A fractional mathematical model for COVID-19 outbreak transmission dynamics with the impact of isolation and social distancing

1.   Introduction

A novel coronavirus (SARS-CoV-2) emerged out of Wuhan city at the end of December, 2019 ^[1,2]. After spread to almost all countries and was declared a pandemic ^[3]. The outbreak has mostly been controlled in some countries since the end of April 2021, but still remains a series of public health and social-economic the problem in all over the world ^[4,5,6]. We know that most people infected with 2019-nCoV will experience mild to moderate respiratory illness, such as difficulty in breathing, sickness, cough, fever, and other symptoms ^{[11,13,14,15]}. When an infected person sneezes, coughs, or comes in touch with a contaminated surface, the virus is mostly spread to healthy people via the mouth, nose, and eyes ^[7,8]. The usual incubation time is 1 to 14 days, according to ^[9]. As the vaccine is not yet available everywhere, the control measures, for example, social distancing, wearing of masks, regular hand sanitation using sanitizer or soaps, and quarantine of the suspected individuals, are effective interventions that control the virus.

Mathematical models are useful tools for analysing outbreak dynamics. Mathematical models can be used to predict the virus's future spread, allowing the government to be prepared and take the necessary actions ^[18,19,20]. Several mathematical models were developed to better understand the behaviour of coronavirus disease. Researchers in ^[21] introduced a novel fractional model of COVID-19 while accounting for the effects of isolation and quarantine. The Mittag-Leffler, exponential kernels and power-law were used to simulate the proposed model. Mathematical modeling is an essential tool for formulating control strategies and forces guessing the spreading of COVID-19. To illustrate the dynamics of COVID-19, Chen and Zhao ^[7] established a susceptible, un-quarantine infected, quarantine infected, confirmed infected (SUQC) model. Similarly, Song et al. ^[17] formulated a mathematical model based on COVID-19 epidemiology, close contact, and the isolation of healthy persons. Acknowledge the means to control the spread of coronavirus in the people has been recorded in various mathematical modeling ^[12,22].

Fractional calculus is the computation of integrals or derivatives of any positive real order. In recent years, there has been a lot of interest in the use of fractional differential equations. Many researchers use fractional differential equations in their research ^[24,27,46]. Therefore, many researchers have shown more interest in studying fractional derivative and fractional integrals. In the investigation of differential equation for optimization, numerical and qualitative importance was contributed by researchers ^[25,26]. It is also notable that fractional differential equation has been characterized in various manners. It is well-known that regular kernels are not included in the definite integral. Therefore types of seeds have been included in different definitions. One crucial definition that has recently attracted the attention of researchers is the ABC derivative presented by Atangana, Baleanu, and Caputo ^[28] in 2016. This derivative exhibits the singular kernel by non-singular kernel, and So much has been studied ^[29,30]. Because most of the non-linear problems are challenging to solve for their analytical or exact solution, different numerical methods have been introduced to solve these problems ^{[10,23,31,32]}. The best numerical procedure has recently been developed to solve partial fractional differential equation under the ABC derivative; see for detail ^[33,34].

The following model investigates the COVID-19 transmission dynamic. The model included seven compartments: susceptible population $\mathscr{S}$, expose population $\mathscr{E}$, infected papulation $\mathscr{I}$, Asymptomatic population $\mathscr{A}$, Quarantined population $\mathscr{Q}$, Hospitalized population $\mathscr{H}$ and the removed class $\mathscr{R}$ as:

with the following initial conditions $\mathscr{S}(0) > 0, \mathscr{E}(0)\geq0, \mathscr{I}(0)\geq0, \mathscr{A}(0)\geq0, \mathscr{Q}(0)\geq, \mathscr{H}(t)\geq0, \mathscr{R}(0)\geq0.$ The COVID-19 transmission dynamics are represented in the Figure 1 flow chart.

Some essential assumptions we put on the model are that all of the model (1.1) involved parameters be non-negative. The above model is examined from three perspectives. First, since the presented model (1.1) is recently defined, we use fixed point theory to demonstrate its existence. However, since stability is vital, we will examine Hyers-Ulam type stability for the proposed model. We refer to ^[43,44,45] for further information on the general models that use ordinary derivatives of fractional order. Furthermore, the model under consideration is nonlinear because it is sometimes impossible to discover an accurate solution to nonlinear situations. Therefore several numerical processes (methods) have been developed in the literature to deal with similar problems, see ^{[39,40,41,42]}. Therefore a Lagrange's interpolation polynomial approach is used to simulate the findings using Matlab-16.

We arranged the paper as Section 1 gives the introduction, section 2 describes preliminaries, section 3 shows the existence of the solution; section 4 offers uniqueness solution to the model, section 5 provides Hyers Ulams Stability analyses, and section 6 gives the numerical solution. Section 7 shows a graphical representation to support the analytical result. Finally, in section 8, we discuss the conclusion.

2.   Basic definitions and theorems

Definition 2.1. Let $\mathcal{F}\in {H}^{1}(a, b), b > a,$ for any $\alpha\in[0, 1]$, the ABC fractional derivative is

where $\mathbb{E}$ is a Mittag-Leffler function and $\varkappa(\alpha)$ satisfied $\varkappa(0) = \varkappa(1) = 1.$

Definition 2.2. Let $\mathcal{F}\in {H}^{1} (a, b), b > a,$ and $0\leq\alpha\leq1$, the ABC fractional integral is

Lemma 2.1. The following Newton Leibniz formula is satisfied for any $\mathcal{F}(t)\in {H}^{1}(a, b)$

3.   Existence of the solution

Applying the Lemma (2.1) to (1.1), we get

Suppose the function $\mathscr{G}_{i}$ for $i = 1, 2... 7$ given below

We defined

Let

● $(p)$, let, for $\mathscr{S}, \mathscr{S}^{*}, \mathscr{E}, \mathscr{E}^{*}, \mathscr{I}, \mathscr{I}^{*}, \mathscr{A}, \mathscr{A}^{*}, \mathscr{Q}, \mathscr{Q}^{*}, \mathscr{H},$ $\mathscr{H}^{*}, \mathscr{R}, \mathscr{R}^{*}\in L[0, 1]$, there exits constants $W_{i} > 0$, for $i = 1, 2, 3, 4, 5, 6, 7$ such that $\left\Vert \mathscr{S}\right\Vert \leq W_{1}, \left\Vert \mathscr{E}\right\Vert \leq W_{2}, \left\Vert \mathscr{I}\right\Vert \leq W_{3}, \left\Vert \mathscr{A}\right\Vert \leq W_{4}, \left\Vert \mathscr{Q}\right\Vert \leq W_{5}, \left\Vert \mathscr{H}\right\Vert \leq W_{6}, \left\Vert \mathscr{R}\right\Vert \leq W_{7}$ and $\eta_{1}, \eta_{2} > 0$.

The $\mathscr{G}_{i}$ for $i\in N_{1}^{7}$ satisfy the Lipschitz condition provided that (p) is obeyed. First, we take for $\mathscr{G}_{1}(t, \mathscr{S})$, we have

For the $\mathscr{G}_{2}(t, \mathscr{E})$, we have

For the $\mathscr{G}_{3}(t, \mathscr{I})$, we have

For the $\mathscr{G}_{4}(t, \mathscr{A})$, we have

Now, for $\mathscr{G}_{5}(t, \mathscr{Q})$, we have

For the $\mathscr{G}_{6}(t, \mathscr{H})$, we have

Similarly, for the $\mathscr{G}_{7}(t, \mathscr{R})$, we have

Hence, the $\mathscr{G}_{i}$, for $i = 1, 2, ...7.$ satisfies the lipschitz condition. We suppose that $\mathscr{S}(0) = \mathscr{E}(0) = \mathscr{A}(0) = \mathscr{I}(0) = \mathscr{Q}(0) = \mathscr{H}(0) = \mathscr{R}(0) = 0$, we have

We defined the following iterative relation for (1.1) as

Using the supposition (P) and (3.1), the model (1.1) has a solution $\Omega = \max\left\{ \varTheta_{1}, \varTheta_{2}, \varTheta_{3}, \varTheta_{4}, \varTheta_{5}, \varTheta_{6}, \varTheta_{7}, \right\}.$ The functions we defined as

Now, we using the definition (2.1) and (3.4), we get

Similarly, using the same procedure for the other compartments of the model (1.1)

Hence, we have $\varPi_{n}(t)\rightarrow0$ as $n\rightarrow \infty$.

4.   Uniqueness of the solution

Theorem 4.1. With the supposition ($P$) the model (1.1) has unique solution if

Proof. We suppose that the solution of (1.1) is not unique, Let $\{\mathscr{S}^{*}(t), \mathscr{E}^{*}(t), \mathscr{I}^{*}(t), \mathscr{A}^{*}(t), \mathscr{Q}^{*}(t), \mathscr{H}^{*}(t), \mathscr{R}^{*}(t)\}$ be another solution of (1.1), such that

Then,

Which implies that

By (4.1), (4.2) is vaild for $\left\Vert \mathscr{S}-\mathscr{S}^{*}\right\Vert = 0$, which means that $\mathscr{S} = \mathscr{S}^{*}$. Similarly, we repeat the same fashion for the remaining compartment of the model (1.1), we have

Which implies that

By using (4.1) which implies the (4.3) is valid for $\left\Vert \mathscr{E}-\mathscr{E}^{*}\right\Vert = 0$, which means that $\mathscr{E} = \mathscr{E}^{*}$. Similarly, for $\mathscr{I}$, we have

Which implies that

From (4.1), (4.4) result is valid for $\left\Vert \mathscr{I}-\mathscr{I}^{*}\right\Vert = 0$, which means that $\mathscr{I} = \mathscr{I}^{*}$. Now, for $\mathscr{A}$, we have

Which implies that

By using (4.1), the (4.5) is valid for $\left\Vert \mathscr{A}-\mathscr{A}^{*}\right\Vert = 0$, which means that $\mathscr{A} = \mathscr{A}^{*}$. Similarly, for $\mathscr{Q}$,

Which implies that

By using (4.1) which implies the (4.6) is valid for $\left\Vert \mathscr{Q}-\mathscr{Q}^{*}\right\Vert = 0$, which means that $\mathscr{Q} = \mathscr{Q}^{*}$. Similarly, for $\mathscr{H}$, we have

Which implies that

By using (4.1), the (4.7) result is valid for $\left\Vert \mathscr{H}-\mathscr{H}^{*}\right\Vert = 0$, which means that $\mathscr{H} = \mathscr{H}^{*}$. Similarly, for $\mathscr{R}$, we have

Which implies that

By using (4.1) which implies that (4.8) results is valid for $\left\Vert \mathscr{R}-\mathscr{R}^{*}\right\Vert = 0$, which shows that $\mathscr{R} = \mathscr{R}^{*}$. Thus the model (1.1) has a unique solution.

5.   Hyers Ulams Stability

The integral system Eqs. (3.4)-(3.4) is Hyers -ulam Stable if for $\eta_{i} > 0, \epsilon_{i} > 0$, for $i\in N_{1}^{7}$ ^[47], with

Now, we have $\dot{\mathscr{S}}, \dot{\mathscr{E}}, \dot{\mathscr{I}}, \dot{\mathscr{A}}, \dot{\mathscr{Q}}, \dot{\mathscr{H}}, \dot{\mathscr{R}}$, which implies that

Such that

Theorem 5.1. We assume that (P), Satisfied then the FOM (1.1) is Hyers Ulam stable.

Proof. By using (4.1), the FOM (1.1) has a unique solution, say $\left(\mathscr{S}, \mathscr{E}, \mathscr{I}, \mathscr{A}, \mathscr{Q}, \mathscr{H}, \mathscr{R}\right)$. Let we suppose that $\left(\dot{\mathscr{S}}, \dot{\mathscr{E}}, \dot{\mathscr{I}}, \dot{\mathscr{A}}, \dot{\mathscr{Q}}, \dot{\mathscr{H}}, \dot{\mathscr{R}}\right)$ be an another solution for the considered model. We go ahead with first equation of the (1.1), satisfying Eqs. (3.4). Then

Now, we taking $\eta_{1} = \Theta_{1}$ and $\frac{1-\alpha}{\varkappa(\alpha)}+\frac{\alpha}{\varkappa(\alpha)\varGamma(\alpha)} = \Delta_{1}$, this implies $\left\Vert \mathscr{S}-\dot{\mathscr{S}}\right\Vert \leq\eta_{1}\Delta_{1}$.

Similarly, for $\mathscr{E}(t), \dot{\mathscr{E}}(t), \mathscr{I}(t), \dot{\mathscr{I}}(t), \mathscr{A}(t), \dot{\mathscr{A}}(t), \mathscr{Q}(t), \dot{\mathscr{Q}}(t)$, $\mathscr{H}(t), \dot{\mathscr{H}}(t), \mathscr{R}(t), \dot{\mathscr{R}}(t)$, we have

Thus, the solution of the considered FOM is stable.

6.   Numerical algorithm

In this section, we find the numerical solution by using the numerical method of Lagrange's interpolation. For the solution of the system (1.1), we consider

Using the Lemma (2.1) and (6.1), we get

Let [0, T] be the interval which we want to find the solution of the system (1.1). For this we divided the given interval as a set of points $t_{m+1}$, for m = 0, 1, 2, ..n, we have

Using lagrange's interpolation, we get

This numerical scheme will be used in the next section.

7.   Graphical representation of the solution

In this part of the paper, we apply the numerical method described in the previous section to demonstrate the graphical results of the model (1.1).

The parameters used in simulations are given in Table.2, most of which are taken from previous published work ^{[35,36,37,38]}. Plotting in the Figure 2, the behavior of the model with two activate parameters, isolation rate ($c\neq0$) and social distancing rate ($\delta\neq 0$), is shown by the bold blue line. In contrast, the bold red line shows the behavior of the model with parameters $c$ and $\delta$ equal to zero.

Individuals in the exposed, asymptomatic, infected, and hospitalized classes declined considerably with isolation and social distance parameters, but the suspectable and recovered population increased fast. These graphs demonstrate that these factors are useful in reducing Covid-19 infection in the population.

From Figures 3-9, we see that the susceptible, infection, asymptotic, quarantined, and hospitalized population will decrease, and consequently, the recovery will increase. We observed a rapid decrease in the population of exposed, asymptomatic, quarantined, hospitalized, and infected classes, which becomes more significant for fractional value compared to integer order $\alpha = 1$.

From Figure 10, we noted that the susceptible, exposed, and asymptomatic individuals are the smallest due to the smallest value of $\Lambda$. Otherwise, the largest value of the $\Lambda$ could be high population density, close contact with the people, improper social distancing, and insufficient preventive measures. This will lead to a relatively high spread of the disease.

8.   Conclusions

This work studied the transmission dynamics of the COVID-19 pandemic with asymptomatic, quarantine, and hospitalization individuals through an ABC fractional model. Initially, we formulated a mathematical model and analyzed the model using the fractional operator with the power-law kernel. We have used fixed point theory to conclude the existence of such a model in the real world. Then we have shown the unique solution of the model. Further, we investigated significant conditions for the Ulam Hyers type stability via non-linear functional analysis. The computational scheme is derived for the numerical simulation and is checked for available data. Further, We simulate the COVID-19 model for different transmission rate values to evaluate the dynamics of the separate compartments in the model. The dynamics of the COVID-19 pandemic and the impact of various control strategies by including additional classes into the current model will be addressed in future research. The vaccinated population and the number of pathogens in the environment are two compartments. The suggested model will include memory characteristics and non-locality and be an expanded version of the current model. This study's findings will help health care centers forecast how the coronavirus may affect the world in the future.

Acknowledgment

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Conflict of interest

We declare that we have no competing interests.