$\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$ model for analyzing $\mathscr {C}\mathscr {O}\mathscr {V}\mathscr {I}\mathscr {D}$-19 pandemic process via $\uppsi$-Caputo fractional derivative and numerical simulation

Abstract

The objective of this study is to develop the $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$ epidemic model for $\mathscr {C}\mathscr {O}\mathscr {V}\mathscr {I}\mathscr {D}$-${\textbf {19}}$ utilizing the $\uppsi$-Caputo fractional derivative. The reproduction number ($\breve{\mathscr {R}}_0$) is calculated utilizing the next generation matrix method. The equilibrium points of the model are computed, and both the local and global stability of the disease-free equilibrium point are demonstrated. Sensitivity analysis is discussed to describe the importance of the parameters and to demonstrate the existence of a unique solution for the model by applying a fixed point theorem. Utilizing the fractional Euler procedure, an approximate solution to the model is obtained. To study the transmission dynamics of infection, numerical simulations are conducted by using MatLab. Both numerical methods and simulations can provide valuable insights into the behavior of the system and help in understanding the existence and properties of solutions. By placing the values $\texttt{t}$, $\ln (\texttt{t})$ and $\sqrt{\texttt{t}}$ instead of $\uppsi$, the derivatives of the Caputo and Caputo–Hadamard and Katugampola appear, respectively, to compare the results of each with real data. Besides, these simulations specifically with different fractional orders to examine the transmission dynamics. At the end, we come to the conclusion that the simulation utilizing Caputo derivative with the order of 0.95 shows the prevalence of the disease better. Our results are new which provide a good contribution to the current research on this field of research.

Introduction

The initial cases of the novel corona virus (nCoV) were identified in China in December 2019, and the virus quickly spread to other countries around the world, resulting in a significant number of casualties¹. The WHO declared it a pandemic on March 11, 2020. Many scientists are trying to find the best way to stop the infection from spreading because it has caused a lot of damage to the world. The $\mathscr {C}$-19 pandemic is a severe acute respiratory disease which no definitive treatment has been found till now^1,2,3. The droplets, airborne, and closed contact transmission leads to virus spreads from person to person^4,5,6,7. With the outbreak of the pandemic, predicting the number of infected cases and the termination of the infection are important.

It should be noted that mathematical modeling plays a critical role in understanding, predicting, and controlling the spread of diseases. Unlike integer-order models, which only consider the current state of the system, fractional-order models take into account past states and interactions. This memory effect allows for a more accurate representation of the disease dynamics, as it considers the cumulative impact of previous events and interventions on the current state of the disease.

In this scientific research, we consider an $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$ dynamics model (1) in Fig. 1, such that it individuals are divided into five classes: ($\mathscr {S}$) susceptible individuals; ($\mathscr {E}$) exposed individuals; ($\mathscr {I}$) individuals with symptoms; ($\mathscr {A}$) individuals without symptoms; ($\mathscr {R}$) recovered individuals.

Figure 1

Proposed model $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$ for $\mathscr {C}$-19.

$$\begin{aligned} \left\{ \begin{array}{l} \mathscr {S}^\prime = \varLambda - \upbeta _1 \mathscr{S}\mathscr{I} - \upbeta _2 \mathscr{S}\mathscr{A} - \upsigma \, \mathscr {S} + \varsigma \mathscr {R},\\ \mathscr {E}^\prime = \upbeta _1 \mathscr{S}\mathscr{I} - (\upupsilon + \upsigma ) \mathscr {E} + \upbeta _2 \mathscr{S}\mathscr{A},\\ \mathscr {I}^\prime (\texttt{t}) = \mathfrak {p} \upupsilon \mathscr {E} - (\upalpha +\upsigma +\updelta ) \mathscr {I},\\ \mathscr {A}^\prime = ( 1- \mathfrak {p} ) \upupsilon \mathscr {E} - (\upgamma + \upsigma ) \mathscr {A},\\ \mathscr {R}^\prime = \upalpha \mathscr {I} - (\upsigma + \varsigma ) \mathscr {R}+ \upgamma \mathscr {A}. \end{array}\right. \end{aligned}$$

(1)

New ways of using math of fractional-order have been created to help predict and control the spread of diseases. These models can help us understand how many people will get sick and die, and also help us slow down the spread of the disease^{8,9,10,11,12,13}. Some mathematicians have also focused their efforts on analyzing the different non-linear dynamics of infection-related problems such as epidemics^{14,15,16,17,18,19,20,21,22,23}. Khan et al. described a mathematical model for dynamics of a novel nCoV (2019) and then developed the it with quarantine and isolation^24,25. A fractional mathematical model to analyze the pandemic trend of the infection was discussed in²⁶. Naik investigated and analyzed a nonlinear fractional-order $\mathscr {S}\mathscr {I}\mathscr {R}$ epidemic model with Crowley–Martin type functional response and Holling type-II treatment rate were established along the memory²². Mathematical modeling and analysis of the $\mathscr {C}$-19 epidemic with reinfection and with vaccine availability were examined in^27,28,29,30. The authors in³¹, provided a $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {R}$ epidemic model, Fig. 2, as form

$$\begin{aligned} {\left\{ \begin{array}{ll} \mathscr {S}' = \omega \mu \mathscr {S} - (\beta _1 \mathscr {E} + \beta _2 \mathscr {I}) \texttt{S},\\ \mathscr {E} = (\beta _1\mathscr {E} + \beta _2 \mathscr {I}) \mathscr {S} - (\lambda + \mu ) \mathscr {E},\\ \mathscr {I} = \lambda \mathscr {E} - ( \tau + \mu + \delta ) \mathscr {I},\\ \mathscr {R}= \tau \mathscr {I} - \mu \mathscr {R}, \end{array}\right. } \end{aligned}$$

(2)

for the spread of $\mathscr {C}$-19 using the CFD where $\omega = n \times N$, N is the total number of individuals and n is the birth rate, $\mu$ is the death rate of people, $\beta _1$, $\beta _2$ are the transmission rate of infection from $\mathscr {E}$ to $\mathscr {S}$, $\mathscr {I}$ to $\mathscr {S}$, respectively, $\lambda$ is the transmission rate of people from $\mathscr {E}$ to $\mathscr {I}$, $\delta$ is the mortality rate due to the disease, and $\tau$ is the rate of recovery of infected people.

Figure 2

$\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {R}$ model of $\mathscr {C}$-19 in³¹.

Infectious diseases mathematical model is a crucial tool that has been used to study the spreading mechanism of many diseases^32,33,34. Gharahasanlou et al. considered the following mathematical biology and dynamical system

$$\begin{aligned} \left\{ \begin{array}{l} \mathscr {X}' = \lambda + \alpha \mathscr {X} \left( 1- \frac{\mathscr {X}}{\mathscr {X}_{\max }} \right) d\mathscr {X} -h(\mathscr {X}, \mathscr {Y}, \mathscr {V})\mathscr {V},\\ \mathscr {Y}' = h(\mathscr {X}, \mathscr {Y}, \mathscr {V}) \mathscr {V}- \beta \mathscr {Y} - \gamma \mathscr {Y} \mathscr {Z},\quad \mathscr {V}' = \zeta \mathscr {Y} - \eta \mathscr {V},\quad \mathscr {Z} = \omega \mathscr {Y}\mathscr {Z} - \varpi \mathscr {Z}, \end{array}\right. \end{aligned}$$

(3)

under initial conditions $\mathscr {X}(0) > 0$, $\mathscr {Y}(0) > 0$, $\mathscr {Z}(0) > 0$ and $\mathscr {V}(0) > 0$³⁵. Recently, in one of the valuable research works, Peter et al.³⁶ studied transmission dynamics model, Fig. 3 of $\mathscr {C}$-19 by the following system of the nonlinear differential equation:

$$\begin{aligned} \left\{ \begin{array}{l} \mathscr {S}^\prime = \vartheta - \varLambda \mathscr {S} -( \upmu + \varphi ) \mathscr {S}+ \uptau \mathscr {V}_1,\\ \mathscr {V}_1^\prime = \varphi \mathscr {S} - (\uptau + \varsigma + \upmu ) \mathscr {V}_1,\\ \mathscr {V}_2^\prime = \varsigma \mathscr {V}_1 - (\upeta + \upmu ) \mathscr {V}_2,\\ \mathscr {E}^\prime = \varLambda \mathscr {S} - (\upepsilon + \upmu ) \mathscr {E},\\ \mathscr {A}^\prime = \upepsilon ( 1 - \varkappa ) \mathscr {E} - (\upmu + \uppsi ) \mathscr {A},\\ \mathscr {I}^\prime = \upepsilon \varkappa \mathscr {E}+ \uppsi (1 - \varrho ) \mathscr {A} - (\upmu + \updelta + \upomega )\mathscr {I},\\ \mathscr {H}^\prime = \upomega ( 1- b ) \mathscr {I} - (\upmu + \updelta + d) \mathscr {H},\\ \mathscr {R}^\prime = \upomega b \mathscr {I} + \uppsi \varrho \mathscr {A} + d \mathscr {H} + \upeta \mathscr {V}_2- \upmu \mathscr {R}. \end{array}\right. \end{aligned}$$

(4)

Figure 3

$\mathscr{S}\mathscr{V}_1\mathscr {V}_2\mathscr {E}\mathscr {A}\mathscr {I}\mathscr {H}\mathscr {R}$ model for $\mathscr {C}$-19 in³⁶.

Musa et al. analyzed a new deterministic co-infection model of $\mathscr {C}$-19³³. We accept that use these references to influence this phenomena and the consolidation of this marvel and its impacts on the co-dynamics of both illnesses will be of awesome intrigued not as it were to open wellbeing specialists but too to analysts within the field of scientific modeling^{9,23,30,33,34}.

The rest of this paper is sorted out within the taking after way. In section “Preliminary definitions”, we review fractional integral and derivative. Then, in section “Model formulation”, the $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$ demonstrate of fractional order for the infection transmission is displayed and the balance focuses and their stability are investigated. The existence and asymptotically stability of the equilibrium points are investigated. The sufficient conditions for the persistence of the disease are provided. First, $\breve{\mathscr {R}}_0$ is obtained which determines the stability of equilibria, then model equilibria are determined and their stability analysis is considered by using fractional Routh–Hurwitz stability criterion. The fractional derivative is taken in Caputo sense and the numerical solution of the model is obtained by (8) scheme which involves the memory trace that can capture and integrate all past activity. Meanwhile, by using Lyapunov functional approach, the global dynamics of the endemic equilibrium point is discussed. Further, some numerical simulations are performed to illustrate the effectiveness of the theoretical results obtained. The outcome of the study reveals that the applied (8) scheme is computationally very strong and effective to analyze fractional-order differential equations arising in disease dynamics. The results show that order of the fractional derivative has a significant effect on the dynamic process. The reproduction number is also calculated and its sensitivity is explored. In section “Existence and uniqueness of solution”, we investigate whether there is only one solution for the system. In section “Numerical results”, we use a math method to solve the model and show a math simulation. Eventually, in last section, conclusion is presented.

Preliminary definitions

Let $\text {J}:= [\grave{\imath }_1, \grave{\imath }_2]$ and consider increasing function $\uppsi :\text {J} \rightarrow \mathbb {R}$ s.t $\uppsi ^\prime (\texttt{t}) \ne 0$, for each $\texttt{t}$. For $\upkappa > 0$, the $\upkappa ^{\text {th}}$ $\uppsi$-Riemann–Liouville fractional ($\uppsi$-RLF) integral for an integrable real function $\upomega$ on $\text {J}$ with respect to $\uppsi$ is illustrated by

$$\begin{aligned} \mathscr {I}_{\grave{\imath }_1^{+}}^{\upkappa ;\uppsi } \upomega (\texttt{t}) = \int _{\grave{\imath }_1}^{\texttt{t}} \frac{ \uppsi ^\prime (\xi )}{ ( \tilde{\uppsi }_{\xi }(\texttt{t}) )^{1 - \upkappa }} \frac{\upomega (\xi ) }{ \Gamma (\upkappa )} \, \text {d}\xi , \qquad \tilde{\uppsi }_{\xi }(\texttt{t}): = \uppsi (\texttt{t})-\uppsi (\xi ), \end{aligned}$$

(5)

where $\Gamma (\upkappa ) = \int _{0}^{+\infty } e^{-\texttt{t}} \texttt{t}^{\upkappa -1} \, \text {d}\texttt{t}$^37,38. Let $\uppsi , \upomega \in C^{ \texttt{n}}(\text {J})$ and $\uppsi$ possesses the same properties referred above. The $\upkappa ^{\text {th}}$ $\uppsi$-RLF derivative of $\upomega$ is defined by

$$\begin{aligned} \mathscr {D}_{\grave{\imath }_1^{+}}^{\upkappa ;\uppsi } \upomega ( \texttt{t})&= \mathscr {Z}^{(\texttt{n})} \mathscr {I}_{\grave{\imath }_1^{+}}^{ n-\upkappa ;\uppsi } \upomega ( \texttt{t}) = \mathscr {Z}^{(\texttt{n})} \int _{\grave{\imath }_1}^{ \texttt{t}} \frac{\uppsi ^\prime (\ddot{\xi })}{( \tilde{\uppsi }_{\ddot{\xi }}( \texttt{t}) )^{\upkappa - \texttt{n} +1}} \frac{ \upomega ( \ddot{\xi })}{ \Gamma (\texttt{n}-\upkappa )} \, \text {d}\ddot{\xi }, \qquad \mathscr {Z} = \left( \frac{1}{\uppsi ^\prime (\texttt{t})} \frac{\text {d}}{\text {d}\texttt{t}} \right) , \end{aligned}$$

where $\texttt{n} = [\upkappa ]+1$^37,38. The $\upkappa ^{\text {th}}$ $\uppsi$-CFD of $\upomega$ is defined by ${}^{\text {C}\!}\mathscr {D}_{\grave{ \imath }_1^{+}}^{\upkappa ;\uppsi } \upomega ( \texttt{t})$ $=\mathscr {I}_{\grave{\imath }_1^{+}}^{ \texttt{n} - \upkappa ; \uppsi } \mathscr {Z}^{(\texttt{n})} \upomega (\texttt{t}),$ where $\texttt{n}=[\upkappa ]+1$ and $\upkappa$ whenever $\upkappa \notin \mathbb {N}$ and $\upkappa \in \mathbb {N}$, respectively³⁹. In other words,

$$\begin{aligned} {}^{\text {C}\!}\mathscr {D}_{\grave{\imath }_1^{+}}^{\upkappa ; \uppsi } \upomega (\texttt{t}) = \left\{ \begin{array}{cl} \displaystyle \mathscr {Z}^\texttt{n}\upomega (\texttt{t}), &{} \upkappa = \texttt{n} \in \mathbb {N},\\ \displaystyle \int _{\grave{\imath }_1}^{\texttt{t}} \frac{ \uppsi ^\prime (\ddot{\xi })}{ \Gamma (\texttt{n}-\upkappa )} \, \left( \tilde{ \uppsi }_{\ddot{\xi }}(\texttt{t}) \right) ^{\texttt{n}-\upkappa -1} \mathscr {Z}^{ ( \texttt{n})} \upomega (\ddot{\xi })\, \text {d} \ddot{\xi }, &{} \upkappa \notin \mathbb {N}. \end{array}\right. \end{aligned}$$

(6)

Extension (6) gives the Caputo and Caputo–Hadamard derivative when $\uppsi (\texttt{t}) = \texttt{t}$ and $\uppsi (\texttt{t})= \ln \texttt{t}$, respectively. The $\uppsi$-CFD of order $\upkappa$ is specified as³⁹, Theorem 3,

$$\begin{aligned} {}^{\text {C}\!}\mathscr {D}_{\grave{\imath }_1^{+}}^{ \upkappa ; \uppsi } \upomega (\texttt{t}) = \mathscr {D}_{ \grave{\imath }_1^{+}}^{\upkappa ; \uppsi } \bigg ( \upomega (\texttt{t}) - \sum _{\ell = 0}^{ \mathtt {\texttt{n}}-1} \frac{\mathscr {Z}^{ (\ell )} \upomega ( \grave{\imath }_1 )}{ \ell ! } \left( \tilde{\uppsi }_{ \grave{\imath }_1} (\texttt{t}) \right) ^{\ell } \bigg ). \end{aligned}$$

Lemma 2.1

(⁴⁰) Let $\upomega \in C^{\texttt{n}} (\text {J})$. Then,

$$\begin{aligned} \mathscr {I}_{\grave{\imath }_1^{+}}^{\upkappa ; \uppsi }~{}^{\text {C}\!}\mathscr {D}_{\grave{\imath }_1^{+}}^{\upkappa ; \uppsi } \upomega (\texttt{t}) = \upomega (\texttt{t}) - \sum _{\ell = 0 }^{\texttt{n}-1} \frac{\mathscr {Z}^{(\ell )} \upomega (\grave{\imath }_1)}{\ell !} \left[ \tilde{\uppsi }_{\grave{\imath }_1} (\texttt{t}) \right] ^{ \ell }, \qquad \forall \, \texttt{t}\in \text {J}, \, \texttt{n}-1<\upkappa <\texttt{n}. \end{aligned}$$

Also, if $\upomega \in C^{\texttt{n} + \texttt{m}}(\text {J} )$ $(\texttt{m}\in \mathbb {N})$, then

$$\begin{aligned} \mathscr {Z}^{(\texttt{m})} \left( {}^{\text {C}\!}\mathscr {D}_{\grave{\imath }_1^{+}}^{\upkappa ; \uppsi } \upomega \right) (\texttt{t}) = ~{}^{\text {C}\!}\mathscr {D}_{\grave{\imath }_1^{+}}^{ \upkappa +\texttt{m}; \uppsi }\upomega (\texttt{t}) + \sum _{\ell =0}^{\texttt{m}-1} \frac{ \left[ \tilde{\uppsi }_{ {\grave{\imath }_1} } (\texttt{t}) \right] ^{\ell + \texttt{n}-\upkappa - \texttt{m}}}{\Gamma (\ell + \texttt{n} - \upkappa - \texttt{m}+1) } \mathscr {Z}^{(\ell + \texttt{n})} \upomega ({ \ell _1}). \end{aligned}$$

(7)

Watch that in the case of $\mathscr {Z}^{ ( \ell ) } \upomega (\grave{\imath }_1) = 0$, $\texttt{n} \le \ell \le \texttt{n} + \texttt{m}-1$, the relationship

$$\begin{aligned} \mathscr {Z}^{(\texttt{m})}\left( {}^{\text {C}\!}\mathscr {D}_{\grave{\imath }_1^{+}}^{\upkappa ;\uppsi } \upomega \right) (\texttt{t}) = {}^{\text {C}\!} \mathscr {D}_{\grave{\imath }_1^{+}}^{\upkappa +\texttt{m}; \uppsi }\upomega (\texttt{t}), \qquad \texttt{t} \in \text {J}, \end{aligned}$$

can be obtained

Lemma 2.2

(⁴⁰) Let $\upkappa , \upnu >0$ and $\upomega \in C(\text {J})$. Then, for all $\texttt{t} \in \text {J}$, (i)

$$\begin{aligned} \mathscr {I}_{\grave{\imath }_1^{+}}^{\upkappa ;\uppsi } \big (\mathscr {I}_{ \grave{\imath }_1^{+}}^{\upnu ; \uppsi } \upomega \big )(\texttt{t}) = \mathscr {I}_{ \grave{\imath }_1^{+}}^{\upkappa + \upnu ;\uppsi } \upomega (\texttt{t}), \qquad {}^{\text {C}\!} \mathscr {D}_{\grave{ \imath }_1^{+}}^{\upkappa ;\uppsi } \big ( \mathscr {I}_{\grave{\imath }_1^{+}}^{\upkappa ; \uppsi }\upomega \big ) (\texttt{t}) = \upomega (\texttt{t}); \end{aligned}$$

(ii) with $\Lambda = \upnu + \upkappa -1$,

$$\begin{aligned} \mathscr {I}_{\grave{\imath }_1^{+}}^{\upkappa ; \uppsi } ( \tilde{\uppsi }_{\grave{\imath }_1}( \texttt{t}))^{\upnu -1} = \frac{\Gamma (\upnu )}{ \Gamma (\upnu +\upkappa )}( \tilde{ \uppsi }_{\grave{\imath }_1} (\texttt{t}))^{\Lambda };\qquad {}^{\text {C}\!}\mathscr {D}_{ \grave{\imath }_1^{+}}^{\upkappa ;\uppsi }( \tilde{ \uppsi }_{ \grave{\imath }_1}(\texttt{ t}))^{\upnu -1} =\frac{\Gamma (\upnu )}{\Gamma (\upnu -\upkappa )}(\tilde{ \uppsi }_{ \grave{\imath }_1} ( \texttt{t}))^{\Lambda }; \end{aligned}$$

(iii) ${}^{\text {C}\!}\mathscr {D}_{\grave{\imath }_1^{+}}^{\upkappa ; \uppsi }( F_{\grave{\imath }_1})^{\ell } = 0, \, \texttt{n} - 1\le \upkappa \le \texttt{n}$, $\ell =0,1, \dots ,\texttt{n}-1, \texttt{n}\in \mathbb {N}$.

Theorem 2.3

(Banach Contraction Principle⁴¹) Let $\mathbb {W}\ne \varnothing$ and $(\mathbb {W},{\rho })$ be a complete metric space via a contraction $\varUpsilon : \mathbb {W}\rightarrow \mathbb {W}$ i.e., ${\rho }(\varUpsilon \upomega , \varUpsilon \upomega ^*) \le \upsigma \rho (\upomega , \upomega ^*)$, for each $\upomega , \upomega ^* \in \mathbb {W},\, \upsigma \in (0, 1)$. Then, $\varUpsilon$ admits a FP uniquely.

Theorem 2.4

(Leray–Schauder⁴¹) Let $\texttt{B}$ be a bounded convex closed set and $\texttt{O}$ be open set contained in $\texttt{B}$ of Banach space $\mathbb {W}$ with $0\in \texttt{O}$. Then, for the continuous and compact mapping $\varUpsilon : \overline{\texttt{O}} \rightarrow \texttt{B}$, either (i) $\varUpsilon$ admits a FP belonging to $\bar{\texttt{O}}$; (ii) $\exists \, \upomega \in \partial \texttt{O}$, $0< \upsigma < 1$ s.t $\upomega = \upsigma \varUpsilon (\upomega )$.

Model formulation

Mathematical models play a crucial role in predicting the behavior of viruses and their transmission among individuals during a viral pandemic. These models are essential for understanding how diseases spread in different parts of the world and for effectively managing the outbreak. Various mathematical models, such as the $\mathscr {S}\mathscr {I}\mathscr {R}$, $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {R}$, $\mathscr {S}\mathscr {C}\mathscr {I}\mathscr {A}\mathscr {R}$, $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {Q}\mathscr {T}\mathscr {S}$, and $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$, etc. are used to evaluate the prevalence of diseases. Based on the data provided by the WHO regarding $\mathscr {C}$-19, there are two categories of individuals infected with the virus: asymptomatic and symptomatic. Both types of individuals can transmit the disease to healthy individuals, and the infected individuals may either recover or succumb to the illness.

We aimed to change the time derivative with the $\uppsi$-CFD in system (1) for the disease in Fig. 1 under the parameters are explained in Table 1, by introducing a parameter $\uptheta$ in the following way

$$\begin{aligned} \left[ \uptheta ^{\upkappa -1}{{}^{\text {C}\!} \mathscr {D}}^{\upkappa ; {\uppsi }} \mathscr {S}(\texttt{t}) \right] = \left[ \frac{\text {d}}{\text {d}\texttt{t}} \right] = \mathfrak {s}^{-1}. \end{aligned}$$

Table 1 The parameters description used in model.

Consequently, the $\mathscr {C}$-19 mathematical model based on fractional derivatives of order $0< \upkappa < 1$ is presented as

$$\begin{aligned} \left\{ \begin{array}{l} \uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {S}= \varLambda - \upsigma \, \mathscr {S} + \varsigma \mathscr {R} - \upbeta _1\mathscr {S}(\texttt{t})\mathscr {I}(\texttt{t}) - \upbeta _2 \mathscr {S} \mathscr {A},\\ \uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{ \upkappa ; {\uppsi }} \mathscr {E} = \upbeta _1 \mathscr {S} \mathscr {I} - (\upupsilon + \upsigma ) \mathscr {E} + \upbeta _2\mathscr {S} \mathscr {A},\\ \uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }} \mathscr {I} = \mathfrak {p}\upupsilon \mathscr {E} -(\upalpha +\upsigma +\updelta )\mathscr {I},\\ \uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }} \mathscr {A} = ( 1-\mathfrak {p})\upupsilon \mathscr {E} - (\upgamma +\upsigma )\mathscr {A},\\ \uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; { \uppsi }}\mathscr {R} = \upalpha \mathscr {I} -(\upsigma + \varsigma ) \mathscr {R} + \upgamma \mathscr {A}, \end{array}\right. \end{aligned}$$

(8)

for $\texttt{t} > 0$ under ICs

$$\begin{aligned} \mathscr {S}(0)=\mathscr {S}_0>0,\quad \mathscr {E}(0) = \mathscr {E}_0>0,\quad \mathscr {I}(0)=\mathscr {I}_0>0,\quad \mathscr {A}(0)=\mathscr {A}_0> 0,\quad \mathscr {R}(0)=\mathscr {R}_0>0. \end{aligned}$$

Lemma 3.1

Let closed set is expressed by

$$\begin{aligned} \tilde{ \uppsi } = \left\{ (\mathscr {S},\mathscr {E},\mathscr {I},\mathscr {A},\mathscr {R})\in \mathbb {R}^+_5 \,:\, \aleph (\texttt{t}) = \mathscr {S}+ \mathscr {E}+ \mathscr {I}+ \mathscr {A}+ \mathscr {R} \le \frac{\varLambda }{\upsigma } \right\} . \end{aligned}$$

Then $\tilde{ \uppsi }$ is positively invariant with respect to system (8).

Proof

Add all of the relations in system (8) to get

$$\begin{aligned} \uptheta ^{\upkappa -1}{\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }} \aleph (\texttt{t})&= \uptheta ^{\upkappa -1} \left( {\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {S} + {\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {E} + {\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {I} + {\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }} \mathscr {A} + {\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {R}\right) = \varLambda -\upsigma \aleph (\texttt{t})-\updelta \mathscr {I}. \end{aligned}$$

Thus,

$$\begin{aligned} \uptheta ^{\upkappa -1}{\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }} \aleph (\texttt{t}) \le \varLambda - \upsigma \aleph (\texttt{t})\Rightarrow {\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }} \aleph (\texttt{t}) \le \uptheta ^{1-\upkappa } \varLambda - \uptheta ^{1-\upkappa }\upsigma \aleph (\texttt{t}). \end{aligned}$$

By applying⁴², Theorem 7.2, we conclude

$$\begin{aligned} \aleph (\texttt{t}) \le \aleph (0)\mathscr {E}_\upkappa (-\upsigma \uptheta ^{1 - \upkappa }\texttt{t}^{\upkappa }) + \int _{0}^{\texttt{t}} \varLambda \uptheta ^{1-\upkappa }\rho ^{\upkappa -1} \mathscr {E}_{\upkappa ,\upkappa }(-\upsigma \uptheta ^{1-\upkappa } \rho ^{\upkappa })\text {d}\rho . \end{aligned}$$

Hence,

$$\begin{aligned} \aleph (\texttt{t})&\le \aleph (0) \mathscr {E}_\upkappa (-\upsigma \uptheta ^{1-\upkappa }\texttt{t}^{\upkappa }) + \int _{0}^{\texttt{t}} \varLambda \uptheta ^{1-\upkappa } \rho ^{\upkappa -1} \sum _{\mathfrak {i}=0}^{\infty } \frac{(-1)^{\mathfrak {i}} \upsigma ^{\mathfrak {i}} \uptheta ^{(1-\upkappa ) \mathfrak {i}} \rho ^{\mathfrak {i}\upkappa }}{ \Gamma (\mathfrak {i} \upkappa + \upkappa )}\text {d}\rho \\&= \aleph (0)\mathscr {E}_\upkappa (-\upsigma \uptheta ^{1-\upkappa } \texttt{t}^{\upkappa }) + \varLambda \uptheta ^{1-\upkappa } \sum _{\mathfrak {i}=0}^{\infty } \frac{(-1)^{\mathfrak {i}} \upsigma ^{\mathfrak {i}} \uptheta ^{(1-\upkappa ) \mathfrak {i}}\texttt{t}^{\mathfrak {i}\upkappa + \upkappa }}{\Gamma (\mathfrak {i}\upkappa +\upkappa +1)}\\&= \aleph (0) \mathscr {E}_\upkappa (-\upsigma \uptheta ^{1-\upkappa }\texttt{t}^{\upkappa }) - \frac{\varLambda }{\upsigma }\sum _{\mathfrak {i}=0}^{ \infty } \frac{(-1)^{\mathfrak {i}} \upsigma ^{\mathfrak {i}} \uptheta ^{(1-\upkappa ) \mathfrak {i}} \texttt{t}^{\mathfrak {i} \upkappa }}{\Gamma (\mathfrak {i} \upkappa +1)}\\&=\aleph (0) \mathscr {E}_\upkappa (-\upsigma \uptheta ^{1- \upkappa }\texttt{t}^{\upkappa }) - \frac{\varLambda }{\upsigma } \left( \mathscr {E}_{ \upkappa } \left( -\upsigma \uptheta ^{ 1 -\upkappa }\texttt{t}^{\upkappa -1} \right) \right) = \frac{\varLambda }{\upsigma }+\mathscr {E}_\upkappa (-\upsigma \uptheta ^{1-\upkappa }\texttt{t}^{\upkappa })\bigg (\aleph (0)-\frac{\varLambda }{\upsigma }\bigg ). \end{aligned}$$

Thus, if $\aleph (0) \le \frac{\varLambda }{\upsigma }$, then $\aleph (\texttt{t}) \le \frac{\varLambda }{\upsigma }$ for each positive real number $\texttt{t}$. This completes the proof. $\square$

Equilibrium points and stability

Equilibrium points (EPs) of system (8) can be determined by solving the following equations.

$$\begin{aligned} \left\{ \begin{array}{l} \varLambda -\upbeta _1\mathscr {S} \mathscr {I} - \upbeta _2 \mathscr {S} \mathscr {A} - \upsigma \, \mathscr {S} + \varsigma \mathscr {R}=0,\\ \upbeta _1\mathscr {S} \mathscr {I} + \upbeta _2\mathscr {S} \mathscr {A} - (\upupsilon + \upsigma ) \mathscr {E} =0,\\ \mathfrak {p} \upupsilon \mathscr {E} - (\upalpha +\upsigma +\updelta )\mathscr {I} =0,\\ (1-\mathfrak {p})\upupsilon \mathscr {E} - (\upgamma +\upsigma ) \mathscr {A}=0,\\ \upalpha \mathscr {I}+ \upgamma \mathscr {A} -(\upsigma + \varsigma ) \mathscr {R}=0. \end{array}\right. \end{aligned}$$

(9)

Clearly, whenever there is no spread of the disease; i.e., $\mathscr {I}=0$, then a disease-free equilibrium (DFE) is occurred. Hence, the DFE point is obtained as $\mathscr {E}_0 = (\frac{\varLambda }{\upsigma },0,0,0, 0 )$. If $\breve{\mathscr {R}}_0>1$, one can find others EPs of the model by solving (9). Therefore, we obtain the endemic EPs of the model whenever $\mathscr {S}$, $\mathscr {E}$, $\mathscr {I}$, $\mathscr {A}$, $\mathscr {R}$ is against zero, and it is in the form: $\mathscr {E}_1 = \left( \mathscr {S}^{\star },\mathscr {E}^{\star },\mathscr {I}^{\star },\mathscr {A}^{\star },\mathscr {R}^{\star } \right)$, where

$$\begin{aligned} \mathscr {S}^{\star }&=\frac{(\upupsilon + \upsigma ) ( \upalpha + \upsigma + \updelta )( \upgamma + \upsigma )}{\upbeta _1 \mathfrak {p} \upupsilon ( \upgamma +\upsigma ) +\upbeta _2(1-\mathfrak {p})\upupsilon (\upalpha +\upsigma +\updelta )},\\ \mathscr {E}^{\star }&=\frac{(\varLambda -\upsigma \mathscr {S}^{\star })(\upalpha +\upsigma +\updelta )(\upgamma +\upsigma ) (\upsigma +\varsigma )}{(\upupsilon +\upsigma )(\upalpha +\upsigma +\updelta )(\upgamma +\upsigma )(\upsigma +\varsigma ) -\mathfrak {p}\upupsilon \upalpha \varsigma (\upgamma +\upsigma )-(1-\mathfrak {p})\upupsilon \upgamma \varsigma (\upalpha +\upsigma +\updelta )},\\ \mathscr {I}^{\star }&=\frac{\mathfrak {p} \upupsilon \mathscr {E}^{\star }}{(\upalpha +\upsigma +\updelta )}, \quad \mathscr {A}^{\star } =\frac{(1-\mathfrak {p}) \upupsilon \mathscr {E}^{\star }}{(\upgamma +\upsigma )}, \quad \mathscr {R}^{\star } =\frac{\upalpha \mathscr {I}^{\star }+\upgamma \mathscr {A}^{\star }}{\upsigma +\varsigma }. \end{aligned}$$

In order to find $\breve{\mathscr {R}}_0$, the system is considered as ${{}^{\text {C}\!}\mathscr {D}}^{\upkappa ; {\uppsi }}\varPhi (\texttt{t}) =\mathfrak {F}(\varPhi (\texttt{t}))-\mathfrak {V}(\varPhi (\texttt{t}))$, where

$$\begin{aligned} \mathfrak {F}(\varPhi (\texttt{t})) = \uptheta ^{1-\upkappa } \begin{pmatrix} (\upbeta _1\mathscr {I}+\upbeta _2\mathscr {A})\mathscr {S}\\ 0\\ 0\\ \end{pmatrix},\qquad \mathfrak {V}(\varPhi (\texttt{t})) = \uptheta ^{1-\upkappa }\begin{pmatrix} (\upupsilon +\upsigma )\mathscr {E}\\ (\upalpha +\upsigma +\updelta )\mathscr {I} -\mathfrak {p}\upupsilon \mathscr {E} \\ (\upgamma +\upsigma )\mathscr {A} -(1-\mathfrak {p})\upupsilon \mathscr {E} \\ \end{pmatrix}. \end{aligned}$$

At $\mathscr {E}_0$, the Jacobian matrix for $\mathfrak {F}$ and $\mathfrak {V}$ is gotten as

$$\begin{aligned} \mathscr {J}_{\mathfrak {F}} ( \mathscr {E}_0) =\uptheta ^{1-\upkappa } \begin{pmatrix} 0&{} \upbeta _1\frac{\varLambda }{\upsigma }&{} \upbeta _2\frac{\varLambda }{\upsigma }\\ 0&{} 0&{} 0\\ 0&{} 0&{} 0\\ \end{pmatrix},\qquad \mathscr {J}_{\mathfrak {V}}(\mathscr {E}_0) = \uptheta ^{1-\upkappa } \begin{pmatrix} \upupsilon +\upsigma &{} 0&{} 0\\ -\mathfrak {p}\upupsilon &{} \upalpha +\upsigma +\updelta &{} 0\\ -(1-\mathfrak {p})\upupsilon &{} 0&{} \upgamma +\upsigma \\ \end{pmatrix}. \end{aligned}$$

The next-generation matrix for system (8) is as

$$\begin{aligned} \mathscr{F}\mathscr{V}^{-1}=\begin{pmatrix} \frac{\upbeta _1\frac{\varLambda }{\upsigma }\mathfrak {p}\upupsilon }{(\upupsilon +\upsigma ) (\upalpha +\upsigma +\updelta )} +\frac{\upbeta _2\frac{\varLambda }{\upsigma } (1-\mathfrak {p})\upupsilon }{(\upupsilon +\upsigma )(\upgamma +\upsigma )}&{} \frac{\upbeta _1\frac{\varLambda }{\upsigma }}{\upalpha +\upsigma +\updelta }&{} \frac{\upbeta _2\frac{\varLambda }{\upsigma }}{\upgamma +\upsigma }\\ 0&{} 0&{} 0\\ 0&{} 0&{} 0\\ \end{pmatrix}, \end{aligned}$$

and the reproduction number is gotten from $\breve{\mathscr {R}}_0=\uprho (\mathscr{F}\mathscr{V}^{-1})$. Therefore,

$$\begin{aligned} \breve{\mathscr {R}}_0=\frac{\mathfrak {p}\upupsilon \upbeta _1\varLambda }{\upsigma (\upupsilon +\upsigma ) (\upalpha +\upsigma +\updelta )}+\frac{(1-\mathfrak {p})\upupsilon \upbeta _2\varLambda }{\upsigma (\upupsilon +\upsigma )(\upgamma +\upsigma )}. \end{aligned}$$

Theorem 3.2

The DFE point $\mathscr {E}_0$ of (8) is locally asymptotically stable if $\breve{\mathscr {R}}_0 < 1$ and it is unstable if $\breve{\mathscr {R}}_0 >1$.

Proof

The Jacobian matrix of (8) at $\mathscr {E}_0=\left( \mathscr {S}^{0},0,0,0,0 \right)$ is

$$\begin{aligned} \mathscr {J}(\mathscr {E}_0)=\uptheta ^{1-\upkappa }\begin{pmatrix} -\upsigma &{} 0&{} -\upbeta _1\mathscr {S}^{0}&{} -\upbeta _2\mathscr {S}^{0} &{} \varsigma \\ 0&{} -(\upupsilon +\upsigma )&{} \upbeta _1\mathscr {S}^{0}&{} \upbeta _2\mathscr {S}^{0} &{} 0 \\ 0&{} \mathfrak {p}\upupsilon &{} -(\upalpha +\upsigma +\updelta ) &{} 0 &{} 0 \\ 0&{} (1-\mathfrak {p})\upupsilon &{} 0 &{} -(\upgamma +\upsigma ) &{} 0 \\ 0&{} 0&{} \upalpha &{} \upgamma &{} -(\upsigma +\varsigma ) \\ \end{pmatrix}, \end{aligned}$$

where $\mathscr {S}^{0}=~\frac{\varLambda }{\upsigma }$. The characteristic equation $|\mathscr {J}(\mathscr {E}_0)-\uplambda \mathscr {I}|=0$, has two eigenvalues $\uplambda _1 = - \upsigma$ and $\uplambda _2 = - ( \upsigma + \varsigma )$. The remaining three eigenvalues are those of the $3 \times 3$ matrix

$$\begin{aligned} \mathscr {J}_1=\uptheta ^{1-\upkappa } \begin{pmatrix} -(\upupsilon +\upsigma )&{} \upbeta _1\mathscr {S}^{0}&{} \upbeta _2\mathscr {S}^{0}\\ \mathfrak {p}\upupsilon &{} -(\upalpha +\upsigma +\updelta ) &{} 0 \\ (1-\mathfrak {p})\upupsilon &{} 0 &{} -(\upgamma +\upsigma ) \\ \end{pmatrix}. \end{aligned}$$

The characteristic equation takes the form

$$\begin{aligned} \begin{vmatrix} -(\upupsilon +\upsigma +\uplambda )&\upbeta _1\mathscr {S}^{0}&\upbeta _2\mathscr {S}^{0}\\ \mathfrak {p}\upupsilon&-(\upalpha +\upsigma +\updelta +\uplambda )&0 \\ (1-\mathfrak {p})\upupsilon&0&-(\upgamma +\upsigma +\uplambda ) \\ \end{vmatrix}=0. \end{aligned}$$

Thus,

$$\begin{aligned} (\upupsilon +\upsigma +\uplambda ) (\upalpha +\upsigma +\updelta +\uplambda )(\upgamma +\upsigma +\uplambda ) -\upbeta _2 \mathscr {S}^{0}(1-\mathfrak {p})\upupsilon (\upalpha +\upsigma +\updelta +\uplambda ) -\upbeta _1\mathscr {S}^{0}\mathfrak {p}\upupsilon (\upgamma +\upsigma +\uplambda )=0. \end{aligned}$$

Therefore, $\mathscr {P}(\uplambda ) : =\uplambda ^3+\mathfrak {b}_1\uplambda ^2+\mathfrak {b}_2\uplambda +\mathfrak {b}_3=0$, where $\mathfrak {b}_1=\upupsilon +\upsigma + \upalpha + \upsigma + \upgamma + \updelta + \upsigma$,

$$\begin{aligned} \mathfrak {b}_2&=(\upalpha +\upsigma +\updelta )(\upupsilon +\upsigma )+(\upalpha +\upsigma +\updelta ) (\upgamma +\upsigma )+(\upgamma +\upsigma )(\upupsilon +\upsigma ) -\upbeta _2\mathscr {S}^{0}(1-\mathfrak {p})\upupsilon -\upbeta _1\mathscr {S}^{0}\mathfrak {p}\upupsilon ,\\ \mathfrak {b}_3&=(\upalpha +\upsigma +\updelta )(\upupsilon +\upsigma )(\upgamma +\upsigma ) -(\upalpha +\upsigma +\updelta )\upbeta _2\mathscr {S}^{0}(1-\mathfrak {p})\upupsilon -(\upgamma +\upsigma )\upbeta _1\mathscr {S}^{0}\mathfrak {p}\upupsilon . \end{aligned}$$

It is clear that if $\mathfrak {b}_3$ s greater than zero, it means that $\breve{\mathscr {R}}_0$ is less than one. In addition, if the value of $\breve{\mathscr {R}}_0$ is greater than 1, then $\mathfrak {b}_3$ is less than zero.

From $\lim _{\uplambda \rightarrow \infty } \mathscr {P}(\uplambda )=\infty$, one can conclude that $\mathscr {P}(\uplambda ) = 0$ has a real positive solution, and the EP $\mathscr {E}_0$ is thus unstable. The EP $\mathscr {E}_0$ is locally asymptotically stable whenever $\breve{\mathscr {R}}_0 < 1$. Specifically, we need to prove that in this situation, the equation $\mathscr {P}(\uplambda ) = 0$ only has solutions that are negative or have a negative real part by employing the Routh–Hurwitz criteria. It follows from $\breve{\mathscr {R}}_0 < 1$ that $\mathfrak {b}_3 > 0$. Expressly, $\mathfrak {b}_1 > 0$. The condition $\breve{\mathscr {R}}_0 < 1$ all so gives

$$\begin{aligned} (\upupsilon +\upsigma )(\upgamma +\upsigma )>\upbeta _2 \mathscr {S}^{0}(1-\mathfrak {p})\upupsilon , \qquad (\upalpha +\upsigma +\updelta )(\upupsilon +\upsigma )>\upbeta _1 \mathscr {S}^{0} \mathfrak {p}\upupsilon . \end{aligned}$$

Hence, $\mathfrak {b}_2>0$. Since $(\upupsilon +\upsigma )(\upgamma +\upsigma )>\upbeta _2 \mathscr {S}^{0} (1-\mathfrak {p})\upupsilon$ and $(\upalpha +\upsigma +\updelta )(\upupsilon +\upsigma )> \upbeta _1 \mathscr {S}^{0} \mathfrak {p}\upupsilon$, we get

$$\begin{aligned} \mathfrak {b}_1\mathfrak {b}_2 > (\upupsilon +\upsigma )(\upgamma +\upsigma )(\upalpha +\upsigma +\updelta ). \end{aligned}$$

On the other hand, $\mathfrak {b}_3<(\upupsilon +\upsigma )(\upgamma +\upsigma )(\upalpha +\upsigma +\updelta )$ and therefore, $\mathfrak {b}_1\mathfrak {b}_2>(\upupsilon +\upsigma )(\upgamma +\upsigma )(\upalpha +\upsigma +\updelta )>\mathfrak {b}_3.$ The Routh–Hurwitz yardstick then connotes that the EP $\mathscr {E}_0$ is locally asymptotically stable whenever $\breve{\mathscr {R}}_0$ is less than 1. $\square$

Theorem 3.3

The DFE point $\mathscr {E}_0$ of (8) is globally asymptotically stable if $\breve{\mathscr {R}}_0 \le 1$.

Proof

To display the result, define a Lyapunov function as $\mathfrak {L}(\texttt{t})=\vartheta _1\mathscr {E}(\texttt{t})+\vartheta _2\mathscr {I}(\texttt{t}) +\vartheta _3\mathscr {A}(\texttt{t})$, where $\vartheta _1$,$\vartheta _2$ and $\vartheta _3$ are positive constants. The $\uppsi$-CFD of the Lyapunov function is given by,

$$\begin{aligned} {{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathfrak {L}(\texttt{t})=\vartheta _1{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {E}(\texttt{t})+\vartheta _2{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {I}(\texttt{t})+\vartheta _3{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }} \mathscr {A}(\texttt{t}). \end{aligned}$$

Apply system (8), to get

$$\begin{aligned} {{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathfrak {L}(\texttt{t})&= \vartheta _1 \left\{ \upbeta _1\mathscr {S}\mathscr {I}+\upbeta _2\mathscr {S}\mathscr {A}-(\upupsilon +\upsigma )\mathscr {E}\right\} \\&\quad + \vartheta _2\left\{ \mathfrak {p}\upupsilon \mathscr {E}-(\upalpha +\upsigma +\updelta )\mathscr {I}\right\} + \vartheta _3 \left\{ (1-\mathfrak {p})\upupsilon \mathscr {E}-(\upgamma +\upsigma )\mathscr {A} \right\} \\&= \left[ \vartheta _1\upbeta _2\dfrac{\varLambda }{\upsigma }-\vartheta _3(\upgamma +\upsigma )\right] \mathscr {A} +\left[ \vartheta _1\upbeta _1\frac{\varLambda }{\upsigma }-\vartheta _2(\upalpha +\upsigma +\updelta ) \right] \mathscr {I} +\big [\vartheta _3(1-\mathfrak {p})\upupsilon + \vartheta _2\mathfrak {p}\upupsilon - \vartheta _1 (\upupsilon +\upsigma )\big ]\mathscr {E}, \end{aligned}$$

where $\vartheta _1=(\upalpha +\upsigma +\updelta )(\upgamma +\upsigma )$, $\vartheta _2=\upbeta _1\frac{\varLambda }{\upsigma }(\upgamma +\upsigma )$ and $\vartheta _3 = \upbeta _2\frac{\varLambda }{\upsigma }(\upalpha +\upsigma +\updelta )$. Therefore,

$$\begin{aligned} {{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathfrak {L}(\texttt{t})= (\upalpha +\upsigma +\updelta )(\upupsilon +\upsigma )(\upgamma +\upsigma )[\breve{\mathscr {R}}_0-1]\mathscr {E}. \end{aligned}$$

Thus, if $\breve{\mathscr {R}}_0 \le 1$, then ${{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathfrak {L}(\texttt{t})\le 0$. So, the DFE point of (8) is globally asymptotically stable whenever $\breve{\mathscr {R}}_0 \le 1$. $\square$

$\breve{\mathscr {R}}_0$ sensitivity analysis

To study the $\breve{\mathscr {R}}_0$ sensitivity, we find the derivatives of it in the following way:

$$\begin{aligned} \frac{\partial \breve{\mathscr {R}}_0 }{\partial \upbeta _1}&=\frac{\mathfrak {p}\upupsilon \varLambda }{\upsigma (\upupsilon +\upsigma )(\upalpha +\upsigma +\updelta )},\qquad \frac{\partial \breve{\mathscr {R}}_0 }{\partial \upbeta _2}=\frac{(1-\mathfrak {p})\upupsilon \varLambda }{\upsigma (\upupsilon +\upsigma )(\upgamma +\upsigma )},\\ \frac{\partial \breve{\mathscr {R}}_0 }{\partial \varLambda }&=\frac{\mathfrak {p}\upupsilon \upbeta _1}{\upsigma (\upupsilon +\upsigma )(\upalpha +\upsigma +\updelta )} +\frac{(1-\mathfrak {p})\upupsilon \upbeta _2}{\upsigma (\upupsilon +\upsigma )(\upgamma +\upsigma )},\\ \frac{\partial \breve{\mathscr {R}}_0 }{\partial \upupsilon }&=\frac{\mathfrak {p}\upbeta _1\varLambda \upsigma }{\upsigma (\upupsilon +\upsigma )^2(\upalpha +\upsigma +\updelta )} +\frac{(1-\mathfrak {p})\upbeta _2\varLambda \upsigma }{\upsigma (\upgamma +\upsigma )(\upupsilon +\upsigma )^2},\\ \frac{\partial \breve{\mathscr {R}}_0 }{\partial \upalpha }&=-\frac{\mathfrak {p} \upupsilon \upbeta _1\varLambda }{\upsigma (\upupsilon +\upsigma )(\upalpha +\upsigma +\updelta )^2},\qquad \frac{\partial \breve{\mathscr {R}}_0 }{\partial \updelta }=-\frac{\mathfrak {p}\upupsilon \upbeta _1 \varLambda }{\upsigma (\upupsilon +\upsigma )(\upalpha +\upsigma +\updelta )^2},\\ \frac{\partial \breve{\mathscr {R}}_0 }{\partial \upgamma }&=-\frac{(1-\mathfrak {p})\upupsilon \upbeta _2\varLambda }{\upsigma (\upupsilon +\upsigma )(\upgamma +\upsigma )^2},\\ \frac{\partial \breve{\mathscr {R}}_0 }{\partial \upsigma }&=-\frac{(\upupsilon \upalpha +2\upsigma \upupsilon +2\upsigma \upalpha +\upupsilon \updelta +3\upsigma ^2+2\upsigma \updelta )(\mathfrak {p}\upupsilon \upbeta _1 \varLambda )}{(\upsigma (\upupsilon +\upsigma )(\upalpha +\upsigma +\updelta ))^2} \\&\quad -\frac{(\upupsilon \upgamma +2\upsigma \upupsilon +2\upsigma \upgamma +3\upsigma ^2)(1-\mathfrak {p}) \upupsilon \upbeta _2\varLambda }{(\upsigma (\upupsilon +\upsigma )(\upgamma +\upsigma ))^2}. \end{aligned}$$

Since all the parameters are positive, so

$$\begin{aligned} \frac{\partial \breve{\mathscr {R}}_0}{\partial \upbeta _1}>0,\ \frac{\partial \breve{\mathscr {R}}_0}{\partial \upbeta _2}>0, \ \frac{\partial \breve{\mathscr {R}}_0}{\partial \varLambda }>0, \ \frac{\partial \breve{\mathscr {R}}_0}{\partial \upupsilon }>0, \ \frac{\partial \breve{\mathscr {R}}_0}{\partial \upalpha }<0, \ \frac{\partial \breve{\mathscr {R}}_0}{\partial \updelta }<0, \ \frac{\partial \breve{\mathscr {R}}_0}{\partial \upgamma }<0, \ \frac{\partial \breve{\mathscr {R}}_0}{\partial \upsigma }<0. \end{aligned}$$

In this way, $\breve{\mathscr {R}}_0$ is increasing with $\upbeta _1$, $\upbeta _2$, $\varLambda$, $\upupsilon$, but it is decreasing with $\upalpha$, $\updelta$, $\upgamma$, $\upsigma$.

Existence and uniqueness of solution

It will be shown here that the system with the IC has a unique solution. To begin with, we compose system (8) as takes after:

$$\begin{aligned} &\uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {S}=\mathscr {H}_1 \left( \texttt{t},\mathscr {S} \right) ,\qquad \uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {E}=\mathscr {H}_2(\texttt{t},\mathscr {E}),\qquad \uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {I} = \mathscr {H}_3(\texttt{t},\mathscr {I}),\\&\uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {A}=\mathscr {H}_4(\texttt{t},\mathscr {A}),\qquad \uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }}\mathscr {R}=\mathscr {H}_5(\texttt{t},\mathscr {R}, \end{aligned}$$

where

$$\begin{aligned} \mathscr {H}_1(\texttt{t},\mathscr {S})&= \varLambda - \upbeta _1 \mathscr {S}(\texttt{t})\mathscr {I} - \upbeta _2 \mathscr {S} \mathscr {A} - \upsigma \, \mathscr {S} + \varsigma \mathscr {R},\\ \mathscr {H}_2(\texttt{t},\mathscr {E})&= \upbeta _1 \mathscr {S}( \texttt{t})\mathscr {I} + \upbeta _2 \mathscr {S} \mathscr {A} - (\upupsilon + \upsigma ) \mathscr {E},\\ \mathscr {H}_3(\texttt{t}, \mathscr {I})&= \mathfrak {p} \upupsilon \mathscr {E} - (\upalpha + \upsigma +\updelta ) \mathscr {I},\\ \mathscr {H}_4(\texttt{t},\mathscr {A})&= ( 1- \mathfrak {p} ) \upupsilon \mathscr {E} - (\upgamma + \upsigma ) \mathscr {A},\\ \mathscr {H}_5(\texttt{t},\mathscr {S})&= \upalpha \mathscr {R} + \upgamma \mathscr {A} - (\upsigma + \varsigma ) \mathscr {R}. \end{aligned}$$

By taking integral, we have

$$\begin{aligned} {\left\{ \begin{array}{ll} \displaystyle \mathscr {S}(\texttt{t})-\mathscr {S}(0) = \uptheta ^{1-\upkappa } \int _{0}^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{(\tilde{\uppsi }_{\xi }(\texttt{t}) )^{1-\upkappa }} \frac{\mathscr {H}_1(\xi ,\mathscr {S})}{\Gamma (\upkappa )} \, \text {d}\xi ,\\ \displaystyle \mathscr {E}(\texttt{t}) - \mathscr {E}(0) = \uptheta ^{1-\upkappa } \int _{0}^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{ (\tilde{\uppsi }_{\xi }(\texttt{t}))^{1-\upkappa }} \frac{\mathscr {H}_2(\xi ,\mathscr {E})}{\Gamma (\upkappa )} \, \text {d}\xi ,\\ \displaystyle \mathscr {I}(\texttt{t})-\mathscr {I}(0) = \uptheta ^{1-\upkappa } \int _{0}^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{(\tilde{\uppsi }_{\xi }(\texttt{t}) )^{1-\upkappa }} \frac{\mathscr {H}_3(\xi ,\mathscr {I})}{\Gamma (\upkappa )} \, \text {d}\xi ,\\ \displaystyle \mathscr {A} ( \texttt{t})-\mathscr {A}(0)=\uptheta ^{1-\upkappa } \int _{0}^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{ (\tilde{\uppsi }_{\xi }(\texttt{t}))^{1 - \upkappa }}\frac{\mathscr {H}_4(\xi ,\mathscr {A})}{\Gamma (\upkappa )} \, \text {d}\xi ,\\ \displaystyle \mathscr {R}(\texttt{t})-\mathscr {R}(0)= \uptheta ^{1-\upkappa } \int _{0}^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{(\tilde{\uppsi }_{\xi }(\texttt{t}))^{1-\upkappa }} \frac{\mathscr {H}_5(\xi ,\mathscr {R})}{\Gamma (\upkappa )} \, \text {d}\xi . \end{array}\right. } \end{aligned}$$

(10)

Theorem 4.1

The kernel of the model will satisfy the Lipschitz condition (LC) if the disparity $0 \le \mathfrak {g}_\imath <1$, $1\le \imath \le 5$, hold, where $\mathfrak {g}_1=\upbeta _1\mathfrak {r}_1+\upbeta _2\mathfrak {r}_2+\upsigma$, $\mathfrak {g}_2=\upupsilon +\upsigma$, $\mathfrak {g}_3=\upalpha +\upsigma +\updelta$, $\mathfrak {g}_4=\upgamma +\upsigma$ and $\mathfrak {g}_5 = \upsigma +\varsigma$.

Proof

We will show for the first part and do the same for the rest. Consider functions $\mathscr {S}(\texttt{t})$ and $\mathscr {S}_1(\texttt{t})$. Then,

$$\begin{aligned} \Vert \mathscr {H}_1(\texttt{t},\mathscr {S}) - \mathscr {H}_1(\texttt{t},\mathscr {S}_1)\Vert&=\Vert -(\upbeta _1\mathscr {I} + \upbeta _2\mathscr {A} ) (\mathscr {S} - \mathscr {S}_1) - \upsigma (\mathscr {S}-\mathscr {S}_1 ) \Vert \\&\le \Vert \upbeta _1\mathscr {I} + \upbeta _2\mathscr {A} \Vert \, \Vert \mathscr {S} - \mathscr {S}_1\Vert + \upsigma \Vert \mathscr {S} - \mathscr {S}_1 \Vert \\&\le (\upbeta _1\Vert \mathscr {I} \Vert + \upbeta _2 \Vert \mathscr {A} \Vert + \upsigma ) \Vert \mathscr {S} - \mathscr {S}_1\Vert \\&\le ( \upbeta _1 \mathfrak {r}_1 + \upbeta _2\mathfrak {r}_2 + \upsigma ) \Vert \mathscr {S} - \mathscr {S}_1\Vert , \end{aligned}$$

where $\Vert \mathscr {I} \Vert \le \mathfrak {r}_1$, $\Vert \mathscr {A} \Vert \le \mathfrak {r}_2$. In a similar manner, we get

$$\begin{aligned} \Vert \mathscr {H}_2(\texttt{t},\mathscr {E})-\mathscr {H}_2(\texttt{t},\mathscr {E}_1)\Vert \le \mathfrak {g}_2 \Vert \mathscr {E} - \mathscr {E}_1\Vert ,\qquad \Vert \mathscr {H}_3(\texttt{t}, \mathscr {I}) - \mathscr {H}_3 ( \texttt{t},\mathscr {I}_1) \Vert \le \mathfrak {g}_3 \Vert \mathscr {I} - \mathscr {I}_1\Vert ,\\ \Vert \mathscr {H}_4(\texttt{t},\mathscr {A})-\mathscr {H}_4(\texttt{t},\mathscr {A}_1 ) \Vert \le \mathfrak {g}_4 \Vert \mathscr {A} - \mathscr {A}_1 \Vert ,\qquad \Vert \mathscr {H}_5(\texttt{t}, \mathscr {R}) - \mathscr {H}_5 ( \texttt{t},\mathscr {R}_1)\Vert \le \mathfrak {g}_5 \Vert \mathscr {R} - \mathscr {R}_1 \Vert . \end{aligned}$$

(11)

$\square$

Let’s look at some recursive versions of system (10),

$$\begin{aligned} \digamma _{1\texttt{n}}(\texttt{t})&= \mathscr {S}_\texttt{n} - \mathscr {S}_{\texttt{n}-1} = \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )} \int _{0}^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{ (\tilde{\uppsi }_{\xi }(\texttt{t}) )^{ 1 - \upkappa }}( \mathscr {H}_1(\xi , \mathscr {S}_{ \texttt{n}-1}) - \mathscr {H}_1(\xi ,\mathscr {S}_{\texttt{n}-2}))\, \text {d}\xi ,\\ \digamma _{2\texttt{n}}(\texttt{t})&= \mathscr {E}_\texttt{n} - \mathscr {E}_{\texttt{n}-1} = \frac{\uptheta ^{ 1 - \upkappa }}{\Gamma (\upkappa )} \int _{0}^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{(\tilde{\uppsi }_{\xi }(\texttt{t}))^{ 1 - \upkappa }}( \mathscr {H}_2(\xi , \mathscr {E}_{\texttt{n}-1}) - \mathscr {H}_2(\xi ,\mathscr {E}_{\texttt{n}-2}))\, \text {d}\xi ,\\ \digamma _{3\texttt{n}}(\texttt{t})&= \mathscr {I}_\texttt{n} - \mathscr {I}_{\texttt{n}-1} = \frac{\uptheta ^{1-\upkappa }}{ \Gamma (\upkappa )} \int _{0}^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{(\tilde{\uppsi }_{\xi }(\texttt{t}))^{ 1 - \upkappa }}(\mathscr {H}_3( \xi ,\mathscr {I}_{ \texttt{n}-1})-\mathscr {H}_3(\xi ,\mathscr {I}_{\texttt{n}-2}))\, \text {d}\xi ,\\ \digamma _{4\texttt{n}}(\texttt{t})&=\mathscr {A}_\texttt{n} - \mathscr {A}_{\texttt{n}-1} = \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )} \int _{0}^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{ (\tilde{\uppsi }_{\xi }(\texttt{t}) )^{ 1 - \upkappa }}(\mathscr {H}_4(\xi ,\mathscr {A}_{\texttt{n}-1})-\mathscr {H}_4(\xi ,\mathscr {A}_{\texttt{n}-2}))\, \text {d}\xi ,\\ \digamma _{5\texttt{n}}(\texttt{t})&=\mathscr {R}_\texttt{n} - \mathscr {R}_{\texttt{n}-1} = \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\int _{0 }^{\texttt{t}} \frac{\uppsi ^\prime (\xi )}{(\tilde{\uppsi }_{\xi }(\texttt{t}))^{ 1 - \upkappa }}(\mathscr {H}_5(\xi ,\mathscr {R}_{\texttt{n}-1})-\mathscr {H}_5(\xi ,\mathscr {R}_{\texttt{n}-2}))\, \text {d}\xi , \end{aligned}$$

with the ICs $\mathscr {S}_0(\texttt{t})=\mathscr {S}(0)$, $\mathscr {E}_0(\texttt{t})=\mathscr {E}(0)$, $\mathscr {I}_0(\texttt{t})=\mathscr {I}(0)$, $\mathscr {A}_0(\texttt{t})=\mathscr {A}(0)$, and $\mathscr {R}_0(\texttt{t})=~\mathscr {R}(0)$. Hence,

$$\begin{aligned} \Vert \digamma _{1\texttt{n}}(\texttt{t})\Vert = \Vert \mathscr {S}_{\texttt{n}} - \mathscr {S}_{\texttt{n}-1} \Vert&=\left\| \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\int _{0}^{ \texttt{t}} \frac{ \uppsi ^\prime (\xi )}{(\tilde{\uppsi }_{\xi }(\texttt{t}))^{1-\upkappa }}(\mathscr {H}_1(\xi ,\mathscr {S}_{\texttt{n}-1}) -\mathscr {H}_1(\xi ,\mathscr {S}_{\texttt{n}-2}))\, \text {d}\xi \right\| \\&\le \frac{ \uptheta ^{1-\upkappa }}{ \Gamma (\upkappa )}\int _{0}^{\texttt{t}} \left\| \frac{ \uppsi ^\prime (\xi )}{(\tilde{\uppsi }_{\xi }(\texttt{t}))^{1-\upkappa }}(\mathscr {H}_1(\xi , \mathscr {S}_{\texttt{n}-1})-\mathscr {H}_1(\xi ,\mathscr {S}_{\texttt{n}-2}))\right\| \, \text {d}\xi . \end{aligned}$$

With LC (11),

$$\begin{aligned} \Vert \digamma _{1\texttt{n}}(\texttt{t})\Vert \le \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )} \mathfrak {g}_1\int _{0}^{\texttt{t}}\Vert \digamma _{1(\texttt{n}-1)}(\xi )\Vert \, \text {d}\xi . \end{aligned}$$

(12)

In a similar manner,

$$\begin{aligned} \Vert \digamma _{2\texttt{n}}\Vert&\le \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )} \mathfrak {g}_2\int _{0}^{\texttt{t}}\Vert \digamma _{2(\texttt{n}-1)}(\xi )\Vert \, \text {d}\xi ,\qquad \Vert \digamma _{3\texttt{n}} \Vert \le \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_3\int _{0}^{\texttt{t}}\Vert \digamma _{3(\texttt{n}-1)}(\xi )\Vert \, \text {d}\xi ,\\ \Vert \digamma _{4\texttt{n}} \Vert&\le \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_4\int _{0}^{\texttt{t}}\Vert \digamma _{4(\texttt{n}-1)}(\xi )\Vert \, \text {d}\xi ,\qquad \Vert \digamma _{5\texttt{n}} \Vert \le \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_5\int _{0}^{\texttt{t}}\Vert \digamma _{5(\texttt{n}-1)}(\xi )\Vert \, \text {d}\xi . \end{aligned}$$

(13)

Thus,

$$\begin{aligned} \mathscr {S}_\texttt{n}(\texttt{t}) = \sum _{\mathfrak {i}=1}^{\texttt{n}}\digamma _{1\mathfrak {i}}(\texttt{t}), \mathscr {E}_\texttt{n}(\texttt{t})=\sum _{\mathfrak {i}=1}^{\texttt{n}}\digamma _{2\mathfrak {i}}(\texttt{t}), \mathscr {I}_\texttt{n}(\texttt{t})=\sum _{\mathfrak {i}=1}^{\texttt{n}}\digamma _{3\mathfrak {i}}(\texttt{t}), \mathscr {A}_\texttt{n}(\texttt{t})=\sum _{\mathfrak {i}=1}^{\texttt{n}}\digamma _{4\mathfrak {i}}(\texttt{t}), \mathscr {R}_\texttt{n}(\texttt{t})=\sum _{\mathfrak {i}=1}^{\texttt{n}}\digamma _{5\mathfrak {i}}(\texttt{t}). \end{aligned}$$

Theorem 4.2

If there exists $\texttt{t}_1$ such that $\frac{1}{ \Gamma (\upkappa )} \uptheta ^{1-\upkappa } \texttt{t}_1 \mathfrak {g}_{\ell }<1$, then a the solution of the system of fractional $\mathscr {C}\mathscr {O}\mathscr {V}\mathscr {I}\mathscr {D}$-${\textbf {19}}$ $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$ model (8) exists.

Proof

From the recessive method and Eqs. (12) and (13) we conclude that

$$\begin{aligned} \Vert \digamma _{1\texttt{n}}\Vert&\le \Vert \mathscr {S}_\texttt{n}(0)\Vert \left[ \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_1\texttt{t}\right] ^\texttt{n},\quad \Vert \digamma _{2\texttt{n}}\Vert \le \Vert \mathscr {E}_\texttt{n}(0)\Vert \left[ \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_2\texttt{t}\right] ^\texttt{n},\quad \Vert \digamma _{3\texttt{n}}\Vert \le \Vert \mathscr {I}_\texttt{n}(0)\Vert \left[ \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_3\texttt{t}\right] ^\texttt{n},\\ \Vert \digamma _{4\texttt{n}}\Vert&\le \Vert \mathscr {A}_\texttt{n}(0)\Vert \left[ \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_4\texttt{t}\right] ^\texttt{n},\quad \Vert \digamma _{5\texttt{n}}\Vert \le \Vert \mathscr {R}_\texttt{n}(0)\Vert \left[ \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_5\texttt{t}\right] ^\texttt{n}. \end{aligned}$$

Hence, the system  (8) has a solution and also it is continuous. Now, we prove that the upper functions fabricate solution for model (10). Let $\mathscr {S}(\texttt{t})-\mathscr {S}(0)$ $=\mathscr {S}_\texttt{n} - G_{1\texttt{n}}$, $\mathscr {E}(\texttt{t})-\mathscr {E}(0)$ $= \mathscr {E}_\texttt{n} - G_{2\texttt{n}}$, $\mathscr {I}(\texttt{t}) - \mathscr {I}(0)$ $=\mathscr {I}_\texttt{n} - G_{3\texttt{n}}$, $\mathscr {A}(\texttt{t})-\mathscr {A}(0)= \mathscr {A}_\texttt{n}- G_{4\texttt{n}}$, $\mathscr {R}(\texttt{t}) - \mathscr {R}(0)$ $=\mathscr {R}_\texttt{n} - G_{5\texttt{n}}$. Thus,

$$\begin{aligned} \Vert G_{1\texttt{n}}\Vert&=\left\| \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\int _{0}^{\texttt{t}}(\mathscr {H}_1(\xi ,\mathscr {S}) -\mathscr {H}_1(\xi ,\mathscr {S}_{\texttt{n}-1})\, \text {d}\xi \right\| \\&\le \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\int _{0}^{\texttt{t}}\Vert (\mathscr {H}_1(\xi ,\mathscr {S}) -\mathscr {H}_1(\xi ,\mathscr {S}_{\texttt{n}-1})\Vert \, \text {d}\xi \le \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_1\Vert \mathscr {S}-\mathscr {S}_{\texttt{n}-1}\Vert \texttt{t}. \end{aligned}$$

With iterate the procedure, we have, at $\texttt{t}_1$,

$$\begin{aligned} \Vert G_{1\texttt{n}}(\texttt{t})\Vert \le \left[ \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )} \texttt{t}_1\right] ^{\texttt{n}+1}\mathfrak {g}_1^{\texttt{n}+1}c. \end{aligned}$$

When we make the value of $\texttt{n}$ go towards infinity, the upper equation gives us a limit, $\Vert G_{ 1 \texttt{n}}\Vert \rightarrow 0$. In a similar manner, one can check that $\Vert G_{\ell \texttt{n}} \Vert \rightarrow 0$, $\ell = 2, 3, 4, 5$. This completes the proof. $\square$

Theorem 4.3

Suppose that $1-\frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_1\texttt{t} >0.$ Then, the solution of $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$ model (8) is unique.

Proof

To show that the solution is one, we assume that there is another solution called $\mathscr {S}_1$, $\mathscr {E}_1$, $\mathscr {I}_1$, $\mathscr {A}_1$, and $\mathscr {R}_1$. Then,

$$\begin{aligned} \mathscr {S}(\texttt{t})-\mathscr {S}_1(\texttt{t})=\frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )} \int _{0}^{\texttt{t}}(\mathscr {H}_1(\xi ,\mathscr {S})-\mathscr {H}_1(\xi ,\mathscr {S}_{1}))\, \text {d}\xi . \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert \mathscr {S}-\mathscr {S}_1\Vert \le \frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\int _{0}^{\texttt{t}}\Vert \mathscr {H}_1(\xi ,\mathscr {S}) -\mathscr {H}_1(\xi ,\mathscr {S}_{1})\Vert \, \text {d}\xi . \end{aligned}$$

It follows from (11) that $\Vert \mathscr {S} - \mathscr {S}_1 \Vert \le \frac{1}{ \Gamma ( \upkappa )} \, \uptheta ^{1-\upkappa } \, \mathfrak {g}_1\texttt{t}\Vert \mathscr {S}(\texttt{t})-\mathscr {S}_1(\texttt{t})\Vert .$ Hence,

$$\begin{aligned} \Vert \mathscr {S}(\texttt{t})-\mathscr {S}_1(\texttt{t})\Vert \left( 1-\frac{\uptheta ^{1-\upkappa }}{\Gamma (\upkappa )}\mathfrak {g}_1\texttt{t}\right) \le 0. \end{aligned}$$

Then, $\Vert \mathscr {S} - \mathscr {S}_1 \Vert =0$. Therefore, $\mathscr {S}=\mathscr {S}_1$. In the same way, we are able to display the same parity for $\mathscr {E}, \mathscr {I}, \mathscr {A}, \mathscr {R}$. $\square$

Numerical results

Utilizing the FEP for $\uppsi$-CFD, approximate solutions for the fractional-order $\mathscr {C}$-19, $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$ model will be provided (see⁴³). Simulations to foreknow the $\mathscr {C}$-19 transmission within the world will also be provided.

Numerical procedure

Let’s think about system (8) in a shorter and simpler way as:

$$\begin{aligned} \uptheta ^{\upkappa -1}{{}^{\text {C}\!}\mathscr {D}}_{\grave{\imath }_1^{+}}^{\upkappa ; {\uppsi }} \uplambda = \upomega (\texttt{t}, \uplambda ), \qquad \uplambda (0)=\uplambda _0=(\mathscr {S}_0,\mathscr {E}_0,\mathscr {I}_0,\mathscr {A}_0,\mathscr {R}_0),\ 0 \le \texttt{t}\le \mathscr {T}<\infty , \end{aligned}$$

(14)

where $\uplambda =(\mathscr {S},\mathscr {E},\mathscr {I},\mathscr {A},\mathscr {R})\in {R}_+^5$ and $\upomega (\texttt{t})\in ~ {R}$ is a continuous vector function satisfying LC

$$\begin{aligned} \Vert \upomega (\uplambda _1(\texttt{t}))-\upomega (\uplambda _2(\texttt{t}))\Vert \le c\Vert \uplambda _1) - \uplambda _2\Vert ,\qquad c>0. \end{aligned}$$

Exerting a fractional integral operator matching to the $\uppsi$-CFD to Eq. (14), we get

$$\begin{aligned} \uplambda \big |_{\texttt{t}} = \uptheta ^{1-\upkappa } \Big [ \uplambda _0 + \mathscr {I}^{\upkappa ; {\uppsi }}\upomega (\uplambda )\big |_{\texttt{t}} \Big ], \qquad 0 \le \texttt{t}\le \mathscr {T}< \infty . \end{aligned}$$

Set $\mathfrak {m}=\frac{\mathscr {T}-0}{\mathscr {N}}$, $\mathscr {N}\in \mathbb {N}$ and $\texttt{t}_{ \texttt{n}}=\texttt{n}\mathfrak {m}$, $\texttt{n}= 0, 1, 2,\dots ,\mathscr {N}$ for $0 \le \texttt{t}\le \mathscr {T}$. Let $\uplambda _{\texttt{n}}=\uplambda \big |_{\texttt{t}_{\texttt{n}}}$. Utilizing the FEP, see⁴³, we obtain

$$\begin{aligned} \uplambda _{\texttt{n}+1}=\uptheta ^{1-\upkappa }\left[ \uplambda _0+\frac{\mathfrak {m}^{\upkappa }}{\Gamma (\upkappa +1)}\sum _{\mathfrak {i}=0}^{\texttt{n}}\texttt{u}_{\texttt{n}+1},_\mathfrak {i} \upomega (\texttt{t}_\mathfrak {i},\uplambda _\mathfrak {i})\right] ,\qquad \mathfrak {i}\in \{0\}\cup \mathbb {N}, \,0\le \mathfrak {i} \le =\mathscr {N}-1, \end{aligned}$$

where $\texttt{u}_{\texttt{n}+1},_\mathfrak {i}=(\texttt{n}+1-\mathfrak {i})^{\upkappa }-(\texttt{n}-\mathfrak {i})^{\upkappa }$, $\mathfrak {i}=0, 1, 2,\dots ,\texttt{n}$. The researchers have proven that the obtained scheme is stable in their work⁴³, Theorem (3.1). As a result, the approximate solution is expressed by

$$\begin{aligned} \mathscr {S}_{\texttt{n}+1}&=\uptheta ^{1-\upkappa }\left[ \mathscr {S}_0+\frac{\mathfrak {m}^{\upkappa }}{\Gamma (\upkappa +1)} \sum _{\mathfrak {i}=0}^{\texttt{n}}\texttt{u}_{\texttt{n}+1},_\mathfrak {i}\mathscr {X}_1(\texttt{t}_\mathfrak {i},\uplambda _\mathfrak {i})\right] ,\\ \mathscr {E}_{\texttt{n}+1}&=\uptheta ^{1-\upkappa }\left[ \mathscr {E}_0+\frac{\mathfrak {m}^{\upkappa }}{\Gamma (\upkappa +1)} \sum _{\mathfrak {i}=0}^{\texttt{n}}\texttt{u}_{\texttt{n}+1},_\mathfrak {i}\mathscr {X}_2(\texttt{t}_\mathfrak {i},\uplambda _\mathfrak {i})\right] ,\\ \mathscr {I}_{\texttt{n}+1}&=\uptheta ^{1-\upkappa }\left[ \mathscr {I}_0+\frac{\mathfrak {m}^{\upkappa }}{\Gamma (\upkappa +1)} \sum _{\mathfrak {i}=0}^{\texttt{n}}\texttt{u}_{\texttt{n}+1},_\mathfrak {i}\mathscr {X}_3(\texttt{t}_\mathfrak {i},\uplambda _\mathfrak {i})\right] ,\\ \mathscr {A}_{\texttt{n}+1}&= \uptheta ^{1-\upkappa }\left[ \mathscr {A}_0+\frac{\mathfrak {m}^{\upkappa }}{\Gamma (\upkappa +1)} \sum _{\mathfrak {i}=0}^{\texttt{n}}\texttt{u}_{\texttt{n}+1},_\mathfrak {i}\mathscr {X}_4(\texttt{t}_\mathfrak {i},\uplambda _\mathfrak {i})\right] , \\ \mathscr {R}_{\texttt{n}+1}&=\uptheta ^{1-\upkappa }\left[ \mathscr {R}_0+\frac{\mathfrak {m}^{\upkappa }}{\Gamma (\upkappa +1)} \sum _{\mathfrak {i}=0}^{\texttt{n}}\texttt{u}_{\texttt{n}+1},_\mathfrak {i}\mathscr {X}_5(\texttt{t}_\mathfrak {i},\uplambda _\mathfrak {i})\right] , \end{aligned}$$

where

$$\begin{aligned} \mathscr {X}_1(\texttt{t}, \uplambda )&= \varLambda - \upbeta _1\mathscr {I} \mathscr {S} - \upbeta _2\mathscr {A} \mathscr {S}- \upsigma \mathscr {S} + \varsigma \mathscr {R},\qquad \mathscr {X}_2 (\texttt{t}, \uplambda ) = \upbeta _1\mathscr {I}\mathscr {S} + \upbeta _2\mathscr {A}\mathscr {S}- (\upupsilon +\upsigma ) \mathscr {E}, \\ \mathscr {X}_3 ( \texttt{t}, \uplambda )&= \mathfrak {p} \upupsilon \mathscr {E}- (\upalpha +\upsigma +\updelta )\mathscr {I},\quad \mathscr {X}_4(\texttt{t}, \uplambda ) = (1 - \mathfrak {p}) \upupsilon \mathscr {E} - (\upgamma +\upsigma ) \mathscr {A}, \quad \mathscr {X}_5(\texttt{t}, \uplambda ) = \upalpha \mathscr {I} +\upgamma \mathscr {A} - (\upsigma + \varsigma ) \mathscr {R}. \end{aligned}$$

The model’s numerical simulations and interpretation

Using MATLAB, model (8) will be simulated for the world’s data. For simulation, the value of the parameters should be first determined. The birth and death rate for the world in 2022 were 17.668 births and 7.678 per 1000 individuals, respectively. The world’s population on 15 June was $\aleph = 7914981120$, so

$$\begin{aligned} \varLambda = \frac{1}{365}\, \texttt{n} \aleph =383128.455, \qquad \upsigma = \frac{7.678}{365\times 1000}=2.10356\times 10^{-5}, \end{aligned}$$

and we choose $\uptheta = 0.99$. Since $\aleph (0) = \mathscr {S}(0) + \mathscr {E}(0) + \mathscr {I}(0) + \mathscr {A}(0) + \mathscr {R}(0),$ and on 15 June $\mathscr {I}(0) = 15\ 315\ 220$, then we can suppose

$$\begin{aligned} \mathscr {E}(0)=20\ 000\ 000,\quad \mathscr {A}(0) =10\ 000\ 000, \quad \mathscr {R}(0)=0,\quad \mathscr {S}(0)=7\ 869\ 665\ 900. \end{aligned}$$

In addition, we consider the number of infection cases in the world in the period of time $\breve{T}$, 15 June to 4 August 2022, so that any part is three day. The parameters values of model (8) are available in Table 2. In this simulation, the EP is

$$\begin{aligned} \mathscr {E}_1=\left( \mathscr {S}^{\star},\ \mathscr {E}^{\star}, \ \mathscr {I}^{\star},\mathscr {A}^{\star}, \ \mathscr {R}^{\star } \right) = \Big (7.3\times 10^{9}, \ 1.9\times 10^{7},\ 1.47\times 10^{7},\ 1.43\times 10^{7},\ 5.5\times 10^{8}\Big ). \end{aligned}$$

Table 2 Details of the model parameters and their numerical value.

Problem (8) will be examined in three cases for $\uppsi (\texttt{t})$ as $\uppsi _1(\texttt{t}) = \texttt{t}$ (Caputo derivative); $\uppsi _2(\texttt{t}) = \ln \texttt{t}$ (Caputo–Hadamard derivative); $\uppsi _3(\texttt{t}) = \sqrt{\texttt{t}}$ (Katugampola derivative).

Case I. Let $\uppsi _1(\texttt{t}) = \texttt{t}$ (Caputo derivative).

The real data for infected cases, as well as the results of model (8) for $\upkappa \in \{0.95, 1\}$ on period $\breve{T}$ can be seen in Fig. 4. Also, in Fig. 5, it is predicted how each of the classes $\mathscr {S}$, $\mathscr {E}$, $\mathscr {I}$, $\mathscr {A}$ and $\mathscr {R}$ will change with $\upkappa \in \{ 0.75, 0.85, 0.95\}$. In addition, Tables 3, 4 and 5 show these results. An important point in this case, over the course of time, $\mathscr {S}$ decreases as the order of the derivative $\kappa$ approaches one (Fig. 5a), and even $\mathscr {E}$, $\mathscr {I}$, $\mathscr {A}$ (Fig. 5b–d) has a faster downward growth. But this is not the case for $\mathscr {R}$. In fact, as the order of the derivative $\kappa$ approaches one, the value of $\mathscr {R}$ increases more (Fig. 5e).

Figure 4

Active cases of $\mathscr {C}\mathscr {O}\mathscr {V}\mathscr {I}\mathscr {D}$-${\textbf {19}}$ on period $\breve{T}$.

Figure 5

Dynamics of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _1(\texttt{t})= \texttt{t}$ and different fractional order $\upkappa = 0.75, 0.85, 0.95$.

Table 3 Obtained results of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _1(\texttt{t})= \texttt{t}$ and fractional order $\upkappa = 0.75$.

Table 4 Obtained results of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _1(\texttt{t})= \texttt{t}$ and fractional order $\upkappa = 0.85$.

Table 5 Obtained results of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _1(\texttt{t})= \texttt{t}$ and fractional order $\upkappa = 0.95$.

Case II. Let $\uppsi _2(\texttt{t}) = \ln \texttt{t}$ (Caputo–Hadamard derivative).

The real data for infected cases, as well as the results of model (8) for $\upkappa \in \{0.95, 1\}$ on period $\breve{T}$ can be seen in Fig. 6. Also, in Fig. 7, it is predicted how each of the classes $\mathscr {S}$, $\mathscr {E}$, $\mathscr {I}$, $\mathscr {A}$ and $\mathscr {R}$ will change with $\upkappa \in \{ 0.75, 0.85, 0.95\}$. In addition, Tables 6, 7 and 8 show these results. The existence of exponential changes in this case can be clearly seen in curves (Fig. 7a–e). What is received from the graphs and numerical results is indicative of the fact that in this case, $\uppsi _2(\texttt{t})= \ln \texttt{t}$, the natural logarithm function is not appropriate in the presented model (1) (Fig. 7).

Figure 6

Active cases of $\mathscr {C}\mathscr {O}\mathscr {V}\mathscr {I}\mathscr {D}$-${\textbf {19}}$ on period $\breve{T}$.

Figure 7

Dynamics of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _2(\texttt{t})= \ln \texttt{t}$ and different fractional order $\upkappa = 0.75, 0.85, 0.95$.

Table 6 Obtained results of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _2(\texttt{t})= \ln \texttt{t}$ and fractional order $\upkappa = 0.75$.

Table 7 Obtained results of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _2(\texttt{t})= \ln \texttt{t}$ and fractional order $\upkappa = 0.85$.

Table 8 Obtained results of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _2(\texttt{t})= \ln \texttt{t}$ and fractional order $\upkappa = 0.95$.

Case III. Let $\uppsi _3(\texttt{t}) = \sqrt{\texttt{t}}$ (Katugampola derivative).

Similar to Case I and Case II, the real data for infected cases, as well as the results of model (8) for $\upkappa \in \{0.95, 1\}$ on period $\breve{T}$ can be seen in Fig. 8. Also, in Fig. 9, it is predicted how each of the classes $\mathscr {S}$, $\mathscr {E}$, $\mathscr {I}$, $\mathscr {A}$ and $\mathscr {R}$ will change with $\upkappa \in \{ 0.75, 0.85, 0.95\}$. In addition, Tables 9, 10 and 11 show these results. The graphs (Fig. 9a–e), in this case are remarkable. Over the course of time, only $\mathscr {S}$ decreases as the order of the derivative $\kappa$ approaches one (Fig. 9a), But the value of rest of the parameters $\mathscr {E}$, $\mathscr {I}$, $\mathscr {A}$ and $\mathscr {R}$ increases more whenever the order of the derivative $\kappa$ approaches one more (Fig. 9b–e).

Figure 8

Active cases of $\mathscr {C}\mathscr {O}\mathscr {V}\mathscr {I}\mathscr {D}$-${\textbf {19}}$ on period $\breve{T}$.

Figure 9

Dynamics of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _3(\texttt{t})= \sqrt{\texttt{t}}$ and different fractional order $\upkappa = 0.75, 0.85, 0.95$.

Table 9 Obtained results of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _3(\texttt{t})= \sqrt{\texttt{t}}$ and fractional order $\upkappa = 0.75$.

Table 10 Obtained results of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _3(\texttt{t})= \sqrt{\texttt{t}}$ and fractional order $\upkappa = 0.85$.

Table 11 Obtained results of $\mathscr {S}(\texttt{t})$, $\mathscr {E}(\texttt{t})$, $\mathscr {I}(\texttt{t})$, $\mathscr {A}(\texttt{t})$ and $\mathscr {R}(\texttt{t})$ whenever $\uppsi _3(\texttt{t})= \sqrt{\texttt{t}}$ and fractional order $\upkappa = 0.95$.

Conclusion

In this paper, an epidemic model $\mathscr {S}\mathscr {E}\mathscr {I}\mathscr {A}\mathscr {R}\mathscr {S}$ for the transmission of infection caused via $\mathscr {C}$-19 is presented utilizing the $\uppsi$-Caputo derivative. The reason for utilizing the derivative of the fractional order is that it provides a more accurate fit than the derivative of the integer order. The reproduction number has been calculated and its sensitivity has also been explored. The EPs have been calculated, and their stability are investigated. We show that the model is locally and globally asymptotically stable if $\breve{\mathscr {R}}_0$ is less than 1. The existence and uniqueness of the solution for the model via the FP theorem has been proven. Utilizing the FEP, an approximate solution to the model has been calculated.

In addition, the number of infection cases in the world in the period of time $\breve{T}$, 15 June to 4 August 2022, is considered to simulate the model to the real information. Also, the behavior of each of the classes after August 4 to the next 150 days is predicted with different cases for $\uppsi$. System (8) have been simulated in three cases for $\uppsi (\texttt{t})$ as $\uppsi _1(\texttt{t}) = \texttt{t}$ (Caputo derivative); $\uppsi _2(\texttt{t}) = \ln \texttt{t}$ (Caputo–Hadamard derivative); $\uppsi _3(\texttt{t}) = \sqrt{\texttt{t}}$ (Katugampola derivative). In the simulation by $\uppsi _1(\texttt{t})$, as can be seen in Fig. 4, the designed model has very good support from the data. In the simulation by $\uppsi _2(\texttt{t})$, as can be seen in Fig. 6, the designed model has acceptable support from the data, and in the simulation by $\uppsi _3(\texttt{t})$, as can be seen in Fig. 8, the designed model has good support from the data. In all simulations, the advantage of utilizing the derivative of the fractional order instead of the utilizing of the integer order can be seen. In Figs. 5c, 7c and 9c, respectively, utilizing $\uppsi _1(\texttt{t})$, $\uppsi _2(\texttt{t})$ and $\uppsi _3(\texttt{t})$, as well as utilizing different values for $\upkappa$, the spread of the disease after August 4 is predicted. At the end, by comparing the simulation results and real data, we come to the conclusion that the simulation utilizing $\uppsi _1(\texttt{t})= \texttt{t}$ (Caputo derivative) with the order of 0.95 shows the prevalence of the disease better.