**7. References**


[4] Esquivel-Flores, O., Benítez-Pérez, H., Méndez Monroy, E. & Menéndez, A. [2010]. Efficient overloading techniques for primary-backup scheduling in real-time systems, *ICI Express Letters Part B: Applications* 1(1): 93–98.

**On New High Order Iterative Schemes for Solving**

Most problems arising from mathematical epidemiology are often described in terms of differential equations. However, it is often very difficult to obtain closed form solutions of such equations, especially those that are nonlinear. In most cases, attempts are made to obtain only approximate or numerical solutions. In this work, we revisit the SIR epidemic model with constant vaccination strategy that was considered in [11], where the Adomian decomposition method was used to solve the governing system of nonlinear initial value

In this work we develop new accurate iterative schemes which are based on extending Taylor series based linearization method to obtain accurate and fast converging sequence of hybrid iteration schemes. At first order, the hybrid iteration scheme reduces to quasilinearization method (QLM) which was originally developed in [1]. More recently Mandelzweig and his co-workers [8–10] have extended the application of the QLM to a wide variety of nonlinear BVPs and established that the method converges quadratically. In this work we demonstrate that the proposed hybrid iteration schemes are more accurate and converge faster than the

To implement the method we consider the SIR model that describes the temporal dynamics of a childhood disease in the presence of a preventive vaccine. In SIR models the population is assumed to be divided into the standard three classes namely, the susceptibles (*S*), who can catch the infection but are so far uninfected, the infectives (*I*), those who have the disease and can transmit it to the susceptibles, and the removed (*R*), who have either died or who have

*S I*

©2012 Motsa and Shateyi, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

*<sup>N</sup>* <sup>−</sup> *<sup>μ</sup>S*, (1)

**Chapter 4**

*<sup>N</sup>* <sup>−</sup> (*<sup>κ</sup>* <sup>+</sup> *<sup>μ</sup>*)*I*, (2)

*dt* = (<sup>1</sup> <sup>−</sup> *<sup>P</sup>*)*π<sup>N</sup>* <sup>−</sup> *<sup>β</sup>*

and reproduction in any medium, provided the original work is properly cited.

**Initial Value Problems in Epidemiology**

Sandile Motsa and Stanford Shateyi

http://dx.doi.org/10.5772/48264

**1. Introduction**

differential equations.

QLM approach.

recovered and are therefore immune.

The governing equations for the problem are described [11] by *dS*

> *dI dt* <sup>=</sup> *<sup>β</sup> S I*

cited.

Additional information is available at the end of the chapter

