Fractional order COVID-19 model with transmission rout infected through environment

1.   Introduction

Mathematical models are known to address and answer specific questions for the disease under consideration. For example, for the prediction of the spread of infectious diseases, epidemiological models are very supportive. To control the disease, it assists the community to notice sensitive features. We aim to explore one of these models to study the behavior of the virus COVID-19 that appeared in early 2020 and is still not fully controlled. For the better understanding of physical phenomena, we look into fractional calculus. Several operators were demonstrated in the literature ^[1,2] with the help of fractional calculus. Few implementations of these operators can be seen in ^[3,4,5,6,7]. In $C^{\eta}$-Calculus, Golmankhaneh et al. ^[8] explained the Sumudu transform and Laplace. In 2019, with the help of the fractional model, Goyal provided an approach ^[9] to control the Lassa hemorrhagic fever disease. In China, Zhao et al. ^[10] formulated a model to control COVID-19. It spreads from person to person via connection with the contaminated person using breathing the same air, coughing, sneezing, touching, and touching the same surfaces. The prime prevention approaches are wearing a mask in public at all times, washing your hands, and maintaining a proper distance from anyone. A well-defined approach of the fractional-order model is described in ^[11]. Some fractional models of COVID-19 are coming up recently, in which Atangana and Khan ^[12] have investigated the condition of China due to pandemics. In ^[13], by considering Fuzzy Caputo and ABC derivative founders look into the dynamical study of the COVID-19 model. Further, the writers measured the model of COVID in the Caputo operator ^[14]. Using actual data from March 02 to April 14 ^[15], Alshammari also contributed to the pandemic. To enforce the fractional model ^[16,17,18], the investigator applied the homotopy analysis transform method (HATM), and the comparison showed that this technique is highly effective. In ^[17], stability of fractal differentials in the sense of Lyapunov is defined. Moreover, based on the fractal set, they generalized the non-local fractional integrals and derivatives. A hybrid method based on operational matrices of the derivative is proposed and successfully applied to explore the solution of the mobile–immobile advection-dispersion problem of variable order ^[19]. similarly, Work with exact solutions and existing methods included numerical and analytical was made ^[20] that showed an excellent level of accuracy for the problems. Some applications of fractional order model with local and non-local nonsingular kernel have also been studied in ^{[21,22,23,24,25,26,27]}. Padmavathi et al. ^[28] also worked on the q-HATM with the latest operator Atangana-Baleanue to get finer recognization, and they expressed their results in the form of visualization. For the set up of operational matrices, a piecewise function was deployed in ^[29]. Also, for implementation in a simple way, researchers converted the design into a linear system. In ^[30], a fractional Biswas-Milovic model was inspected through the technique of fractional complex transform. To analyze the nonlinear oscillatory fractional-order differential equation, A spectral approach through the Chelyshkov polynomial method (CPM) and Picard iterative (PI) was used ^[31]. Mehmood et al. ^[32] used the techniques Galerkin-Petrove (G-P) and Rung Kutta (RK) to study the model and they concluded that in comparison, the G-P technique is better. The effect of vaccination of COVID-19 through different values of parameters had studied in ^[33]. The common SEIR model is generalized in order to show the dynamics of COVID-19 transmission taking into account the ABO blood group of the infected people. Fractional order Caputo derivative are used in the proposed model ^[34].

Some basic definitions of fractional calculus are described in Section 2. Later, a fractional-order COVID-19 model is presented in Section 3. Moreover, sensitivity analysis of the model and some theorems are added. Further in Section 4, the simulations are demonstrated. The last section consists of a conclusion.

2.   Materials and methods

In this section, some primary notions have been described that are helpful to analyze the system.

Definition 1. For $z\in H^{1}(x, y)$ and $\kappa\in(0, 1)$. The Caputo-Fabrizio fractional derivative ^[35] is defined as

where $M(\kappa)$ is a normalization function.

Definition 2. Antagana-Baleanu in Caputo sense (ABC) can be defined ^[36] as

where $E_\alpha$ is the Mittag-Leffler function and $AB(\alpha)$ is a normalization function. For Eq (2.2), a Laplace transformation is presented as:

For using ST for (2.2), we obtain

Definition 3. Atangana-Baleanu fractional integral of order $\mu$ of a function $\psi (t)$ can be expressed as ^[37]

3.   Fractional order COVID-19 model

The current section investigates the displaying of the novel Coronavirus. We start the demonstrating interaction be meaning the host populace by $N(t)$ dividing into five totally unrelated epidemiological classes based on dynamics of COVID-19 contamination. These classes comprise of susceptible $S$, exposed $U$, irresistible appearance indications of disease $C$, contaminated with no illness symptoms $C_{a}$ and the people recuperated are denoted $R$ respectively. Classical order Covid-19 model is given in ^[38] and fractional order form with ABC sense is shown in followings equation

With initial condintion

Here $H$, $b$, and $\mu$ are represents birth rate, passing rate in each class and infected rate respectively. The parameters $\mu_{1}, \; \mu_{2}$ and $\mu_{3}$ are the viable transmission paces of contamination due to exposed, symptomatically-infected and symptomatically-infected individuals, respectively. $\mu_{4}$ denotes the age of infection due to climate. We denote the incubation period of the individuals by $\rho$. $d_{1}$ and $d_{2}$ are represents the recovery rates from infected population. The contribution of the virus to the environment due to exposed, and the population of both infected compares (i.e., $C$ and $C_{a}$) are shown respectively by $\phi_{1}, \; \phi_{2}$ and $\phi_{3}$. We have

From above, we have

3.1.   Equilibria and sensitivity analysis

We define corona free equilibria as:

and the corona existing equilibria as:

we get $S^{*}, \; U^{*}, \; C^{*}, \; C_{a}^{*}, \; R^{*}, \; B^{*}$ as:

where

By substituting (3.5) into (3.6), we get $\lambda^{*}$ as:

where

For the derivation, the reproduction number $R_0$ is the spectral radius of $Y*Z^{-1}$, where $Y$ and $Z$ are the transmission and the transition matrices respectively. Thus, we get

or

Sensitivity of $R_0$ can be analyzed by taking the partial derivatives of reproductive number for the involved parameters as follows

Clearly, in case of change in parameter $R_0$ is very sensitive. In this manuscript, $\rho, \; \tau, \; l_{3}, \; \mu_{1}, \; \mu_{2}, \; \mu_{3}, \; \mu_{4}$, $\phi_{1}, \; \phi_{2}$ are growing while $l_{1}, \; l_{2}, \; \sigma$ are reducing. Thus, based on sensitivity analysis, we can say that prevention is better to control the disease.

Theorem 1. For a Banach Space $(G, |.|)$ and $F$ is a self-map of $K$ fulfilling

$\forall \; G, \; r \in G$ where $0 \leq \theta \leq 1$. Let $F$ be a Picard F-stable.

We take into consideration Eq (3.2) and get

Theorem 2. We describe $F$ as a self-map by

Then, we reach

$F$ satisfies the condition associated with Theorem 1 if

We add that $F$ is Picard $k$-stable.

Theorem 3. If we use the technique of recurrence then the system (3.2) posses a particular singular solution.

Proof. Assume Hilbert space $H = Z^2 (x, w)\times(0, T))$ which can be defined as

Then, we take into consideration:

We establish that the inner product of

where

are the special solution of the system. Then, we have

In case of large number ${e_1}$, ${e_2}, \; {e_3}, \; {e_4}, \; {e_5}$ and ${e_6}$, both solutions happen to be converged to the exact solution. By applying the concept of topology, we can get six positive very small variables \big($X_{e_1}, \; X_{e_2}, \; X_{e_3}, \; X_{e_4}, \; X_{e_5}$ and $X_{e_6}.$\big)

where

But it is obvious that

where

Therefore, we have

which yields that

This completes the proof of uniqueness. If we use the Adams-Moulton rule for Atangana-Baleanu fractional integral to indicate the mathematical strategies then

where

We obtain the following for the system 3.1:

An operator $\hslash : E \rightarrow E$ can be set out as:

For $\mho(t, \Delta(t))$ that attains the development and Lipscitz condition, so for $\Delta\; \in \; Z \; \exists$ positive constants $E_{\mho }, \; F_{\mho }$ such that

Also, for $\Delta, \; \hat{\Delta}\; \in \; Z \; \exists$ constant $G_{\mho } > 0$ such that

Theorem 4. If the state defined in (3.18) is true and by assuming the continuous function $\mho:[0, \tau] \times X\rightarrow R$ then it has unique outcome.

Proof. In the beginning, we reveal that $\hslash$ is completely continuous that are explained in (3.17). while $\mho$ is a continuous function, so $\hslash$ is also continuous function. Consider $J = \{ \Delta\; \in \; X \; : \| \Delta\| \leq R, \; R > 0 \}$. For any $\Delta\; \in \; X$, we have

Hence, $\hslash$ is uniformly bounded, and $J(\varpi + \vartheta)$ representing the beta function.

For equicontinuity of $\hslash$, we take $t_1 \; < \; t_2 \; \leq \; \tau.$ Then consider

If $t_1 \; \rightarrow \; t_2$ then $\|\hslash(\Delta)(t_2) \; - \; \hslash(\Delta)(t_1) \; \rightarrow \; 0 \|.$

Consequently $\|\hslash(\Delta)(t_2) \; - \; \hslash(\Delta)(t_1) \; \rightarrow \; 0 \|$, as $t_1 \; \rightarrow \; t_2$. Thus, $\hslash$ is completely continuous by theorem of Arzela-Ascoli

Thus, $\hslash$ is equicontinuous and under the condition of Arzela-Ascoli theorem it is completely continuous. Consequently, the following result of Schauder's fixed point, it has at least one solution.

Theorem 5. By assuming the condition (3.19) is accurate and $\rho \; = \; (\frac{\vartheta \tau^{\vartheta -1} (1-\varpi}{AB(\varpi)} \; + \; \frac{\varpi \vartheta}{AB(\varpi) \Gamma(\varpi)} \; \tau^{\varpi + \vartheta -1} J(\varpi + \vartheta)) I_{\mho}$ then for $\rho \; < \; 1$ gives a sole outcome.

For $\Delta, \; \hat{\Delta}\; \in \; X$, we have

Thus, $\hslash$ has contraction and by using the Banach contraction principle, it has a special outcome.

4.   Simulation and Discussion

In the simulation of a model, parameters have a key contribution. Here we use the actual data of Saudi Arabia having parameters values given in ^[38]. After that, we have spreading rates of the uncovered ($\mu_{1}$), symptomatic ($\mu_{2}$), asymptomatic ($\mu_{3}$), and environment ($\mu_{4}$) are 0.2259, 0.1298, 0.4579, 0.0969, respectively. Further, we got the values of the incubation period ($\rho = 0.1141$), asymptotically infected persons ($\tau = 0.3346$), the Improvement rate of C ($d_{1}0.3346$), and the improvement rate of $C_{a}\; (d_{2} = 0.0867)$. The parameters $\phi_{1} = 0.2616, \; \phi_{1} = 0.0100, \; \phi_{1} = 0.1815, \; \sigma = 0.2786$ represents the contribution caused by $U, \; C, \; Ca,$ destruction of virus. We presented the simulation data in Figures 1–6 with several values of fractional order ($\alpha$). In Figures 1–3, 5 and 6, we will observe that $S(t)$, $U(t)$, $C(t)$, $C_a(t)$ and $B(t)$ population increase with decreasing the fractional order $\alpha$ where as $R(t)$ start decreasing by decreasing fractional values. It is observed that the decrease in pandemic peaks is comparatively faster for smaller values of $\alpha$ as shown in Figures 1–6. Fractional parameter shows the study of COVID-19 outbreak continuously from starting place to their end boundary. It can be easily seen that the COVID-19 spread is more and more and recovery rate reduces from the actual data of Saudi Arabia when study it by every smaller region. This type study will help us locate the rate of actual number of infected people. The impact of memory index $\alpha$ on the dynamics of virus concentration in the environment at $\alpha$ is analyzed to check the outbreak of this pandemic.

5.   Conclusions

Mathematical modeling plays an important role to control, planning and reduce the bad impact of infectious disease in the society from last decays. Results of fractional order model have a memory effect on epidemic model as compared to classical model. In this formation, results show that contaminated parts are reduced by lowering the fractional order. Some notional outcomes are produced for the model to demonstrate the productivity of the created procedures.Graphical representation indicates that we can reduce the cases rapidly if communities of the country follow some rules, including social distance, cleaning their hands, keeping away from the throng. Theoretical results are investigated for the fractional-order model, which proved the efficiency of the developed schemes. Numerical simulation has been made to check the actual behavior of the COVID-19 outbreak. Such type of study will be helpful in future to understand the outbreak of this epidemic and to control the disease in a community. The power of these component operators is their non-local features that are not in the integer separator operator. Separated features of differentiated statistics define the memory and transfer structures of many mathematical models. As a reality that fractional order models are more practical and beneficial than classical integer order models. Fractional order findings produce a greater degree of freedom in these models. Unnecessary order outscoring is powerful tools for understanding the dynamic behavior of various bio objects and systems.

Acknowledgments

National Natural Science Foundation of China (No. 71601072), Key Scientific Research Project of Higher Education Institutions in Henan Province of China (No. 20B110006) and the Fundamental Research Funds for the Universities of Henan Province, China (No. NS-FRF210314).

Conflict of interest

The authors declare that they have no competing interests.