In this paper, a noninteger order Brucellosis model is developed by employing the Caputo–Fabrizio noninteger order operator. The approach of noninteger order calculus is quite new for such a biological phenomenon. We establish the existence, uniqueness, and stability conditions for the model via the fixed-point theory. The initial approachable approximate solutions are derived for the proposed model by applying the iterative Laplace transform technique. Finally, numerical simulations are conducted for the analytical results to visualize the effect of various parameters that govern the dynamics of infection, and the results are presented using plots.

Brucellosis is a contagious zoonotic infection that is attributable to diverse species of Brucella. Although the disease is well contained in Australia and the UK, the annual global incidence of Brucellosis is estimated above 500 000 [

The quantification of intricate and complex biological systems in terms of mathematical structures has posed a serious challenge to contemporary scientists worldwide. A good number of models with soliton solutions that are designed in terms of integer-order derivatives had been extensively studied in the last two decades [

Scientific and technical models of Brucellosis dynamics can be used to interpret the primary outcomes and condense the discoveries to the best advantage of protection at various stages; still, a number of those models in the form of nonfractional derivative were thoroughly investigated [

As time goes by, advancing mathematical models using noninteger order became a significant area of study since the evolution-related realities and evidence are represented more effectively in terms of noninteger calculus [

There are four compartments into which the Brucellosis model in [

Motivated by the previously mentioned literature, we have applied Caputo–Fabrizio fractional order

The typical model in [

The remainder of this script is arranged as follows. In Section

(see [

(see [

(see [

Generally, we have

This section deals with the existence and uniqueness of the fractional Brucellosis model using fixed-point theory [

For clarity, we consider the following kernels:

The kernels

Let

By applying the properties of the norm on (

We construct the following recursive formula:

Furthermore, by applying triangular inequality, we have

The kernels

This proves the result.

The

Following the result in (

Hence, (

We make use of norm and Lipschitz condition to obtain

Using

Therefore,

In the same way, as

Using the same approach, in (

Similarly, from (

The

Proof We adopt another solution (

By applying the norm on (

By considering Theorems

The solution functions in (

Going by the last equation, we conclude that

This section deals with the application of iterative Laplace transformation technique on the fractional Brucellosis model, obtaining the stability condition for the approximate solution.

We consider the Brucellosis model in (

By rearranging the following, we have

The inverse Laplace transform of (

We obtain the following infinite series solution for the

In this section, we analyze the numerical simulations of the Caputo–Fabrizio Brucellosis model (

Details of parameters [

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

3.0 | Annual natural mortality of Brucella | |

2.0 | Annual natural elimination rate | |

0.8 | Annual elimination rate for the positive cows | |

4.0 | Clinical outcome rate | |

1.0 | The annual birth rate of dairy cows | |

0.2 | Indirect transmission rate | |

The annual introduction number of dairy cows | ||

0.2 | The annual quantity of Brucella | |

0.52 | The rate of sterilization in a disinfection | |

0.5 | The yearly number of disinfections |

Figures

Approximate solution of susceptible cows.

Approximate solution of exposed cows.

Approximate solution of infectious cows.

Approximate solution of concentration of bacteria in the environment.

Figures

Approximate solution of susceptible cows for

Approximate solution of exposed cows for

Approximate solution of infectious cows for

Approximate solution of concentration of bacteria for

In this paper, we have analyzed the Caputo–Fabrizio noninteger order Brucellosis model. Using fixed-point theorem, the steady results for the existence and uniqueness of solutions and the stability of the proposed noninteger order Brucellosis model have been derived. The iterative Laplace transform technique is applied to conduct the simulations by using a set of values whose sources are from the literature for the model parameters. The effects of various values for the order of the Caputo–Fabrizio noninteger order derivative Brucellosis model on the transmission dynamics of the disease are revealed by the results of the simulations. The results of the simulations, therefore, confirm the validity and efficiency of the Caputo–Fabrizio noninteger derivative in quantifying dynamics of Brucellosis disease and other related kind of problems.

The data used to support the findings of this study are available from the corresponding author upon request.

The author declares that they have no conflicts of interest.