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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 136263, 9 pages
http://dx.doi.org/10.1155/2014/136263
Research Article

On Fractional SIRC Model with Salmonella Bacterial Infection

1Department of Mathematical Sciences, College of Science, UAE University, P.O. Box 15551, Al Ain, UAE
2Faculty of Science, Helwan University, Cairo, Egypt
3Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 80204, Saudi Arabia
4Department of Mathematics and Computer Sciences, Cankaya University, 06530 Ankara, Turkey
5Institute of Space Sciences, P.O. Box MG-23, 76900 Magurele-Bucharest, Romania
6Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu 641046, India

Received 3 March 2014; Accepted 25 March 2014; Published 22 April 2014

Academic Editor: Ali H. Bhrawy

Copyright © 2014 Fathalla A. Rihan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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., [110] 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., [1419]). 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 [2325]. 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.

136263.fig.001
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 ifHowever, 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.

136263.fig.002
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.

Of course this numerical technique can be used both for linear and for nonlinear problems, and it may be extended to multiterm FODEs.

3.2. Numerical Simulations

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

tab1
Table 1: List of parameters.
136263.fig.003
Figure 3: Numerical simulation of fractional order epidemic model (6), when , and (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.
fig4
Figure 4: Phase plane portrait for fractional order endemic model (6), in absence of and components, when (a) and (b) with . We note that solution paths approach the disease-free equilibrium .
fig5
Figure 5: Phase plane portrait for classic fractional order endemic model (6) when (a) and (b) with . We note that solution paths approach the endemic equilibrium given by (7).
fig6
Figure 6: Sensitivity of with respect to the transmission coefficients and .

4. 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) 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 integer-order counterpart steady states, when .

Conflict of Interests

The authors declare that they have no competing interests in this paper.

Acknowledgments

This work was supported by UAE University (UAEU-NRF-7-20886). The authors would like to thank Dr. Ali Bhrawy for his valuable comments.

References

  1. F. A. Rihan, “Numerical modeling of fractional-order biological systems,” Abstract and Applied Analysis, vol. 2013, Article ID 816803, 11 pages, 2013. View at Google Scholar · View at MathSciNet
  2. E. Ahmed, A. Hashish, and F. A. Rihan, “On fractional order cancer model,” Fractional Calculus and Applied Analysis, vol. 3, no. 2, pp. 1–6, 2012. View at Google Scholar
  3. F. Mainardi and R. Gorenflo, “On Mittag-Leffler-type functions in fractional evolution processes,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 283–299, 2000. View at Google Scholar
  4. W.-C. Chen, “Nonlinear dynamics and chaos in a fractional-order financial system,” Chaos, Solitons and Fractals, vol. 36, no. 5, pp. 1305–1314, 2008. View at Google Scholar
  5. L. Debnath, “Recent applications of fractional calculus to science and engineering,” International Journal of Mathematics and Mathematical Sciences, no. 54, pp. 3413–3442, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  6. A. M. A. El-Sayed, “Nonlinear functional-differential equations of arbitrary orders,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal A: Theory and Methods, vol. 33, no. 2, pp. 181–186, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  7. A. M. A. El-Sayed, A. E. M. El-Mesiry, and H. A. A. El-Saka, “On the fractional-order logistic equation,” Applied Mathematics Letters. An International Journal of Rapid Publication, vol. 20, no. 7, pp. 817–823, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  8. E. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, River Edge, NJ, USA, 2000.
  9. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus, Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, 2012, Models and numerical methods. View at Publisher · View at Google Scholar · View at MathSciNet
  10. W. Lin, “Global existence theory and chaos control of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 709–726, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  11. K. Diethelm, N. J. Ford, and A. D. Freed, “A predictor-corrector approach for the numerical solution of fractional differential equations,” Nonlinear Dynamics. An International Journal of Nonlinear Dynamics and Chaos in Engineering Systems, vol. 29, no. 1–4, pp. 3–22, 2002, Fractional order calculus and its applications. View at Publisher · View at Google Scholar · View at MathSciNet
  12. F. A. Rihan, “Computational methods for delay parabolic and time-fractional partial differential equations,” Numerical Methods for Partial Differential Equations. An International Journal, vol. 26, no. 6, pp. 1556–1571, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  13. P. Bouvet, “Human Salmonellosis surveillance in france: recent data from the national reference center,” Salmonella and Salmonellosis, pp. 411–416, 2002. View at Google Scholar
  14. L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhäuser Mathematics Series, Random House, New York, NY, USA, 1988. View at MathSciNet
  15. D. S. Jones, M. J. Plank, and B. D. Sleeman, Differential Equations and Mathematical Biology, Mathematical & Computational Biology, 2008.
  16. D. Kaplan and L. Glass, Understanding nonlinear dynamics, vol. 19 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  17. J. D. Murray, Mathematical Biology. II, Interdisciplinary Applied Mathematics, Springer, 3rd edition, 2003, Spatial models and biomedical applications. View at MathSciNet
  18. M. Safan and F. A. Rihan, “Mathematical analysis of an SIS model with imperfect vaccination and backward bifurcation,” Mathematics and Computers in Simulation, vol. 96, pp. 195–206, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  19. F. A. Rihan, Numerical Treatment of Delay Differential Equations in Bioscience [Ph.D. thesis], University of Manchester, 2000.
  20. W. O. Kermack and A. G. McKendrick, “Contributions to the mathematical theory of epidemics,” Proceedings of Royal Society of London, vol. 115, no. 722, pp. 700–721, 1927. View at Publisher · View at Google Scholar
  21. R. Casagrandi, L. Bolzoni, S. A. Levin, and V. Andreasen, “The SIRC model and influenza A,” Mathematical Biosciences, vol. 200, no. 2, pp. 152–169, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  22. M. El-Shahed and A. Alsaedi, “The fractional SIRC model and influenza A,” Mathematical Problems in Engineering, vol. 2011, Article ID 480378, 9 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  23. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  24. I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, 1999, An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. View at MathSciNet
  25. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, 1993. View at MathSciNet
  26. Z. M. Odibat and N. T. Shawagfeh, “Generalized Taylor's formula,” Applied Mathematics and Computation, vol. 186, no. 1, pp. 286–293, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  27. L. Jódar, R. J. Villanueva, A. J. Arenas, and G. C. González, “Nonstandard numerical methods for a mathematical model for influenza disease,” Mathematics and Computers in Simulation, vol. 79, no. 3, pp. 622–633, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  28. H. Xu, “Analytical approximations for a population growth model with fractional order,” Communications in Nonlinear Science and Numerical Simulation, vol. 14, no. 5, pp. 1978–1983, 2009. View at Publisher · View at Google Scholar
  29. O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,” Journal of Mathematical Biology, vol. 28, no. 4, pp. 365–382, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  30. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” Computational Engineering in System Application, vol. 2, pp. 963–968, 1996. View at Google Scholar
  31. E. Ahmed, A. M. A. El-Sayed, and H. A. A. El-Saka, “On some Routh-Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems,” Physics Letters A, vol. 358, no. 1, pp. 1–4, 2006. View at Publisher · View at Google Scholar · View at MathSciNet
  32. T. Suzuki, “A generalized Banach contraction principle that characterizes metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  33. R. Anguelov and J. M.-S. Lubuma, “Nonstandard finite difference method by nonlocal approximation,” Mathematics and Computers in Simulation, vol. 61, no. 3–6, pp. 465–475, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  34. K. Diethelm, “An algorithm for the numerical solution of differential equations of fractional order,” Electronic Transactions on Numerical Analysis, vol. 5, pp. 1–6, 1997. View at Google Scholar · View at MathSciNet
  35. K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  36. P. Chapagain, J. S. Van Kessel, J. K. Karns, et al., “A mathematical model of the dynamics of Salmonella cerro infection in a us dairy herd,” Epidemiology & Infection, vol. 136, pp. 263–272, 2008. View at Google Scholar