Mathematical modelling with computational fractional order for the unfolding dynamics of the communicable diseases

Abstract

Mathematical models based on computational fractional orders, employed for accurate modelling of complex dynamic systems, can ensure the implementation of various analytical, numerical and computing methods encompassing their applications to emerging and ever-varying real-world problems. Tracking, managing and controlling communicable diseases, one being monkeypox with different features, virological and taxonomic attributes, are oriented towards high-risk groups concerning global public health. This study, accordingly, is devoted to the presentation of the piecewise global derivative model of the monkeypox virus by applying the Caputo and Atangana Baleanu fractional-order derivatives in the partitioned two sub-intervals. The model includes eight compartments with two categories of human and rodent populations. The cases which take part in some sense for the said infection are investigated along with connection in this format. The existence and uniqueness of the solution in the framework of the piecewise global derivative are analyzed for both sub-intervals using fixed point theory. The detailed investigation of the dynamics of fractional-order systems and among many other dynamic features, stability is addressed. The stability of the solution is, thus, examined using the idea of Ulam Hyers concept. For the best fitting values of the parameters, the results are simulated using the monkeypox data. Using the method of Newton polynomial, different piecewise dynamics of each compartment are simulated on different fractional orders and time durations. This kind of a proposed approach is thought to lay a foundation where the transmission takes place to control epidemic events and other infectious medical conditions through vaccines or taking preventive measures to maintain and advance global public health while fully optimizing the clinical care of the diseases to manage complications, alleviate symptoms as well as prevent the long-term sequelae. This analysis also deals with sudden variation in monkeypox dynamics and also for crossover dynamics along with removal of discontinuity through modification of piecewise global analysis.

Keywords:

• Computational fractional orders
• fractional calculus
• piecewise global derivative
• Caputo operator
• ABC operator
• Newton polynomial technique
• Ulam–Hyers stability analysis
• global public health
• taxonomic attributes
• epidemic events

Maths:

• 34C60
• 34D20
• 65E05

1. Introduction

Modern calculus derivatives, with their extensive applications in mathematical models, help for the understanding of viral infections, epidemic events and other unpredictable phenomena since they have more degrees of freedom than integer-order models. For this reason, fractional-order models are considered to be more precise and reliable, with fractional-order differential equation models appearing to be more compatible with the varying conditions compared to the integer-order ones [1]. Compartmental models, furthermore, are regarded as one of the most frequently employed epidemic model types, related to both experimental and observation data. In these kinds of models, having a finite number of discrete states means that individuals manifest varying aspects, with some of the related states being merely labels which identify the various traits of the individuals affected under epidemiologically relevant conditions [2]. Another perspective is that models based on computational fractional orders can be employed for accurate modelling of complex dynamic systems, which allows one to implement different analytical, numerical and computing methods that include the related applications to emerging and ever-varying real-world problems. As such, tracking, managing and controlling communicable diseases, one being monkeypox (Mpox) which has different features, virological and taxonomic attributes, inclined towards high-risk groups concerning global public health conditions.

The monkeypox is a disease present in animals which has spread the monkeypox infections into human society. As animal species are present all over the globe, it can be found and is communicable over the globe. It was first discovered in the forests of Africa, where forest species specially monkeys are higher compared to the number in the rest of the world. Subsequently, two infections of the said disease matching the pox in community of monkeys, which were then used for the investigation, led to the testing of monkeypox in 1958. For about 6 months of that year, many countries tested different cases of the monkeypox reporting to the World Health Organization (WHO). On 15 June 2022, 2104 cases were tested with clinical records having one death occurred and reported by WHO [3,4]. The said virus may be transmitted directly with contact of an infected scab, rash or fluids of the body are other means the virus may be transmitted from one body of human to another one. The transmission process occurs also through respiration secretion at the time of prolonged face-to-face connection or in sexual activities or any other abrupt physical connections [3,5,6]. Initially, forest animals in Africa like rats and monkeys spread the said viruses to human population, and yet, person-to-person transmission is also mostly present in the reported cases. The transmission from the species to human population occurred as a result of bites or scratches, the process of bush meat, directly connected with body fluids, or used food eaten by rodents seem to be the main causes of the spread of the disease. Direct connection with sores and body fluids from infectious persons can also lead to the transmission of the disease in question. Accordingly, substantial amount of research has analyzed different approaches for the transmission of the said infection by semen and vaginal fluids or in respiration process drops [3,7]. While the signs of monkeypox indicate human to human transmission, it also includes other main signs such as headache, muscle pain, fever, backbone pain, swollen lymph nodes, chills and exhaustion. Mostly, the situation of monkeypox is said to be a commonly mature illness and most people are able to recover on their means after several weeks. Those who have a weak immune defensive system may have more challenging and tougher signs though [3,8]. The smallpox vaccines, anti-viral and vaccine immune globulin established to provide protection against smallpox may also be used as a cure to stop the transmission of monkeypox but there is currently no proven on time curve for the monkeypox virus disease. The vaccine is presently unavailable due to the fact that smallpox has been extinct worldwide [9,10].

Analysis of the new Monkeypox virus for the forecast and infectivity, the authors in [11] established a deep learning algorithm and found the two animals of this virus which is bat and minks. Mostly, mathematical models have been an important role in the direct transmission between humans-to-humans in the outbreak. The infected people have a long incubation period, yet, it is likely that they may not be aware of the symptoms and do not know about the quarantine time. This new disease can easily spread to other people and many researchers have documented models on monkeypox on different levels [12–14].

Much of the research has been geared towards the investigation of the dynamics of fractional-order systems and among many other dynamic features, stability has been reported [15]. Concerned upon our work, we have re-considered the monkeypox mathematical model [16], which has not been investigated under novel piecewise derivative and integrals operator before. The considered model has eight compartments having two categories namely: susceptible humans ${\mathbb{S}}_h\left(t\right)$, exposed humans ${\mathbb{E}}_h\left(t\right)$, infected humans ${\mathbb{I}}_h\left(t\right)$, clinical infected humans ${\mathbb{C}}_h\left(t\right)$, recovered human ${\mathbb{R}}_h\left(t\right)$, susceptible rodents ${\mathbb{S}}_r\left(t\right)$, exposed rodents ${\mathbb{E}}_r\left(t\right)$, infected rodents ${\mathbb{I}}_r\left(t\right)$.

The monkeypox model is taken from [16] which has the following form as indicated below: (1) $\begin{array}{rl}& \dot{{\mathbb{S}}_h}={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ & \dot{{\mathbb{E}}_h}={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ & \dot{{\mathbb{I}}_h}=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ & \dot{{\mathbb{C}}_h}=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ & \dot{{\mathbb{R}}_h}=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ & \dot{{\mathbb{S}}_r}={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ & \dot{{\mathbb{E}}_r}={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ & \dot{{\mathbb{I}}_r}=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r,\end{array}$(1) along with initial conditions $\begin{array}{rl}{\mathbb{S}}_h\left(0\right)& ={\mathbb{S}}_{h_0},{\mathbb{E}}_h\left(0\right)={\mathbb{E}}_{h_0},{\mathbb{I}}_h\left(0\right)={\mathbb{I}}_{h_0},{\mathbb{C}}_h\left(0\right)={\mathbb{C}}_{h_0},\\ {\mathbb{R}}_h\left(0\right)& ={\mathbb{R}}_{h_0},{\mathbb{S}}_r\left(0\right)={\mathbb{S}}_{r_0},{\mathbb{E}}_r\left(0\right)={\mathbb{E}}_{r_0},{\mathbb{I}}_r\left(0\right)={\mathbb{I}}_{r_0},\end{array}$where the given parameters in the model with descriptions are given in Table 1. The calculus of fractional order (FC) has attracted more interesting results from the researchers and scientist over the last 20 years [17,18]. FC, when compared with traditional integer-order models, is said to provide novel, accurate and deeper information on the complex activities of many infectious diseases and epidemic events with the mathematical models used as applicable schemes [19–24]. Because of genetic properties and memory behaviours, integer order problems are not superior than FC problems, and fractional order differential equations employ nonlocal operators, which can be considered to be ideal tools in the mathematical modelling of systems with memory. Many kinds of integer order equations are used in mathematical models of the real world with the real phenomena possible to be analyzed for a higher degree of choices and precision by applying the fractional differential equations [25–31]. The superiority of fractional-order models to classical ones has been demonstrated by several researchers from various branches of science, engineering and life sciences [32–37]. Correspondingly, authors in [38] used the Mittag–Leffler derivative and investigated the fractional infectious disease model. In [39], the authors used Atangana–Baleanu–Caputo fractional derivative and studied a mathematical model of COVID-19. The authors in [40] applied the Caputo–Fabrizio operator along with double Laplace transform and find the series solution for the fractional biological model. The scholars in [41] applied a novel fractional order Lagrangian scheme to show the motion of the beam on nanowire. Liaqat et al. [42] also established a new scheme to obtain the approximate and exact solution in the sense of Caputo fractional partial differential equation along with variable. Odibat and Baleanu [43] studied a novel system of fractional differential equations with involving generalized fractional Caputo operator. The researchers have investigated many number of groundbreaking works employing various operators and applied them to infectious disease and real models [44–50]. The framework of piecewise Caputo and Atangana–Baleanu operator is as follows: A novel operator for piecewise global derivative and integrals was given by Atangana and Araz [51].

Table 1. Parameters and their description according to model 1(1) $\begin{array}{rl}& \dot{{\mathbb{S}}_h}={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ & \dot{{\mathbb{E}}_h}={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ & \dot{{\mathbb{I}}_h}=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ & \dot{{\mathbb{C}}_h}=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ & \dot{{\mathbb{R}}_h}=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ & \dot{{\mathbb{S}}_r}={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ & \dot{{\mathbb{E}}_r}={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ & \dot{{\mathbb{I}}_r}=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r,\end{array}$(1) .

Parameter	Description
${\mathrm{\Phi }}_h$	Rate of birth of humans susceptible population
${\mathrm{\Phi }}_r$	Rate of birth of rodents susceptible population
${\nu }_h$	Rate of natural occurring in human population
${\nu }_r$	Rate of natural occurring in rodent population
α	Rate of infection progressive from exposed to infected human population
${\mathrm{\Delta }}_1$	Rate of induced infected human individuals
${\mathrm{\Delta }}_2$	Rate of induced clinical infected human individuals
γ	Rate of clinically ill individuals
ρ	Rate of recovery for clinical human individuals
Ω	Rate of recovery because of immune system
${\alpha }_1$	Rate of contact between infectious and susceptible humans
${\alpha }_2$	Rate of contact for infected rats and susceptible humans
σ	Rate of contact for infected rats and susceptible rodents
${\mathrm{\Lambda }}_h={\displaystyle \frac{{\alpha }_1{\mathbb{I}}_h}{N_h}}+{\displaystyle \frac{{\alpha }_2{\mathbb{I}}_r}{N_r}}$	infection influence
${\mathrm{\Lambda }}_r={\displaystyle \frac{\sigma {\mathbb{I}}_r}{N_r}}$	infection influence

The piecewise derivative is divided into two subintervals, the first interval solution is find out in the sense of one fractional operator while the second interval solution is under the other fractional operator. To overcome cross-over behaviour challenges, one of the unique ways of piece-wise derivative has been proposed in [51]. A new window of cross-over behaviours using these operators has been studied by researchers. Several uses of the fractional operators are investigated in the literature by different researchers [52–56]. Inspired from the above novel operator, we will investigate the model taken from [16] under the framework of piecewise Caputo and Atangana–Baleanu operator in the current study as follows: (2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) In more depth, Equation (2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) ) can be expressed as follows: (3) $\begin{array}{rl}{}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_1({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^d\left({\mathbb{E}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_2({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_2({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_3({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_3({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{C}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{C}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_4({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{C}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_4({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{R}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{R}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_5({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{R}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_5({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_r\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_r\right(t\left)\right){=}^C{\mathbb{Y}}_6({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_r\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_6({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_r\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_r\right(t\left)\right){=}^C{\mathbb{Y}}_7({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_r\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_7({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_r\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^d\left({\mathbb{I}}_r\right(t\left)\right){=}^C{\mathbb{Y}}_8({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^d\left({\mathbb{I}}_r\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_8({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T.\end{array}\end{array}$(3) The organization of the rest of this paper is as follows. The preliminaries and basic definitions are introduced in Section 2. Section 3 presents qualitative analysis: existence and uniqueness. Stability analysis with varying forms of function is handled in Section 4. Section 5 describes the numerical scheme for the fractional piecewise for the monkeypox model and Section 6 focuses on numerical simulations and analyses. Finally, Section 7 concludes the study with future directions.

2. Preliminaries and basic definitions

Certain definitions regarding Caputo and ABC fractional as well as piecewise derivatives and also integration are provided in this section of the study.

Definition 2.1

The definition of $ABC$ operator of function $\mathbb{F}\left(t\right)$ with condition $\mathbb{F}\left(t\right)\in {\mathcal{H}}^1(0,T)$ [14] is (4) ${}^{ABC}{{}_{0}D}_t^{\kappa }\left(\mathbb{F}\right(t\left)\right)=\frac{ABC\left(\kappa \right)}{1-\kappa }{\int }_0^t\frac{\mathrm{d}}{\mathrm{d}\mathrm{\wp }}\mathbb{F}\left(\mathrm{\wp }\right)E_{\kappa }[\frac{-\kappa (t-\mathrm{\wp }{)}^{\kappa }}{1-\kappa }]\mathrm{d}\mathrm{\wp }.$(4) One can replace $E_{\kappa }\left[\frac{-\kappa }{1-\kappa }\right(t-\mathrm{\wp }{)}^{\mathrm{\wp }}]$ by $E_1=\exp \left[\frac{-\kappa }{1-\kappa }\right(t-\mathrm{\wp }\left)\right],$ in (4(4) ${}^{ABC}{{}_{0}D}_t^{\kappa }\left(\mathbb{F}\right(t\left)\right)=\frac{ABC\left(\kappa \right)}{1-\kappa }{\int }_0^t\frac{\mathrm{d}}{\mathrm{d}\mathrm{\wp }}\mathbb{F}\left(\mathrm{\wp }\right)E_{\kappa }[\frac{-\kappa (t-\mathrm{\wp }{)}^{\kappa }}{1-\kappa }]\mathrm{d}\mathrm{\wp }.$(4) ), obtain the Caputo–Fabrizio operator.

Definition 2.2

Consider $\mathbb{F}\left(t\right)\in P[0,T]$, then $\mathbb{A}\mathbb{B}\mathbb{C}$ integral [14] is (5) ${}^{ABC}{{}_{0}I}_t^{\kappa }\mathbb{F}\left(t\right)=\frac{1-\kappa }{ABC\left(\kappa \right)}\mathbb{F}\left(t\right)+\frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^t\mathbb{F}\left(\mathrm{\wp }\right)(t-\mathrm{\wp }{)}^{\kappa -1}\mathrm{d}\mathrm{\wp }.$(5)

Definition 2.3

The Caputo operator of function $\mathbb{F}\left(t\right)$ [15] is ${}_0^C{D}_t^{\kappa }\mathbb{F}\left(t\right)=\frac{1}{\mathrm{\Gamma }(1-\kappa )}{\int }_0^t{\mathbb{F}}^{\mathrm{\prime }}\left(\mathrm{\wp }\right)(t-\mathrm{\wp }{)}^{n-\kappa -1}\mathrm{d}\mathrm{\wp }.$

Definition 2.4

Suppose $\mathbb{F}\left(t\right)$ is piecewise differentiable, then piecewise derivative with Caputo and ABC operators [38] is $\begin{array}{r}{}_0^{PCABC}{D}_t^{\kappa }\mathbb{F}\left(t\right)=\{\begin{array}{l}{}_0^C{D}_t^{\kappa }\mathbb{F}\left(t\right),0<t\le t_1,\\ {}_0^{ABC}{D}_t^{\kappa }\mathbb{F}\left(t\right)t_1<t\le T,\end{array}\end{array}$where ${}_0^{PCABC}{D}_t^{\kappa }$ represents piecewise differential operator, where Caputo operator is in interval $0<t\le t_1$ and ABC operator in interval $t_1<t\le T$.

Definition 2.5

Suppose $\mathbb{F}\left(t\right)$ is piecewise integrable, then piecewise derivative with Caputo and ABC operators [38] is $\begin{array}{r}{}_0^{PCABC}{I}_t\mathbb{F}\left(t\right)=\{\begin{array}{l}{\displaystyle \frac{1}{\mathrm{\Gamma }\kappa }}{\displaystyle {\int }_{t_1}^t\mathbb{F}\left(\mathrm{\wp }\right)(t-\mathrm{\wp }{)}^{\kappa -1}d(\mathrm{\wp }),0<t\le t_1,}\\ {\displaystyle \frac{1-\kappa }{ABC\kappa }}\mathbb{F}\left(t\right)+{\displaystyle \frac{\kappa }{ABC\kappa \mathrm{\Gamma }\kappa }}{\displaystyle {\int }_{t_1}^t\mathbb{F}\left(\mathrm{\wp }\right)(t-\mathrm{\wp }{)}^{\kappa -1}d(\mathrm{\wp })t_1<t\le T,}\end{array}\end{array}$where ${}_0^{PCABC}{I}_t^{\kappa }$ represents piecewise integral operator, where Caputo operator is in interval $0<t\le t_1$ and ABC operator in interval $t_1<t\le T$.

3. Qualitative analysis: existence and uniqueness

The qualitative analysis of the proposed model in the format of piecewise derivative can be found in this section of the paper where we shall now determine whether a solution exists for the hypothetical piecewise derivable function as well as its specific solution attribute or not. To do this, we may use the system (3(3) $\begin{array}{rl}{}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_1({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^d\left({\mathbb{E}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_2({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_2({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_3({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_3({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{C}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{C}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_4({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{C}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_4({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{R}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{R}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_5({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{R}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_5({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_r\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_r\right(t\left)\right){=}^C{\mathbb{Y}}_6({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_r\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_6({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_r\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_r\right(t\left)\right){=}^C{\mathbb{Y}}_7({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_r\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_7({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_r\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^d\left({\mathbb{I}}_r\right(t\left)\right){=}^C{\mathbb{Y}}_8({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^d\left({\mathbb{I}}_r\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_8({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T.\end{array}\end{array}$(3) ), and we can also write the following by way of further explanation: ${}_0^{\mathrm{P}CABC}{\mathbf{D}}_t^{\kappa }\mathcal{L}\left(t\right)=\mathcal{Z}(t,\mathcal{L}(t\left)\right),0<\kappa \le 1$is defined as (6) $\mathcal{L}\left(t\right)=\{\begin{array}{l}{\mathcal{L}}_0+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^t\mathcal{Z}(\mathrm{\wp },\mathcal{L}(\mathrm{\wp }\left)\right)(t-\mathrm{\wp }{)}^{\kappa -1}\mathrm{d}\mathrm{\wp },0<t\le t_1}\\ \mathcal{L}\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}\mathcal{Z}(t,\mathcal{L}(t\left)\right)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right)\left)d\right(\mathrm{\wp }),t_1<t\le T,}\end{array}$(6) where (7) $\begin{array}{rl}\mathcal{L}\left(t\right)& =\{\begin{array}{l}{\mathbb{S}}_h\left(t\right)\\ {\mathbb{E}}_h\left(t\right)\\ {\mathbb{I}}_h\left(t\right)\\ {\mathbb{C}}_h\left(t\right)\\ {\mathbb{R}}_h\left(t\right)\\ {\mathbb{S}}_r\left(t\right)\\ {\mathbb{E}}_r\left(t\right)\\ {\mathbb{I}}_r\left(t\right)\end{array}{\mathcal{L}}_0=\{\begin{array}{l}{\mathbb{S}}_{h_0}\\ {\mathbb{E}}_{h_0}\\ {\mathbb{I}}_{h_0}\\ {\mathbb{C}}_{h_0}\\ {\mathbb{R}}_{h_0}\\ {\mathbb{S}}_{r_0}\\ {\mathbb{E}}_{r_0}\\ {\mathbb{I}}_{r_0}\end{array}{\mathcal{L}}_{t_1}=\{\begin{array}{l}{\mathbb{S}}_h\left(t_1\right)\\ {\mathbb{E}}_h\left(t_1\right)\\ {\mathbb{I}}_h\left(t_1\right)\\ {\mathbb{C}}_h\left(t_1\right)\\ {\mathbb{R}}_h\left(t_1\right)\\ {\mathbb{S}}_r\left(t_1\right)\\ {\mathbb{E}}_r\left(t_1\right)\\ {\mathbb{I}}_r\left(t_1\right)\end{array}\\ \mathcal{Z}(t,\mathcal{L}(t\left)\right)& =\{\begin{array}{l}{\mathbb{Y}}_i=\{\begin{array}{l}{}^C{\mathbb{Y}}_1({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t)\\ {}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t)\end{array},\end{array}\end{array}$(7) where $i=1,2,3,\dots 8$. Taking $0<t\le \mathbb{T}<\mathrm{\infty }$ and the Banach space $X_1=C[0,\mathbb{T}]$ with a norm $\Vert \mathcal{L}\Vert =\underset{t\in [0,\mathbb{T}]}{max}|\mathcal{L}\left(t\right)|.$We assume the following growth condition:

1. ∃${\mathcal{L}}_{\mathcal{L}}>0$; ∀$\mathcal{Z},\overline{\mathcal{L}}\in X$ we have $|\mathcal{Z}(t,\mathcal{L})-\mathcal{Z}(t,\overline{\mathcal{L}})|\le {\mathcal{L}}_{\mathcal{Z}}|\mathcal{L}-\overline{\mathcal{L}}|,$

2. ∃$C_{\mathcal{Z}}>0$$\mathrm{\&}$$M_{\mathcal{Z}}>0$; $|\mathcal{Z}(t,\mathcal{L}(t\left)\right)|\le C_{\mathcal{Z}}|\mathcal{L}|+M_{\mathcal{Z}}.$

Theorem 3.1

If $\mathcal{Z}$ be piece-wise continuous on $(0,t_1]$ and $[t_1,T]$ on $[0,\mathcal{T}]$, also satisfy $\left(\mathrm{E}2\right)$, then (3(3) $\begin{array}{rl}{}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_1({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^d\left({\mathbb{E}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_2({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_2({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_3({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_3({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{C}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{C}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_4({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{C}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_4({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{R}}_h\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{R}}_h\right(t\left)\right){=}^C{\mathbb{Y}}_5({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{R}}_h\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_5({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_r\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_r\right(t\left)\right){=}^C{\mathbb{Y}}_6({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{S}}_r\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_6({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_r\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_r\right(t\left)\right){=}^C{\mathbb{Y}}_7({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{E}}_r\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_7({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T,\end{array}\\ {}_0^{CABC}{\mathbf{D}}_t^{\kappa }\left({\mathbb{I}}_r\right(t\left)\right)& =\{\begin{array}{l}{}_0^C{\mathbf{D}}_t^d\left({\mathbb{I}}_r\right(t\left)\right){=}^C{\mathbb{Y}}_8({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),0<t\le t_1,\\ {}_0^{ABC}{\mathbf{D}}_t^d\left({\mathbb{I}}_r\right(t\left)\right){=}^{ABC}{\mathbb{Y}}_8({\mathbb{S}}_h,{\mathbb{E}}_h,{\mathbb{I}}_h,{\mathbb{C}}_h,{\mathbb{R}}_h,{\mathbb{S}}_r,{\mathbb{E}}_r,{\mathbb{I}}_r,t),t_1<t\le T.\end{array}\end{array}$(3) ) has at least one solution.

Proof.

Let's use the Schauder theorem to define a closed sub-set as $\mathfrak{B}$ and X in both subintervals of $[0,\mathcal{F}]$. $B=\{\mathcal{L}\in X:\Vert \mathcal{L}\Vert \le R_{1,2},R_{1,2}>0\},$Suppose $\mathcal{F}:\mathfrak{B}\to \mathfrak{B}$ and using (6(6) $\mathcal{L}\left(t\right)=\{\begin{array}{l}{\mathcal{L}}_0+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^t\mathcal{Z}(\mathrm{\wp },\mathcal{L}(\mathrm{\wp }\left)\right)(t-\mathrm{\wp }{)}^{\kappa -1}\mathrm{d}\mathrm{\wp },0<t\le t_1}\\ \mathcal{L}\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}\mathcal{Z}(t,\mathcal{L}(t\left)\right)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right)\left)d\right(\mathrm{\wp }),t_1<t\le T,}\end{array}$(6) ) as (8) $\mathcal{F}\left(\mathcal{L}\right)=\{\begin{array}{l}{\mathcal{L}}_0+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{{\mathrm{t}}_1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}(\mathrm{\wp }\left)\right)(\mathrm{t}-\mathrm{\wp }{)}^{\kappa -1}\mathrm{d}\mathrm{\wp },0<\mathrm{t}\le {\mathrm{t}}_1}\\ \mathcal{L}\left({\mathrm{t}}_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}\mathcal{Z}(\mathrm{t},\mathcal{L}(\mathrm{t}\left)\right)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{{\mathrm{t}}_1}^{\mathrm{t}}(\mathrm{t}-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right)\left)d\right(\mathrm{\wp }),{\mathrm{t}}_1<\mathrm{t}\le T.}\end{array}$(8) Any $\mathcal{L}\in B$, we have $\begin{array}{rl}|\mathcal{F}\left(\mathcal{L}\right)\left(t\right)|& \le \{\begin{array}{l}|{\mathcal{L}}_0|+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{{\mathrm{t}}_1}(\mathrm{t}-\mathrm{\wp }{)}^{\kappa -1}|\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))|\mathrm{d}\mathrm{\wp },}\\ |{\mathcal{L}}_{(}{t}_1)|+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}|\mathcal{Z}(t,\mathcal{L}\left(t\right))|\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{{\mathrm{t}}_1}^{\mathrm{t}}(\mathrm{t}-\mathrm{\wp }{)}^{\kappa -1}|\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right)\left)|d\right(\mathrm{\wp }),}\end{array}\\ & \le \{\begin{array}{l}|{\mathcal{L}}_0|+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{{\mathrm{t}}_1}(\mathrm{t}-\mathrm{\wp }{)}^{\kappa -1}[C_{\mathcal{Z}}|\mathcal{L}|+M_{\mathcal{Z}}]\mathrm{d}\kappa ,}\\ |{\mathcal{L}}_{(}{\mathrm{t}}_1)|+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}[C_{\mathcal{Z}}|\mathcal{L}|+M_{\mathcal{Z}}]\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{{\mathrm{t}}_1}^{\mathrm{t}}(\mathrm{t}-\mathrm{\wp }{)}^{\kappa -1}[C_{\mathcal{Z}}|\mathcal{L}|+M_{\mathcal{Z}}\left]d\right(\kappa ),}\end{array}\\ & \le \{\begin{array}{l}|{\mathcal{L}}_0|+{\displaystyle \frac{{\mathbf{T}}^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}}[C_H|\mathcal{L}|+M_{\mathcal{Z}}]=R_1,0<\mathrm{t}\le {\mathrm{t}}_1,\\ |{\mathcal{L}}_{(}{t}_1)|+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}[C_{\mathcal{Z}}|\mathcal{L}|+M_{\mathcal{Z}}]\\ +{\displaystyle \frac{\kappa (T-\mathbf{T}{)}^{\kappa }}{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)+1}}[C_{\mathcal{Z}}|\mathcal{L}|+M_{\mathcal{Z}}]d\left(\kappa \right)=R_2,{\mathrm{t}}_1<\mathrm{t}\le T,\end{array}\\ & \le \{\begin{array}{l}{R}_1,0<\mathrm{t}\le {\mathrm{t}}_1,\\ R_2,{\mathrm{t}}_1<\mathrm{t}\le T.\end{array}\end{array}$As we derived that $\mathcal{L}\in \mathbf{\text{B}}$, so $\mathcal{F}\left(\mathbf{\text{B}}\right)\mathbb{S}\subset \mathbf{\text{B}}.$ This shows that $\mathcal{F}$ is bounded. For complete continuity, we proceed as by taking ${\mathrm{t}}_i<{\mathrm{t}}_j\in [0,{\mathrm{t}}_1]$, the first interval in the Caputo sense as (9) $\begin{array}{rl}|\mathcal{F}\left(\mathcal{L}\right)\left({\mathrm{t}}_j\right)-\mathcal{F}\left(\mathcal{L}\right)\left({\mathrm{t}}_i\right)|& =|\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{{\mathrm{t}}_j}({\mathrm{t}}_j-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp },\\ & -\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_i}({\mathrm{t}}_i-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp }|\\ & \le \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_i}\left[\right(t_i-\mathrm{\wp }{)}^{\kappa -1}-({\mathrm{t}}_j-\mathrm{\wp }{)}^{\kappa -1}]|\mathcal{Z}(\mathrm{\wp },\mathcal{L}(\mathrm{\wp }\left)\right)|\mathrm{d}\mathrm{\wp }\\ & +\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_{t_i}^{t_j}(t_j-\mathrm{\wp }{)}^{\kappa -1}|\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))|\mathrm{d}\mathrm{\wp }\\ & \le \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}[{\int }_0^{t_i}\left[\right(t_i-\mathrm{\wp }{)}^{\kappa -1}-(t_j-\mathrm{\wp }{)}^{\kappa -1}]\mathrm{d}\mathrm{\wp }\\ & +{\int }_{t_i}^{t_j}(t_j-\mathrm{\wp }{)}^{\kappa -1}\mathrm{d}\mathrm{\wp }](C_H|\mathcal{L}|+M_{\mathcal{Z}})\\ & \le \frac{(C_{\mathcal{Z}}\mathcal{L}+M_{\mathcal{Z}})}{\mathrm{\Gamma }(\kappa +1)}[t_j^{\mathrm{\wp }}-t_i^{\kappa }+2(t_j-t_i{)}^{\kappa }].\end{array}$(9) Next (9(9) $\begin{array}{rl}|\mathcal{F}\left(\mathcal{L}\right)\left({\mathrm{t}}_j\right)-\mathcal{F}\left(\mathcal{L}\right)\left({\mathrm{t}}_i\right)|& =|\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{{\mathrm{t}}_j}({\mathrm{t}}_j-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp },\\ & -\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_i}({\mathrm{t}}_i-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp }|\\ & \le \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_i}\left[\right(t_i-\mathrm{\wp }{)}^{\kappa -1}-({\mathrm{t}}_j-\mathrm{\wp }{)}^{\kappa -1}]|\mathcal{Z}(\mathrm{\wp },\mathcal{L}(\mathrm{\wp }\left)\right)|\mathrm{d}\mathrm{\wp }\\ & +\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_{t_i}^{t_j}(t_j-\mathrm{\wp }{)}^{\kappa -1}|\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))|\mathrm{d}\mathrm{\wp }\\ & \le \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}[{\int }_0^{t_i}\left[\right(t_i-\mathrm{\wp }{)}^{\kappa -1}-(t_j-\mathrm{\wp }{)}^{\kappa -1}]\mathrm{d}\mathrm{\wp }\\ & +{\int }_{t_i}^{t_j}(t_j-\mathrm{\wp }{)}^{\kappa -1}\mathrm{d}\mathrm{\wp }](C_H|\mathcal{L}|+M_{\mathcal{Z}})\\ & \le \frac{(C_{\mathcal{Z}}\mathcal{L}+M_{\mathcal{Z}})}{\mathrm{\Gamma }(\kappa +1)}[t_j^{\mathrm{\wp }}-t_i^{\kappa }+2(t_j-t_i{)}^{\kappa }].\end{array}$(9) ), we obtain $t_i\to t_j$, then $|\mathcal{F}\left(\mathcal{L}\right)\left(t_j\right)-\mathcal{F}\left(\mathcal{L}\right)\left(t_i\right)|\to 0,\mathrm{a}\mathrm{s}{t}_i\to t_j.$So $\mathcal{F}$ is equi-continuous in $[0,t_1]$. Consider $t_i,t_j\in [t_1,T]$ in ABC sense for the second interval as (10) $\begin{array}{rl}|\mathcal{F}\left(\mathcal{L}\right)\left(t_j\right)-\mathcal{F}\left(\mathcal{L}\right)\left({\mathrm{t}}_i\right)|& =|\frac{1-\kappa }{ABC\left(\kappa \right)}\mathcal{Z}(t,\mathcal{L}(t\left)\right)\\ & +\frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}{\int }_{t_1}^{{\mathrm{t}}_j}({\mathrm{t}}_j-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp },\\ & -\frac{1-\kappa }{ABC\left(\kappa \right)}\mathcal{Z}(t,\mathcal{L}(t\left)\right)\\ & +\frac{\left(\kappa \right)}{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}{\int }_{{\mathrm{t}}_1}^{{\mathrm{t}}_i}({\mathrm{t}}_i-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp }|\\ & \le \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}{\int }_{t_1}^{t_i}\left[\right(t_i-\mathrm{\wp }{)}^{\kappa -1}-(t_j-\mathrm{\wp }{)}^{\kappa -1}]\\ & \times |\mathcal{Z}(\mathrm{\wp },\mathcal{L}(\mathrm{\wp }\left)\right)|\mathrm{d}\mathrm{\wp }\\ & +\frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}{\int }_{t_i}^{t_j}(t_j-\mathrm{\wp }{)}^{\kappa -1}|\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))|\mathrm{d}\mathrm{\wp }\\ & \le \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}[{\int }_{t_1}^{t_i}\left[\right(t_i-\mathrm{\wp }{)}^{\kappa -1}-(t_j-\mathrm{\wp }{)}^{\kappa -1}]\mathrm{d}\mathrm{\wp }\\ & +{\int }_{t_i}^{t_j}(t_j-\mathrm{\wp }{)}^{\kappa -1}\mathrm{d}\kappa ](C_{\mathcal{Z}}|\mathcal{L}|+M_{\mathcal{Z}})\\ & \le \frac{\kappa (C_{\mathcal{Z}}\mathcal{L}+M_{\mathcal{Z}})}{ABC\left(\kappa \right)\mathrm{\Gamma }(\kappa +1)}[t_j^{\kappa }-t_i^{\kappa }+2(t_j-t_i{)}^{\kappa }].\end{array}$(10) If $t_i\to {\mathrm{t}}_j$, then $|\mathcal{F}\left(\mathcal{L}\right)\left({\mathrm{t}}_j\right)-\mathcal{F}\left(\mathcal{L}\right)\left(t_i\right)|\to 0,\mathrm{a}\mathrm{s}{t}_i\to t_j.$This shows that $\mathcal{F}$ is equi-continuous, so by Arzel'a theorem $\mathcal{F}$ is completely continuous and bounded. Thus our derivation shows that the proposed piecewise model has at least one solution in both subintervals.

Theorem 3.2

If the operator $\mathcal{F}$ is contraction along with assumption $\left(\mathrm{E}1\right)$ then the proposed system has unique solution.

Proof.

As $\mathcal{F}:\mathbf{\text{B}}\to \mathbf{\text{B}}$ piece-wise continuous, consider $\mathcal{L}$ and $\overline{\mathcal{L}}\in B$ on $[0,t_1]$ in sense of Caputo as (11) $\begin{array}{rl}\Vert \mathcal{F}\left(\mathcal{L}\right)-\mathcal{L}\left(\overline{\mathcal{L}}\right)\Vert & =\underset{t\in [0,t_1]}{max}|\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^t(t-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp }\\ & -\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^t(t-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\overline{\mathcal{L}}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp }|\\ & \le \frac{{\mathbf{T}}^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}{L}_{\mathcal{Z}}\Vert \mathcal{L}-\overline{\mathcal{L}}\Vert .\end{array}$(11) From (11(11) $\begin{array}{rl}\Vert \mathcal{F}\left(\mathcal{L}\right)-\mathcal{L}\left(\overline{\mathcal{L}}\right)\Vert & =\underset{t\in [0,t_1]}{max}|\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^t(t-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\mathcal{L}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp }\\ & -\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^t(t-\mathrm{\wp }{)}^{\kappa -1}\mathcal{Z}(\mathrm{\wp },\overline{\mathcal{L}}\left(\mathrm{\wp }\right))\mathrm{d}\mathrm{\wp }|\\ & \le \frac{{\mathbf{T}}^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}{L}_{\mathcal{Z}}\Vert \mathcal{L}-\overline{\mathcal{L}}\Vert .\end{array}$(11) ), we have (12) $\Vert \mathcal{F}\left(\mathcal{L}\right)-\mathcal{L}\left(\overline{\mathcal{L}}\right)\Vert \le \frac{{\mathbf{T}}^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}{L}_{\mathcal{Z}}\Vert \mathcal{L}-\overline{\mathcal{L}}\Vert .$(12) This shows that $\mathcal{L}$ is contraction and the proposed system has unique solution.

Also $\mathrm{t}\in [{\mathrm{t}}_1,T]$ in the sense of ABC derivative as (13) $\Vert \mathcal{F}\left(\mathcal{L}\right)-\mathcal{L}\left(\overline{\mathcal{L}}\right)\Vert \le \frac{1-\kappa }{ABC\left(\kappa \right)}{L}_{\mathcal{Z}}\Vert \mathcal{L}-\overline{\mathcal{L}}\Vert +\frac{\kappa \left({\mathbf{T}-T}^{\kappa }\right)}{ABC\left(\kappa \right)\mathrm{\Gamma }(\kappa +1)}{L}_{\mathbb{F}}\Vert \mathcal{L}-\overline{\mathcal{L}}\Vert$(13) or (14) $\Vert \mathcal{F}\left(\mathcal{L}\right)-\mathcal{L}\left(\overline{\mathcal{L}}\right)\Vert \le L_{\mathcal{Z}}[\frac{1-\kappa }{ABC\left(\kappa \right)}+\frac{\kappa (T-\mathbf{T}{)}^{\kappa }}{ABC\left(d\right)\mathrm{\Gamma }(\kappa +1)}]\Vert \mathcal{L}-\overline{\mathcal{L}}\Vert .$(14) This is why $\mathcal{F}$ is contraction. As a result, the issue under consideration has a singleton solution in the provided subinterval in light of the Banach result. So (12(12) $\Vert \mathcal{F}\left(\mathcal{L}\right)-\mathcal{L}\left(\overline{\mathcal{L}}\right)\Vert \le \frac{{\mathbf{T}}^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}{L}_{\mathcal{Z}}\Vert \mathcal{L}-\overline{\mathcal{L}}\Vert .$(12) ) and (14(14) $\Vert \mathcal{F}\left(\mathcal{L}\right)-\mathcal{L}\left(\overline{\mathcal{L}}\right)\Vert \le L_{\mathcal{Z}}[\frac{1-\kappa }{ABC\left(\kappa \right)}+\frac{\kappa (T-\mathbf{T}{)}^{\kappa }}{ABC\left(d\right)\mathrm{\Gamma }(\kappa +1)}]\Vert \mathcal{L}-\overline{\mathcal{L}}\Vert .$(14) ) the suggested problem has unique solution on each sub-intervals.

4. Stability analysis with varying forms of function

The proving of the H–U stability and different forms is provided herein for our considered model.

Definition 4.1

Our proposed model (1(1) $\begin{array}{rl}& \dot{{\mathbb{S}}_h}={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ & \dot{{\mathbb{E}}_h}={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ & \dot{{\mathbb{I}}_h}=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ & \dot{{\mathbb{C}}_h}=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ & \dot{{\mathbb{R}}_h}=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ & \dot{{\mathbb{S}}_r}={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ & \dot{{\mathbb{E}}_r}={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ & \dot{{\mathbb{I}}_r}=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r,\end{array}$(1) ) is U–H stable, if for each $\mathrm{\Phi }>0$, and the inequality (15) $\left|{}^{PCABC}{\mathbf{D}}_t^{\kappa }\mathrm{\Phi }\right(t)-\mathbb{F}(t,\mathrm{\Phi }\left(t\right)\left)\right|<\mathrm{\Phi },\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l},t\in \mathcal{T},$(15) unique solution $\overline{\mathrm{\Phi }}\in Z$ exists with a constant $\mathcal{H}>0$, (16) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathcal{H}\mathrm{\Phi },\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l},t\in \mathcal{T},$(16) In addition, if a nondecreasing function $\mathrm{\Phi }:[0,\mathrm{\infty })\to R^{+}$ for the inequality presented above (17) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathcal{H}\mathrm{\Phi }\left(\mathrm{\Phi }\right),\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y},t\in \mathcal{T},$(17) it is fated $\mathrm{\Phi }\left(0\right)=0$, then obtained solution is generally U–H stable.

Definition 4.2

Our considered model 2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) is H–U–R stable if $\mathrm{\Psi }:[0,\mathrm{\infty })\to R^{+}$, for each $\mathrm{\Phi }>0$, and inequality (18) $\left|{}^{PCABC}{\mathbf{D}}_t^{\kappa }\mathrm{\Phi }\right(t)-\mathbb{F}(t,\mathrm{\Phi }\left(t\right)\left)\right|<\mathrm{\Phi }\mathrm{\Psi }\left(t\right),\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{a}\mathrm{l}\mathrm{l},t\in \mathcal{T},$(18) unique solution $\overline{\mathrm{\Phi }}\in Z$ with constant ${\mathcal{H}}_{\mathrm{\Psi }}>0$, so that (19) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le {\mathcal{H}}_{\mathrm{\Psi }}\mathrm{\Phi }\mathrm{\Psi }\left(t\right),t\in \mathcal{T}.$(19) Anew, if $\mathrm{\Psi }:[0,\mathrm{\infty })\to R^{+}$ is existent, for the inequality (20) $\left|{}^{PCABC}{\mathbf{D}}_t^{\kappa }\mathrm{\Phi }\right(t)-\mathbb{F}(t,\mathrm{\Phi }\left(t\right)\left)\right|<\mathrm{\Psi }\left(t\right),t\in \mathcal{T},$(20) unique solution exists $\overline{\mathrm{\Phi }}\in Z$ with constant ${\mathcal{H}}_{\mathrm{\Psi }}>0$, so (21) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le {\mathcal{H}}_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right),t\in \mathcal{T}.$(21) then the obtained solution is generally H–U–R stable.

Remark 4.1

Suppose a function $\varphi \in C\left(\mathcal{T}\right)$ does not depend upon $\mathrm{\Phi }\in \mathcal{Z}$, and $\varphi \left(0\right)=0$, then $\begin{array}{rl}\left|\varphi \right(t\left)\right|& \le \mathrm{\Phi },t\in \mathcal{T};\\ {}^{PCABC}{\mathbf{D}}_t^{\kappa }\mathrm{\Phi }\left(t\right)& =\mathbb{F}(t,\mathrm{\Phi }(t\left)\right)+\varphi \left(t\right),t\in \mathcal{T}.\end{array}$

Lemma 4.3

Suppose the function (22) ${}_0^{PCABC}{\mathbf{D}}_t^{\varrho }\mathrm{\Phi }\left(t\right)=\mathbb{F}(t,\mathrm{\Phi }(t\left)\right),0<\varrho \le 1$(22) The solution of (22(22) ${}_0^{PCABC}{\mathbf{D}}_t^{\varrho }\mathrm{\Phi }\left(t\right)=\mathbb{F}(t,\mathrm{\Phi }(t\left)\right),0<\varrho \le 1$(22) ) is (23) $\begin{array}{rl}\mathrm{\Phi }\left(t\right)& =\{\begin{array}{l}{\mathrm{\Phi }}_0+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^t\mathbb{F}(\varrho ,\mathrm{\Phi }(\varrho \left)\right)(t-\varrho {)}^{\kappa -1}\mathrm{d}\varrho ,0<t\le t_1}\\ \mathrm{\Phi }\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}\mathbb{F}(t,\mathrm{\Phi }(t\left)\right)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\varrho {)}^{\kappa -1}\mathbb{F}(\varrho ,\mathrm{\Phi }\left(\varrho \right)\left)d\right(\varrho ),t_1<t\le T,}\end{array}\end{array}$(23) (24) $\begin{array}{rl}\left|\left|F\right(\mathrm{\Phi })-F(\overline{\mathrm{\Phi }}\left)\right|\right|& \le \{\begin{array}{l}{\displaystyle \frac{{\mathcal{T}}_1^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}}\mathrm{\Phi },t\in {\mathcal{T}}_1\\ \left[{\displaystyle \frac{(1-\kappa )\mathrm{\Gamma }\left(\kappa \right)+\left(T_2^{\kappa }\right)}{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}\right]\mathrm{\Phi }=\mathrm{\Lambda }\mathrm{\Phi },t\in {\mathcal{T}}_2.\end{array}\end{array}$(24)

Theorem 4.4

In the light of Lemma (4.3) if the condition $\frac{L_f{\mathcal{T}}^{\kappa }}{\mathrm{\Gamma }\left(\kappa \right)}<1$ satisfies, then the solution of our considered model (2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) ) is H–U as well as generalized H–U stable.

Proof.

Let us suppose $\mathrm{\Phi }\in Z$ is the solution of 2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) also $\overline{\mathrm{\Phi }}\in Z$ is a unique solution of (2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) ), so we have

Case:1 for $t\in \mathcal{T}$, we have $\begin{array}{rl}\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|& =\underset{t\in \mathcal{T}}{sup}|\mathrm{\Phi }-({\mathrm{\Phi }}_{\circ }+\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp })|\\ & \le \underset{t\in \mathcal{T}}{sup}|\mathrm{\Phi }-({\mathrm{\Phi }}_{\circ }+\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp })|\\ & +\underset{t\in \mathcal{T}}{sup}|+\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\mathrm{\Phi }(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp }\\ & -\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp }|\\ & \le \frac{{{\mathcal{T}}_{\mathcal{1}}}^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}\mathrm{\Phi }+\frac{L_f{\mathcal{T}}_{\mathcal{1}}}{\mathrm{\Gamma }(\kappa +1)}\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|.\end{array}$On further simplification (25) $\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|\le \left(\frac{\frac{{\mathcal{T}}_{\mathcal{1}}}{\mathrm{\Gamma }(\kappa +1)}}{1-\frac{L_f{\mathcal{T}}_{\mathcal{1}}}{\mathrm{\Gamma }(\kappa +1)}}\right)\mathrm{\Phi }.$(25) Case:2 $\begin{array}{rl}\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|& \le \underset{t\in \mathcal{T}}{sup}|\mathrm{\Phi }-\left[\mathrm{\Phi }\right(t_1)+\frac{1-\kappa }{ABC\left(\kappa \right)}[\mathbb{F}(t,\mathrm{\Phi }(t\left)\right),]\\ & +\frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}\left[{\int }_{t_1}^t\right(t-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)d\left(\mathrm{\wp }\right)]]|\\ & +\underset{t\in \mathcal{T}}{sup}\frac{1-\kappa }{ABC\left(\kappa \right)}|\mathbb{F}(t,\mathrm{\Phi }(t\left)\right)-\mathbb{F}(t,\overline{\mathrm{\Phi }}(t),)|\\ & +\underset{t\in \mathcal{T}}{sup}\frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}{\int }_{t_1}^t(t-\mathrm{\wp }{)}^{\kappa -1}|\mathbb{F}(\mathrm{\wp },\mathrm{\Phi }(\mathrm{\wp }\left)\right)-\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)|\mathrm{d}s.\end{array}$By further simplification and using $\mathrm{\Lambda }=\left[\frac{(1-\kappa )\mathrm{\Gamma }\left(\kappa \right)+T_2^{\kappa }}{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}\right],$ we have ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathrm{\Lambda }\mathrm{\Phi }+\mathrm{\Lambda }{L}_f{\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z$we have ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \left(\frac{\mathrm{\Lambda }}{1-\frac{\mathrm{\Lambda }}{L_f}}\right)\mathrm{\Phi }{\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z.$using $\mathcal{H}=max\{\left(\frac{\frac{{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}{1-\frac{L_f{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}\right),\frac{\mathrm{\Lambda }}{1-\frac{\mathrm{\Lambda }{L}_f}{1-M_f}}\}$At this point, from Equations (25(25) $\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|\le \left(\frac{\frac{{\mathcal{T}}_{\mathcal{1}}}{\mathrm{\Gamma }(\kappa +1)}}{1-\frac{L_f{\mathcal{T}}_{\mathcal{1}}}{\mathrm{\Gamma }(\kappa +1)}}\right)\mathrm{\Phi }.$(25) ) and (26(26) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathcal{H}\mathrm{\Phi },\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}t\in \mathcal{T}\mathcal{.}$(26) ), we have (26) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathcal{H}\mathrm{\Phi },\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}t\in \mathcal{T}\mathcal{.}$(26) So the solution of model 2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) is H–U stable. Also if we replace Φ by $\mathrm{\Phi }\left(\mathrm{\Phi }\right)$ then from (26(26) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathcal{H}\mathrm{\Phi },\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}t\in \mathcal{T}\mathcal{.}$(26) ), ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathcal{H}\mathrm{\Phi }\left(\mathrm{\Phi }\right),\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}t\in \mathcal{T}.$Now $\mathrm{\Phi }\left(0\right)=0$ shows that the solution of our proposed model 2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) is generalized H–U stable.

We define the remark to conclude the Rassias stability results and also the generalized form.

Remark 4.2

Suppose a function $\varphi \in C\left(\mathcal{T}\right)$ does not depend upon $\mathrm{\Phi }\in \mathcal{Z}$, and $\varphi \left(0\right)=0$, then $\begin{array}{rl}\left|\varphi \right(t\left)\right|& \le \mathrm{\Psi }\left(t\right)\mathrm{\Phi },t\in \mathcal{T};\\ {}^{PCABC}{\mathcal{D}}_t^{\kappa }\mathrm{\Phi }\left(t\right)& =\mathbb{F}(t,\mathrm{\Phi }(t\left)\right)+\varphi \left(t\right),t\in \mathcal{T};\\ {\int }_0^t\mathrm{\Psi }\left(\mathrm{\wp }\right)\mathrm{d}s& \le C_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right),t\in \mathcal{T}.\end{array}$

Lemma 4.5

Solution to the model $\begin{array}{rl}{}^{PCABC}{\mathcal{D}}_t^{\kappa }\mathrm{\Phi }\left(t\right)& =\mathbb{F}(t,\mathrm{\Phi }(t\left)\right)+\varphi \left(t\right),\\ \mathrm{\Phi }\left(0\right)& ={\mathrm{\Phi }}_{\circ },\end{array}$hold the relation given below (27) $\begin{array}{r}\left|\left|F\right(\mathrm{\Phi })-F(\overline{\mathrm{\Phi }}\left)\right|\right|\le \{\begin{array}{l}{\displaystyle \frac{{\mathcal{T}}_1^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}}{C}_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right)\mathrm{\Phi },t\in {\mathcal{T}}_1\\ \left[{\displaystyle \frac{(1-\kappa )\mathrm{\Gamma }\left(\kappa \right)+\left(T_2^{\kappa }\right)}{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}\right]C_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right)\mathrm{\Phi }=\mathrm{\Lambda }{C}_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right)\mathrm{\Phi },t\in {\mathcal{T}}_2.\end{array}\end{array}$(27) where ${\mathcal{H}}_{f,\mathrm{\Psi },\mathrm{\Lambda }}=\mathrm{\Lambda }{\mathcal{H}}_{f,\mathrm{\Psi }}$.

With the help of Remark 4.2, one can get Equation (27(27) $\begin{array}{r}\left|\left|F\right(\mathrm{\Phi })-F(\overline{\mathrm{\Phi }}\left)\right|\right|\le \{\begin{array}{l}{\displaystyle \frac{{\mathcal{T}}_1^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}}{C}_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right)\mathrm{\Phi },t\in {\mathcal{T}}_1\\ \left[{\displaystyle \frac{(1-\kappa )\mathrm{\Gamma }\left(\kappa \right)+\left(T_2^{\kappa }\right)}{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}\right]C_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right)\mathrm{\Phi }=\mathrm{\Lambda }{C}_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right)\mathrm{\Phi },t\in {\mathcal{T}}_2.\end{array}\end{array}$(27) ).

Theorem 4.6

The solution of model (27(27) $\begin{array}{r}\left|\left|F\right(\mathrm{\Phi })-F(\overline{\mathrm{\Phi }}\left)\right|\right|\le \{\begin{array}{l}{\displaystyle \frac{{\mathcal{T}}_1^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}}{C}_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right)\mathrm{\Phi },t\in {\mathcal{T}}_1\\ \left[{\displaystyle \frac{(1-\kappa )\mathrm{\Gamma }\left(\kappa \right)+\left(T_2^{\kappa }\right)}{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}\right]C_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right)\mathrm{\Phi }=\mathrm{\Lambda }{C}_{\mathrm{\Psi }}\mathrm{\Psi }\left(t\right)\mathrm{\Phi },t\in {\mathcal{T}}_2.\end{array}\end{array}$(27) ) is H–U–R stable if the following conditions hold:

$\left(H_1\right)$ For each $\mathrm{\Phi },v\in \mathcal{Z}$ and a constant $C_{\mathrm{\Phi }}>0$, we get $\left|\mathrm{\Phi }\right(\mathrm{\Phi })-\mathrm{\Phi }(v\left)\right|\le C_{\mathrm{\Phi }}|\mathrm{\Phi }-v|;$$\left(H_2\right)$ For each $\mathrm{\Phi },v,\overline{\mathrm{\Phi }},\overline{v}\in \mathcal{Z}$ and constant $L_f>0,0<M_f<1$, we get $\begin{array}{rl}\left|\mathbb{F}\right(t,\mathrm{\Phi },v)-\mathbb{F}(t,\overline{\mathrm{\Phi }},\overline{v}\left)\right|& \le L_f|\mathrm{\Phi }-\overline{\mathrm{\Phi }}|+M_f|v-\overline{v}|\\ M_f& <1.\end{array}$

Proof.

We prove these results in two cases.

Case:1 for $t\in \mathcal{T}$, we have $\begin{array}{rl}\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|& =\underset{t\in \mathcal{T}}{sup}|\mathrm{\Phi }-({\mathrm{\Phi }}_{\circ }+\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp })|\\ & \le \underset{t\in \mathcal{T}}{sup}|\mathrm{\Phi }-({\mathrm{\Phi }}_{\circ }+\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp })|\\ & +\underset{t\in \mathcal{T}}{sup}|+\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\mathrm{\Phi }(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp }\\ & -\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp }|\\ & \le \frac{{\mathcal{T}}_1^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}{C}_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\mathrm{\Phi }+\frac{L_f{\mathcal{T}}_{\mathcal{1}}}{\mathrm{\Gamma }(\kappa +1)}\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|.\end{array}$On further simplification, (28) $\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|\le \left(\frac{C_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\frac{{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}{1-\frac{L_f{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}\right)\mathrm{\Phi }.$(28) Case:2 $\begin{array}{rl}\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|& \le \underset{t\in \mathcal{T}}{sup}|\mathrm{\Phi }-\left[\mathrm{\Phi }\right(t_1)+\frac{1-\kappa }{ABC\left(\kappa \right)}[\mathbb{F}(t,\mathrm{\Phi }(t\left)\right),]\\ & +\frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}\left[{\int }_{t_1}^t\right(t-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)d\left(\mathrm{\wp }\right)]]|\\ & +\underset{t\in \mathcal{T}}{sup}\frac{1-\kappa }{ABC\left(\kappa \right)}|\mathbb{F}(t,\mathrm{\Phi }(t\left)\right)-\mathbb{F}(t,\overline{\mathrm{\Phi }}(t),)|\\ & +\underset{t\in \mathcal{T}}{sup}\frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}{\int }_{t_1}^t(t-\mathrm{\wp }{)}^{\kappa -1}|\mathbb{F}(\mathrm{\wp },\mathrm{\Phi }(\mathrm{\wp }\left)\right)-\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)|\mathrm{d}s.\end{array}$By further simplification and using $\mathrm{\Lambda }=\left[\frac{(1-\kappa )\mathrm{\Gamma }\left(\kappa \right)+T_2^{\kappa }}{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}\right],$ we have (29) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathrm{\Lambda }{C}_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\mathrm{\Phi }+\mathrm{\Lambda }{L}_f{\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z,$(29) we have ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \left(\frac{\mathrm{\Lambda }{C}_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)}{1-\frac{\mathrm{\Lambda }}{L_f}}\right)\mathrm{\Phi }{\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z$using ${\mathcal{H}}_{\mathrm{\Lambda },C_{\mathrm{\Phi }}}=max\{\left(\frac{\frac{{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}{1-\frac{L_f{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}\right),\frac{C_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\mathrm{\Lambda }}{1-\frac{\mathrm{\Lambda }{L}_f}{1-M_f}}\}$At that point, from equations (28(28) $\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|\le \left(\frac{C_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\frac{{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}{1-\frac{L_f{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}\right)\mathrm{\Phi }.$(28) ) and (29(29) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathrm{\Lambda }{C}_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\mathrm{\Phi }+\mathrm{\Lambda }{L}_f{\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z,$(29) ), we have ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le {\mathcal{H}}_{\mathrm{\Lambda },C_{\mathrm{\Phi }}}\mathrm{\Phi },ateacht\in \mathcal{T}\mathcal{.}$So the solution of model 2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) is H–U–R stable.

Remark 4.3

Suppose a function $\varphi \in C\left(\mathcal{T}\right)$ does not depend upon $\mathrm{\Phi }\in \mathcal{Z}$, and $\varphi \left(0\right)=0$, then $\left|\varphi \right(t\left)\right|\le \mathrm{\Psi }\left(t\right),t\in \mathcal{T}.$

Theorem 4.7

In light of $H_1$, $H_2$, Remark 4.3 and 4.5, the solution of model 2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) is generalized H–U–R stable, if $M_f<1$.

Where

$\left(H_1\right)$ For each $\mathrm{\Phi },v\in \mathcal{Z}$ and constant $C_{\mathrm{\Phi }}>0$, we get $\left|\mathrm{\Phi }\right(\mathrm{\Phi })-\mathrm{\Phi }(v\left)\right|\le C_{\mathrm{\Phi }}|\mathrm{\Phi }-v|$and

$\left(H_2\right)$ For each $\mathrm{\Phi },v,\overline{\mathrm{\Phi }},\overline{v}\in \mathcal{Z}$ and constant $L_f>0,0<M_f<1$, we get $\left|\mathbb{F}\right(t,\mathrm{\Phi },v)-\mathbb{F}(t,\overline{\mathrm{\Phi }},\overline{v}\left)\right|\le L_f|\mathrm{\Phi }-\overline{\mathrm{\Phi }}|+M_f|v-\overline{v}|.$

Proof.

We have obtained our results in two cases:

Case 1 for $t\in \mathcal{T}$, we have $\begin{array}{rl}\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|& =\underset{t\in \mathcal{T}}{sup}|\mathrm{\Phi }-({\mathrm{\Phi }}_{\circ }+\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp })|\\ & \le \underset{t\in \mathcal{T}}{sup}|\mathrm{\Phi }-({\mathrm{\Phi }}_{\circ }+\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp })|\\ & +\underset{t\in \mathcal{T}}{sup}|+\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\mathrm{\Phi }(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp }\\ & -\frac{1}{\mathrm{\Gamma }\left(\kappa \right)}{\int }_0^{t_1}(t_1-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)\mathrm{d}\mathrm{\wp }|\\ & \le \frac{{\mathcal{T}}_1^{\kappa }}{\mathrm{\Gamma }(\kappa +1)}{C}_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\mathrm{\Phi }+\frac{L_f{\mathcal{T}}_{\mathcal{1}}}{\mathrm{\Gamma }(\kappa +1)}\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|.\end{array}$On further simplification, (30) $\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|\le \left(\frac{C_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\frac{{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}{1-\frac{L_f{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}\right)\mathrm{\Phi }$(30) Case 2 $\begin{array}{rl}\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|& \le \underset{t\in \mathcal{T}}{sup}|\mathrm{\Phi }-\left[\mathrm{\Phi }\right(t_1)+\frac{1-\kappa }{ABC\left(\kappa \right)}\left[\mathbb{F}(t,\mathrm{\Phi }(t\left)\right)\right]\\ & +\frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}\left[{\int }_{t_1}^t\right(t-\mathrm{\wp }{)}^{\kappa -1}\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)d\left(\mathrm{\wp }\right)]]|\\ & +\underset{t\in \mathcal{T}}{sup}\frac{1-\kappa }{ABC\left(\kappa \right)}|\mathbb{F}(t,\mathrm{\Phi }(t\left)\right)-\mathbb{F}(t,\overline{\mathrm{\Phi }}(t),)|\\ & +\underset{t\in \mathcal{T}}{sup}\frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}{\int }_{t_1}^t(t-\mathrm{\wp }{)}^{\kappa -1}|\mathbb{F}(\mathrm{\wp },\mathrm{\Phi }(\mathrm{\wp }\left)\right)-\mathbb{F}(\mathrm{\wp },\overline{\mathrm{\Phi }}(\mathrm{\wp }\left)\right)|\mathrm{d}s.\end{array}$By further simplification and using $\mathrm{\Lambda }=\left[\frac{(1-\kappa )\mathrm{\Gamma }\left(\kappa \right)+T_2^{\kappa }}{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}\right],$ we have (31) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathrm{\Lambda }{C}_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\mathrm{\Phi }+\mathrm{\Lambda }{L}_f{\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z$(31) we have ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \left(\frac{\mathrm{\Lambda }{C}_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)}{1-\mathrm{\Lambda }{L}_f}\right){\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z,$using ${\mathcal{H}}_{\mathrm{\Lambda },C_{\mathrm{\Phi }}}=max\{\left(\frac{\frac{{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}{1-\frac{L_f{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}\right),\frac{C_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\mathrm{\Lambda }}{1-\mathrm{\Lambda }{L}_f}\}.$At this juncture, from Equations (30(30) $\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|\le \left(\frac{C_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\frac{{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}{1-\frac{L_f{\mathcal{T}}_1}{\mathrm{\Gamma }(\kappa +1)}}\right)\mathrm{\Phi }$(30) ) and (31(31) ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le \mathrm{\Lambda }{C}_{\mathrm{\Phi }}\mathrm{\Phi }\left(t\right)\mathrm{\Phi }+\mathrm{\Lambda }{L}_f{\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z$(31) ), we have ${\left||\mathrm{\Phi }-\overline{\mathrm{\Phi }}|\right|}_Z\le {\mathcal{H}}_{\mathrm{\Lambda },C_{\mathrm{\Phi }}},\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{c}\mathrm{h}t\in \mathcal{T}\mathcal{.}$So the solution of the model (2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) ) is generalized H–U–R stable.

5. Numerical scheme for the fractional piecewise for the monkeypox model

In this section, the numerical scheme for the following monkeypox according to model (2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) ) is given. (32) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(32) By applying the piece-wise integral to the Caputo and ABC derivative, we obtain (33) $\begin{array}{rl}{\mathbb{S}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{S}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_1(t,{\mathbb{S}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{S}}_h\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_1(t,{\mathbb{S}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_1(t,{\mathbb{S}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{E}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{E}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_2(t,{\mathbb{E}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{E}}_h\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_2(t,{\mathbb{E}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_2(t,{\mathbb{E}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{I}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{I}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_3(t,{\mathbb{I}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{I}}_h\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_3(t,{\mathbb{I}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_3(t,{\mathbb{I}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{C}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{C}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_4(t,{\mathbb{C}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ \mathbb{C}\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_4(t,{\mathbb{C}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_4(t,{\mathbb{C}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{R}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{R}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_5(t,{\mathbb{R}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{R}}_h\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_5(t,{\mathbb{R}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_5(t,{\mathbb{R}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{S}}_r\left(t\right)& =\{\begin{array}{l}{\mathbb{S}}_r\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_6(t,{\mathbb{S}}_r)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{S}}_r\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_6(t,{\mathbb{S}}_r)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_6(t,{\mathbb{S}}_r)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{E}}_r\left(t\right)& =\{\begin{array}{l}{\mathbb{E}}_r\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_7(t,{\mathbb{E}}_r)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{E}}_r\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_7(t,{\mathbb{E}}_r)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_7(t,{\mathbb{E}}_r)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{I}}_r\left(t\right)& =\{\begin{array}{l}{\mathbb{I}}_r\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_8(t,{\mathbb{I}}_r)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{I}}_r\left(t_1\right)+{\displaystyle \frac{1-\kappa }{AB\left(\kappa \right)}}{\mathbb{Y}}_8(t,{\mathbb{I}}_r)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^t(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_8(t,{\mathbb{I}}_r)\mathrm{d}\rho t_1<t\le T.}\end{array}\end{array}$(33) At $t=t_{n+1}$ (34) $\begin{array}{rl}{\mathbb{S}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{S}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_1(t,{\mathbb{S}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{S}}_h\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_1(t,{\mathbb{S}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^{t_{n+1}}(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_1(t,{\mathbb{S}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{E}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{E}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_2(t,{\mathbb{E}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{E}}_h\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_2(t,{\mathbb{E}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^{t_{n+1}}(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_2(t,{\mathbb{E}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{I}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{I}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_3(t,{\mathbb{I}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{I}}_h\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_3(t,{\mathbb{I}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^{t_{n+1}}(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_3(t,{\mathbb{I}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{C}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{C}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_4(t,{\mathbb{C}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ \mathbb{C}\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_4(t,{\mathbb{C}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^{t_{n+1}}(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_4(t,{\mathbb{C}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{R}}_h\left(t\right)& =\{\begin{array}{l}{\mathbb{R}}_h\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_5(t,{\mathbb{R}}_h)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{R}}_h\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_5(t,{\mathbb{R}}_h)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^{t_{n+1}}(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_5(t,{\mathbb{R}}_h)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{S}}_r\left(t\right)& =\{\begin{array}{l}{\mathbb{S}}_r\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_6(t,{\mathbb{S}}_r)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{S}}_r\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_6(t,{\mathbb{S}}_r)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^{t_{n+1}}(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_6(t,{\mathbb{S}}_r)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{E}}_r\left(t\right)& =\{\begin{array}{l}{\mathbb{E}}_r\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_7(t,{\mathbb{E}}_r)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{E}}_r\left(t_1\right)+{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{\mathbb{Y}}_7(t,{\mathbb{E}}_r)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^{t_{n+1}}(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_7(t,{\mathbb{E}}_r)\mathrm{d}\rho t_1<t\le T}\end{array}\\ {\mathbb{I}}_r\left(t\right)& =\{\begin{array}{l}{\mathbb{I}}_r\left(0\right)+{\displaystyle \frac{1}{\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_0^{t_1}(t-\rho {)}^{\kappa -1c}{\mathbb{Y}}_8(t,{\mathbb{I}}_r)\mathrm{d}\rho 0<t\le t_1,}\\ {\mathbb{I}}_r\left(t_1\right)+{\displaystyle \frac{1-\kappa }{AB\left(\kappa \right)}}{\mathbb{Y}}_8(t,{\mathbb{I}}_r)\mathrm{d}\rho \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)\mathrm{\Gamma }\left(\kappa \right)}}{\displaystyle {\int }_{t_1}^{t_{n+1}}(t-\rho {)}^{\kappa -1}{\mathbb{Y}}_8(t,{\mathbb{I}}_r)\mathrm{d}\rho t_1<t\le T.}\end{array}\end{array}$(34) We put the Newton polynomials, so the following is obtained: (35) $\begin{array}{r}{\mathbb{S}}_{\mathbb{h}}\left(t_{n+1}\right)=\{\begin{array}{l}{\mathbb{S}}_{{\mathbb{h}}_{\mathbb{0}}}+\left\{\begin{array}{l}{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1}){-}^C{\mathbb{f}}_1({\mathbb{S}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa (\mathrm{\Delta }t{)}^{\kappa -1}}{2\mathrm{\Gamma }(\kappa +3)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^C{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^C{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\\ {\mathbb{S}}_h\left(t_1\right)+\left\{\begin{array}{l}{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h^n,t_n)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\delta t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})+ABC{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{\kappa (\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +3)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\end{array}\end{array}$(35)

Similarly, for the remaining compartments we get (36) $\begin{array}{rl}{\mathbb{E}}_{\mathbb{h}}\left(t_{n+1}\right)& =\{\begin{array}{l}{\mathbb{E}}_{{\mathbb{h}}_{\mathbb{0}}}+\left\{\begin{array}{l}{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_2({\mathbb{S}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1}){-}^C{\mathbb{f}}_1({\mathbb{S}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa (\mathrm{\Delta }t{)}^{\kappa -1}}{2\mathrm{\Gamma }(\kappa +3)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^C{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^C{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\\ {\mathbb{E}}_h\left(t_1\right)+\left\{\begin{array}{l}{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{}^{ABC}{\mathbb{Y}}_2({\mathbb{E}}_h^n,t_n)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\delta t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})+ABC{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{\kappa (\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +3)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^{ABC}{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^{ABC}{\mathbb{Y}}_2({\mathbb{E}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\end{array}\end{array}$(36) (37) $\begin{array}{rl}{\mathbb{I}}_{\mathbb{h}}\left(t_{n+1}\right)& =\{\begin{array}{l}{\mathbb{I}}_{{\mathbb{h}}_{\mathbb{0}}}+\left\{\begin{array}{l}{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1}){-}^C{\mathbb{f}}_1({\mathbb{I}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa (\mathrm{\Delta }t{)}^{\kappa -1}}{2\mathrm{\Gamma }(\kappa +3)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^C{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^C{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\\ {\mathbb{I}}_h\left(t_1\right)+\left\{\begin{array}{l}{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{}^{ABC}{\mathbb{Y}}_3({\mathbb{I}}_h^n,t_n)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\delta t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ +ABC{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{\kappa (\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +3)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^{ABC}{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^{ABC}{\mathbb{Y}}_3({\mathbb{I}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\end{array}\end{array}$(37) (38) $\begin{array}{rl}{\mathbb{C}}_{\mathbb{h}}\left(t_{n+1}\right)& =\{\begin{array}{l}{\mathbb{C}}_{{\mathbb{h}}_{\mathbb{0}}}+\left\{\begin{array}{l}{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1}){-}^C{\mathbb{f}}_1({\mathbb{C}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa (\mathrm{\Delta }t{)}^{\kappa -1}}{2\mathrm{\Gamma }(\kappa +3)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^C{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^C{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\\ {\mathbb{C}}_h\left(t_1\right)+\left\{\begin{array}{l}{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{}^{ABC}{\mathbb{Y}}_4({\mathbb{C}}_h^n,t_n)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\delta t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ +ABC{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{\kappa (\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +3)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n}\\ {[}^{ABC}{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^{ABC}{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})\\ {+}^{ABC}{\mathbb{Y}}_4({\mathbb{C}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\end{array}\end{array}$(38) (39) $\begin{array}{rl}{\mathbb{R}}_{\mathbb{h}}\left(t_{n+1}\right)& =\{\begin{array}{l}{\mathbb{R}}_{{\mathbb{h}}_{\mathbb{0}}}+\left\{\begin{array}{l}{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1}){-}^C{\mathbb{f}}_1({\mathbb{R}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa (\mathrm{\Delta }t{)}^{\kappa -1}}{2\mathrm{\Gamma }(\kappa +3)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^C{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^C{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\\ {\mathbb{R}}_h\left(t_1\right)+\left\{\begin{array}{l}{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{}^{ABC}{\mathbb{Y}}_5({\mathbb{R}}_h^n,t_n)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\delta t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ +ABC{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{\kappa (\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +3)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}},t_{\mathbf{k}})-2^{ABC}{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^{ABC}{\mathbb{Y}}_5({\mathbb{R}}_h^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\end{array}\end{array}$(39) (40) $\begin{array}{rl}{\mathbb{S}}_{\mathbb{r}}\left(t_{n+1}\right)& =\{\begin{array}{l}{\mathbb{S}}_{{\mathbb{h}}_{\mathbb{0}}}+\left\{\begin{array}{l}{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1}){-}^C{\mathbb{f}}_1({\mathbb{S}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa (\mathrm{\Delta }t{)}^{\kappa -1}}{2\mathrm{\Gamma }(\kappa +3)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}},t_{\mathbf{k}})-2^C{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^C{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\\ {\mathbb{S}}_r\left(t_1\right)+\left\{\begin{array}{l}{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{}^{ABC}{\mathbb{Y}}_6({\mathbb{S}}_r^n,t_n)+{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\delta t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ +ABC{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{\kappa (\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +3)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}},t_{\mathbf{k}})-2^{ABC}{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^{ABC}{\mathbb{Y}}_6({\mathbb{S}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\end{array}\end{array}$(40) (41) $\begin{array}{rl}{\mathbb{E}}_{\mathbb{r}}\left(t_{n+1}\right)& =\{\begin{array}{l}{\mathbb{S}}_{{\mathbb{h}}_{\mathbb{0}}}+\left\{\begin{array}{l}{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1}){-}^C{\mathbb{f}}_1({\mathbb{E}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa (\mathrm{\Delta }t{)}^{\kappa -1}}{2\mathrm{\Gamma }(\kappa +3)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}},t_{\mathbf{k}})-2^C{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^C{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\\ {\mathbb{E}}_r\left(t_1\right)+\left\{\begin{array}{l}{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{}^{ABC}{\mathbb{Y}}_7({\mathbb{E}}_r^n,t_n)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\delta t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1})+ABC{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{\kappa (\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +3)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}},t_{\mathbf{k}})-2^{ABC}{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^{ABC}{\mathbb{Y}}_7({\mathbb{E}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\end{array}\end{array}$(41) (42) $\begin{array}{rl}{\mathbb{I}}_{\mathbb{r}}\left(t_{n+1}\right)& =\{\begin{array}{l}{\mathbb{S}}_{{\mathbb{h}}_{\mathbb{0}}}+\left\{\begin{array}{l}{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{(\mathrm{\Delta }t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1}){-}^C{\mathbb{f}}_1({\mathbb{I}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge }\\ +{\displaystyle \frac{\kappa (\mathrm{\Delta }t{)}^{\kappa -1}}{2\mathrm{\Gamma }(\kappa +3)}}{\displaystyle \sum _{\mathbf{k}=2}^i{[}^C{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}},t_{\mathbf{k}})-2^C{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^C{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\\ {\mathbb{I}}_r\left(t_1\right)+\left\{\begin{array}{l}{\displaystyle \frac{1-\kappa }{ABC\left(\kappa \right)}}{}^{ABC}{\mathbb{Y}}_8({\mathbb{I}}_r^n,t_n)\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\delta t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +1)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Pi }}\\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{(\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +2)}}{\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ +ABC{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\bigwedge \\ +{\displaystyle \frac{\kappa }{ABC\left(\kappa \right)}}{\displaystyle \frac{\kappa (\kappa t{)}^{\kappa -1}}{\mathrm{\Gamma }(\kappa +3)}}\\ {\displaystyle \sum _{\mathbf{k}=i+3}^n{[}^{ABC}{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}},t_{\mathbf{k}})-2^{ABC}{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}-1},t_{\mathbf{k}-1})}\\ {+}^{ABC}{\mathbb{Y}}_8({\mathbb{I}}_r^{\mathbf{k}-2},t_{\mathbf{k}-2})]\mathrm{\Delta }\end{array}\right\}\end{array}\end{array}$(42)

Here $\begin{array}{rl}\mathrm{\Pi }& =\left[\begin{array}{l}(1-\mathbf{k}+n{)}^{\kappa }(2(-\mathbf{k}+n{)}^2+(3\kappa +10)(-\mathbf{k}+n)+2d^2+9\kappa +12)\\ -(-\mathbf{k}+n)(2(-\mathbf{k}+n{)}^2+(5\kappa +10\left)\right(n-\mathbf{k})+6{\kappa }^2+18\kappa +12)\end{array}\right],\\ \bigwedge & =\left[\begin{array}{l}(1-\mathbf{k}+n{)}^{\kappa }(3+n+2\kappa -\mathbf{k})\\ -(-\mathbf{k}+n)(n+3\kappa -\mathbf{k}+3)\end{array}\right],\\ \mathrm{\Delta }& =\left[\right(1-\mathbf{k}+n{)}^{\kappa }-(-\mathbf{k}+n{)}^{\kappa }]\end{array}$and $\begin{array}{rl}{}^C{\mathbb{Y}}_1({\mathbb{S}}_h,t)& ={}^{ABC}{\mathbb{Y}}_1({\mathbb{S}}_h,t)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h,\\ {}^C{\mathbb{Y}}_3(\mathbb{I},t)& ={}^{ABC}{\mathbb{f}}_3(\mathbb{I},t)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h,\\ {}^C{\mathbb{Y}}_2(\mathbb{A},t)& ={}^{ABC}{\mathbb{f}}_2(\mathbb{A},t)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h,\\ {}^C{\mathbb{Y}}_4(\mathbb{D},t)& ={}^{ABC}{\mathbb{f}}_4(\mathbb{D},t)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h,\\ {}^C{\mathbb{Y}}_5(\mathbb{R},t)& ={}^{ABC}{\mathbb{f}}_5(\mathbb{R},t)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h,\\ {}^C{\mathbb{Y}}_6({\mathbb{I}}_r,t)& ={}^{ABC}{\mathbb{f}}_5({\mathbb{I}}_r,t)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r,\\ {}^C{\mathbb{Y}}_7(\mathbb{T},t)& ={}^{ABC}{\mathbb{f}}_5(\mathbb{T},t)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r,\\ {}^C{\mathbb{Y}}_8({\mathbb{H}}_d,t)& ={}^{ABC}{\mathbb{f}}_5({\mathbb{H}}_d,t)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}$With the aforementioned application steps in line with the considered model, we have obtained our results accordingly. All the analyses and numerical simulation results obtained have been performed with MATLAB software [57].

6. Numerical simulations and analysis

In this section, the development of the numerical simulation is provided for all the eight agents of the considered problem along with the initial data and parameters values mentioned in Table 2 and taken from [16]. The initial population is ${\mathbb{S}}_h\left(0\right)=2000,{\mathbb{E}}_h\left(0\right)=80,{\mathbb{I}}_h\left(0\right)=50,{\mathbb{C}}_h\left(0\right)=45,\mathbb{R}\left(0\right)=40,{\mathbb{S}}_r\left(0\right)=2000,{\mathbb{E}}_r\left(0\right)=100,{\mathbb{I}}_r\left(0\right)=400$. The graphical representation of all the eight quantities of the considered problem has been shown on three different data of fractional orders and time under piecewise Caputo and ABC derivative. The first agent of susceptible human population is shown in Figure 1(a)–1(c) respectively, on two sub intervals with the bending terminal point of the first interval. The said class population decreases slowly in the first interval while in second interval it becomes stable. On small fractional orders, it becomes stable quickly and piece wise behaviours are negligible in this case as shown in Figure 1(c). The second agent of expose human population is shown in Figure 2(a)–(c), respectively on two subintervals with the bending terminal point of the first interval. The said class population first increases slowly in the first interval reaching to its peak value and then moving towards stability. On small fractional orders, it becomes stable quickly and piecewise behaviours are negligible in this case as shown in Figure 2(a) and (c). The third quantity of infected human population is shown in Figure 3(a)–(c), respectively on two subintervals with the bending terminal point of the first interval. The said class population first increases slowly in the first interval reaching to its peak value and then moving towards stability. On small fractional orders, it becomes stable quickly and piece wise behaviours are negligible in this case as shown in Figure 3(c). On large fractional orders, the curves are far away from each other while on small they are close to each other. Next we simulate the fourth quantity of clinically human infected population is shown in Figure 4(a)–(c), respectively on two subintervals with the bending curves on terminal point of the first interval. The said class population first increases quickly in the first interval reaching to its peak value and then moving towards stability. On small fractional orders, it becomes stable quickly and piece wise behaviours are negligible in this case as shown in Figure 4(c). On large fractional orders, the curves are far away from each other while on small they are close to each other.

Table 2. Parameters and their description according to model (2(2) $\begin{array}{r}\{\begin{array}{l}{}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_h\left(t\right)={\mathrm{\Phi }}_h-({\mathrm{\Lambda }}_h+{\nu }_h){\mathbb{S}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_h\left(t\right)={\mathrm{\Lambda }}_h{\mathbb{S}}_h-({\nu }_h+\alpha ){\mathbb{E}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_h\left(t\right)=\alpha {\mathbb{E}}_h-(\mathrm{\Omega }+\gamma +{\nu }_h+{\mathrm{\Delta }}_1){\mathbb{I}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{C}}_h\left(t\right)=\gamma {\mathbb{I}}_h-(\rho +{\nu }_h+{\mathrm{\Delta }}_2){\mathbb{C}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{R}}_h\left(t\right)=\rho {\mathbb{C}}_h+\mathrm{\Omega }{\mathbb{I}}_h-{\nu }_h{\mathbb{R}}_h\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{S}}_r\left(t\right)={\mathrm{\Phi }}_r-{\mathrm{\Lambda }}_r{\mathbb{S}}_r-{\nu }_r{\mathbb{S}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{E}}_r\left(t\right)={\mathrm{\Lambda }}_r{\mathbb{S}}_r-(ϵ+{\nu }_r){\mathbb{E}}_r\\ {}_0^{PCABC}{\mathbf{D}}_t^{\kappa }{\mathbb{I}}_r\left(t\right)=ϵ{\mathbb{E}}_r-{\nu }_r{\mathbb{I}}_r.\end{array}\end{array}$(2) ).

Parameter	Description	Parameter	Description
${\mathrm{\Phi }}_h$	64850	${\mathrm{\Phi }}_r$	0.2
${\nu }_h$	0.0003	${\nu }_r$	0.002
α	0.016744	${\mathrm{\Delta }}_1$	0.003
${\mathrm{\Delta }}_2$	0.055	γ	0.05
ρ	0.036	Ω	0.088
${\alpha }_1$	0.032	${\alpha }_2$	0.022
σ	0.012

Figure 1. Dynamical behaviours of susceptible human individuals ${\mathbb{S}}_h\left(t\right)$ on different arbitrary fractional orders $\kappa$ and time durations on sub interval $[0,t_1]$ and $[t_1,T]$ of $[0,T]$.

Figure 2. Dynamical behaviours of Exposed human individuals ${\mathbb{E}}_h\left(t\right)$ on different arbitrary fractional orders $\kappa$ and time durations on sub interval $[0,t_1]$ and $[t_1,T]$ of $[0,T]$.

Figure 3. Dynamical behaviours of infected human individuals ${\mathbb{I}}_h\left(t\right)$ on different arbitrary fractional orders $\kappa$ and time durations on sub interval $[0,t_1]$ and $[t_1,T]$ of $[0,T]$.

Figure 4. Dynamical behaviours of clinical ill human individuals ${\mathbb{C}}_h\left(t\right)$ on different arbitrary fractional orders $\kappa$ and time durations on sub interval $[0,t_1]$ and $[t_1,T]$ of $[0,T]$.

Next we simulate the fifth quantity of the Recovered human population is shown in Figure 5(a)–(c) respectively on two subintervals with the bending curves on terminal point of the first interval. The said class population increases gradually in the first as well as in the second interval when the infected human recovered. On large fractional orders, the curves are far away from each other while on small they are close to each other. As the last related stage, we simulate the last quantity of infected rodent population as shown in Figure 8(a)–(c) respectively on two subintervals with bending curves on terminal point of the first interval. The said class population first increases quickly in the first interval reaching to its peak value and then decreasing moving towards stability. On small fractional orders, it becomes stable quickly and piecewise behaviours are negligible in this case as shown in Figure 8(c). On large fractional orders, the curves are far away from each other while on small they are close to each other. Next, we simulate the sixth quantity of susceptible rodent population is shown in Figure 6(a)–(c), respectively on two subintervals with the bending curves on terminal point of the first interval. The said class population first declines quickly in the first interval and in the second interval and then moves towards stability. On small fractional orders it becomes stable quickly and piecewise behaviours are negligible in this case as shown in Figure 6(c). On large fractional orders, the curves are far away from each other while on small they are close to each other. Next, we simulate the seventh quantity of exposed rodent population as shown in Figure 7(a)–7(c), respectively on two subintervals with the bending curves on terminal point of the first interval. The said class population first increases quickly in the first interval reaching to its peak value and then moving towards stability. On small fractional orders, it becomes stable quickly and piecewise behaviours are negligible in this case as shown in Figure 7(c). On large fractional orders, the curves are far away from each other while on small they are close to each other.

Figure 5. Dynamical behaviours of Recovered human individuals ${\mathbb{R}}_h\left(t\right)$ on different arbitrary fractional orders $\kappa$ and time durations on sub interval $[0,t_1]$ and $[t_1,T]$ of $[0,T]$.

Figure 6. Dynamical behaviours of susceptible rodent individuals ${\mathbb{S}}_r\left(t\right)$ on different arbitrary fractional orders $\kappa$ and time durations on subinterval $[0,t_1]$ and $[t_1,T]$ of $[0,T]$.

Figure 7. Dynamical behaviours of exposed rodent individuals ${\mathbb{E}}_r\left(t\right)$ on different arbitrary fractional orders $\kappa$ and time durations on sub interval $[0,t_1]$ and $[t_1,T]$ of $[0,T]$.

7. Conclusion and future directions

The investigated work developed for addressing the considered problem of piecewise monkeypox infection, which is composed of eight agents for monkeypox virus under the fractional derivatives of Caputo and Atangana Baleanu in the sub-partitioned scheme. Through our study, we have proven that all the quantities are converging and quickly stabilized on small fractional orders as given in the numerical simulation section. The important theoretical, numerical and simulation-related properties have been presented for the considered model. By the application of fixed point results, we have derived results which deal with existence and uniqueness of solution for both sub intervals in the sense of Caputo and Atangana Baleanu operators. The Ulam–Hyers stability concept on both interval has also been derived. The Newton Polynomial technique has been utilized for computing numerical solution of the piecewise fractional model of monkeypox virus. Consequently, we have observed that the piecewise data provide more information describing crossover dynamics for different fractional orders, and the graphical results present thought-provoking information regarding both piecewise and fractional order analysis. Based on these derived results, control measures and strategies to curb the disease and prevent future occurrences can be proposed as one of the future directions. Another direction can be concerned with the further investigation of the new dynamics of the viruses under examination through newly designed computational fractional models. Furthermore, the efficient numerical techniques can be sought and adapted to verify the theoretical aspects as well as findings related to the dynamic behaviour of diseases. The kind of schemes we used can allow one to implement various analytical, numerical and computing methods by including their applications in emerging real-world problems. The effective management and control of communicable diseases, like monkeypox handled in this study, with different features, virological attributes and varying characteristics concern high-risk groups concerning broader society and global public health.

Figure 8. Dynamical behaviours of infected rodent individuals ${\mathbb{I}}_r\left(t\right)$ on different arbitrary fractional orders $\kappa$ and time durations on sub interval $[0,t_1]$ and $[t_1,T]$ of $[0,T]$.

Disclosure statement

No potential conflict of interest was reported by the author(s).