We propose a theoretical study investigating the spread of the novel coronavirus (COVID-19) reported in Wuhan City of China in 2019. We develop a mathematical model based on the novel corona virus's characteristics and then use fractional calculus to fractionalize it. Various fractional order epidemic models have been formulated and analyzed using a number of iterative and numerical approaches while the complications arise due to singular kernel. We use the well-known

Corona-viruses family causes illnesses in humans, starting with the usual cold and leading to SARS. In the previous twenty years, two corona-virus epidemics have been reported [

Fractional computing is a growing field of applied mathematics and has attracted the attention of several researchers [

Corona-virus disease (COVID-2019) has been recognized as a global threat and therefore got the attention of various researchers due to its novel nature. Modelling the dynamics of multiple infections disease has a rich literature [

It could be noted that the coronavirus disease spreading rises globally from human-to-human transmission, while the initial source of the disease was an animal/reservoir. It has also been confirmed from the characteristic of SARS-CoV-2 that various phases of the infections are very significant and influence the transmission of the disease. The latent individuals are notable because of having no symptoms while transmitting the infection. So a small number of latent individuals leads to a significant disaster. We formulate the model keeping in view the above aesthetic of the SARS-CoV-2 virus and study the temporal dynamics of the disease. Once to develop the model, we then fractionalize because of increased development that the epidemiological models having fractional order are more significant than integer-order. Therefore, the fractionalization of the model to its associated fractional-order version will be accorded with the application of fractional calculus. We prove the existence with uniqueness and discuss the feasibility of the developed epidemic problem with the help of the fixed point theory. We will also investigate whether the proposed model is bounded and possesses positive solutions. We perform the numerical visualization of the analytical results to verify the theocratical parts. We also show the difference between integer and non-integer order epidemiological cases.

We formulate the proposed problem by considering the characteristic of the novel coronavirus disease. We classify the total human population

The proposed model represents the dynamical population problem, so all the variables, parameters, and constants are positive.

Three different transmission routes transmit the disease, i.e., from a latent population, infected population to susceptible, and from the reservoir.

We assume that individuals with a strong immune system will recover in the latent period.

It is also assumed that there are two types of recovery from infection, i.e., naturally and due to treatment.

The death rate due to disease is assumed to be only in the infected compartment.

Moreover the prorogation of novel corona virus disease transmission is demonstrated by

In the proposed problem,

In the above

Using the theory of fractional calculus to take the associated fractional version of the considered model. Since

To discuss the feasibility of the above epidemiological system

We exploit fixed point theory to show the model's existence and uniqueness under-considered, as Equation reported

We apply the definition of

Let us assume that

Upon, the application of

We then obtain recursively the following relations:

The difference of two successive terms with the application of norm and majorizing, one may obtain

It could be noted that the kernels

So, the relations as described by the above equation are smooth and exists, however to investigate that the functions in these relations are the solutions for system

The application of norm on both sides of the above system with utilization of the

The application of

Majorizing one may leads to the assertions as given by

We now use the result stated by Theorems 2.1 and 2.2, we obtain

For all

Now we are going to discuss the biological as well as mathematical feasibility of the problem under consideration. Notably, we discuss the positivity and boundedness of the reported model

Solving the above

It could be also noted that

The solution of

In

We discuss the temporal dynamics of the considered model for the long run and present the significance of the fractional parameter. We find the numerical simulation to verify the theocratical work carried out for the fractional-order SARS-CoV-2 transmission epidemiological model

Furthermore, we have chosen the value of epidemic parameters biologically as given in

Parameter | Value | Parameter | Value |
---|---|---|---|

0.4000 | 0.0110 | ||

0.0050 | 0.0160 | ||

0.8631 | 0.0100 | ||

0.0100 | 0.0500 | ||

0.0500 | 0.0050 | ||

0.0020 | 0.0010 | ||

0.0600 | 0.0600 |

We investigated the dynamics of SARS-CoV-2 with latent and infected individuals using an epidemic model. First, the formulation of the model is proposed and then consequently fractionalized due to the increased development in fractional calculus. Mainly, we used the well-known

In the near future, we will use the operators Atangana Baleanu Caputo, Atangana bi order, Atangana Gomez and fractal-fractional operator to study the complex dynamics of novel corona virus disease transmission. We will also apply the optimal control theory to the model reported in this study to present the control mechanism for the novel corona virus disease transmission.