A vigorous study of fractional order COVID-19 model via ABC derivatives

Abstract

The newly arose irresistible sickness known as the Covid illness (COVID-19), is a highly infectious viral disease. This disease caused millions of tainted cases internationally and still represent a disturbing circumstance for the human lives. As of late, numerous mathematical compartmental models have been considered to even more likely comprehend the Covid illness. The greater part of these models depends on integer-order derivatives which cannot catch the fading memory and crossover behavior found in many biological phenomena. Along these lines, the Covid illness in this paper is studied by investigating the elements of COVID-19 contamination utilizing the non-integer Atangana–Baleanu–Caputo derivative. Using the fixed-point approach, the existence and uniqueness of the integral of the fractional model for COVID is further deliberated. Along with Ulam–Hyers stability analysis, for the given model, all basic properties are studied. Furthermore, numerical simulations are performed using Newton polynomial and Adams Bashforth approaches for determining the impact of parameters change on the dynamical behavior of the systems.

Introduction

An exceptionally infectious viral illness brought about by a Covid (SARS- CoV-2) [1], [2]. Over 10.27 million affirmed cases and a larger number of than 0.5 million combined passings all around the world are recorded till June 30, 2020 [3], [4]. The death rate is about 7% and the recuperation rate is 93% globally [1], [3], [5], [6], [7]. The drug organizations, administrative specialists are doing all that could be within reach for the accessibility of a protected and powerful enemy of viral antibody against this original disease. The pandemic ceaselessly arches extreme general wellbeing, hit the worker’s local area and economy all through the world, remembering for Pakistan. The proposed the brooding time frame for this viral contamination is in the scope of 2 to 10 days by WHO [6], [8], [9]. The manifestations of a COVID-19 tainted individual incorporate fever, myalgia or weakness, dry hack, windedness or dyspnoea, pneumonia, chills, sore throat, different organ disappointment, anorexia or lungs disappointment, intense respiratory pain condition (ARDS), lymphopenia (a decrease of lymphocytes in the circling blood), loss of smell or taste furthermore, an indication of cytokine storm [6], [8]. Most of the passings have happened in other persistent sicknesses like diabetes, hypertension and cardiovascular disease. Without treatments, supported immunization furthermore, antiviral the entire world is zeroing in on non-pharmaceutical interventions to destroy this COVID-19. The most widely recognized NPIs rehearses these days are social actual distance, presented to be an isolate, disconnection of contaminated, wearing a cover, really testing office, conclusion of the school, unnecessary business, hand washing or disinfecting, lock down (stay at home and work from home) and defensive pack for clinical individual and staying away from pointless get-togethers.

Pakistan is one of the focal points of COVID-19 in Asia with an expected populace of more than 220 million [3], [10], [11]. Presently, Pakistan is the third country in Asia and twelfth on the planet with high affirmed tainted cases [3], [12]. Starting at 30 June 2020, 209,337 complete affirmed cases are recorded and around 4304 lost their lives. Albeit in Pakistan the death rate is exceptionally low when contrasted with the recuperation rate, still this infection has a adverse consequence on the economy. The primary case detailed in Karachi on 26 Feb 2020, the infection spread rapidly inside three weeks in the whole country. Additionally, the accounted cases are not exactly the complete cases because of restricting testing. Right now, the circumstance is more awful, because of the open lock down on 9 May, the public authority cannot keep up with severe lock down due to serious monetary difficulty. Yet, the public authority forced keen lock down and facilitating limitations, by permitting workplaces, organizations, markets with restricted hours and staff limit, five days seven days to return the economy of the country.

Fractional calculus is the generalization of classical calculus. To get a better insight into a mathematical model and to deeply understand phenomena, non-integer order operators can be used. Moreover, models involving fractional-order derivatives provide a greater degree of accuracy and are able to abduct the fading memory and spanning behavior. Fractional order differential equations models give more understanding about a disease under consideration [11], [13], [14], [15]. Literature has suggested a number of fractional operators with singular and nonsingular kernel [9], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26] and their applications can be found in some recent studies [13], [15], [24]. In [27], the authors have considered co-dynamics for cancer and hepatitis using a mathematical model with fractional derivative and examined its results.

To describe the inherited traits of diverse substances and manners, we need proper tools. Fractional derivatives are wonderful means for doing such an aim. The reason for the current article is to research the elements of the original COVID-19 pandemic utilizing a compartmental model through the ABC fractional order derivatives. The remainder of this paper is organized: The mathematical model with fractional order derivative is formulated in Section “Mathematical models”. In Section “Stability analysis for system (1)” we find the equilibrium points and basic reductive number $R_0$, and the local stability of the DFE and EE points for the proposed model. The existence results for the fractional order model is presented in Section “Existence results”. Section “Hyers-UlamsStability” contains the Hyers-Ulam Stability of fractional order model in ABC sense. The numerical schemes is presented in Section “Numerical schemes & graphical results”. In Section “Discussion” presented the discussions of obtained figures, and the last section we presented the concluding remarks.

Mathematical models

In this work, we consider [28] model. The model consist six compartment susceptible group $\mathcal{S}$, exposed group $\mathcal{E}$, symptomatic infected group $\mathcal{I}$, asymptomatic infected group $\mathcal{A}$, hospitalized group $\mathcal{H}$, recovered group $\mathcal{R}$. The COVID-19 pandemic model in its classical version as available in [28] and the assumption therein can be presented as:

$$\left\{\begin{array}{cc}\hfill & \dot{\mathcal{S}}=\mathcal{G}-\frac{{\mathcal{G}}_1\left({\mathcal{G}}_2\mathcal{I}+\mathcal{A}\right)\mathcal{S}}{N}-{\mathcal{G}}_8\mathcal{S}\hfill \\ \hfill & \dot{\mathcal{E}}=\frac{{\mathcal{G}}_1\left({\mathcal{G}}_2\mathcal{I}+\mathcal{A}\right)\mathcal{S}}{N}-\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\mathcal{E}\hfill \\ \hfill & \dot{\mathcal{I}}=\left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4\mathcal{E}-\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right)\mathcal{I}\hfill \\ \hfill & \dot{\mathcal{A}}={\mathcal{G}}_3{\mathcal{G}}_4\mathcal{E}-\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)\mathcal{A}\hfill \\ \hfill & \dot{\mathcal{H}}={\mathcal{G}}_5\mathcal{I}+{\mathcal{G}}_{10}\mathcal{A}-\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)\mathcal{H}\hfill \\ \hfill & \dot{\mathcal{R}}={\mathcal{G}}_6\mathcal{I}+{\mathcal{G}}_7\mathcal{A}+\left({\mathcal{G}}_{11}+{\mathcal{G}}_9\right)\mathcal{H}-{\mathcal{G}}_8\mathcal{R}\hfill \end{array}\right)$$ (1)

with initial condition

$$\mathcal{S}\left(0\right),\mathcal{E}\left(0\right),\mathcal{I}\left(0\right),\mathcal{A}\left(0\right),\mathcal{H}\left(0\right),\mathcal{R}\left(0\right)\ge 0.$$ (2)

The parameters value defined in Table 1.

Table 1.

Values and descriptions of the parameters (based on data [28]).

Parameter	Name of the parameter	values/day	Ref.
$\mathcal{G}$	Influx rate	80.89	[28]
${\mathcal{G}}_1$	Rate of transmission from $\mathcal{A}$ to $\mathcal{S}$ cases	0.25	[28]
${\mathcal{G}}_2$	Rate of transmission from $\mathcal{I}$ to $\mathcal{S}$ cases	1	[28]
${\mathcal{G}}_3$	The proportion of $\mathcal{A}$ cases	0.80	[28]
${\mathcal{G}}_4$	The incubation period of Coronavirus	0.1923	[28]
${\mathcal{G}}_5$	The rate at which $\mathcal{I}$ cases are transferred to $\mathcal{H}$ cases	0.6000	[28]
${\mathcal{G}}_6$	The cure rate of $\mathcal{I}$ cases	0.05	[28]
${\mathcal{G}}_7$	The cure rate of $\mathcal{A}$ cases	0.0714	[28]
${\mathcal{G}}_8$	Natural mortality rate	0.0004563	[28]
${\mathcal{G}}_9$	The rate at which $\mathcal{H}$ cases are transferred to $\mathcal{R}$ case	0.04255	[28]
${\mathcal{G}}_{10}$	The rate at which $\mathcal{A}$ cases are transformed into $\mathcal{H}$ cases	0.03	[28]
${\mathcal{G}}_{11}$	Coronavirus induced death rate	0.0018	[28]

We develop the fractional model of the considered systems (1) by replacing the fractional derivative with the classical ones. So, with regard to the fractional operators give in the next section we have as follows. By considering the Atangana–Baleanu–Caputo sense operator for (1), we have

$$\left\{\begin{array}{cc}\hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{S}\right]=\mathcal{G}-\frac{{\mathcal{G}}_1\left({\mathcal{G}}_2\mathcal{I}+\mathcal{A}\right)\mathcal{S}}{N}-{\mathcal{G}}_8\mathcal{S},\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{E}\right]=\frac{{\mathcal{G}}_1\left({\mathcal{G}}_2\mathcal{I}+\mathcal{A}\right)\mathcal{S}}{N}-\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\mathcal{E},\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{I}\right]=\left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4\mathcal{E}-\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right)\mathcal{I},\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{A}\right]={\mathcal{G}}_3{\mathcal{G}}_4\mathcal{E}-\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)\mathcal{A},\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{H}\right]={\mathcal{G}}_5\mathcal{I}+{\mathcal{G}}_{10}\mathcal{A}-\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)\mathcal{H},\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{R}\right]={\mathcal{G}}_6\mathcal{I}+{\mathcal{G}}_7\mathcal{A}+\left({\mathcal{G}}_{11}+{\mathcal{G}}_9\right)\mathcal{H}-{\mathcal{G}}_8\mathcal{R}.\hfill \end{array}\right)$$ (3)

Under the initial conditions

$$\mathcal{S}\left(0\right),\mathcal{E}\left(0\right),\mathcal{I}\left(0\right),\mathcal{A}\left(0\right),\mathcal{H}\left(0\right),\mathcal{R}\left(0\right)\ge 0.$$ (4)

Stability analysis for system (1)

We know that for model (1) two equilibria points are obtained, use the method [29], [30]:

Disease-free equilibrium (DFE):$N_0=\left(\frac{\mathcal{G}}{{\mathcal{G}}_8},0,0,0,0,0\right)$.

Endemic equilibrium (EE): ${\mathcal{N}}^{\ast }=\left({\mathcal{S}}^{\ast },{\mathcal{E}}^{\ast },{\mathcal{I}}^{\ast },{\mathcal{A}}^{\ast },{\mathcal{H}}^{\ast },{\mathcal{R}}^{\ast }\right)$, where

$${\mathcal{S}}^{\ast }=\frac{\mathcal{G}}{{\mathcal{G}}_8}+\frac{{\mathcal{Y}}_1}{{\mathcal{G}}_8}{\mathcal{E}}^{\ast },$$$${\mathcal{I}}^{\ast }=-\frac{{\mathcal{Y}}_2}{{\mathcal{Y}}_3}{\mathcal{E}}^{\ast },$$$${\mathcal{A}}^{\ast }=-\frac{{\mathcal{G}}_3{\mathcal{G}}_4}{{\mathcal{Y}}_4}{\mathcal{E}}^{\ast },$$$${\mathcal{H}}^{\ast }=\frac{1}{{\mathcal{Y}}_5}\left(\frac{{\mathcal{G}}_5{\mathcal{Y}}_2}{{\mathcal{Y}}_3}+\frac{{\mathcal{G}}_{10}{\mathcal{G}}_4{\mathcal{G}}_3}{{\mathcal{Y}}_4}\right){\mathcal{E}}^{\ast },$$$${\mathcal{R}}^{\ast }=\frac{1}{{\mathcal{G}}_8}\left(-{\mathcal{G}}_6\frac{{\mathcal{Y}}_2}{{\mathcal{Y}}_3}-{\mathcal{G}}_7\frac{{\mathcal{G}}_3{\mathcal{G}}_4}{{\mathcal{Y}}_4}+\frac{{\mathcal{Y}}_6}{{\mathcal{Y}}_5}\left(\frac{{\mathcal{G}}_5{\mathcal{Y}}_2}{{\mathcal{Y}}_3}+\frac{{\mathcal{G}}_{10}{\mathcal{G}}_4{\mathcal{G}}_3}{{\mathcal{Y}}_4}\right)\right){\mathcal{E}}^{\ast },$$ (5)

and ${\mathcal{E}}^{\ast }=\frac{-m_2}{m_1}=\frac{{\mathcal{G}}_{8}N-\mathcal{G}{\mathcal{R}}_0}{{\mathcal{Y}}_1{\mathcal{R}}_0}$, which is a solution of a quadratic equation $m_1{\mathcal{E}}^{\star 2}+m_2{\mathcal{E}}^{\ast }=0$, where,

$$m_1=\frac{-{\mathcal{Y}}_1{\mathcal{Y}}_1}{N{\mathcal{G}}_8}{\mathcal{R}}_0,m_2=\frac{{\mathcal{Y}}_1\left(N{\mathcal{G}}_8-\mathcal{G}{\mathcal{R}}_o\right)}{N{\mathcal{G}}_8},{\mathcal{R}}_0={\mathcal{R}}_1+{\mathcal{R}}_2,$$$${\mathcal{R}}_1=\frac{{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{Y}}_2}{{\mathcal{Y}}_1{\mathcal{Y}}_3},{\mathcal{R}}_2=\frac{{\mathcal{G}}_1{\mathcal{G}}_3{\mathcal{G}}_4}{{\mathcal{Y}}_1{\mathcal{Y}}_4},{\mathcal{Y}}_1=-\left({\mathcal{G}}_4+{\mathcal{G}}_8\right),$$$${\mathcal{Y}}_2=\left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4,{\mathcal{Y}}_3=-\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right),{\mathcal{Y}}_4=-\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right),$$$${\mathcal{Y}}_5=-\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right),{\mathcal{Y}}_6=\left({\mathcal{G}}_{11}+{\mathcal{G}}_9\right).$$ (6)

As a result, a positive endemic equilibrium point exists only for ${\mathcal{R}}_0>1$ taking into account the assumption that $\mathcal{G}={\mathcal{G}}_8\mathcal{N}$.

Basic reproductive number${\mathcal{R}}_0$: The basic reproductive number, denoted by ${\mathcal{R}}_0$ is obtained by establishing the next generation matrix [29], [30], [31] as a spectral radius of the matrix $T{V}^{-1}$ at $N_0$. The matrices $T$ and $V^{-1}$ are obtained by linearizing the mathematical model (3) about DFE, which results in the Jacobian matrix $J\left({\mathcal{E}}^0\right)$ given in (7).

$$J\left({\mathcal{E}}^0\right)=\left(\begin{array}{cccccc}\hfill -{\mathcal{G}}_8\hfill & \hfill 0\hfill & \hfill -{\mathcal{G}}_1{\mathcal{G}}_2\hfill & \hfill -{\mathcal{G}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill -\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\hfill & \hfill {\mathcal{G}}_1{\mathcal{G}}_2\hfill & \hfill {\mathcal{G}}_1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill J_{32}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathcal{G}}_3{\mathcal{G}}_4\hfill & \hfill 0\hfill & \hfill -\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{G}}_5\hfill & \hfill {\mathcal{G}}_{10}\hfill & \hfill -\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{G}}_6\hfill & \hfill {\mathcal{G}}_7\hfill & \hfill \left({\mathcal{G}}_{11}+{\mathcal{G}}_9\right)\hfill & \hfill -{\mathcal{G}}_8\hfill \end{array}\right),$$ (7)

where $J_{32}=\left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4-\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right)$.

From the matrix, $J\left({\mathcal{E}}^0\right)$ we construct a matrix $M$ such that $M=T-V$, where

$$M=\left(\begin{array}{cccc}\hfill -\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\hfill & \hfill {\mathcal{G}}_1{\mathcal{G}}_2\hfill & \hfill {\mathcal{G}}_1\hfill & \hfill 0\hfill \\ \hfill \left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4\hfill & \hfill -\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right)\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\mathcal{G}}_3{\mathcal{G}}_4\hfill & \hfill 0\hfill & \hfill -\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathcal{G}}_5\hfill & \hfill {\mathcal{G}}_{10}\hfill & \hfill -\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)\hfill \end{array}\right).$$

$$T=\left(\begin{array}{cccc}\hfill 0\hfill & \hfill {\mathcal{G}}_1{\mathcal{G}}_2\hfill & \hfill {\mathcal{G}}_1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right),$$

and

$$V^{-1}=\left(\begin{array}{cccc}\hfill \frac{1}{\left({\mathcal{G}}_4+a_8\right)}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \frac{\left(1-{\mathcal{G}}_3\right)\phi }{\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\left({\mathcal{G}}_5+{\mathcal{G}}_6+a_8\right)}\hfill & \hfill \frac{1}{\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \frac{{\mathcal{G}}_3{\mathcal{G}}_4}{\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\left({\mathcal{G}}_{10}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)}\hfill & \hfill 0\hfill & \hfill \frac{1}{\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)}\hfill & \hfill 0\hfill \\ \hfill V_{41}\hfill & \hfill \frac{{\mathcal{G}}_5}{\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_5\right)\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)}\hfill & \hfill \frac{10}{\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)}\hfill & \hfill \frac{1}{\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)}\hfill \end{array}\right)$$

where, $V_{41}=\frac{{\mathcal{G}}_5\left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)+{\mathcal{G}}_{10}\left({\mathcal{G}}_3{\mathcal{G}}_4\right)\left(a_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right)}{\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right)\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)}$.

The basic reproductive number is the spectral radius $\rho \left(T{V}^{-1}\right)$ and is given by

$${\mathcal{R}}_0=\frac{{\mathcal{G}}_1{\mathcal{G}}_2\left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4}{\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right)}+\frac{{\mathcal{G}}_1{\mathcal{G}}_3{\mathcal{G}}_4}{\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)}$$

${\mathcal{R}}_0$ can be written as ${\mathcal{R}}_0={\mathcal{R}}_1+{\mathcal{R}}_2,{\mathcal{R}}_1=\frac{{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{Y}}_2}{{\mathcal{Y}}_1{\mathcal{Y}}_3},{\mathcal{R}}_2=\frac{{\mathcal{G}}_1{\mathcal{G}}_3{\mathcal{G}}_4}{{\mathcal{Y}}_1{\mathcal{Y}}_4}$ where ${\mathcal{Y}}_i,i=1\dots 6$ are as defined above.

Local stability analysis

Theorem 1

Let $m>0$ and $n>0$ are the integers such that $gcd(m,n)=1$ for $\mathcal{G}=\frac{m}{n}$ and $M=n$ , then the DFE of the model in fractional derivative is locally asymptotically stable if $|arg\left(\lambda \right)|>\frac{\pi }{2M}$ for all roots $\lambda$ of the associated characteristic equation,

$$det\left(diag\left[{\mathcal{F}}^{M\mathcal{G}}{\mathcal{F}}^{M\mathcal{G}}{\mathcal{F}}^{M\mathcal{G}}{\mathcal{F}}^{M\mathcal{G}}{\mathcal{F}}^{M\mathcal{G}}{\mathcal{F}}^{M\mathcal{G}}\right]-J\left(E_0^{\ast \ast }\right)\right)=0$$ (8)

Proof

The characteristic equation of Jacobian matrix (7) at the disease free point takes the following from;

$${\left({\mathcal{F}}^M+{\mathcal{G}}_8\right)}^2\left({\mathcal{F}}^M+{\mathcal{G}}_8+{\mathcal{G}}_9+{\mathcal{G}}_{11}\right)({\mathcal{F}}^{3M}+a_2{\mathcal{F}}^{2M}+a_1{\mathcal{F}}^M+a_0)=0,$$ (9)

where

$$a_2=-\left({\mathcal{Y}}_1+{\mathcal{Y}}_3+{\mathcal{Y}}_4\right)$$

$$a_1={\mathcal{Y}}_1{\mathcal{Y}}_3+{\mathcal{Y}}_1{\mathcal{Y}}_4+{\mathcal{Y}}_3{\mathcal{Y}}_4-{\mathcal{G}}_1{\mathcal{G}}_4{\mathcal{G}}_3-{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{Y}}_2,$$

$$a_0=-{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4+{\mathcal{G}}_1{\mathcal{G}}_4{\mathcal{G}}_3{\mathcal{Y}}_3+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{Y}}_2{\mathcal{Y}}_4,$$

where ${\mathcal{Y}}_i,i=1\dots 6$ are as defined above in Eq. (6).

From (9), we have $a_2>0$ and $a_1={\mathcal{Y}}_1{\mathcal{Y}}_3\left(1-{\mathcal{R}}_1\right)+{\mathcal{Y}}_1{\mathcal{Y}}_4\left(1-{\mathcal{R}}_2\right)+{\mathcal{Y}}_3{\mathcal{Y}}_4>0$ for ${\mathcal{R}}_0<1$ as ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ are positive and ${\mathcal{R}}_1+{\mathcal{R}}_2={\mathcal{R}}_0$ Besides, $a_0=-{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4+{\mathcal{G}}_1{\mathcal{G}}_4{\mathcal{G}}_3{\mathcal{Y}}_3+{\mathcal{G}}_1{\mathcal{G}}_2{\mathcal{Y}}_2{\mathcal{Y}}_4={\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4\left({\mathcal{R}}_0-1\right)>0$ for ${\mathcal{R}}_0<1$, since ${\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4<0$.

All the eigenvalues of the characteristic equation $\Psi \left(\mathcal{F}\right)=\left({\mathcal{F}}^{3M}+a_2{\mathcal{F}}^{2M}+a_1{\mathcal{F}}^M+a_0\right)=0$, have a negative real part by Routh–Hurwitz stability criteria as it can easily be shown that $a_0-a_2{a}_1<0$. That is,

$$a_0-a_1{a}_2={\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4\left({\mathcal{R}}_0-1\right)-\left[-\left({\mathcal{Y}}_1+{\mathcal{Y}}_3+{\mathcal{Y}}_4\right)\left({\mathcal{Y}}_1{\mathcal{Y}}_3\left(1-{\mathcal{R}}_1\right)\right)\right)$$

$$\left(\left(+{\mathcal{Y}}_1{\mathcal{Y}}_4\left(1-{\mathcal{R}}_2\right)+{\mathcal{Y}}_3{\mathcal{Y}}_4\right)\right]=2{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4+\left({\mathcal{Y}}_1+{\mathcal{Y}}_3\right)\left({\mathcal{Y}}_1{\mathcal{Y}}_3\right)\left(1-{\mathcal{R}}_1\right)+\left({\mathcal{Y}}_1+{\mathcal{Y}}_4\right)\left({\mathcal{Y}}_1{\mathcal{Y}}_4\right)\left(1-{\mathcal{R}}_2\right)+\left({\mathcal{Y}}_4+{\mathcal{Y}}_3\right)\left({\mathcal{Y}}_4{\mathcal{Y}}_3\right)<0.$$

for ${\mathcal{R}}_1<1,{\mathcal{R}}_2<1,{\mathcal{R}}_0<1$. The argument of the root of equations

$$\left({\mathcal{F}}_1^M+{\mathcal{G}}_8\right)=0,\left({\mathcal{F}}_2^M+{\mathcal{G}}_8\right)=0\text{and}\left({\mathcal{F}}_3^M+{\mathcal{G}}_8+{\mathcal{G}}_9+{\mathcal{G}}_{11}\right)=0$$

are similar, that is:

$$\left[\left|arg\left({\vartheta }_k\right)\right|>\frac{\pi }{m}+k\frac{2\pi }{m}>\frac{\pi }{M}>\frac{\pi }{2M},\right]\text{where}k=0,1,2,3,\dots ,(m-1).$$

In similar fashion, we find out the arguments of the roots of equation

$$\left({\mathcal{F}}^{3M}+a_2{\mathcal{F}}^{2M}+a_1{\mathcal{F}}^M+a_0\right)=0$$

are all greater than $\frac{\pi }{2M}$ if ${\mathcal{R}}_0<1$, having an argument less than $\frac{\pi }{2M}$ for ${\mathcal{R}}_0>1$. Thus, the DFE is locally asymptotically stable for ${\mathcal{R}}_0<1$.

Local stability at EEP

Theorem 2

If ${\mathcal{R}}_0>1$ , then the EEP ${\mathcal{N}}^{\ast }=\left({\mathcal{S}}^{\ast },{\mathcal{E}}^{\star },{\mathcal{I}}^{\ast },{\mathcal{A}}^{\ast },{\mathcal{H}}^{\ast },{\mathcal{R}}^{\ast }\right)$ of Atangana–Baleanu–Caputo fractional model is locally asymptotically stable.

Proof

Suppose ${\mathcal{R}}_0>1$ so that the EEP exists. Now the Jacobian matrix $J_{EEP}$ evaluated at the EEP is given by

$$J_{EEP}=\left(\begin{array}{cccccc}\hfill -{\mathcal{Y}}_7-{\mathcal{G}}_8\hfill & \hfill 0\hfill & \hfill -{\mathcal{G}}_2{\mathcal{Y}}_8\hfill & \hfill -{\mathcal{Y}}_8\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill {\mathcal{Y}}_7\hfill & \hfill {\mathcal{Y}}_1\hfill & \hfill {\mathcal{G}}_2{\mathcal{Y}}_8\hfill & \hfill {\mathcal{Y}}_8\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathcal{Y}}_2\hfill & \hfill {\mathcal{Y}}_3\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill {\mathcal{G}}_3{\mathcal{G}}_4\hfill & \hfill 0\hfill & \hfill {\mathcal{Y}}_4\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{G}}_5\hfill & \hfill {\mathcal{G}}_{10}\hfill & \hfill {\mathcal{Y}}_5\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill {\mathcal{G}}_6\hfill & \hfill {\mathcal{G}}_7\hfill & \hfill {\mathcal{Y}}_6\hfill & \hfill -{\mathcal{G}}_8\hfill \end{array}\right),$$ (10)

where ${\mathcal{Y}}_1=-\left({\mathcal{G}}_4+{\mathcal{G}}_8\right),{\mathcal{Y}}_2=\left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4,{\mathcal{Y}}_3=-\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right),{\mathcal{Y}}_4=-\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)$

$${\mathcal{Y}}_5=-\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right),{\mathcal{Y}}_6=\left({\mathcal{G}}_{11}+{\mathcal{G}}_9\right),{\mathcal{Y}}_7={\mathcal{G}}_1\frac{\left({\mathcal{G}}_2{I}^{\ast }+{\mathcal{A}}^{\ast }\right)}{N},{\mathcal{Y}}_8={\mathcal{G}}_1\frac{{\mathcal{S}}^{\ast }}{N}.$$

The two eigenvalues ${\lambda }_1=-{\mathcal{G}}_8$ and ${\lambda }_2={\mathcal{Y}}_5=-\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)$ of the matrix (10) are negative. The remaining four eigenvalues are determined if they have a negative real part or not by the method of Routh–Hurwitz stability criteria from a characteristic Eq. (11) given below.

$$f\left(\lambda \right)={\lambda }^4+{\mathcal{B}}_3{\lambda }^3+{\mathcal{B}}_2{\lambda }^2+{\mathcal{B}}_1\lambda +{\mathcal{B}}_0,$$ (11)

where

$${\mathcal{B}}_3={\mathcal{G}}_8-{\mathcal{Y}}_1-{\mathcal{Y}}_3-{\mathcal{Y}}_4+{\mathcal{Y}}_7$$

$${\mathcal{B}}_2={\mathcal{Y}}_1{\mathcal{Y}}_3+{\mathcal{Y}}_1{\mathcal{Y}}_4+{\mathcal{Y}}_3{\mathcal{Y}}_4-{\mathcal{Y}}_1{\mathcal{Y}}_7-{\mathcal{Y}}_3{\mathcal{Y}}_7-{\mathcal{Y}}_4{\mathcal{Y}}_7-{\mathcal{G}}_8{\mathcal{Y}}_1-{\mathcal{G}}_8{\mathcal{Y}}_3-{\mathcal{G}}_8{\mathcal{Y}}_4-{\mathcal{G}}_4{\mathcal{G}}_3{\mathcal{Y}}_8-{\mathcal{G}}_2{\mathcal{Y}}_2{\mathcal{Y}}_8$$

$${\mathcal{B}}_1={\mathcal{G}}_8{\mathcal{Y}}_1{\mathcal{Y}}_3+{\mathcal{G}}_8{\mathcal{Y}}_1{\mathcal{Y}}_4+{\mathcal{G}}_8{\mathcal{Y}}_3{\mathcal{Y}}_4-{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4+{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_7+{\mathcal{Y}}_1{\mathcal{Y}}_4{\mathcal{Y}}_7+{\mathcal{Y}}_3{\mathcal{Y}}_4{\mathcal{Y}}_7-{\mathcal{G}}_4{\mathcal{G}}_8{\mathcal{G}}_3{\mathcal{Y}}_8-{\mathcal{G}}_2{\mathcal{G}}_8{\mathcal{Y}}_2{\mathcal{Y}}_8+{\mathcal{G}}_4{\mathcal{G}}_3{\mathcal{Y}}_3{\mathcal{Y}}_8+{\mathcal{G}}_2{\mathcal{Y}}_2{\mathcal{Y}}_4{\mathcal{Y}}_8,$$

$${\mathcal{B}}_0={\mathcal{G}}_8{\mathcal{G}}_4{\mathcal{G}}_3{\mathcal{Y}}_3{\mathcal{Y}}_8-{\mathcal{G}}_8{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4-{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4{\mathcal{Y}}_7+{\mathcal{G}}_2{\mathcal{G}}_8{\mathcal{Y}}_2{\mathcal{Y}}_4{\mathcal{Y}}_8.$$

The coefficient ${\mathcal{B}}_3$ can easily be shown to be positive and ${\mathcal{B}}_2,{\mathcal{B}}_1,{\mathcal{B}}_0$ are also positive as shown below:

$${\mathcal{B}}_2=\left(\left({\mathcal{Y}}_1{\mathcal{Y}}_3{R}_2+{\mathcal{Y}}_1{\mathcal{Y}}_4{R}_1\right)/R_0\right)+{\mathcal{Y}}_3{\mathcal{Y}}_4-{\mathcal{Y}}_1{\mathcal{Y}}_7-{\mathcal{Y}}_3{\mathcal{Y}}_7$$

$$-{\mathcal{Y}}_4{\mathcal{Y}}_7-{\mathcal{G}}_8{\mathcal{Y}}_1-{\mathcal{G}}_8{\mathcal{Y}}_3-{\mathcal{G}}_8{\mathcal{Y}}_4>0$$

$${\mathcal{B}}_1={\mathcal{G}}_8{\mathcal{Y}}_1{\mathcal{Y}}_3\left(R_2/R_0\right)+{\mathcal{G}}_8{\mathcal{Y}}_1{\mathcal{Y}}_4\left(R_1/R_0\right)+{\mathcal{G}}_8{\mathcal{Y}}_3{\mathcal{Y}}_4-2{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4$$

$$+{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_7+{\mathcal{Y}}_1{\mathcal{Y}}_4{\mathcal{Y}}_7+{\mathcal{Y}}_3{\mathcal{Y}}_4{\mathcal{Y}}_7>0,$$

$${\mathcal{B}}_0={\mathcal{G}}_8{\mathcal{G}}_4{\mathcal{G}}_3{\mathcal{Y}}_3{\mathcal{Y}}_8-{\mathcal{G}}_8{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4-{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4{\mathcal{Y}}_7+{\mathcal{G}}_2{\mathcal{G}}_8{\mathcal{Y}}_2{\mathcal{Y}}_4$$

$${\mathcal{Y}}_8=-{\mathcal{Y}}_1{\mathcal{Y}}_3{\mathcal{Y}}_4{\mathcal{Y}}_7>0.$$

Since it is not hard to show that ${\mathcal{B}}_0{\mathcal{B}}_3^2{\mathcal{B}}_1^2-{\mathcal{B}}_1{\mathcal{B}}_2{\mathcal{B}}_3<0$, one can be inferred that, all the eigenvalues of the Eq. (11) have a negative real part. Accordingly, the EEP ${\mathcal{N}}^{\ast }=\left({\mathcal{S}}^{\ast },{\mathcal{E}}^{\ast },I^{\ast },{\mathcal{A}}^{\ast },{\mathcal{H}}^{\ast },{\mathcal{R}}^{\ast }\right)$ is locally asymptotically stable for $R_0>1$.

It should be noticed that EEP exists if $R_0>1$ as displayed in Section “Stability analysis for system (1)”.

Existence results

The existence of the solution as well as its uniqueness will be established for (3). Assume that a continuous real-valued function denoted by $B\left(J\right)$ containing the sup norm property is a Banach space on $J=[0,b]$ and $P=B\left(J\right)\times B\left(J\right)\times B\left(J\right)\times B\left(J\right)\times B\left(J\right)\times B\left(J\right)$ with norm $\Vert \left(\dot{\mathcal{S}},\dot{\mathcal{E}},\dot{\mathcal{I}},\dot{\mathcal{A}},\dot{\mathcal{H}},\dot{\mathcal{R}}\right)\Vert =\Vert \dot{\mathcal{S}}\Vert +\Vert \dot{\mathcal{E}}\Vert +\Vert \dot{\mathcal{I}}\Vert +\Vert \dot{\mathcal{A}}\Vert +\Vert \dot{\mathcal{H}}\Vert +\Vert \dot{\mathcal{R}}\Vert$, where $\Vert \dot{\mathcal{S}}\Vert ={sup}_{t\in J}\left|\dot{\mathcal{S}}\left(t\right)\right|,\Vert \dot{\mathcal{E}}\Vert ={sup}_{t\in j}\left|\dot{\mathcal{E}}\left(t\right)\right|,\Vert \dot{\mathcal{I}}\Vert ={sup}_{t\in j}\left|\dot{\mathcal{I}}\left(t\right)\right|,\Vert \dot{\mathcal{A}}\Vert ={sup}_{t\in j}\left|\dot{\mathcal{A}}\left(t\right)\right|,\Vert \dot{\mathcal{H}}\Vert ={sup}_{t\in j}\left|\dot{\mathcal{H}}\left(t\right)\right|,\Vert \dot{\mathcal{R}}\Vert ={sup}_{t\in j}\left|\dot{\mathcal{R}}\left(t\right)\right|$. Applying the Atangana–Baleanu–Caputo fractional integral operator to the both sides of Eq. (3), we obtain

$$\left\{\begin{array}{cc}\hfill & \dot{\mathcal{S}}\left(t\right)-\dot{\mathcal{S}}\left(0\right){=}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\dot{\mathcal{S}}\right]\left\{\mathcal{G}-\frac{{\mathcal{G}}_1\left({\mathcal{G}}_2\mathcal{I}+\mathcal{A}\right)\mathcal{S}}{N}-{\mathcal{G}}_8\mathcal{S}\right\},\hfill \\ \hfill & \dot{\mathcal{E}}\left(t\right)-\dot{\mathcal{E}}\left(0\right){=}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\dot{\mathcal{E}}\right]\left\{\frac{{\mathcal{G}}_1\left({\mathcal{G}}_2\mathcal{I}+\mathcal{A}\right)\mathcal{S}}{N}-\left({\mathcal{G}}_4+{\mathcal{G}}_8\right)\mathcal{E}\right\}\hfill \\ \hfill & \dot{\mathcal{I}}\left(t\right)-\dot{\mathcal{I}}\left(0\right){=}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\dot{\mathcal{I}}\right]\left\{\left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4\mathcal{E}-\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right)\mathcal{I}\right\},\hfill \\ \hfill & \dot{\mathcal{A}}\left(t\right)-\dot{\mathcal{A}}\left(0\right){=}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\dot{\mathcal{A}}\right]\left\{{\mathcal{G}}_3{\mathcal{G}}_4\mathcal{E}-\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right)\mathcal{A}\right\},\hfill \\ \hfill & \dot{\mathcal{H}}\left(t\right)-\dot{\mathcal{H}}\left(0\right){=}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\dot{\mathcal{H}}\right]\left\{{\mathcal{G}}_5\mathcal{I}+{\mathcal{G}}_{10}\mathcal{A}-\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right)\mathcal{H}\right\},\hfill \\ \hfill & \dot{\mathcal{R}}\left(t\right)-\dot{\mathcal{R}}\left(0\right){=}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\dot{\mathcal{R}}\right]\left\{{\mathcal{G}}_6\mathcal{I}+{\mathcal{G}}_7\mathcal{A}+\left({\mathcal{G}}_{11}+{\mathcal{G}}_9\right)\mathcal{H}-{\mathcal{G}}_8\mathcal{R}\right\}.\hfill \end{array}\right\}.$$ (12)

Now using definition 1, we can write

$$\dot{\mathcal{S}}\left(t\right)-\dot{\mathcal{S}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_1(\Phi ,t,\dot{\mathcal{S}})+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_1(\Phi ,\vartheta ,\dot{\mathcal{S}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{E}}\left(t\right)-\dot{\mathcal{E}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_2(\Phi ,t,\dot{\mathcal{E}})+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_2(\Phi ,\vartheta ,\dot{\mathcal{E}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{I}}\left(t\right)-\dot{\mathcal{I}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_3(\Phi ,t,\dot{\mathcal{I}})+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_3(\Phi ,\vartheta ,\dot{\mathcal{I}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{A}}\left(t\right)-\dot{\mathcal{A}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_4(\Phi ,t,\dot{\mathcal{A}})+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_4(\Phi ,\vartheta ,\dot{\mathcal{A}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{H}}\left(t\right)-\dot{\mathcal{H}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_5(\Phi ,t,\dot{\mathcal{H}})+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_5(\Phi ,\vartheta ,\dot{\mathcal{H}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{R}}\left(t\right)-\dot{\mathcal{R}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_6(\Phi ,t,\dot{\mathcal{R}})+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_6(\Phi ,\vartheta ,\dot{\mathcal{R}}\left(\vartheta \right))d\vartheta .$$ (13)

where The symbols ${\mathfrak{K}}_1,{\mathfrak{K}}_2,{\mathfrak{K}}_3,{\mathfrak{K}}_4,{\mathfrak{K}}_5,{\mathfrak{K}}_6$ have to hold for the Lipschitz condition only if $\dot{\mathcal{S}},\dot{\mathcal{E}},\dot{\mathcal{I}},\dot{\mathcal{A}},\dot{\mathcal{H}}$, $\dot{\mathcal{R}}$ possess an upper bound. Surmising that $\dot{\mathcal{S}}$ and ${\dot{\mathcal{S}}}^{\ast }$ are couple functions, we reach

$$\Vert {\mathfrak{K}}_1(\lambda ,t,\dot{\mathcal{S}})-{\mathfrak{K}}_1\left(\lambda ,t,{\dot{\mathcal{S}}}^{\ast }\right)\Vert =\Vert -\left(\frac{{\mathcal{G}}_1({\mathcal{G}}_{2}I+A)}{N}+{\mathcal{G}}_8\right)\left(\dot{\mathcal{S}}-\dot{{\mathcal{S}}^{\ast }}\right)\Vert$$ (14)

Taking into account

$${\eta }_1≔\left(\frac{{\mathcal{G}}_1({\mathcal{G}}_{2}I+A)}{N}+{\mathcal{G}}_8\right),$$

one reaches

$$\Vert {\mathfrak{K}}_1(\Phi ,t,\dot{\mathcal{S}})-{\mathfrak{K}}_1\left(\Phi ,t,{\dot{\mathcal{S}}}^{\ast }\right)\Vert \le {\eta }_1\Vert \dot{\mathcal{S}}-{\dot{\mathcal{S}}}^{\ast }\Vert .$$ (15)

Continuing in the same way, one gets

$$\begin{array}{c}\Vert {\mathfrak{K}}_2(\Phi ,t,\dot{\mathcal{E}})-{\mathfrak{K}}_2\left(\Phi ,t,{\dot{\mathcal{E}}}^{\ast }\right)\Vert \le {\eta }_2\Vert \dot{\mathcal{E}}-{\dot{\mathcal{E}}}^{\ast }\Vert ,\hfill \\ \hfill \\ \Vert {\mathfrak{K}}_3\left(\Phi ,t,\dot{\mathcal{I}}\right)-{\mathfrak{K}}_3\left(\Phi ,t,{\dot{\mathcal{I}}}^{\ast }\right)\Vert \le {\eta }_3\Vert \dot{\mathcal{I}}-{\dot{\mathcal{I}}}^{\ast }\Vert ,\hfill \\ \hfill \\ \Vert {\mathfrak{K}}_4\left(\Phi ,t,\dot{\mathcal{A}}\right)-{\mathfrak{K}}_4\left(\Phi ,t,{\dot{\mathcal{A}}}^{\ast }\right)\Vert \le {\eta }_4\Vert \dot{\mathcal{A}}-{\dot{\mathcal{A}}}^{\ast }\Vert ,\hfill \\ \hfill \\ \Vert {\mathfrak{K}}_5(\Phi ,t,\dot{\mathcal{H}})-{\mathfrak{K}}_5\left(\Phi ,t,{\dot{\mathcal{H}}}^{\ast }\right)\Vert \le {\eta }_5\Vert \dot{\mathcal{H}}-{\dot{\mathcal{H}}}^{\ast }\Vert \hfill \\ \hfill \\ \Vert {\mathfrak{K}}_6\left(\Phi ,t,\dot{\mathcal{R}}\right)-{\mathfrak{K}}_4\left(\Phi ,t,{\dot{\mathcal{R}}}^{\ast }\right)\Vert \le {\eta }_6\Vert \dot{\mathcal{R}}-{\dot{\mathcal{R}}}^{\ast }\Vert .\hfill \end{array}$$ (16)

Where

$${\eta }_2=\left({\mathcal{G}}_4+{\mathcal{G}}_8\right),{\eta }_3=\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right),{\eta }_4=\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right),$$

$${\eta }_5=\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right),$$

$${\eta }_5=\left({\mathcal{G}}_8\right).$$

This implies that the Lipschitz condition has held for all the functions. Going in a recursive manner, the expressions in (13) yields

$${\dot{\mathcal{S}}}_n\left(t\right)-\dot{\mathcal{S}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_1(\Phi ,t,{\dot{\mathcal{S}}}_{n-1}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_1(\Phi ,\vartheta ,{\dot{\mathcal{S}}}_{n-1}\left(\vartheta \right))d\vartheta ,$$$${\dot{\mathcal{E}}}_n\left(t\right)-\dot{\mathcal{E}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_2(\Phi ,t,{\dot{\mathcal{E}}}_{n-1}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_2(\Phi ,\vartheta ,{\dot{\mathcal{E}}}_{n-1}\left(\vartheta \right))d\vartheta ,$$$${\dot{\mathcal{I}}}_n\left(t\right)-\dot{\mathcal{I}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_3(\Phi ,t,{\dot{\mathcal{I}}}_{n-1}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_3(\Phi ,\vartheta ,{\dot{\mathcal{I}}}_{n-1}\left(\vartheta \right))d\vartheta ,$$$${\dot{\mathcal{A}}}_n\left(t\right)-\dot{\mathcal{A}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_4(\Phi ,t,{\dot{\mathcal{A}}}_{n-1}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_4(\Phi ,\vartheta ,{\dot{\mathcal{A}}}_{n-1}\left(\vartheta \right))d\vartheta ,$$$${\dot{\mathcal{H}}}_n\left(t\right)-\dot{\mathcal{H}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_5(\Phi ,t,{\dot{\mathcal{H}}}_{n-1}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_5(\Phi ,\vartheta ,{\dot{\mathcal{H}}}_{n-1}\left(\vartheta \right))d\vartheta ,$$$${\dot{\mathcal{R}}}_n\left(t\right)-\dot{\mathcal{R}}\left(0\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_6(\Phi ,t,{\dot{\mathcal{R}}}_{n-1}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_6(\Phi ,\vartheta ,{\dot{\mathcal{R}}}_{n-1}\left(\vartheta \right))d\vartheta ,$$ (17)

together with ${\dot{\mathcal{S}}}_0\left(t\right)=\dot{\mathcal{S}}\left(0\right)$, ${\dot{\mathcal{E}}}_0\left(t\right)=\dot{\mathcal{E}}\left(0\right)$, ${\dot{\mathcal{I}}}_0\left(t\right)=\dot{\mathcal{I}}\left(0\right)$, ${\dot{\mathcal{A}}}_0\left(t\right)=\dot{\mathcal{A}}\left(0\right)$, ${\dot{\mathcal{H}}}_0\left(t\right)=\dot{\mathcal{H}}\left(0\right)$, ${\dot{\mathcal{R}}}_0\left(t\right)=\dot{\mathcal{R}}\left(0\right)$. When the successive terms difference is taken, we get

$${\Xi }_{\dot{\mathcal{S}},n}={\dot{\mathcal{S}}}_n-{\dot{\mathcal{S}}}_{n-1}=\frac{1-\Phi }{B\left(\Phi \right)}({\mathfrak{K}}_1(\Phi ,t,{\dot{\mathcal{S}}}_{n-1})-{\mathfrak{K}}_1(\Phi ,t,{\dot{\mathcal{S}}}_{n-2}))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\left({\mathfrak{K}}_1\left(\Phi ,\vartheta ,{\dot{\mathcal{S}}}_{n-1}\left(\vartheta \right)\right)\right)-\left({\mathfrak{K}}_1\left(\Phi ,\vartheta ,{\dot{\mathcal{S}}}_{n-2}\left(\vartheta \right)\right)\right)d\vartheta$$$${\Xi }_{\dot{\mathcal{E}},n}={\dot{\mathcal{E}}}_n-{\dot{\mathcal{E}}}_{n-1}=\frac{1-\Phi }{B\left(\Phi \right)}({\mathfrak{K}}_2(\Phi ,t,{\dot{\mathcal{E}}}_{n-1})-{\mathfrak{K}}_2(\Phi ,t,{\dot{\mathcal{E}}}_{n-2}))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^l{(t-\vartheta )}^{\Phi -1}\left({\mathfrak{K}}_2\left(\Phi ,\vartheta ,{\dot{\mathcal{E}}}_{n-1}\left(\vartheta \right)\right)\right)-\left({\mathfrak{K}}_2\left(\Phi ,\vartheta ,{\dot{\mathcal{E}}}_{n-2}\left(\vartheta \right)\right)\right)d\vartheta$$$${\Xi }_{\dot{\mathcal{I}},n}={\dot{\mathcal{I}}}_{1n}-{\dot{\mathcal{I}}}_{n-1}=\frac{1-\Phi }{B\left(\Phi \right)}({\mathfrak{K}}_3(\Phi ,t,{\dot{\mathcal{I}}}_{n-1})-{\mathfrak{K}}_3(\Phi ,t,{\dot{\mathcal{I}}}_{n-2}))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\left({\mathfrak{K}}_3\left(\Phi ,\vartheta ,{\dot{\mathcal{I}}}_{n-1}\left(\vartheta \right)\right)\right)-\left({\mathfrak{K}}_3\left(\Phi ,\vartheta ,{\dot{\mathcal{I}}}_{n-2}\left(\vartheta \right)\right)\right)d\vartheta$$$${\Xi }_{\dot{\mathcal{A}},n}={\dot{\mathcal{A}}}_{2n}-{\dot{\mathcal{A}}}_{n-1}=\frac{1-\Phi }{B\left(\Phi \right)}({\mathfrak{K}}_4(\Phi ,t,{\dot{\mathcal{A}}}_{n-1})-{\mathfrak{K}}_4(\Phi ,t,{\dot{\mathcal{A}}}_{n-2}))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\left({\mathfrak{K}}_4\left(\Phi ,\vartheta ,{\dot{\mathcal{A}}}_{n-1}\left(\vartheta \right)\right)\right)-\left({\mathfrak{K}}_4\left(\Phi ,\vartheta ,{\dot{\mathcal{A}}}_{n-2}\left(\vartheta \right)\right)\right)d\vartheta$$$${\Xi }_{\dot{\mathcal{H}},n}={\dot{\mathcal{H}}}_n-{\dot{\mathcal{H}}}_{n-1}=\frac{1-\Phi }{B\left(\Phi \right)}({\mathfrak{K}}_5(\Phi ,t,{\dot{\mathcal{H}}}_{n-1})-{\mathfrak{K}}_5(\Phi ,t,{\dot{\mathcal{H}}}_{n-2}))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\left({\mathfrak{K}}_5\left(\Phi ,\vartheta ,{\dot{\mathcal{H}}}_{n-1}\left(\vartheta \right)\right)\right)-\left({\mathfrak{K}}_5\left(\Phi ,\vartheta ,{\dot{\mathcal{H}}}_{n-2}\left(\vartheta \right)\right)\right)d\vartheta$$$${\Xi }_{\dot{\mathcal{R}},n}={\dot{\mathcal{R}}}_{1n}-{\dot{\mathcal{R}}}_{n-1}=\frac{1-\Phi }{B\left(\Phi \right)}({\mathfrak{K}}_6(\Phi ,t,{\dot{\mathcal{R}}}_{n-1})-{\mathfrak{K}}_3(\Phi ,t,{\dot{\mathcal{R}}}_{n-2}))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\left({\mathfrak{K}}_6\left(\Phi ,\vartheta ,{\dot{\mathcal{R}}}_{n-1}\left(\vartheta \right)\right)\right)-\left({\mathfrak{K}}_6\left(\Phi ,\vartheta ,{\dot{\mathcal{R}}}_{n-2}\left(\vartheta \right)\right)\right)d\vartheta$$ (18)

It is vital to observe that

$${\dot{\mathcal{S}}}_n=\sum _{i=0}^n{\Xi }_{\dot{\mathcal{S}},i},{\dot{\mathcal{E}}}_n=\sum _{i=0}^n{\Xi }_{\dot{\mathcal{E}},i},{\dot{\mathcal{I}}}_n=\sum _{i=0}^n{\Xi }_{\dot{\mathcal{I}},i},{\dot{\mathcal{A}}}_n=\sum _{i=0}^n{\Xi }_{\dot{\mathcal{A}},i}$$

$${\dot{\mathcal{H}}}_n=\sum _{i=0}^n{\Xi }_{\dot{\mathcal{H}},i},{\dot{\mathcal{R}}}_n=\sum _{i=0}^n{\Xi }_{\dot{\mathcal{R}},i}.$$

Additionally, by using Eqs. (15)–(16) and considering that

$${\Xi }_{\dot{\mathcal{S}},n-1}={\dot{\mathcal{S}}}_{n-1}-{\dot{\mathcal{S}}}_{n-2},{\Xi }_{\dot{\mathcal{E}},n-1}={\dot{\mathcal{E}}}_{n-1}-{\dot{\mathcal{E}}}_{n-2},{\Xi }_{\dot{\mathcal{I}},n-1}={\dot{\mathcal{I}}}_{n-1}-{\dot{\mathcal{I}}}_{n-2},$$

$${\Xi }_{\dot{\mathcal{A}},n-1}={\dot{\mathcal{A}}}_{n-1}-{\dot{\mathcal{A}}}_{n-2},{\Xi }_{\dot{\mathcal{H}},n-1}={\dot{\mathcal{H}}}_{n-1}-{\dot{\mathcal{H}}}_{n-2},{\Xi }_{\dot{\mathcal{R}},n-1}={\dot{\mathcal{R}}}_{n-1}-{\dot{\mathcal{R}}}_{n-2},$$

we reach

$$\begin{array}{c}\Vert {\Xi }_{\dot{\mathcal{S}},n}\left(t\right)\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}{\eta }_1\Vert {\Xi }_{\dot{\mathcal{S}},n-1}\left(t\right)\Vert \frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_1\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\Xi }_{\dot{\mathcal{S}},n-1}\left(\vartheta \right)\Vert d\vartheta \hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{E}},n}\left(t\right)\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}{\eta }_2\Vert {\Xi }_{\dot{\mathcal{E}},n-1}\left(t\right)\Vert \frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_2\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\Xi }_{\dot{\mathcal{E}},n-1}\left(\vartheta \right)\Vert d\vartheta \hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{I}},n}\left(t\right)\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}{\eta }_3\Vert {\Xi }_{\dot{\mathcal{I}},n-1}\left(t\right)\Vert \frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_3\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\Xi }_{\dot{\mathcal{I}},n-1}\left(\vartheta \right)\Vert d\vartheta \hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{A}},n}\left(t\right)\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}{\eta }_4\Vert {\Xi }_{\dot{\mathcal{A}},n-1}\left(t\right)\Vert \frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_4\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\Xi }_{\dot{\mathcal{A}},n-1}\left(\vartheta \right)\Vert d\vartheta \hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{H}},n}\left(t\right)\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}{\eta }_5\Vert {\Xi }_{\dot{\mathcal{H}},n-1}\left(t\right)\Vert \frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_5\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\Xi }_{\dot{\mathcal{H}},n-1}\left(\vartheta \right)\Vert d\vartheta \hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{R}},n}\left(t\right)\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}{\eta }_6\Vert {\Xi }_{\dot{\mathcal{R}},n-1}\left(t\right)\Vert \frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_6\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\Xi }_{\dot{\mathcal{R}},n-1}\left(\vartheta \right)\Vert d\vartheta .\hfill \end{array}$$ (19)

Theorem 3

Surmising that the following condition holds

$$\frac{1-\Phi }{B\left(\Phi \right)}{\eta }_i+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{b}^{\Phi }{\eta }_i<1,i=1,2,\dots ,8.$$ (20)

Then, (3) has a unique solution for $t\in [0,b]$ .

Proof

It is shown $\dot{\mathcal{S}}\left(t\right),\dot{\mathcal{E}}\left(t\right),\dot{\mathcal{I}}\left(t\right),\dot{\mathcal{A}}\left(t\right),\dot{\mathcal{H}}\left(t\right),\dot{\mathcal{R}}\left(t\right)$ are bounded functions. In Addition, as can be seen from Eqs. (15), (16), the symbols ${\mathfrak{K}}_1,{\mathfrak{K}}_2,{\mathfrak{K}}_3,{\mathfrak{K}}_4$, ${\mathfrak{K}}_5,{\mathfrak{K}}_6$ hold for Lipschitz condition. Therefore, utilizing Eq. (19) together with a recursive hypothesis, we arrive at

$$\begin{array}{c}\Vert {\Xi }_{\dot{\mathcal{S}},n}\left(t\right)\Vert \le \Vert {\dot{\mathcal{S}}}_0\left(t\right)\Vert {\left(\frac{1-\Phi }{B\left(\Phi \right)}{\eta }_1+\frac{\Phi b^{\Phi }}{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_1\right)}^n\hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{E}},n}\left(t\right)\Vert \le \Vert {\dot{\mathcal{E}}}_0\left(t\right)\Vert {\left(\frac{1-\Phi }{B\left(\Phi \right)}{\eta }_3+\frac{\Phi b^{\Phi }}{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_2\right)}^n\hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{I}},n}\left(t\right)\Vert \le \Vert {\dot{\mathcal{I}}}_0\left(t\right)\Vert {\left(\frac{1-\Phi }{B\left(\Phi \right)}{\eta }_3+\frac{\Phi b^{\Phi }}{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_3\right)}^n\hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{A}},n}\left(t\right)\Vert \le \Vert {\dot{\mathcal{A}}}_0\left(t\right)\Vert {\left(\frac{1-\Phi }{B\left(\Phi \right)}{\eta }_4+\frac{\Phi b^{\Phi }}{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_4\right)}^n\hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{H}},n}\left(t\right)\Vert \le \Vert {\dot{\mathcal{H}}}_0\left(t\right)\Vert {\left(\frac{1-\Phi }{B\left(\Phi \right)}{\eta }_5+\frac{\Phi b^{\Phi }}{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_5\right)}^n\hfill \\ \hfill \\ \Vert {\Xi }_{\dot{\mathcal{R}},n}\left(t\right)\Vert \le \Vert {\dot{\mathcal{R}}}_0\left(t\right)\Vert {\left(\frac{1-\Phi }{B\left(\Phi \right)}{\eta }_6+\frac{\Phi b^{\Phi }}{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\eta }_6\right)}^n\hfill \end{array}$$ (21)

Hence, it can be noted that for $n\to \infty$, all sequences exists and satisfies

$$\Vert {\Xi }_{\dot{\mathcal{S}},n}\Vert \to 0,\Vert {\Xi }_{\dot{\mathcal{E}},n}\Vert \to 0,\Vert {\Xi }_{\dot{\mathcal{I}},n}\Vert \to 0,\Vert {\Xi }_{\dot{\mathcal{A}},n}\Vert \to 0,\Vert {\Xi }_{\dot{\mathcal{H}},n}\Vert \to 0,\Vert {\Xi }_{\dot{\mathcal{H}},n}\Vert \to 0.$$

Moreover, from Eq. (21) and imposing the triangle inequality, for any $k$, we have

$$\begin{array}{c}\Vert {\dot{\mathcal{S}}}_{n+k}-{\dot{\mathcal{S}}}_n\Vert \le \sum _{j=n+1}^{n+k}{Z}_1^j=\frac{Z_1^{n+1}-Z_1^{n+k+1}}{1-Z_1}\hfill \\ \hfill \\ \Vert {\dot{\mathcal{E}}}_{n+k}-{\dot{\mathcal{E}}}_n\Vert \le \sum _{j=n+1}^{n+k}{Z}_2^j=\frac{Z_2^{n+1}-Z_2^{n+k+1}}{1-Z_2}\hfill \\ \hfill \\ \Vert {\dot{\mathcal{I}}}_{n+k}-{\dot{\mathcal{I}}}_n\Vert \le \sum _{j=n+1}^{n+k}{Z}_3^j=\frac{Z_3^{n+1}-Z_3^{n+k+1}}{1-Z_3}\hfill \\ \hfill \\ \Vert {\dot{\mathcal{A}}}_{n+k}-{\dot{\mathcal{A}}}_n\Vert \le \sum _{j=n+1}^{n+k}{Z}_4^j=\frac{Z_4^{n+1}-Z_4^{n+k+1}}{1-Z_4}\hfill \\ \hfill \\ \Vert {\dot{\mathcal{H}}}_{n+k}-{\dot{\mathcal{H}}}_n\Vert \le \sum _{i=n+1}^{n+k}{Z}_5^j=\frac{Z_5^{n+1}-Z_5^{n+k+1}}{1-Z_5}\hfill \\ \hfill \\ \Vert {\dot{\mathcal{R}}}_{n+k}-{\dot{\mathcal{R}}}_n\Vert \le \sum _{j=n+1}^{n+k}{Z}_3^j=\frac{Z_3^{n+1}-Z_3^{n+k+1}}{1-Z_3},\hfill \end{array}$$ (22)

with $Z_i=\frac{1-\Phi }{B\left(\Phi \right)}{\eta }_i+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{b}^{\Phi }{\eta }_i<1$ by hypothesis. Therefore, ${\dot{\mathcal{S}}}_n,{\dot{\mathcal{E}}}_n,{\dot{\mathcal{I}}}_n,{\dot{\mathcal{A}}}_n,{\dot{\mathcal{H}}}_n,{\dot{\mathcal{R}}}_n$ can be seen as a Cauchy sequences in the Banach space $B\left(J\right)$. This has shown that they are uniformly convergent [32]. Imposing the limit theorem in Eq. (18) as $n\to \infty$ affirms that the limit of these sequences is the unique solution of (3). This guarantee the existence of a unique solution for Eq. (3) under the condition (20).

Hyers-Ulam stability

Definition [33] —

The AB fractional integral system given by Eqs. (13) is said to be Hyers-Ulam stable if exist constants ${\Delta }_i>0,i\in {\mathbf{N}}^6$ satisfying: For every${\gamma }_i>0,i\in {\mathbf{N}}^8$, for

$$|\mathcal{S}\left(t\right)-\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_1(\Phi ,t,\mathcal{S}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_1(\Phi ,\vartheta ,\mathcal{S}\left(\vartheta \right))d\vartheta |$$

$$\le {\gamma }_1,|\mathcal{E}\left(t\right)-\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_2(\Phi ,t,\mathcal{E}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_2(\Phi ,\vartheta ,\mathcal{E}\left(\vartheta \right))d\vartheta |$$

$$\le {\gamma }_2,$$

$$|\mathcal{I}\left(t\right)-\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_3(\Phi ,t,\mathcal{I}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_3(\Phi ,\vartheta ,\mathcal{I}\left(\vartheta \right))d\vartheta |$$$$\le {\gamma }_3,|\mathcal{A}\left(t\right)-\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_4(\Phi ,t,\mathcal{A}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_4(\Phi ,\vartheta ,\mathcal{A}\left(\vartheta \right))d\vartheta |$$$$\le {\gamma }_4,|\mathcal{H}\left(t\right)-\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_5(\Phi ,t,\mathcal{H}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_5(\Phi ,\vartheta ,\mathcal{H}\left(\vartheta \right))d\vartheta |$$$$\le {\gamma }_5,|\mathcal{R}\left(t\right)-\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_6(\Phi ,t,\mathcal{R}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_6(\Phi ,\vartheta ,\mathcal{R}\left(t\right)\left(\vartheta \right))d\vartheta |$$$$\le {\gamma }_6.$$ (23)

there exist $\left(\dot{\mathcal{S}},\dot{\mathcal{E}},\dot{\mathcal{I}},\dot{\mathcal{A}},\dot{\mathcal{H}},\dot{\mathcal{R}}\right)$ which are satisfying

$$\dot{\mathcal{S}}\left(t\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_1(\Phi ,t,\mathcal{S}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_1(\Phi ,\vartheta ,\dot{\mathcal{S}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{E}}\left(t\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_2(\Phi ,t,\mathcal{Es}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_2(\Phi ,\vartheta ,\dot{\mathcal{E}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{I}}\left(t\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_3(\Phi ,t,\mathcal{I}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_3(\Phi ,\vartheta ,\dot{\mathcal{I}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{A}}\left(t\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_4(\Phi ,t,\mathcal{A}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_4(\Phi ,\vartheta ,\dot{\mathcal{A}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{H}}\left(t\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_5(\Phi ,t,\mathcal{H}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_5(\Phi ,\vartheta ,\dot{\mathcal{H}}\left(\vartheta \right))d\vartheta ,$$$$\dot{\mathcal{R}}\left(t\right)=\frac{1-\Phi }{B\left(\Phi \right)}{\mathfrak{K}}_6(\Phi ,t,\mathcal{R}\left(t\right))+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\times {\int }_0^t{(t-\vartheta )}^{\Phi -1}{\mathfrak{K}}_6(\Phi ,\vartheta ,\dot{\mathcal{R}}\left(\vartheta \right))d\vartheta .$$ (24)

Such that

$$|\mathcal{S}-\dot{\mathcal{S}}|\le {\zeta }_1{\gamma }_1,|\mathcal{E}-\dot{\mathcal{E}}|\le {\zeta }_2{\gamma }_2,|\mathcal{I}-\dot{\mathcal{I}}|\le {\zeta }_3{\gamma }_3,|\mathcal{A}-\dot{\mathcal{A}}|\le {\zeta }_4{\gamma }_4,|\mathcal{H}-\dot{\mathcal{H}}|\le {\zeta }_5{\gamma }_5,|\mathcal{R}-\dot{\mathcal{R}}|\le {\zeta }_6{\gamma }_6.$$

Theorem 4

With assumption $J$ , the suggested model of fractional order (3) is Hyers-Ulam stable.

Proof

With the help of theorem 3, the proposed AB fractional model (3) has a unique solution $(\mathcal{S},\mathcal{E},\mathcal{I},\mathcal{A},\mathcal{H},\mathcal{R})$ satisfying equations of system (13). Then, we have

$$\Vert \mathcal{S}-\dot{\mathcal{S}}\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}\Vert {\mathfrak{K}}_1(\Phi ,t,\mathcal{S})-{\mathfrak{K}}_1(\Phi ,t,\dot{\mathcal{S}})\Vert +\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\mathfrak{K}}_1(\Phi ,t,\mathcal{S})-{\mathfrak{K}}_1(\Phi ,t,\dot{\mathcal{S}})\Vert d\vartheta \le \left[\frac{1-\Phi }{B\left(\Phi \right)}+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\right]{\varphi }_1\Vert \mathcal{S}-\dot{\mathcal{S}}\Vert$$ (25)

$$\Vert \mathcal{E}-\dot{\mathcal{E}}\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}\Vert {\mathfrak{K}}_2(\Phi ,t,\mathcal{E})-{\mathfrak{K}}_2(\Phi ,t,\dot{\mathcal{E}})\Vert +\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\mathfrak{K}}_2(\Phi ,t,\mathcal{E})-{\mathfrak{K}}_2(\Phi ,t,\dot{\mathcal{E}})\Vert d\vartheta \le \left[\frac{1-\Phi }{B\left(\Phi \right)}+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\right]{\varphi }_2\Vert \mathcal{E}-\dot{\mathcal{E}}\Vert$$ (26)

$$\Vert \mathcal{I}-\dot{\mathcal{I}}\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}\Vert {\mathfrak{K}}_3(\Phi ,t,\mathcal{I})-{\mathfrak{K}}_3(\Phi ,t,\dot{\mathcal{I}})\Vert +\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\mathfrak{K}}_3(\Phi ,t,\mathcal{I})-{\mathfrak{K}}_3(\Phi ,t,\dot{\mathcal{I}})\Vert d\vartheta \le \left[\frac{1-\Phi }{B\left(\Phi \right)}+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\right]{\varphi }_3\Vert \mathcal{I}-\dot{\mathcal{I}}\Vert$$ (27)

$$\Vert \mathcal{A}-\dot{\mathcal{A}}\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}\Vert {\mathfrak{K}}_4(\Phi ,t,\mathcal{A})-{\mathfrak{K}}_4(\Phi ,t,\dot{\mathcal{A}})\Vert +\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\mathfrak{K}}_4(\Phi ,t,\mathcal{A})-{\mathfrak{K}}_4(\Phi ,t,\dot{\mathcal{A}})\Vert d\vartheta \le \left[\frac{1-\Phi }{B\left(\Phi \right)}+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\right]{\varphi }_4\Vert \mathcal{A}-\dot{\mathcal{A}}\Vert$$ (28)

$$\Vert \mathcal{H}-\dot{\mathcal{H}}\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}\Vert {\mathfrak{K}}_5(\Phi ,t,\mathcal{H})-{\mathfrak{K}}_5(\Phi ,t,\dot{\mathcal{H}})\Vert +\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\mathfrak{K}}_5(\Phi ,t,\mathcal{H})-{\mathfrak{K}}_5(\Phi ,t,\dot{\mathcal{H}})\Vert d\vartheta \le \left[\frac{1-\Phi }{B\left(\Phi \right)}+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\right]{\varphi }_5\Vert \mathcal{H}-\dot{\mathcal{H}}\Vert$$ (29)

$$\Vert \mathcal{R}-\dot{\mathcal{R}}\Vert \le \frac{1-\Phi }{B\left(\Phi \right)}\Vert {\mathfrak{K}}_6(\Phi ,t,\mathcal{R})-{\mathfrak{K}}_6(\Phi ,t,\dot{\mathcal{R}})\Vert +\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}{\int }_0^t{(t-\vartheta )}^{\Phi -1}\Vert {\mathfrak{K}}_6(\Phi ,t,\mathcal{R})-{\mathfrak{K}}_6(\Phi ,t,\dot{\mathcal{R}})\Vert d\vartheta \le \left[\frac{1-\Phi }{B\left(\Phi \right)}+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}\right]{\varphi }_6\Vert \mathcal{R}-\dot{\mathcal{R}}\Vert$$ (30)

Taking, ${\gamma }_i={\varphi }_i,{\Delta }_i=\frac{1-\Phi }{B\left(\Phi \right)}+\frac{\Phi }{B\left(\Phi \right)\Gamma \left(\Phi \right)}$, this implies

$$\Vert \mathcal{S}-\dot{\mathcal{S}}\Vert \le {\gamma }_1{\Delta }_1$$ (31)

Similarly, we have the followings

$$\left\{\begin{array}{c}\Vert \mathcal{E}-\dot{\mathcal{E}}\Vert \le {\gamma }_2{\Delta }_2\hfill \\ \Vert \mathcal{I}-\dot{\mathcal{I}}\Vert \le {\gamma }_3{\Delta }_3\hfill \\ \Vert \mathcal{A}-\dot{\mathcal{A}}\Vert \le {\gamma }_4{\Delta }_4\hfill \\ \Vert \mathcal{H}-\dot{\mathcal{H}}\Vert \le {\gamma }_5{\Delta }_5\hfill \\ \Vert \mathcal{R}-\dot{\mathcal{R}}\Vert \le {\gamma }_6{\Delta }_6.\hfill \end{array}\right)$$ (32)

Hence the proof is accomplished.

Numerical schemes & graphical results

In this section we presented numerical scheme for our model with Atangana–Baleanu Fractional derivative.

Numerical solution by Newton polynomial

$$\left\{\begin{array}{cc}\hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{S}\right]=\mathcal{G}-\frac{{\mathcal{G}}_1\left({\mathcal{G}}_2{\mathcal{I}}^{\ast }+{\mathcal{A}}^{\ast }\right){\mathcal{S}}^{\ast }}{N}-{\mathcal{G}}_8{\mathcal{S}}^{\ast },\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{E}\right]=\frac{{\mathcal{G}}_1\left({\mathcal{G}}_2{\mathcal{I}}^{\ast }+{\mathcal{A}}^{\ast }\right)\ast \mathcal{S}}{N}-\left({\mathcal{G}}_4+{\mathcal{G}}_8\right){\mathcal{E}}^{\ast },\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{I}\right]=\left(1-{\mathcal{G}}_3\right){\mathcal{G}}_4{\mathcal{E}}^{\ast }-\left({\mathcal{G}}_5+{\mathcal{G}}_6+{\mathcal{G}}_8\right){\mathcal{I}}^{\ast },\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{A}\right]={\mathcal{G}}_3{\mathcal{G}}_4{\mathcal{E}}^{\ast }-\left({\mathcal{G}}_{10}+{\mathcal{G}}_7+{\mathcal{G}}_8\right){\mathcal{A}}^{\ast },\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{H}\right]={\mathcal{G}}_5{\mathcal{I}}^{\ast }+{\mathcal{G}}_{10}{\mathcal{A}}^{\ast }-\left({\mathcal{G}}_{11}+{\mathcal{G}}_9+{\mathcal{G}}_8\right){\mathcal{H}}^{\ast },\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{R}\right]={\mathcal{G}}_6{\mathcal{I}}^{\ast }+{\mathcal{G}}_7{\mathcal{A}}^{\ast }+\left({\mathcal{G}}_{11}+{\mathcal{G}}_9\right){\mathcal{H}}^{\ast }-{\mathcal{G}}_8{\mathcal{R}}^{\ast }.\hfill \end{array}\right)$$ (33)

For simplicity, we write the above equation as follows;

$$\left\{\begin{array}{cc}\hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{S}\left(t\right)\right]={\mathcal{S}}^{\ast }(t,\mathcal{S},\mathcal{E},\mathcal{I},\mathcal{A},\mathcal{H},\mathcal{R}),\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{E}\left(t\right)\right]={\mathcal{E}}^{\ast }(t,\mathcal{S},\mathcal{E},\mathcal{I},\mathcal{A},\mathcal{H},\mathcal{R}),\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{I}\left(t\right)\right]={\mathcal{I}}^{\ast }(t,\mathcal{S},\mathcal{E},\mathcal{I},\mathcal{A},\mathcal{H},\mathcal{R}),\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{A}\left(t\right)\right]={\mathcal{A}}^{\ast }(t,\mathcal{S},\mathcal{E},\mathcal{I},\mathcal{A},\mathcal{H},\mathcal{R}),\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{H}\left(t\right)\right]={\mathcal{H}}^{\ast }(t,\mathcal{S},\mathcal{E},\mathcal{I},\mathcal{A},\mathcal{H},\mathcal{R}),\hfill \\ \hfill & {}^{\mathrm{ABC}}{\mathbb{D}}_{0,t}^{\Phi }\left[\mathcal{R}\left(t\right)\right]={\mathcal{R}}^{\ast }(t,\mathcal{S},\mathcal{E},\mathcal{I},\mathcal{A},\mathcal{H},\mathcal{R}).\hfill \end{array}\right)$$ (34)

After applying fractional integral with Mittag-Leffler kernel and putting Newton polynomial into these equations, we can solve our model as follows;

$${\mathcal{S}}^{a+1}=\frac{1-\Phi }{AB\left(\Phi \right)}+{\mathcal{S}}^{\ast }(t_a,{\mathcal{S}}^a,{\mathcal{E}}^a,{\mathcal{I}}^a,{\mathcal{A}}^a,{\mathcal{H}}^a,{\mathcal{R}}^a)+\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +1)}\sum _{\mu =2}^a{\mathcal{S}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\Pi +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +2)}\sum _{\mu =2}^a\left[\begin{array}{c}\hfill {\mathcal{S}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill -{\mathcal{S}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right]\Sigma +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{2AB\left(\Phi \right)\Gamma (\Phi +3)}\sum _{\mu =2}^a\left\{\begin{array}{c}\hfill {\mathcal{S}}^{\ast }(t_{\mu },{\mathcal{S}}^{\mu },{\mathcal{E}}^{\mu },{\mathcal{I}}^{\mu },{\mathcal{A}}^{\mu },{\mathcal{H}}^{\mu },{\mathcal{R}}^{\mu })\hfill \\ \hfill -2{\mathcal{S}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill +{\mathcal{S}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right\}\Delta$$

$${\mathcal{E}}^{a+1}=\frac{1-\Phi }{AB\left(\Phi \right)}+{\mathcal{E}}^{\ast }(t_a,{\mathcal{S}}^a,{\mathcal{E}}^a,{\mathcal{I}}^a,{\mathcal{A}}^a,{\mathcal{H}}^a,{\mathcal{R}}^a)+\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +1)}\sum _{\mu =2}^a{\mathcal{E}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\Pi +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +2)}\sum _{\mu =2}^a\left[\begin{array}{c}\hfill {\mathcal{E}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill -{\mathcal{E}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right]\Sigma +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{2AB\left(\Phi \right)\Gamma (\Phi +3)}\sum _{\mu =2}^a\left\{\begin{array}{c}\hfill {\mathcal{E}}^{\ast }(t_{\mu },{\mathcal{S}}^{\mu },{\mathcal{E}}^{\mu },{\mathcal{I}}^{\mu },{\mathcal{A}}^{\mu },{\mathcal{H}}^{\mu },{\mathcal{R}}^{\mu })\hfill \\ \hfill -2{\mathcal{E}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill +{\mathcal{E}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right\}\Delta$$

$${\mathcal{I}}^{a+1}=\frac{1-\Phi }{AB\left(\Phi \right)}+{\mathcal{I}}^{\ast }(t_a,{\mathcal{S}}^a,{\mathcal{E}}^a,{\mathcal{I}}^a,{\mathcal{A}}^a,{\mathcal{H}}^a,{\mathcal{R}}^a)+\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +1)}\sum _{\mu =2}^a{\mathcal{I}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\Pi +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +2)}\sum _{\mu =2}^a\left[\begin{array}{c}\hfill {\mathcal{I}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill -{\mathcal{I}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right]\Sigma +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{2AB\left(\Phi \right)\Gamma (\Phi +3)}\sum _{\mu =2}^a\left\{\begin{array}{c}\hfill {\mathcal{I}}^{\ast }(t_{\mu },{\mathcal{S}}^{\mu },{\mathcal{E}}^{\mu },{\mathcal{I}}^{\mu },{\mathcal{A}}^{\mu },{\mathcal{H}}^{\mu },{\mathcal{R}}^{\mu })\hfill \\ \hfill -2{\mathcal{I}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill +{\mathcal{I}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right\}\Delta$$

$${\mathcal{A}}^{a+1}=\frac{1-\Phi }{AB\left(\Phi \right)}+{\mathcal{A}}^{\ast }(t_a,{\mathcal{S}}^a,{\mathcal{E}}^a,{\mathcal{I}}^a,{\mathcal{A}}^a,{\mathcal{H}}^a,{\mathcal{R}}^a)+\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +1)}\sum _{\mu =2}^a{\mathcal{A}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\Pi +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +2)}\sum _{\mu =2}^a\left[\begin{array}{c}\hfill {\mathcal{A}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill -{\mathcal{A}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right]\Sigma +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{2AB\left(\Phi \right)\Gamma (\Phi +3)}\sum _{\mu =2}^a\left\{\begin{array}{c}\hfill {\mathcal{A}}^{\ast }(t_{\mu },{\mathcal{S}}^{\mu },{\mathcal{E}}^{\mu },{\mathcal{I}}^{\mu },{\mathcal{A}}^{\mu },{\mathcal{H}}^{\mu },{\mathcal{R}}^{\mu })\hfill \\ \hfill -2{\mathcal{A}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill +{\mathcal{A}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right\}\Delta$$

$${\mathcal{H}}^{a+1}=\frac{1-\Phi }{AB\left(\Phi \right)}+{\mathcal{H}}^{\ast }(t_a,{\mathcal{S}}^a,{\mathcal{E}}^a,{\mathcal{I}}^a,{\mathcal{A}}^a,{\mathcal{H}}^a,{\mathcal{R}}^a)+\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +1)}\sum _{\mu =2}^a{\mathcal{H}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\Pi +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +2)}\sum _{\mu =2}^a\left[\begin{array}{c}\hfill {\mathcal{H}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill -{\mathcal{H}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right]\Sigma +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{2AB\left(\Phi \right)\Gamma (\Phi +3)}\sum _{\mu =2}^a\left\{\begin{array}{c}\hfill {\mathcal{H}}^{\ast }(t_{\mu },{\mathcal{S}}^{\mu },{\mathcal{E}}^{\mu },{\mathcal{I}}^{\mu },{\mathcal{A}}^{\mu },{\mathcal{H}}^{\mu },{\mathcal{R}}^{\mu })\hfill \\ \hfill -2{\mathcal{H}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill +{\mathcal{H}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right\}\Delta$$

$${\mathcal{R}}^{a+1}=\frac{1-\Phi }{AB\left(\Phi \right)}+{\mathcal{R}}^{\ast }(t_a,{\mathcal{S}}^a,{\mathcal{E}}^a,{\mathcal{I}}^a,{\mathcal{A}}^a,{\mathcal{H}}^a,{\mathcal{R}}^a)+\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +1)}\sum _{\mu =2}^a{\mathcal{R}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\Pi +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{AB\left(\Phi \right)\Gamma (\Phi +2)}\sum _{\mu =2}^a\left[\begin{array}{c}\hfill {\mathcal{R}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill -{\mathcal{R}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right]\Sigma +\frac{\Phi {\left(\Delta t\right)}^{\Phi }}{2AB\left(\Phi \right)\Gamma (\Phi +3)}\sum _{\mu =2}^a\left\{\begin{array}{c}\hfill {\mathcal{R}}^{\ast }(t_{\mu },{\mathcal{S}}^{\mu },{\mathcal{E}}^{\mu },{\mathcal{I}}^{\mu },{\mathcal{A}}^{\mu },{\mathcal{H}}^{\mu },{\mathcal{R}}^{\mu })\hfill \\ \hfill -2{\mathcal{R}}^{\ast }(t_{\mu -1},{\mathcal{S}}^{\mu -1},{\mathcal{E}}^{\mu -1},{\mathcal{I}}^{\mu -1},{\mathcal{A}}^{\mu -1},{\mathcal{H}}^{\mu -1},{\mathcal{R}}^{\mu -1})\hfill \\ \hfill +{\mathcal{R}}^{\ast }(t_{\mu -2},{\mathcal{S}}^{\mu -2},{\mathcal{E}}^{\mu -2},{\mathcal{I}}^{\mu -2},{\mathcal{A}}^{\mu -2},{\mathcal{H}}^{\mu -2},{\mathcal{R}}^{\mu -2})\hfill \end{array}\right\}\Delta$$

Where

$$\Delta =\left[\begin{array}{c}\hfill {(a-\mu +1)}^{\Phi }\left[\begin{array}{c}\hfill 2{(a-\mu )}^2+(3\Phi +10)(a-\mu )\hfill \\ \hfill +2\Phi {\mathcal{I}}^2+9\Phi +12\hfill \end{array}\right]\hfill \\ \hfill -{(a-\mu )}^{\Phi }\left[\begin{array}{c}\hfill 2{(a-\mu )}^2+(5\Phi +10)(a-\mu )\hfill \\ \hfill +6\Phi {\mathcal{I}}^2+18\Phi +12\hfill \end{array}\right]\hfill \end{array}\right],$$

$$\Sigma =\left[\begin{array}{c}\hfill {(a-\mu +1)}^{\Phi }(a-\mu +3+2\Phi )\hfill \\ \hfill -{(a-\mu )}^{\Phi }(a-\mu +3+3\Phi )\hfill \end{array}\right],$$

$$\Pi =[{(a-\mu +1)}^{\Phi }-{(a-\mu )}^{\Phi }].$$

Numerical solution by Adams–Bashforth method

This section examines the numerical simulation results for COVID-19 pandemic disease model (3). The numerical method employed on system (3) hinged on Adams–Bashforth Rule.

$$\mathcal{S}\left(t_{n+1}\right)=\mathcal{S}\left(0\right)+\frac{1-\Phi }{ABC\left(\Phi \right)}\left[{\Psi }_1(\mathcal{S}\left(t_n\right),\mathcal{E}\left(t_n\right),\mathcal{I}\left(t_n\right),\mathcal{A}\left(t_n\right),\mathcal{H}\left(t_n\right),\mathcal{R}\left(t_n\right),t_n)\right]+\frac{\Phi }{ABC\left(\Phi \right)}\sum _{q=0}^n(\frac{{\Psi }_1(\mathcal{S}\left(t_q\right),\mathcal{E}\left(t_q\right),\mathcal{I}\left(t_q\right),\mathcal{A}\left(t_q\right),\mathcal{H}\left(t_q\right),\mathcal{R}\left(t_q\right),t_q)}{\Gamma (\Phi +2)}$$$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi }(n-q+2+\Phi )-{(n-q)}^{\Phi }(n-q+2+2\Phi )]-\frac{{\Psi }_1(\mathcal{S}\left(t_{q-1}\right),\mathcal{E}\left(t_{q-1}\right),\mathcal{I}\left(t_{q-1}\right),\mathcal{A}\left(t_{q-1}\right),\mathcal{H}\left(t_{q-1}\right),\mathcal{R}\left(t_{q-1}\right),t_{q-1})}{\Gamma (\Phi +2)}$$$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi +1}-{(n-1)}^{\Phi }(n-q+1+\Phi )])$$$$\mathcal{E}\left(t_{n+1}\right)=\mathcal{E}\left(0\right)+\frac{1-\Phi }{ABC\left(\Phi \right)}\left[{\Psi }_2(\mathcal{S}\left(t_n\right),\mathcal{E}\left(t_n\right),\mathcal{I}\left(t_n\right),\mathcal{A}\left(t_n\right),\mathcal{H}\left(t_n\right),\mathcal{R}\left(t_n\right),t_n)\right]+\frac{\Phi }{ABC\left(\Phi \right)}\sum _{q=0}^n(\frac{{\Psi }_2(\mathcal{S}\left(t_q\right),\mathcal{E}\left(t_q\right),\mathcal{I}\left(t_q\right),\mathcal{A}\left(t_q\right),\mathcal{H}\left(t_q\right),\mathcal{R}\left(t_q\right),t_q)}{\Gamma (\Phi +2)}$$$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi }(n-q+2+\Phi )-{(n-q)}^{\Phi }(n-q+2+2\Phi )]-\frac{{\Psi }_2(\mathcal{S}\left(t_{q-1}\right),\mathcal{E}\left(t_{q-1}\right),\mathcal{I}\left(t_{q-1}\right),\mathcal{A}\left(t_{q-1}\right),\mathcal{H}\left(t_{q-1}\right),\mathcal{R}\left(t_{q-1}\right),t_{q-1})}{\Gamma (\Phi +2)}$$$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi +1}-{(n-1)}^{\Phi }(n-q+1+\Phi )])$$$$\mathcal{I}\left(t_{n+1}\right)=\mathcal{I}\left(0\right)+\frac{1-\Phi }{ABC\left(\Phi \right)}\left[{\Psi }_3(\mathcal{S}\left(t_n\right),\mathcal{E}\left(t_n\right),\mathcal{I}\left(t_n\right),\mathcal{A}\left(t_n\right),\mathcal{H}\left(t_n\right),\mathcal{R}\left(t_n\right),t_n)\right]+\frac{\Phi }{ABC\left(\Phi \right)}\sum _{q=0}^n(\frac{{\Psi }_3(\mathcal{S}\left(t_q\right),\mathcal{E}\left(t_q\right),\mathcal{I}\left(t_q\right),\mathcal{A}\left(t_q\right),\mathcal{H}\left(t_q\right),\mathcal{R}\left(t_q\right),t_q)}{\Gamma (\Phi +2)}$$$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi }(n-q+2+\Phi )-{(n-q)}^{\Phi }(n-q+2+2\Phi )]-\frac{{\Psi }_3(\mathcal{S}\left(t_{q-1}\right),\mathcal{E}\left(t_{q-1}\right),\mathcal{I}\left(t_{q-1}\right),\mathcal{A}\left(t_{q-1}\right),\mathcal{H}\left(t_{q-1}\right),\mathcal{R}\left(t_{q-1}\right),t_{q-1})}{\Gamma (\Phi +2)}$$$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi +1}-{(n-1)}^{\Phi }(n-q+1+\Phi )])$$ (35)

$$\mathcal{A}\left(t_{n+1}\right)=\mathcal{A}\left(0\right)+\frac{1-\Phi }{ABC\left(\Phi \right)}\left[{\Psi }_4(\mathcal{S}\left(t_n\right),\mathcal{E}\left(t_n\right),\mathcal{I}\left(t_n\right),\mathcal{A}\left(t_n\right),\mathcal{H}\left(t_n\right),\mathcal{R}\left(t_n\right),t_n)\right]+\frac{\Phi }{ABC\left(\Phi \right)}\sum _{q=0}^n(\frac{{\Psi }_4(\mathcal{S}\left(t_q\right),\mathcal{E}\left(t_q\right),\mathcal{I}\left(t_q\right),\mathcal{A}\left(t_q\right),\mathcal{H}\left(t_q\right),\mathcal{R}\left(t_q\right),t_q)}{\Gamma (\Phi +2)}$$

$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi }(n-q+2+\Phi )-{(n-q)}^{\Phi }(n-q+2+2\Phi )]-\frac{{\Psi }_4(\mathcal{S}\left(t_{q-1}\right),\mathcal{E}\left(t_{q-1}\right),\mathcal{I}\left(t_{q-1}\right),\mathcal{A}\left(t_{q-1}\right),\mathcal{H}\left(t_{q-1}\right),\mathcal{R}\left(t_{q-1}\right),t_{q-1})}{\Gamma (\Phi +2)}$$

$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi +1}-{(n-1)}^{\Phi }(n-q+1+\Phi )])$$

$$\mathcal{H}\left(t_{n+1}\right)=\mathcal{H}\left(0\right)+\frac{1-\Phi }{ABC\left(\Phi \right)}\left[{\Psi }_5(\mathcal{S}\left(t_n\right),\mathcal{E}\left(t_n\right),\mathcal{I}\left(t_n\right),\mathcal{A}\left(t_n\right),\mathcal{H}\left(t_n\right),\mathcal{R}\left(t_n\right),t_n)\right]+\frac{\Phi }{ABC\left(\Phi \right)}\sum _{q=0}^n(\frac{{\Psi }_5(\mathcal{S}\left(t_q\right),\mathcal{E}\left(t_q\right),\mathcal{I}\left(t_q\right),\mathcal{A}\left(t_q\right),\mathcal{H}\left(t_q\right),\mathcal{R}\left(t_q\right),t_q)}{\Gamma (\Phi +2)}$$

$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi }(n-q+2+\Phi )-{(n-q)}^{\Phi }(n-q+2+2\Phi )]-\frac{{\Psi }_5(\mathcal{S}\left(t_{q-1}\right),\mathcal{E}\left(t_{q-1}\right),\mathcal{I}\left(t_{q-1}\right),\mathcal{A}\left(t_{q-1}\right),\mathcal{H}\left(t_{q-1}\right),\mathcal{R}\left(t_{q-1}\right),t_{q-1})}{\Gamma (\Phi +2)}$$

$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi +1}-{(n-1)}^{\Phi }(n-q+1+\Phi )])$$

$$\mathcal{R}\left(t_{n+1}\right)=\mathcal{R}\left(0\right)+\frac{1-\Phi }{ABC\left(\Phi \right)}\left[{\Psi }_6(\mathcal{S}\left(t_n\right),\mathcal{E}\left(t_n\right),\mathcal{I}\left(t_n\right),\mathcal{A}\left(t_n\right),\mathcal{H}\left(t_n\right),\mathcal{R}\left(t_n\right),t_n)\right]+\frac{\Phi }{ABC\left(\Phi \right)}\sum _{q=0}^n(\frac{{\Psi }_6(\mathcal{S}\left(t_q\right),\mathcal{E}\left(t_q\right),\mathcal{I}\left(t_q\right),\mathcal{A}\left(t_q\right),\mathcal{H}\left(t_q\right),\mathcal{R}\left(t_q\right),t_q)}{\Gamma (\Phi +2)}$$$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi }(n-q+2+\Phi )-{(n-q)}^{\Phi }(n-q+2+2\Phi )]-\frac{{\Psi }_6(\mathcal{S}\left(t_{q-1}\right),\mathcal{E}\left(t_{q-1}\right),\mathcal{I}\left(t_{q-1}\right),\mathcal{A}\left(t_{q-1}\right),\mathcal{H}\left(t_{q-1}\right),\mathcal{R}\left(t_{q-1}\right),t_{q-1})}{\Gamma (\Phi +2)}$$$$\Delta {\mathcal{A}}^{\Phi }[{(n+1-q)}^{\Phi +1}-{(n-1)}^{\Phi }(n-q+1+\Phi )])$$ (36)

Discussion

The COVID-19 numerical model (3) in Atangana–Baleanu–Caputo sense is addressed by Newton polynomial and Adams–Bashforth methods. The values of the corresponding parameters are utilized from Table 1 which are acquired from reported contaminated cases are utilized in the numerical results. In Figs. 1(a)–1(f), the effect of arbitrary fractional order $\Phi$ (i.e., memory record) is portrayed graphically. The susceptible class diminishes for all values of $\Phi$ to a particular positive density as can be found in Fig. 1, Fig. 2. A similar conduct is found for all values of $\Phi$. Fig. 1, Fig. 2 portrays the elements of the exposed group for changing values of $\Phi$. Fig. 1, Fig. 2 portrays the elements of the asymptomatic class for changing values of $\Phi$. It is seen that the peaks of the contaminated curves slightly diminished and happened relatively throughout a more drawn out time frame for more modest values of $\Phi$. Similar interpretations are acquired for the population in the leftover tainted classes as introduced in Fig. 1, Fig. 2. The graphical results for the different fractional order values of hospitalized individuals are displayed in Fig. 1, Fig. 2, it shows that the hospitalized individuals are increased as the covid infection increased. The elements of recovered or eliminated population for different values of $\Phi$ are investigated in Fig. 1, Fig. 2 (see Fig. 3, Fig. 4, Fig. 5, Fig. 6, Fig. 7, Fig. 8).

Fig. 1.

Numerical simulation for COVID-19 pandemic model (3) via Mittag-Leffler Generalized Function by Newton polynomial.

Fig. 2.

Numerical simulation for COVID-19 pandemic model (3) via Mittag-Leffler Generalized Function by Adams–Bashforth method.

Fig. 3.

Solution of susceptible group by newton polynomial (L) and Adams–Bashforth (R).

Fig. 4.

Solution of exposed group by newton polynomial (L) and Adams–Bashforth (R).

Fig. 5.

Solution of symptomatic Infected group by newton polynomial (L) and Adams–Bashforth (R).

Fig. 6.

Solution of asymptomatic Infected group by newton polynomial (L) and Adams–Bashforth (R).

Fig. 7.

Solution of Hospitalized group by newton polynomial (L) and Adams–Bashforth (R).

Fig. 8.

Solution of Recovered group by newton polynomial (L) and Adams–Bashforth (R).

Conclusion

Non-integer order Atangana–Baleanu–Caputo derivative was used to discuss the dynamical behavior of the modified fractional non-linear $\mathcal{S}\mathcal{E}\mathcal{I}\mathcal{A}\mathcal{H}\mathcal{R}$ model. The controlling and stabilizing of COVID infection was investigated from the given model. Using certain well-known results from the non-linear functional analysis mentioned therein, the qualitative analysis has been achieved. The detailed stability results for DFE and EE are explored using fractional stability approaches. For simulating the model, we have used fractional Newton polynomials and Adams–Bashforth techniques were used which showed the dynamical behavior for each compartment at different fractional order. The graphical representation of our model gives the compartmental density between two different integer values (0,1].

CRediT authorship contribution statement

Xiao-Ping Li: Conceptualization, Data curation, Validation, Formal analysis, Writing – original draft. Hilal Al Bayatti: Supervision, Project administration, Funding acquisition, Visualization, Review & editing. Anwarud Din: Visualization, Writing – review & editing. Anwar Zeb: Supervision, Methodology, Software, Visualization, Review & editing.

Data availability

Data sharing is not applicable to this article as no new data were created or analyzed in this study.