Various epidemics have occurred throughout history, which has led to the investigation and understanding of their transmission dynamics. As a result, non-local operators are used for mathematical modeling in this study. Therefore, this research focuses on developing a dysentery diarrhea model with the use of a fractional operator using a one-parameter Mittag–Leffler kernel. The model consists of three classes of the human population, whereas the fourth one belongs to the pathogen population. The model carefully deals with the dimensional homogeneity among the parameters and the fractional operator. In addition, the model was validated by fitting the actual number of dysentery diarrhea infected cases covering 52 weeks in 2017, which occurred in Ethiopia. The biological parameters were fitted, and fractional order

The literature provides mathematical models for the transmission of infectious diseases. These models play a significant role in quantifying and evaluating the effective control and preventive measures of infectious diseases [

It is popularly known that mathematical models can predict the emergence of infectious diseases and epidemics, which are beneficial for public health planning and initiatives. By using compartmental models as a simple mathematical structure, the complex dynamics of epidemiological processes can be examined [

Additionally, other populations denoted by R, which is the image of the immune/removed/ recovered compartment, are considered to make these models more practical. A significant challenge here is to obtain sufficient parameters for a particular disease, which would determine the factors affecting potential control measures, such as medication or vaccination. The crucial question is about the execution of such measures from an optimal viewpoint. Several notable attempts have recently been made to introduce this research program for various diseases with integer compartmental models [

Over the past few decades, many scientists have shown that fractional models can effectively represent natural phenomena compared to integer-order differential equations. Therefore, fractional calculus has gained more importance and popularity for modeling realistic cases, especially memory effects [

This section demonstrates the formulation of the dysentery diarrhea disease in terms of a deterministic model based on a nonlinear system of ODEs over a finite time interval [0,

The model is designed based on the following assumptions:

Transmission of the dysentery diarrhea disease occurs through multiple pathways.

There is a homogeneously mixed population

Standard incidence is assumed in the human to human interaction

Logistic phenomenon is taken into consideration in the human to environment interaction, which is represented below:

where

After losing immunity, individuals return to

Infected individuals cause concentration of

Rate of recovery of infected humans is

Natural mortality rate of humans is

Non-negativity is assumed for all biological parameters introduced within the model

Hence, after incorporating all the above assumptions and considering the Atangana–Baleanu differential operator taken in the Caputo sense [

subject to the following initial conditions:

where

This section presents the existence and uniqueness of solutions of the proposed model using the techniques of fixed point theory. Here, we denote

where

Thus, the proposed fractional model takes the forms shown below:

We then obtain:

where,

The kernels in

where

Thus,

Repeating the same procedure above, yields:

Subsequently,

where

_{A}_{S}_{1}, _{2}, _{3}, _{4}, _{5}, _{6} from

Hence, the sequences above exist as

In this section, the stability of the ailment-free equilibrium and its analytic conditions will be discussed. From the derivation of the existence of equilibria in [

_{0}

_{0} can be achieved by:

The associated eigenvalues are _{0} is locally asymptotically stable. On the other hand, when

By employing the approach used in [

where

_{1} _{2} _{0} = (^{*}, 0)

For the validation of an epidemiological model, it is extremely important to compare the results of simulations with the actual data of infected individuals. This increases the reliability of the proposed disease model. Similar values from the simulations and actual data give better information on the disease being investigated. In addition, unknown values of the working parameters that contribute to the model can be determined.

There are different techniques including maximum likelihood estimation, Bayesian technique, nonlinear least-squares approach, and probability plotting, which can be used to obtain the best parameters. In this research, we utilized the nonlinear least-squares approach for computing the best-fitted parameters, including

Parameters | Interpretation | ||
---|---|---|---|

Recruitment rate | 6692.3677 | 6692 | |

Transmission rate of disease for human to human interaction | 0.113003 | 1.324811e −01 | |

Transmission rate of disease for environment to human interaction | 0.001013 | 1.007441e −03 | |

200 | 200 | ||

Humans’ natural mortality rate | 0.000457 | 0.000457 | |

Death rate due to disease | 0.05279 | 0.05279 | |

Rate of recovery | 0.094724 | 1.149954e −01 | |

Relapse rate for recovered to susceptible | 2.096175 | 0.08537 | |

Shedding rate of the pathogen | 0.00028 | 2.420341e −04 | |

0.117504 | 1.117099e −01 | ||

ABC fractional order | 1 | 9.9410e −01 |

When the least-squares technique is utilized, we need to minimize the objective function. This is achieved by tuning the system’s parameters to fit the available data points accurately. Real data cases for the dysentery diarrhea disease in this study are denoted by _{n}_{n}

In the present study, we obtained the best fit by measuring the difference between the real data and the simulations. This is shown below:

Finally, the optimal set of parameters is obtained, as shown in

where

In this section, the concept of sensitivity analysis is used to discover the robust significance of the generic parameters present in the base reproduction number

If the dynamics follow the model (1), the analytical expressions can be used to explain the process of tracking the model’s onset at various locations. The threshold value,

where

The numerical values indicating the relative significance of _{0} are given in

Parameter | Baseline value | Elasticity index |
---|---|---|

1.324811 |
0.4585555129 | |

6692 | 0.5355444870 | |

2.420341 |
0.5355444870 | |

1.007441 |
0.5355444870 | |

200 | −0.5387229527 | |

1.117099 |
−0.5355444870 | |

0.05279 | −0.3128781515 | |

1.149954 |
−0.6784363099 | |

0.000457 | −0.5383300266 |

In this section, an algorithm is first developed to obtain the approximate solution of the ABC dysentery diarrhea model, wherein the operator uses non-local and non-singular types of the kernel. The algorithm being developed is discussed in [

This leads to:

At

With the help of interpolation polynomial, we approximate function

By solving the above integrals, we obtain the approximate solution shown below:

Hence, the proposed dysentery diarrhea model becomes:

where,

Using numerical simulations, the ABC dysentery diarrhea model (2) uses the developed algorithm shown above. Different values of the key parameters are chosen to investigate their effects on the dynamics of the disease. These parameter values are taken from

This shows that diarrhea disease is principally due to the environment to human interaction. It means that humans must take care of their hygiene and surroundings in order to avoid the spread of dysentery. In order to investigate the effect of the concentration of

In order to investigate the effect of the pathogen shedding rate of infected humans (

Finally, in order to investigate the effects of different parameters on the basic reproductive number,

In the present research, one of the robust non-local and non-singular fractional operator, called Atangana-Baleanu, was used to model dysentery diarrhea. The employed fractional operator was suitable for the investigation of transmission dynamics of a disease from the literature. The fractionalized order is

Furthermore, in order to shed more light on the features of the model, various numerical simulations were carried out using an effective numerical scheme. In future studies, we plan to apply the techniques used in [

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under Grant No. RG-6-135-40. The authors, therefore, gratefully acknowledge DSR technical and financial support.