is witnessed that the symptomatic population I(t) increases as ${\beta }_3$ increases. This scenario biologically implies that the symptomatic population depends on ${\beta }_3$, i.e., the symptomatic population is directly proportional to the value of ${\beta }_3$. So, this Fig. 10 conveys to us that the symptomatic population can be controlled once the transmission rate ${\beta }_3$ 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 ${\beta }_3$ can be reduced, automatically the symptomatic population.

According to the above analysis of disease transmission rates ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$, 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 COVID-19 when they contact an asymptomatic individual, as they tend to spread disease in the absence of symptoms. Thus, the above scenario cites the biological reason for COVID-19’s widespread.

Fig. 8.

The above figure has been plotted with same initial conditions and parameters used for Fig. 5 with $\alpha =0.98$, which denotes the variation of parameter ${\beta }_1$ among symptomatic phase I(t) of system (2), where $R_0=2.9285>1$

Fig. 9.

The above figure has been plotted with same initial conditions and parameters used for Fig. 5 with $\alpha =0.98$, which denotes the variation of parameter ${\beta }_2$ among symptomatic phase I(t) of system (2), where $R_0=2.9285>1$

Fig. 10.

The above figure has been plotted with same initial conditions and parameters used for Fig. 5 with $\alpha =0.98$, which denotes the variation of parameter ${\beta }_3$ among symptomatic phase I(t) of system (2), where $R_0=2.9285>1$

Dynamics of the system (32) with control strategies

This subsection deals with the analysis of impact of the control strategies $v_1,v_2$ and $v_3$. Figures 11, 12, 13 and 14 has been plotted with the same parameters and initial condition used for Fig. 5 with $\alpha =0.95$.

• Strategy.1. when ${\mathit{v}}_{\mathbf{1}}\mathbf{\ne }\mathbf{0}\mathbf{,}{\mathit{v}}_{\mathbf{2}}\mathbf{\ne }\mathbf{0}$ and ${\mathit{v}}_{\mathbf{3}}\mathbf{=}\mathbf{0}$.

  In strategy 1, we set the control measure $v_1\ne 0$ (non-pharmaceutical interventions), $v_2\ne 0$ (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 time. The infected population tends to zero as the control measure increases, whereas the susceptible population progressively increases.

• Strategy.2. when ${\mathit{v}}_{\mathbf{1}}\mathbf{=}\mathbf{0}\mathbf{,}{\mathit{v}}_{\mathbf{2}}\mathbf{\ne }\mathbf{0}$ and ${\mathit{v}}_{\mathbf{3}}\mathbf{\ne }\mathbf{0}$.

  In strategy 2, the control measures are $v_1=0$ (non-pharmaceutical interventions), $v_2\ne 0$ (rapid test for infectives at an asymptomatic stage), and $v_3\ne 0$ (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 ${\beta }_1$, ${\beta }_2$, and ${\beta }_3$ 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.

• Strategy.3. when ${\mathit{v}}_{\mathbf{1}}\mathbf{\ne }\mathbf{0}\mathbf{,}{\mathit{v}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$ and ${\mathit{v}}_{\mathbf{3}}\mathbf{\ne }\mathbf{0}$.

  In strategy 3, we adopted the control measures $v_1\ne 0$ (non- pharmaceutical interventions), $v_2=0$ (rapid test to infectives at asymptomatic stage) and $v_3\ne 0$ (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 ($v_1\ne 0$) and the infectives are treated properly ($v_3\ne 0$), 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.

• Strategy.4. when ${\mathit{v}}_{\mathbf{1}}\mathbf{\ne }\mathbf{0}\mathbf{,}{\mathit{v}}_{\mathbf{2}}\mathbf{\ne }\mathbf{0}$ and ${\mathit{v}}_{\mathbf{3}}\mathbf{\ne }\mathbf{0}$.

  In strategy 4, we set the control measure $v_1\ne 0$ (non-pharmaceutical interventions), $v_2\ne 0$ (rapid test to infectives at asymptomatic stage) and $v_3\ne 0$ (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.

Fig. 11.

The above figure denotes the impact of control on infectives for strategy No.1: ($v_1\ne 0,v_2\ne 0,v_3=0$)

Fig. 12.

The above figure denotes the impact of control on infectives for strategy No.2: ($v_1=0,v_2\ne 0,v_3\ne 0$)

Fig. 13.

The above figure denotes the impact of control on infectives for strategy No.3: ($v_1\ne 0,v_2=0,v_3\ne 0$)

Fig. 14.

The above figure denotes the impact of control on infectives for strategy No.4: ($v_1\ne 0,v_2\ne 0,v_3\ne 0$)

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.

Parameter estimation

The estimated resulting point values that are obtained through curve fitting are presented in Table 4. The parameters are considered as follows: $\mathrm{\Lambda }=2274,{\beta }_1=0.00003,{\beta }_2=0.000111,{\beta }_3=0.0001197,{\sigma }_1=0.094,{\sigma }_2=0.09,{\delta }_1=0.00004,r_1=0.9999,$ $r_2=0.16,r_3=0.2553,r_4=0.4449,h_1=0.10001,h_2=0.4129,d_1=0.002,d_2=0.001\text{and}\mu =0.3349$. These parameters are used in the curve fitting with the initial condition listed in Table 3.

Table 4.

Estimated parameter values of the model (2) for the countries China, USA, UK, Italy, France, Germany, Spain, Iran for phase 1, phase 2, phase 3 and phase 4

Parameters	China	USA	UK	Italy	France	Germany	Spain	Iran
Phase 1
${\beta }_1$	0.00004	0.00026	0.00029	0.00009	0.00009	0.00012	0.00001	0.00015
${\beta }_2$	0.00101	0.00026	0.03892	0.0069	0.00837	0.00616	0.00806	0.00537
${\beta }_3$	0.00359	0.00273	0.002	0.00125	0.00226	0.00133	0.00423	0.00853
${\sigma }_1$	1.38805	0.11143	0.61477	0.14328	0.24014	0.06381	0.01139	1.05788
${\sigma }_2$	0.00808	0.37684	0.1581	0.06478	0.03731	0.01397	0.04956	0.00008
${\delta }_1$	0.26933	0.1563	0.15992	0.33229	0.33307	0.3705	0.73241	0.285
$h_1$	0.52123	0.51084	0.49615	0.49901	0.50428	0.50053	0.50254	0.00859
$d_1$	0.02553	0.00209	0.00168	0.00277	0.00494	0.00281	0.00209	0.00909
Phase 2
${\beta }_1$	0.00012	0.00036	0.00002	0.00004	0.0006	0.00009	0.00009	0.001
${\beta }_2$	0.00935	0.00035	0.00971	0.01167	0.00044	0.01805	0.00379	0.01855
${\beta }_3$	0.00159	0.0038	0.00098	0.00173	0.01159	0.00517	0.00112	0.00891
${\sigma }_1$	0.1935	0.21821	0.00698	0.19984	0.10599	0.19697	0.24783	0.24977
${\sigma }_2$	0.00708	0.42354	0.09491	0.07572	1.09817	0.38457	0.28016	0.04624
${\delta }_1$	0.4085	0.10666	0.4861	0.32141	0.44031	1.09854	0.20582	0.00276
$h_1$	0.49797	0.50591	0.5155	0.51799	0.47546	0.52017	0.50517	0.51668
$d_1$	0.00253	0.00081	0.00453	0.004	0.01848	0.00617	0.00005	0.00971
Phase 3
${\beta }_1$	0.00027	0.57291	0.22401	0.38324	0.00011	0.00017	0.00011	0.00035
${\beta }_2$	0.00759	0.00297	0.00359	0.00010	0.00317	0.00405	0.00390	0.00297
${\beta }_3$	0.00745	0.00150	0.00171	0.00270	0.00021	0.00076	0.00140	0.00880
${\sigma }_1$	0.83848	0.47551	0.21380	0.36216	0.16487	0.03739	0.05561	0.47057
${\sigma }_2$	0.01161	0.09362	0.29435	0.01072	0.00388	0.06884	0.31240	0.00496
${\delta }_1$	0.20023	0.11127	0.15637	0.40755	0.40792	0.33231	0.15320	0.15613
$h_1$	0.49091	0.50863	0.50450	0.51040	0.50795	0.50400	0.50283	0.50291
$d_1$	0.00431	0.01081	0.00689	0.01005	0.00660	0.00427	0.00434	0.00109
Phase 4
${\beta }_1$	0.00012	0.00037	0.00029	0.00072	0.00015	0.00009	0.00038	0.00024
${\beta }_2$	0.00917	0.00032	0.0027	0.01917	0.00585	0.00735	0.00921	0.00711
${\beta }_3$	0.00213	0.00443	0.00478	0.0117	0.00387	0.00011	0.00878	0.00223
${\sigma }_1$	0.03085	0.86723	0.10458	5.3965	0.37149	0.36012	0.65545	0.18798
${\sigma }_2$	0.00739	0.00556	0.0804	0.00266	0.0613	0.03337	0.00581	0.0004
${\delta }_1$	0.66585	0.53597	0.20224	5.19249	0.45202	0.34285	0.33273	0.25036
$h_1$	0.49506	0.50793	0.50417	0.56561	0.49765	0.50092	0.49241	0.49909
$d_1$	0.0002	0.00569	0.00262	0.07002	0.00127	0.00532	0.00826	0.00086

Table 3.

Initial population values used for data calibration of the model (2) for the countries China, USA, UK, Italy, France, Germany, Spain and Iran for phase 1, phase 2, phase 3 and phase 4

Initial population	China	USA	UK	Italy	France	Germany	Spain	Iran
Phase 1
S(0)	90000	1113000	187500	209500	200000	169000	240000	98000
E(0)	88000	1112000	186500	209000	190000	166000	230000	98000
A(0)	86000	1111000	185500	208000	180000	165000	220000	97000
I(0)	84000	1110000	183500	207000	170000	164000	200000	96000
Q(0)	82000	1109000	182500	206000	160000	163000	190000	94000
H(0)	80000	1108000	181500	205000	150000	162000	180000	93000
Phase 2
S(0)	92000	6100000	370000	300000	190000	276000	500000	410000
E(0)	91000	6090000	360000	290000	180000	266000	490000	400000
A(0)	90000	6080000	350000	280000	170000	256000	480000	390000
I(0)	89780	6070000	340000	270000	169367	246000	470973	376894
Q(0)	89000	6060000	330000	260000	150000	236000	460000	360000
H(0)	88000	6050000	320000	250000	140000	226000	450000	350000
Phase 3
S(0)	99000	20700000	2900000	2357000	2997000	2132150	2230000	1534000
E(0)	98000	20600000	2800000	2446000	2840000	2093400	2130000	1430000
A(0)	97000	20500000	2700000	2525000	2743000	1966500	2030000	1320000
I(0)	96023	20400000	2600000	2129376	2637018	1762637	1930000	1231429
Q(0)	95000	20300000	2500000	2000000	2540000	1656000	1830000	1120000
H(0)	94000	20200000	2400000	1945000	2430000	1554000	1730000	1100000
Phase 4
S(0)	130000	35520000	4700000	7110000	6100000	3700000	3800000	2815000
E(0)	120000	34560000	4600000	6220000	5920000	3600000	3700000	2702400
A(0)	110000	33540000	4500000	5330000	5820000	3500000	3600000	2622000
I(0)	102517	32463748	4434157	4035617	5703499	3423900	3524077	2516157
Q(0)	90000	31510000	4300000	3240000	5630000	3323900	3424077	2316159
H(0)	80000	30520000	4200000	2205000	5540000	3200000	3324077	2216157

Data

The data were obtained from datahub (https://datahub.io/core/covid-19) [47]. The used data represent daily new COVID-19 cases of eight countries, i.e, China, USA, UK, Italy, France, Germany, Spain and Iran. Data are collected routinely on a daily basis and was retrieved for the period beginning from May 2020 to August 2021. The pictorial representation of the raw data for eight countries is plotted in Figs. 19 and 20.

Fig. 19.

Denote real data of COVID-19 infected cases for countries China, UK, Italy, France, Germany, Spain, Iran for phase 1, phase 2, phase 3 and phase 4

Fig. 20.

Denote real data of COVID-19 infected cases for USA for phase 1, phase 2, phase 3 and phase 4

Curve fitting

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,

$$\begin{array}{c}\hfill RSS=\sum _{i=1}^n{\theta }_i^2=\sum _{i=1}^n{(Y_i-\widehat{Y})}^2,\end{array}$$

where ${\theta }_i=(Y_i-\widehat{Y})$ 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.

Fig. 15.

Denote the COVID-19 infected cases at symptomatic phase I(t) of system (2) fitted to real data of eight countries for phase 1

Fig. 16.

Denote the COVID-19 infected cases at symptomatic phase I(t) of system (2) fitted to real data of eight countries for phase 2

Fig. 17.

Denote the COVID-19 infected cases at symptomatic phase I(t) of system (2) fitted to real data of eight countries for phase 3

Fig. 18.

Denote the COVID-19 infected cases at symptomatic phase I(t) of system (2) fitted to real data of eight countries for phase 4

Table 5.

Value of $R_0$ using estimated parameter values

Phase no.	China	USA	UK	Italy	France	Germany	Spain	Iran
Phase 1	3.792	3.869	1.0065	6.7417	11.7359	3.6395	4.912	17.834
Phase 2	10.653	6.204	5.564	13.8	6.234	5.564	6.1977	4.778
Phase 3	2.49687	7.9916	5.3619	3.970	3.7441	2.6921	3.2020	9.9458
Phase 4	2.1795	2.3160	5.4796	48.5951	10.3758	13.2520	25.1879	10.8987

Phase 1: MAY 2020 TO AUGUST 2020

• Daily new infected cases of China, USA, UK, Italy, France, Germany, Spain, Iran were, respectively, 83959, 1115991, 183501, 207428, 169387, 164077, 215216, 95646, 83959 at MAY 2020 and 89914, 6025632, 338083, 269214, 321160, 244802, 462858, 375212 at AUG 2020.

• During phase 1, infectives of China, USA, UK, Italy, France, Germany, Spain, Iran have been increased by $7\%,439\%,84\%,$ $30\%,90\%,49\%,115\%,292\%,$ respectively. First, the USA has a high percentage of increased infected cases among all eight countries. Secondly Iran, thirdly Spain have more infectives. Throughout phase 1, however, China has a relatively low rate of infectives.

Phase 2: SEPTEMBER 2020 TO DECEMBER 2020

• Daily new infected cases of China, USA, UK, Italy, France, Germany, Spain, Iran were, respectively, 89933, 6068478, 339415, 270189, 326264, 246015, 470973, 376894 at SEP 2020 and 95963, 20161386, 2496235, 2107166, 2677660, 1760520, 1928265, 1225142 at DEC 2020.

• During phase 2, infectives of China, USA, UK, Italy, France, Germany, Spain, Iran have been increased by $7\%,232\%,635\%,$ $680\%,720\%,615\%,309\%,225\%,$ respectively. France has a very high percentage of increased infected cases among all eight countries during phase 2. Secondly Italy, thirdly the UK have more infectives. However, China has a very low percentage of infectives throughout phase 2.

Phase 3: JANUARY 2021 TO APRIL 2021

• Daily new infected cases of China, USA, UK, Italy, France, Germany, Spain, Iran were, respectively, 96023, 20326157, 2549693, 2129376, 2697018, 1762637, 1928265, 1231429 at SEP 2020 and 102494, 32417504, 4432246, 4022653, 5677829, 3405365, 3524077, 2499077 at DEC 2020.

• During phase 3, infectives of China, USA, UK, Italy, France, Germany, Spain, Iran have been increased by $7\%,60\%,74\%,$ $89\%,110\%,93\%,83\%,103\%,$ respectively. France has a very high percentage of increased infections among all eight countries during phase 3. Secondly Iran, thirdly Germany have more infectives. However, China has a very low percentage of infectives throughout phase 3.

Phase 4: MAY 2021 TO AUGUST 2021

• Daily new infected cases of China, USA, UK, Italy, France, Germany, Spain, Iran were, respectively, 102517, 32463748, 4434157, 4035617, 5703499, 3423900, 3524077, 2516157 at SEP 2020 and 107073, 39341669, 6821356, 4539991, 6835022, 3965681, 4855065, 4992063 at DEC 2020.

• During phase 4, infectives of China, USA, UK, Italy, France, Germany, Spain, Iran have been increased by $5\%,21\%,54\%,$ $12\%,20\%,16\%,38\%,98\%,$ respectively. France has a very high percentage of increased infections among all eight countries during phase 4. Secondly Iran, thirdly Germany have more infected cases. However, China has a very low percentage of infectives throughout phase 4.

It can be observed from Figs. 19 and 20 that all the eight countries have recorded a gradual increase in daily new infected COVID-19 cases during May 2020 to Aug 2021. The study emphasis that all the eight countries have faced drastic epidemic outbreak during phase 2 when compare to phase 1, phase 3 and phase 4.

Discussion and conclusion

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 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.

Future challenges

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 [54–56], 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 [58–60], 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.

Data Availability Statement

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