**Charge Separation and Electric Field at a Cylindrical Plasma Edge**

Magdi Shoucri

12 Will-be-set-by-IN-TECH

In this work, a sequence of new iteration schemes for solving nonlinear differential equations is used to solve the SIR epidemic model with constant vaccination strategy. The proposed iteration schemes are derived as an extension to the quasi-linearization method to obtain hybrid iteration schemes which converge very rapidly. The accuracy and validity of the proposed schemes is confirmed by comparing with the ode45 MATLAB routine for solving initial value problems. It is hoped that the proposed method of solution will spawn further interest in computational analysis of differential equations in epidemiology and other areas of

*School of Mathematics, Computer Science and Statistics, University of KwaZulu Natal, Private Bag*

*University of Venda, Department of Mathematics, Private Bag X5050, Thohoyandou 0950, South*

[1] R.E. Bellman and R.E. Kalaba, (1965) Quasilinearization and Nonlinear Boundary-Value

[2] C. Canuto, M.Y. Hussaini, A. Quarteroni, and T.A. Zang (1988). Spectral Methods in

[3] C. Chun, (2005) Iterative methods: Improving Newton's method by the Decomposition method, Computers and Mathematics with Applications, Vol 50, pg 1559–1568. [4] H. Khan, R.N. Mohapatra, K.Vajravelu, and S.J. Liao, Explicit series solution of SIR and SIS epidemic models, Appl. Math.Comput, doi:10.1016/j.amc.2009.05.051(2009). [5] W.O. Kermack and A.G. McKendrick,(1927) Contribution to the mathematical theory of

[6] H. W. Hethcote, (2000) The mathematics of infectious diseases, SIAM Review,

[7] R. Casagrandi, L. Bolzoni, S.A. Levin. V. Andreasen, (2006) The SIRC model and

[8] R. Krivec, V.B. Mandelzweig, (2001) Numerical investigation of quasilinearization method in quantum mechanics, Computer Physics Communications Vol 138, pg 69–79 [9] V.B. Mandelzweig, F. Tabakin, (2001) Quasilinearization approach to nonlinear problems in physics with application to nonlinear ODEs, Computer Physics Communications,

[10] V. B. Mandelzweig, (2005) Quasilinearization Method: Nonperturbative Approach to

[11] O. D. Makinde, (2007) Adomian decomposition approach to a SIR epidemic model with

Physical Problems, Physics of Atomic Nuclei, Vol 68(7) pg 1227–1258.

constant vaccination strategy, Appl. Math.Comput, Vol 184, pg 842-848.

[12] Murray, J.D. Mathematical Biology, Springer-Verlag, 3rd.ed., 2001. [13] Trefethen,L.N., (2000) Spectral Methods in MATLAB, SIAM.

**5. Conclusion**

science.

*Africa*

**Author details**

Stanford Shateyi

**6. References**

42(4):599–653.

Vol 141, pg 268–281

Sandile Sydney Motsa

*X01, Scottsville 3209, Pietermaritzburg, South Africa*

Problems, Elsevier, New York.

Fluid Dynamics, Springer-Verlag, Berlin.

epidemics, Proc. Roy. Soc, A115:700-721.

influenza A, Math. BioSc. Vol 200, pg 152–169.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48353
