## 1. Introduction

Mathematical epidemic models, for *Salmonella* infections, provide a comprehensive framework for understanding the disease transmission behaviors and for evaluating the effectiveness of different intervention strategies [1, 2]. We recall here that the *Salmonella* infection, a major zoonotic disease, is transmitted between humans and other animals. Reports conducted by the National Center for Emerging and Zoonotic Infectious Diseases (NCEZID) revealed that the number of people infected by *Salmonella*, over the past few years, has remained increasing. The most commonly developed symptoms of *Salmonella* include diarrhea, fever, and abdominal cramps that appear 12–72 hours after infection. The infected people usually recover without medical aid within a period of 4–7 days [3, 4]. However, hospitalization may be needed for some infected people in the case of severe diarrhea. *Salmonella* is found living in the intestinal tracts of not only humans but also other creatures such as birds. The transmission of bacterium to humans occurs through the ingestion of food that has been contaminated with animal feces. These contaminated foods are commonly from an animal source, such as beef, poultry, milk, or eggs [5]. However, vegetables and other foods may also become contaminated. Additionally, foods that have been contaminated are almost impossible to detect while eating, due to their normal taste and smell. Therefore, *Salmonella* is considered as a serious problem for the public health throughout the world. There are no doubts that mathematical modeling of *Salmonella* infection plays an important role in gaining understanding of the transmission of the disease in a specific environment and to predict the behavior of any outbreak. Furthermore, mathematical analysis leads to determining the nature of equilibrium states and to suggest recommended actions to be taken by decision makers to control the spreading of the disease. The objective of this work is to adopt the fractional‐order epidemic model to describe the dynamics of *Salmonella* infections in animal herds.

Fractional‐order (or free‐order) differential models have been successfully applied to system biology, physics, chemistry, and biochemistry, hydrology, medicine, and finance (see, e.g., [6–12] and the references therein). In many cases, they are more contestant with the real phenomena than the integer‐order models, because the fractional derivatives and integrals enable the description of the memory and hereditary properties inherent in various materials and processes. Hence, there is a growing need to study and use the fractional‐order differential and integral equations in epidemiology and biological systems with memory [13]. However, analytical and closed solutions of these types of fractional equations cannot generally be obtained. As a consequence, approximate and numerical techniques are playing an important role in identifying the solution behavior of such fractional equations and exploring their applications (see, e.g., [14–16] and the references therein).

A large number of work done on modeling biological systems have been restricted to integer‐order ordinary (or delay) differential equations (see, e.g., [17–22]). In Ref. [23], the authors proposed the classical *Susceptible‐Infected‐Recovered* (SIR) model. The authors in Ref. [24] introduced a new compartment into the SIR model, which is called cross‐immune compartment to be called SIRC model. The added compartment cross‐immune *Salmonella* infection in animal herds. However, we will take into account the disease‐induced mortality rate

Definitions of fractional‐order integration and fractional‐order differentiation/integration are given in Appendix.

## 2. Construction of the model

Assume that the *Salmonella* infection spreads in animal herds which are grouped as four compartments, according to their infection status:

(1) |

Here **Figure 1**.

### 2.1. Fractional‐order SIRC epidemic model

Most of biological systems have long‐range temporal memory. Modeling of such systems by fractional‐order (or arbitrary order) models provides the systems with long‐time memory and gains them extra degrees of freedom [27]. A large number of mathematical models, based on ordinary and delay differential equations with integer‐orders, have been proposed in modeling the dynamics of epidemiological diseases [18, 20, 28, 29]. In recent years, it has turned out that many phenomena in different fields can be described very successfully by models using *fractional‐order differential equations* (FODEs) [13, 6, 27]. This is due to the fact that fractional derivatives enable the description of the memory and hereditary properties inherent in various processes. Herein, we replace the integer‐order of the model (1) into a fractional‐order (or free‐order) and assume that

(2) |

Here,

When

(The initial conditions

### 2.2. Stability criteria for the epidemic SIRC model (2)

To find the equilibria of the model (2), we put

where

(6) |

The positive endemic equilibrium

(7) |

The Jacobian matrix of the model (2) is

### 2.3. The reproduction number R 0

The basic reproduction number
[1] -

where the matrices

From the model (2), we have

(11) |

Since we have only two distinct stages namely

The following theorem states that

**Theorem 1** *The disease‐free equilibrium is locally asymptotically stable and the infection will die out if* *and is unstable if* *. Conversely, the endemic equilibrium* *is stable when* *and*

*where*

(14) |

and

Proof The disease‐free equilibrium is locally asymptotically stable if all the eigenvalues,

where

The eigenvalues of the Jacobian matrix

Hence

Now, we extend the analysis to endemic equilibrium

with characteristic equation

Using Routh‐Hurwitz stability criteria [31], the endemic equilibrium

This completes the proof.

## 3. Numerical method and simulations

Since most of the FODEs do not have exact analytic solutions, so approximation and numerical techniques must be used. In addition, most of resulting biological systems are stiff, [2] - therefore, efficient use of a reliable numerical method for dealing with such problems is necessary. In this section, we provide an implicit scheme to approximate the solutions of the fractional‐order epidemic model. We also verify that the approximate solution is stable and convergent.

Consider a biological system, with fractional‐order, of the form

Here,

where

**Theorem 2** *The FODE (22) has a unique solution if Lipschitz condition (23) is satisfied and*

Proof One can apply the fractional integral operator (given in the Appendix) to the differential Eq. (22) and incorporate the initial conditions. Thus, Eq. (22) can be expressed as

which is a Volterra equation of the second kind. Define the operator

Then, we have

(27) |

Then, we have

Using the Banach contraction principle, we can prove that that

Many efficient numerical methods have been proposed to solve the FODEs [14, 32]. Among them, the so‐called predictor‐corrector algorithm is a powerful technique for solving the FODEs, and considered as a generalization of the Adams‐Bashforth‐Moulton method. The modification of the Adams‐Bashfourth‐Moulton algorithm is proposed by Diethelm [14, 33–34] to approximate the fractional‐order derivative. However, the converted Volterra integral equation (25) is with a weakly singular kernel, such that regularization is not necessary anymore. In our case, the kernel may not be continuous, and therefore the classical numerical algorithms for the integral part of Eq. (25) are unable to handle the solution of Eq. (22). Therefore, we implement the implicit Euler's scheme to approximate the fractional‐order derivative.

Given fractional‐order model (Eq. (22)) and mesh points *implicit* Euler's approximation as follows (see [15]):

(29) |

Setting

then the first‐order approximation method for the computation of Caputo's fractional derivative is then given by the expression

From the above analysis and numerical approximation, one arrives at the following Remark.

**Remark 1** *The presence of a fractional differential order in a differential equation can lead to a notable increase in the complexity of the observed behavior, and the solution continuously depends on all the previous states.*

### 3.1. Stability and convergence

Here, we prove that the suggested numerical scheme of implicit difference approximation (Eq. (31)) is unconditionally stable. It follows then that the numerical solution converges to the exact solution as

In order to study the stability of the numerical method, let us consider a test problem of linear scaler fractional differential equation

such that

**Theorem 3** *The fully implicit numerical approximation (31), to test problem (32) for all* *, is consistent and unconditionally stable.*

Proof We assume that the approximate solution of Eq. (32) is of the form

Or

Since

Thus, for

Using the inequality (35) and the positivity of the coefficients

Repeating the process, we have from Eq. (36)

Since each term in the summation is negative. Thus

The above numerical technique can then be used both for both linear and nonlinear problems, and it may be extended to multiterm FODEs.

### 3.2. Numerical simulations

The approximate solutions of epidemic model (2) are displayed in **Figures 2**–**4**, and sensitivity of **Figure 5**. The numerical simulations are performed by Euler's implicit scheme discussed in Section 3. We choose different fractional‐order values (**Table 1**. The displayed solutions in **Figure 4** confirm that the fractional order of the derivative plays the role of time‐delay (or memory) in the system.

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

Replacement and exit rate (day | 0.011 | [35] | |

Transmission rate of susceptible to be infected (animal | 0.15 | [35] | |

Recovery rate of infected animals day | 0.16 | Assumed | |

Disease‐induced mortality rate (day | 0.041 | Assumed | |

Cross‐immune period | 0.5 | [36] | |

The average reinfection probability of | 0.06 | Assumed | |

The average time of appearance of new dominant clusters | 1 | Assumed | |

The total number of population | 345 | Assumed |

## 4. Discussion and conclusion

In this chapter, we provided a fractional‐order SIRC epidemic model with *Salmonella* infection. The model provides a comprehensive framework for understanding the disease transmission behaviors, as well as for evaluating the effectiveness of different intervention strategies. We derived the sufficient conditions to preserve the asymptotical stability of disease‐free and endemic steady states. The threshold parameter (reproduction number)

It has been found that fractional‐order dynamical models are more suitable to model biological systems with memory than their integer‐orders. The presence of a fractional differential order into a corresponding differential equation leads to a notable increase in the complexity of the observed behavior. However, fractional‐order differential models are as stable as their integer‐order counterpart.