New fractional operators, the COVID-19 model has been studied in this paper. By using different numerical techniques and the time fractional parameters, the mechanical characteristics of the fractional order model are identified. The uniqueness and existence have been established. The model’s Ulam-Hyers stability analysis has been found. In order to justify the theoretical results, numerical simulations are carried out for the presented method in the range of fractional order to show the implications of fractional and fractal orders. We applied very effective numerical techniques to obtain the solutions of the model and simulations. Also, we present conditions of existence for a solution to the proposed epidemic model and to calculate the reproduction number in certain state conditions of the analyzed dynamic system. COVID-19 fractional order model for the case of Wuhan, China, is offered for analysis with simulations in order to determine the possible efficacy of Coronavirus disease transmission in the Community. For this reason, we employed the COVID-19 fractal fractional derivative model in the example of Wuhan, China, with the given beginning conditions. In conclusion, again the mathematical models with fractional operators can facilitate the improvement of decision-making for measures to be taken in the management of an epidemic situation.

Coronavirus (COVID-19) is a new phenomenon in recent days, which affected the entire world while it was emerging. According to reference [

In order to address problems in the real world, fractional calculus (FC) is essential. It is widely utilized in a variety of scientific, engineering, and financial sectors. The key characteristics of FC are fractional integrals and derivatives of fractional order. Researchers’ interest in fractional calculus and the numerous aspects of that study under inquiry has grown in recent years. This is due to the fact that genetic mutations are a crucial tool for characterizing the dynamic operation of diverse biological systems. These component operators’ non-local properties, which are absent from the integer separator operator, give them their power [

In this section, we consider some definition related to fractal fractional operator given in [

where

where

We suppose the COVID-19 model formulated by Ahmad et al. [

with initial conditions are

In this section, we will discuss the equilibrium points of the given COVID-19 model ^{’} represents disease free equilibrium and endemic equilibrium is represented by E^{*}. If we take both of our equilibriums, we have

We obtain the basic reproductive number

We consider [

where

We can write system

By replacing

where

We describe a Banach space

Define as operator

We suppose that

For each

Considering

Suppose that

Therefore, we get

When

Thus,

Then the solution of the system is unique.

Therefore,

We denote by:

Let

We give the following theorem, for the hypothesis that

then, the semi linear inclusion system:

has at least one solution in

The demonstration of the theorem uses elements from the fixed-point theory and results from the fact that

And there exists unique solution

satisfies the following relation:

We note:

Consequently, one can write

We can write the above relation is

where

Consider:

We construct the numerical scheme at

Applying the approximation of the integrals on the right hand side of system

We consider

Then, we have

COVID-19 fractional order model for the case of Wuhan, China, is offered for analysis with simulations in order to determine the possible efficacy of Coronavirus disease transmission in the Community. For this reason, we employed the COVID-19 fractal fractional derivative model in the example of Wuhan, China, with the given beginning conditions. The parameters of actual data are described in detail in [

In

In

In

In this paper, the fractal-fractional differential equation model for COVID-19 disease has been investigated with fractal fractional operator. The steady state and fundamental characteristics of the model equilibria are investigated. Fixed point theory is used to demonstrate in detail the existence and uniqueness of solutions for the model with FFM derivative. The Ulam-Hyers technique is used to conduct the stability analysis of the system which fulfills all properties. The two-step fractional Lagrange polynomial approach with FFM derivative is used to generate the model’s numerical solution. The numerical simulations are obtained and briefly described by choosing various values of the fractional order and dimension. We applied very effective numerical techniques to obtain the solutions of the model. We analyzed our obtained results and concluded that they are effective for the proposed model. Some theoretical results were also discussed for the model. This model turns out to be quite trustworthy when precise estimations of transmission structures are provided in real-time. The new suggested improvement will shed some modeling-related light on problems with and without singularity at the origin.

The analysis of the following problems forms further directions of research and developments: creating and exploring families of other epidemiological models based on fractal-fractional differential equations for diseases; the exploration of distinctions, taking into account the types of distinctions between fractional models; creating the output information for subsequent methodological recommendations, for diseases expansions analysis and for anti-diseases interventions plans.

The authors would like to acknowledge the Lucian Blaga University of Sibiu & Hasso Plattner Foundation’s research grant.

Lucian Blaga University of Sibiu & Hasso Plattner Foundation Research Grants LBUS-IRG-2020-06.

Conceptualization, M.F., A.A. and M.S.H.; Methodology, M.F., A.A. and M.S.H.; Formal analysis, M.F., A.A., M.S.H., A.B. and L.G.; Investigation, M.F., A.A. and M.S.H.; Supervision, A.B. and L.G.; Project administration, A.B.; Funding acquisition, A.B. All authors have read and agreed to the published version of the manuscript.

Not applicable.

The authors declare that they have no conflicts of interest to report regarding the present study.