On Fractional SIRC Model with Salmonella Bacterial Infection

and Applied Analysis 3


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][2][3][4][5][6][7][8][9][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 However, the fractional derivative of order  ∈ ( − 1, ) 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  = 0 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].

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. −1 is the crossimmune period,  −1 is the infectious period,  −1 is the total immune period, and  is the fraction of the exposed crossimmune 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.

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 (0) =  0 , (0) =  0 , and (0) =  0 .

Stability Criteria for the Fractional Order SIRC Model.
In model ( 6), assume that  1 =  2 =  3 =  4 = .Then, to find the equilibria, we put    =    =    =    = 0. Consider where The positive endemic equilibrium E + = ( * ,  * ,  * ,  * ) satisfies ( 6) and  * is the positive root of  1  * 2 +  2  * +  3 , where The Jacobian matrix of model ( 6) is ) . (10) 2.3.The Reproduction Number R 0 .The basic reproduction number (the number of individuals infected by a single infected individual placed in a totally susceptible population) R 0 that includes the indirect transmission may be obtained using next-generation matrix method.The spectral radius of the next-generation matrix ( −1 ), which is the dominant eigenvalue of the same matrix, gives the value of R 0 [29].
Then, the basic reproductive number R 0 is obtained by the form where the matrices  = [F  ()/  ] = 0 and  = [V  ()/  ] = 0 .F  (), 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 V  = V −  − V +  , where V −  is the rate of transfer of individuals out of compartment , and V +  is the rate of transfer of individuals into compartment  by all other means.Therefore, the disease transmission model consists of nonnegative initial conditions,   (0), together with the following system of equations: From model ( 6), we have dominant eigenvalue of the matrix  −1 , which is obtained by solving the characteristic equation ( −1 ) − Λ = 0, where Λ is the eigenvalue and () is the identity matrix.At the disease-free equilibrium E 0 = (1, 0, 0, 0), we have The following theorem states that R 0 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 R 0 < 1 and is unstable if R 0 > 1.
Proof.The disease-free equilibrium is locally asymptotically stable if all the eigenvalues,   , = 1, 2, 3, 4, of Jacobian matrix (E 0 ) satisfy condition [30]: where ) . ( Figure 2 depicts the stability region of the fractional order system, according to condition (15).The eigenvalues of Jacobian matrix Now, we extend the analysis to endemic equilibrium E + .Jacobian matrix (E + ) evaluated at the endemic equilibrium is with characteristic equation where If (Φ) denotes the discriminant of the polynomial: From [31], we have the proposition.
(5)  4 > 0 is the necessary condition for the equilibrium point E + to be locally asymptotically stable.

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: where () is the solution of the perturbed system.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 L, such that Then, we have By Banach contraction principle [32], we can deduce that L 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 T = { 0 ,  1 , . . .,   }, such that  0 = 0 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 , 0 <  ≤ 1 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.6), when  = 0.8, and R 0 > 1 (each infected individual infects more than one other member of the population and a self-sustaining group of infectious individuals will propagate), with parameter values of Table 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 ℎ → 0. In order to study the stability of the numerical method, let us consider a test problem of linear scaler fractional differential equation such that 0 <  ≤ 1, and  0 < 0,  1 > 0 are constants.
Proof.We assume that the approximate solution of ( 31) is of the form (  ) ≈   ≡   ; then (31) can be reduced to or Since (1 − ( 0 /G ,ℎ )) ≥ 1, for all G ,ℎ , then Figure 4: Phase plane portrait for fractional order endemic model (6), in absence of () and () components, when  = 0.7 (a) and  = 0.9 (b) with R 0 = 0.5 < 1.We note that solution paths approach the disease-free equilibrium E 0 = (1, 0, 0).Thus, for  = 2, the above inequality implies Using relation (34) and the positivity of the coefficients  2 , we get Repeating the process, we have, from (35), Of course this numerical technique can be used both for linear and for nonlinear problems, and it may be extended to multiterm FODEs.

Numerical Simulations.
The approximate solutions of epidemic model (6) are displayed in Figures 3, 4, and 5, and sensitivity of R 0 to transmission coefficients is displayed in Figure 6.The numerical simulations are performed by Euler's implicit scheme.We choose commensurate fractional order that  1 =  2 =  3 =  4 = , with different fractional order values and the parameter values given in Table 1.

Conclusions
In this paper, we provided a fractional order SIRC epidemic model with Salmonella bacteria infection.We derived the sufficient conditions to preserve the asymptotic stability of infection-free and endemic steady states.The threshold parameter (reproduction number) R 0 has been evaluated in terms of contact rate, recovery rate, and other parameters in the model.We provided unconditionally stable method, using Euler's implicit method for the fractional order differential system.The solution of a fractional order model at any time  * continuously depends on all the previous states at  ≤  * .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 and enlarges the stability region of the solutions.However, fractional order differential models have the same integerorder counterpart steady states, when  > 0.5.

8 Figure 3 :
Figure3: Numerical simulation of fractional order epidemic model(6), when  = 0.8, and R 0 > 1 (each infected individual infects more than one other member of the population and a self-sustaining group of infectious individuals will propagate), with parameter values of Table1.

Table 1 :
List of parameters.