In this study, we classify the genera of COVID-19 and provide brief information about the root of the spread and the transmission from animal (natural host) to humans. We establish a model of fractional-order differential equations to discuss the spread of the infection from the natural host to the intermediate one, and from the intermediate one to the human host. At the same time, we focus on the potential spillover of bat-borne coronaviruses. We consider the local stability of the co-existing critical point of the model by using the Routh–Hurwitz Criteria. Moreover, we analyze the existence and uniqueness of the constructed initial value problem. We focus on the control parameters to decrease the outbreak from pandemic form to the epidemic by using both strong and weak Allee Effect at time t. Furthermore, the discretization process shows that the system undergoes Neimark–Sacker Bifurcation under specific conditions. Finally, we conduct a series of numerical simulations to enhance the theoretical findings.

In the last few months, nature has showed its laws in establishing the environment of the 21st century. It is out of our primary objective whether the coronavirus (COVID-19) is used as a biological weapon or not. The main point is now that humans are fighting against something to survive that has a genome size of 27 to 34 kilobases. Coronaviruses are members of the sub-family coronavirinae in the family coronaviridae and the order Nidovirales [

Coronavirinae genera | ||||
---|---|---|---|---|

Pathogenic class | Mammals | Mammals | Both non-mammal and mammals | Both non-mammal and mammals |

The natural host of SARS-CoV, MERS-CoV, HCoV-NL63, and HCoV-229e are bats, while HCoV-OC43 and HKU1 have originated from rodents [

Covid-19 was characterized by two members of β-coronavirus; the human-origin coronavirus (SARS-CoV Tor2) and bat-origin coronavirus (bat-SL-CoVZC45). Intensive studies show that it was most closely related to the bat-origin coronavirus [

Domestic animals, like snakes in that area, were hunted for the food market in Wuhan, which played an intermediate host role in the transmission. Finally, this virus spillover from the intermediate hosts to cause several diseases in human. A virus that started with an endemic pathogenic behavior in China (Wuhan) reaches somehow to a pandemic point worldwide with the infection from human-to-human.

It has been realized that the dynamics of many biological and medical phenomena can be characterized via mathematical models. Over the years, many models are formulated mathematically to analyze events in biological and medicine such as infections, treatments, or environmental phenomena [

In this paper, we establish a model that describes the pandemic infection, which occurs when the virus is transmitted from the human body to the intermediate host and continues to spread from human-to-human. The model consists of five fractional differential equations. The first three equations show an SI (susceptible-infected) model to explain the transmission from human-to-human, where

Indeed, the mathematical model of this biological phenomena has the form:

where

represents the Holling type II function and all the parameters of the model

The susceptible

The

Parameter | Symbol | rate |
---|---|---|

The growth rate of |
0.012 | |

The growth rate of |
0.009 | |

The growth rate of |
0.014 | |

The growth rate of |
0.01 | |

Logistic rate of |
0.05 | |

Logistic rate of |
0.1 | |

Logistic rate of |
0.15 | |

Logistic rate of |
0.01 | |

Logistic rate of |
0.01 | |

Rate of the |
1.6 | |

Parametric lost from class |
0.00134, 0.00044 | |

Rate of interaction between |
0.0001 | |

Predation rate | 0.0044 | |

Rate of screening | [0.01, 0.05] | |

Recognition of infection | [0.1, 0.4] | |

A conversion factor of |
0.0045 | |

The attack rate of |
0.15 | |

Rate of average time on infecting |
0.15 | |

Potential infectivity of |

defined on

with

Consider the model

To analyze the stability of model

Thus, we have

Thus, we obtain a linearized system about the equilibrium point of the form

where

where the co-existing equilibrium point is

and

whose solutions are given by Mittag–Leffler functions

and

By using the result of [

Evaluating the Jacobian matrix

where

and

The characteristic equation of the matrix

and

if

From

and

where

(i)

(ii)

(iii)

(iv)

(v)

where

then all roots of

and

From (ii) and

if

where

In considering both (iii) and

where

Let us consider now the case for

we obtain

and

From (v) and

and

Since the discriminant of

and

Then all roots of

and

From (ii) and

if

where

where

Additionally, we get

Similarly, let us consider the case for

we obtain

and

From (v) and

and

Moreover, we get

This completes the proof.

If

where

then the

If

then the

In

In

This awareness of the people through media and health organizations let them go to hospitals for screening so that the class who does not know they are infected decreases.

We considered in these examples the infection from human-to-human since the pandemic case reaches from the human transmission. We want to emphasize the strong coordination between health organizations and the media which is an essential tool for two critical parameters, which are

The design of nature keeps the natural host and intermediate host in a stable dynamical system in the habitat. The intermediate host had only a transmission role from animal to human, while the main spread happens through human to human from the

Considering system

for

Let us assume that

when

when

Operating

Define the operator

It follows that

This implies that

and

which implies

from which we can deduce that

It follows that

which implies

and thus

Therefore, this IVP is equivalent to

In 1838, Pierre Verhulst [

The Allee effect can be divided into two main types:

(i) strong Allee effect and

(ii) weak Allee effect.

A population with a strong Allee effect will have a critical population size, which is the threshold of the population, and any size that is less than the threshold will go to extinction without any further aid. However, a population with a weak Allee effect will reduce the

Let us incorporate an Allee function to the

where

is a function of Holling Type II and

Let

where we obtain

and

where

The characteristic equation of system

and

where

From

and

where

For a strong Allee effect, let us assume that the Allee function is given by

where

The following Theorem is given without proof since it is similar to the stability analysis of Section 3.

where

then all the roots of the system are real or complex conjugates with negative real parts.

If

and the ratio between the susceptible and intermediate host is given by

and

Thus, all roots of the system are complex conjugates with positive real parts.

If

where

then the

If

then the

In this section, we consider the discretization process to analyze Neimark–Sacker bifurcation. We will modify our system in

where

The solution of system

If we repeat the discretization process

For

The Jacobian matrix of (55) around the co-existing equilibrium point

where

We obtain the characteristic equation of the matrix such as

and

where (i)-(v) hold and

To analyze the conditions for Neimark-Sacker Bifurcation, we use the following Theorem.

a pair of complex conjugate roots of

where

where

which holds for

where

Finally, from (d) we obtain

which gives

where

In considering both

which completes the proof of the

The characteristic equation

then

and

Finally, from (d) we get

which holds for

where

This completes the proof.

In this paper, we classified the coronaviruses and their spread from the natural host to the human host. We proposed a model of the novel coronavirus, which is known as COVID-19, as a system of fractional-order differential equations. We divided the system into five sub-classes:

the susceptible class

the infected class

the intermediate domestic host

the natural host

We consider the pandemic infection case; animal to human and human to human. Therefore, the first three equations in the constructed model show human to human transmission. The spillover from the intermediate infected class to the human host denotes a predator-prey mathematical model, and the transmission from the natural host to intermediate host

In Sections 3 and 4, we analyzed the local stability of the co-existing equilibrium point by using the Routh–Hurwitz Criteria. We proved the existence and the uniqueness of the initial value problem.

Theorem 3.1., shows that among the human hosts, those who do not know they are infected are the control class in the spread. While between the animal hosts, the intermediate class plays a dominant role in the spread since that class has an essential role in transmitting the virus from animal to human. The transmission potential for both

In Theorem 3.2., we emphasized that

In Section 5, we incorporate the Allee function at time

In Section 6, we deduced that the system demonstrates a Neimark–Sacker bifurcation under specific conditions.

F. B. acknowledges the support of Erciyes University for the research study.