Abstract

We propose a fractional order SIRC epidemic model to describe the dynamics of Salmonella bacterial infection in animal herds. The infection-free and endemic steady sates, of such model, are asymptotically stable under some conditions. The basic reproduction number is calculated, using next-generation matrix method, in terms of contact rate, recovery rate, and other parameters in the model. The numerical simulations of the fractional order SIRC model are performed by Caputo’s derivative and using unconditionally stable implicit scheme. The obtained results give insight to the modelers and infectious disease specialists.

1. Introduction

During the past three decades, the subject of fractional calculus has gained popularity and importance, mainly due to its demonstrated applications in numerous diverse and widespread fields of science and engineering. For example, fractional calculus has been successfully applied to system biology, physics, chemistry and biochemistry, hydrology, medicine, and finance (see, e.g., [1–10] and the references therein). In many cases, the fractional order differential/integral equations models are more consistent 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. However, analytical and closed solutions of these types of fractional equations cannot generally be obtained. As a consequence, approximate and numerical techniques are playing important role in identifying the solution behavior of such fractional equations and exploring their applications (see, e.g., [9, 11, 12] and the references therein).

We recall that the Salmonella infection is a major zoonotic disease which is transmitted between humans and other animals. Most persons infected with Salmonella develop diarrhea, fever, and abdominal cramps 12 to 72 hours after infection. The illness usually lasts 4 to 7 days, and most persons recover without treatment. However, in some persons, the diarrhea may be so severe that the patient needs to be hospitalized. Salmonella live in the intestinal tracts of humans and other animals, including birds. Salmonella are usually transmitted to humans by eating foods contaminated with animal feces. Contaminated foods usually look and smell normal. Contaminated foods are often of animal origin, such as beef, poultry, milk, or eggs, but any food, including vegetables, may become contaminated [13]. Therefore, Salmonella is considered as a serious problem for the public health throughout the world. There are no doubts that mathematical modeling of Salmonella bacterial infection plays an important role in gaining understanding of the transmission of the disease in specific environment and in predicting the behavior of any outbreak. Furthermore, mathematical analysis leads to determining the nature of equilibrium states and to suggesting 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.

A large amount of work done on modelling biological systems has been restricted to integer-order ordinary (or delay) differential equations (see, e.g., [14–19]). In [20], the authors proposed the classical Susceptible-Infected-Recovered (SIR) model. The authors in [21] introduced a new compartment into SIR model, which is called cross-immune compartment to be called SIRC model. The new compartment cross-immune describes an intermediate state between the fully susceptible and the fully protected one. Recently, the fractional order SIRC model of influenza, a disease in human population, was discussed in [22]. In the present paper, we consider the fractional order SIRC model associated with evolution of Salmonella bacterial infection in animal herds. However, we will take into account the disease-induced mortality rate in the model. Qualitative behavior of the fractional order SRIC model is then investigated. Numerical simulations of the fractional order SRIC model are provided to demonstrate the effectiveness of the proposed method by using implicit Euler’s method.

We first give the definition of fractional order integration and fractional order differentiation [23–25]. Let be the class of Lebesgue integrable functions on , .

Definition 1. The fractional integral of order of the function , () is defined by However, the fractional derivative of order of is defined by two ways.(i)Riemann-Liouville fractional derivative: take fractional integral of order and then take th derivative: (ii)Caputo’s fractional derivative: take th derivative and then take a fractional integral of order :

We notice that the definition of time-fractional derivative of a function at involves an integration and calculating time-fractional derivative that requires all the past history, that is, all the values of from to . Caputo’s definition, which is a modification of the Riemann-Liouville definition, has the advantage of dealing properly with initial value problems. For more properties of the fractional derivatives and integrals, we refer to [8, 9, 24, 25] and references therein. The generalized mean value theorem is defined in the following Remark [26].

Remark 2. (i) Suppose that and for ; then we have
(ii) If (i) holds and , then is nondecreasing for each . If , then is nonincreasing for each .

2. The SIRC Epidemic Model

Assume that the Salmonella infection spreads in animal herds which are grouped as four compartments, according to their infection status: is the proportion of susceptible individuals at time (individuals that do not have the bacterial infection), is the proportion of infected individuals (that have the bacterial infection), is the proportion of recovered individuals (that recovered from the infection and have temporary immunity), and is the proportion of cross-immune individuals at time . The total number of animals in the herd is given by . We consider that initially all the animals are susceptible to the infection. Once infected, a susceptible individual leaves the susceptible compartment and enters the infectious compartment where it then becomes infectious. The infected animals pass into the recovered compartment. The individuals who have recovered from the disease have temporary immunity and grouped into compartment. Therefore, we consider that the disease transmission model consists of nonnegative initial conditions together with system of equations: Here, parameter denotes the mortality rate in every compartment and is assumed to be equal to the rate of newborns in the population. is the contact rate and also called transmission from susceptible to infected. is the cross-immune period, is the infectious period, is the total immune period, and is the fraction of the exposed cross-immune individuals who are recruited in a unit time into the infective subpopulation [21, 27]. We also assume that the disease induces mortality rate ; see the diagram of Figure 1.

Figure 1 

Schematic diagram of SIRC model.

2.1. Fractional Order of SIRC Epidemic Model

Although a large number of work has been done in modeling the dynamics of epidemiological diseases, it has been restricted to integer-order (delay) differential equations. 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) [1, 5, 28]. Now, we introduce fractional order into model (5) and assume that , , , , where is the total number of population. Then the model takes the form with initial conditions , , and .

2.2. Stability Criteria for the Fractional Order SIRC Model

In model (6), assume that . Then, to find the equilibria, we put .

Consider where The positive endemic equilibrium satisfies (6) and is the positive root of , where The Jacobian matrix of model (6) is

2.3. The Reproduction Number

The basic reproduction number (the number of individuals infected by a single infected individual placed in a totally susceptible population) that includes the indirect transmission may be obtained using next-generation matrix method. The spectral radius of the next-generation matrix , which is the dominant eigenvalue of the same matrix, gives the value of [29]. Then, the basic reproductive number is obtained by the form where the matrices and . , where is the set of all disease free states in the compartment , is the rate of appearance of new infections in compartment , and is net transfer rate (other than infections) of compartment . The net transfer rate is given by , where is the rate of transfer of individuals out of compartment , and is the rate of transfer of individuals into compartment by all other means. Therefore, the disease transmission model consists of nonnegative initial conditions, , together with the following system of equations: From model (6), we have Since we have only two distinct stages, namely, and , it follows that both and are square matrices. Further, it can be noticed that is nonnegative and is nonsingular. The basic reproductive number is the dominant eigenvalue of the matrix , which is obtained by solving the characteristic equation , where is the eigenvalue and is the identity matrix. At the disease-free equilibrium , we have The following theorem states that is a threshold parameter for the stability of model (6).

Theorem 3. The disease-free equilibrium is locally asymptotically stable and the infection will die out if and is unstable if .

Proof. The disease-free equilibrium is locally asymptotically stable if all the eigenvalues, , , of Jacobian matrix satisfy condition [30]: where Figure 2 depicts the stability region of the fractional order system, according to condition (15). The eigenvalues of Jacobian matrix are , , , . Hence, is locally asymptotically stable if and is unstable if .
Now, we extend the analysis to endemic equilibrium . Jacobian matrix evaluated at the endemic equilibrium iswith characteristic equation where If denotes the discriminant of the polynomial: , then denote From [31], we have the proposition.
Proposition 4. Assume that exists in .(1) Let be the Routh-Hurwitz determinants: , , . Therefore, when , the equilibrium point is locally asymptotically stable if However, conditions (21) are sufficient (not necessary) conditions for to be locally asymptotically stable for all .(2) If , , , and , then the equilibrium point is unstable.(3) If , , , , , and , then the equilibrium is locally asymptotically stable. Also, if , , , , , then the equilibrium point is unstable.(4) If , , , , , and , then the equilibrium point is locally asymptotically stable for all . (5) is the necessary condition for the equilibrium point to be locally asymptotically stable.

Figure 2 

Stability region of the fractional order system with .

3. Implicit Euler’s Scheme for FODEs

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 (one definition of the stiffness is that the global accuracy of the numerical solution is determined by stability rather than local error and implicit methods are more appropriate for it). The stiffness often appears due to the differences in speed between the fastest and slowest components of the solutions and stability constraints. In addition, the state variables of these types of models are very sensitive to small perturbations (or changes) in the parameters which occur in the model. Therefore, efficient use of a reliable numerical method for dealing with stiff problems is necessary.

Consider the following fractional order differential equation: Here, and satisfy the Lipschitz condition in variable : where is the solution of the perturbed system.

Theorem 5. Problem (22) has a unique solution provided that Lipschitz condition (23) is satisfied and .

Proof. Using the definitions of Section 1, we can apply a fractional integral operator to the differential equation (22) and incorporate the initial conditions, thus converting the equation into the equivalent equation: which also is a Volterra equation of the second kind. Define operator , such that Then, we have So, we obtain By Banach contraction principle [32], we can deduce that has a unique fixed point which implies that our problem has a unique solution.

Several numerical methods have been proposed to solve the FODEs [11, 33]. Recently, the predictor-corrector algorithm is an efficient and powerful technique for solving the FODEs, which is a generalization of the Adams-Bashforth-Moulton method. The modification of Adams-Bashforth-Moulton algorithm is proposed by Diethelm [34, 35] to approximate the fractional order derivative. However, converted Volterra integral equation (24) is with a weakly singular kernel, such that a regularization is not necessary any more. It seems that there exist only a very small number of software packages for nonlinear Volterra equations. In our case, the kernel may not be continuous, and therefore the classical numerical algorithms for the integral part of (24) are unable to handle the solution of (22). Therefore, we implement the implicit Euler’s scheme to approximate the fractional order derivative.

Given model (22) and mesh points , such that and , then a discrete approximation to the fractional derivative can be obtained by a simple quadrature formula, using Caputo’s fractional derivative (3) of order , and using implicit Euler’s approximation as follows (see [12]): Setting then the first-order approximation method for the computation of Caputo’s fractional derivative is then given by the expression From the analysis and numerical approximation, we also arrive at the following proposition.

Proposition 6. 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

We here prove that fractional order implicit difference approximation (30) 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 , and , are constants.

Theorem 7. The fully implicit numerical approximation (30), to test problem (31) for all , is consistent and unconditionally stable.

Proof. We assume that the approximate solution of (31) is of the form ; then (31) can be reduced to or Since , for all , then Thus, for , the above inequality implies Using relation (34) and the positivity of the coefficients , we get Repeating the process, we have, from (35), since each term in the summation is negative. Thus, . With the assumption that , which entails , we have stability.