Analytic algorithms for Some Models of Nonlinear Age–Structured Population Dynamics and Epidemiology

ABSTRACT

Three analytic algorithms based on Adomian decomposition, homotopy perturbation and homotopy analysis methods are proposed to solve some models of nonlinear age-structured population dynamics and epidemiology. Truncating the resulting convergent infinite series, we obtain numerical solutions of high accuracy for these models. Three numerical examples are given to illustrate the simplicity and accuracy of the methods.

Three analytic algorithms based on Adomian decomposition, homotopy perturbation and homotopy analysis methods are proposed to solve some models of nonlinear age-structured population dynamics and epidemiology. Truncating the resulting convergent infinite series, we obtain numerical solutions of high accuracy for these models. Three numerical examples are given to illustrate the simplicity and accuracy of the methods.

KEYWORDS

Age-Structured Population Models, Population Dynamics, SIS Epidemic Models, Adomian Decomposition Method, Homotopy Perturbation and Homotopy Analysis Methods

Age-Structured Population Models, Population Dynamics, SIS Epidemic Models, Adomian Decomposition Method, Homotopy Perturbation and Homotopy Analysis Methods

Cite this paper

nullV. Baranwal, R. Pandey, M. Tripathi and O. Singh, "Analytic algorithms for Some Models of Nonlinear Age–Structured Population Dynamics and Epidemiology,"*Journal of Modern Physics*, Vol. 2 No. 4, 2011, pp. 236-247. doi: 10.4236/jmp.2011.24033.

nullV. Baranwal, R. Pandey, M. Tripathi and O. Singh, "Analytic algorithms for Some Models of Nonlinear Age–Structured Population Dynamics and Epidemiology,"

References

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[2] T. R. Malthus, “An Essay on the Principle of Population, St. Paul’s, London,” 1798, In: T. R. Malthus, “An Essay on the Principle of Population and A Summary View of the Principle of Population,” Penguin, Harmondsworth, England, 1970.

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[5] M. E. Gurtin and R. C. MacCamy, “Nonlinear Age-Dependent Population Dynamics,” Archive for Rational Mechanics and Analysis, Vol. 54, No. 3, 1974, pp. 281- 300.

[6] M. Iannelli, “Mathematical Theory of Age-Structured Population Dynamics,” Applied Mathematics Monographs, Vol. 7, Consiglio Nazionale delle Ricerche, Pisa, 1995.

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[8] X. Y. Li, “Variational Iteration Method for Nonlinear Age-Structured Population Models,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2177-2181. doi:10.1016/j.camwa.2009.03.060

[9] L. M. Abia and J. C. Lopez-Marcos, “Runge-Kutta Methods for Age-Structured Population Models,” Applied Numerical Mathematics, Vol. 17, No. 1, 1995, pp. 1-17. doi:10.1016/0168-9274(95)00010-R

[10] L. M. Abia and J. C. Lopez-Marcos, “On the Numerical Integration of Non-Local Terms for Age-Structured Population Model,” Mathematical Biosciences, Vol. 157, No.1, 1999, pp. 147-167. doi:10.1016/S0025-5564(98)10080-9

[11] L. M. Abia, O. Angulo and J. C. Lopez-Marcos, “Age- Structured Population Models and Their Numerical Solution,” Ecological Modelling, Vol. 188, No. 1, 2005, pp. 112-136. doi:10.1016/j.ecolmodel.2005.05.007

[12] M. Y. Kim and E. J. Park, “An Upwind Scheme for a Nonlinear Model in Age-Structured Population Dynamics,” Computers and Mathematics with Applications, Vol. 30, No. 8, 1995, pp. 5-17. doi:10.1016/0898-1221(95)00132-I

[13] M. Iannelli, M. Y. Kim and E. J. Park, “Splitting Method for the Numerical Approximation of Some Models of Age-Structured Population Dynamics and Epidemiology,” Applied Mathematics and Computation, Vol. 87, No. 1, 1997, pp. 69-93. doi:10.1016/S0096-3003(96)00222-6

[14] M. G. Cui and C. Chen, “The Exact Solution of Nonlinear Age-Structured Population Model,” Nonlinear Analysis: Real World Applications, Vol. 8, No. 4, 2007, pp. 1096-1112. doi:10.1016/j.nonrwa.2006.06.004

[15] P. Krzyzanowski, D. Wrzosek and D. Wit, “Discontinuous Galerkian Method for Piecewise Regular Solution to the Nonlinear Age-Structured Population Model,” Mathematical Biosciences, Vol. 203, No. 2, 2006, pp. 277-300. doi:10.1016/j.mbs.2006.05.005

[16] Norhayati and G. C. Wake, “The Solution and the Stability of a Nonlinear Age-Structured Population Model,” Journal of the Australian Mathematical Society, Vol. 45, 2003, pp. 153-165. doi:10.1017/S1446181100013237

[17] G. Adomian, “A Review of the Decomposition Method in Applied Mathematics,” Journal of Mathematical Analysis and Applications,” Vol. 135, No. 2, 1988, pp. 501-544. doi:10.1016/0022-247X(88)90170-9

[18] G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Boston, 1999.

[19] J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3

[20] S. J. Liao, “Beyond Perturbation: Introduction to Homotopy Analysis Method,” Chapman & Hall/CRC Press, Bosca Raton, December 2003.

[21] M. Dehghan and R. Salehi, “Solution of a Nonlinear Time-Delay Model in Biology via Semi-Analytical Approaches,” Computer Physics Communication, Vol. 181, No. 7, 2010, pp. 1255-1265. doi:10.1016/j.cpc.2010.03.014

[22] S. J. Liao and Y. Tan, “A General Approach to Obtain Series Solutions of Nonlinear Differential Equations,” Studies in Applied Mathematics, Vol. 119, No. 4, 2007, pp. 297-355. doi:10.1111/j.1467-9590.2007.00387.x

[23] S. Busenberg, K. Cooke and M. Iannelli, “Endemic Thresholds and Stability in a Class of Age-Structured Epidemics,” SIAM Journal Applied Mathematics, Vol. 48, No. 6, December 1988, pp. 1379-1395. doi:10.1137/0148085

[24] S. Busenberg, M. Iannelli and H. Thieme, “Global Behaviour of an Age Structured Epidemic Model,” SIAM Journal on Mathematical Analysis, Vol. 22, No. 4, July 1991, pp. 1065-1080. doi:10.1137/0522069

[25] M. Iannelli, F. Milner and A. Pugliese, “Analytical and Numerical Results for the Age Structured SIS Epidemic Model with Mixed Inter-Intracohort Transmission,” SIAM Journal on Mathematical Analysis, Vol. 23, No. 3, May 1992, pp. 662-688. doi:10.1137/0523034

[1] L. Sigler, F. L. Abaci, “A Translation into Modern English of Leonardo Pisano’s Book of Calculation,” Springer-Verlag, New-York, 2002.

[2] T. R. Malthus, “An Essay on the Principle of Population, St. Paul’s, London,” 1798, In: T. R. Malthus, “An Essay on the Principle of Population and A Summary View of the Principle of Population,” Penguin, Harmondsworth, England, 1970.

[3] F. R. Sharpe and A. J. Lotka, “A Problem in Age Distributions,” Philosophical Magzine, Vol. 21, No. 124, 1911, pp. 435-438.

[4] A. G. McKendrick, “Applications of Mathematics to Medical Problems,” Proceedings of Edinburgh Mathematical Society, Vol. 44, 1926, pp. 98-130. doi:10.1017/S0013091500034428

[5] M. E. Gurtin and R. C. MacCamy, “Nonlinear Age-Dependent Population Dynamics,” Archive for Rational Mechanics and Analysis, Vol. 54, No. 3, 1974, pp. 281- 300.

[6] M. Iannelli, “Mathematical Theory of Age-Structured Population Dynamics,” Applied Mathematics Monographs, Vol. 7, Consiglio Nazionale delle Ricerche, Pisa, 1995.

[7] G. F. Webb, “Theory of Nonlinear Age-Dependent Population Dynamics,” Marcel Dekker, New York, January 1985.

[8] X. Y. Li, “Variational Iteration Method for Nonlinear Age-Structured Population Models,” Computers and Mathematics with Applications, Vol. 58, No. 11-12, 2009, pp. 2177-2181. doi:10.1016/j.camwa.2009.03.060

[9] L. M. Abia and J. C. Lopez-Marcos, “Runge-Kutta Methods for Age-Structured Population Models,” Applied Numerical Mathematics, Vol. 17, No. 1, 1995, pp. 1-17. doi:10.1016/0168-9274(95)00010-R

[10] L. M. Abia and J. C. Lopez-Marcos, “On the Numerical Integration of Non-Local Terms for Age-Structured Population Model,” Mathematical Biosciences, Vol. 157, No.1, 1999, pp. 147-167. doi:10.1016/S0025-5564(98)10080-9

[11] L. M. Abia, O. Angulo and J. C. Lopez-Marcos, “Age- Structured Population Models and Their Numerical Solution,” Ecological Modelling, Vol. 188, No. 1, 2005, pp. 112-136. doi:10.1016/j.ecolmodel.2005.05.007

[12] M. Y. Kim and E. J. Park, “An Upwind Scheme for a Nonlinear Model in Age-Structured Population Dynamics,” Computers and Mathematics with Applications, Vol. 30, No. 8, 1995, pp. 5-17. doi:10.1016/0898-1221(95)00132-I

[13] M. Iannelli, M. Y. Kim and E. J. Park, “Splitting Method for the Numerical Approximation of Some Models of Age-Structured Population Dynamics and Epidemiology,” Applied Mathematics and Computation, Vol. 87, No. 1, 1997, pp. 69-93. doi:10.1016/S0096-3003(96)00222-6

[14] M. G. Cui and C. Chen, “The Exact Solution of Nonlinear Age-Structured Population Model,” Nonlinear Analysis: Real World Applications, Vol. 8, No. 4, 2007, pp. 1096-1112. doi:10.1016/j.nonrwa.2006.06.004

[15] P. Krzyzanowski, D. Wrzosek and D. Wit, “Discontinuous Galerkian Method for Piecewise Regular Solution to the Nonlinear Age-Structured Population Model,” Mathematical Biosciences, Vol. 203, No. 2, 2006, pp. 277-300. doi:10.1016/j.mbs.2006.05.005

[16] Norhayati and G. C. Wake, “The Solution and the Stability of a Nonlinear Age-Structured Population Model,” Journal of the Australian Mathematical Society, Vol. 45, 2003, pp. 153-165. doi:10.1017/S1446181100013237

[17] G. Adomian, “A Review of the Decomposition Method in Applied Mathematics,” Journal of Mathematical Analysis and Applications,” Vol. 135, No. 2, 1988, pp. 501-544. doi:10.1016/0022-247X(88)90170-9

[18] G. Adomian, “Solving Frontier Problems of Physics: The Decomposition Method,” Kluwer Academic Publishers, Boston, 1999.

[19] J. H. He, “Homotopy Perturbation Technique,” Computer Methods in Applied Mechanics and Engineering, Vol. 178, No. 3, 1999, pp. 257-262. doi:10.1016/S0045-7825(99)00018-3

[20] S. J. Liao, “Beyond Perturbation: Introduction to Homotopy Analysis Method,” Chapman & Hall/CRC Press, Bosca Raton, December 2003.

[21] M. Dehghan and R. Salehi, “Solution of a Nonlinear Time-Delay Model in Biology via Semi-Analytical Approaches,” Computer Physics Communication, Vol. 181, No. 7, 2010, pp. 1255-1265. doi:10.1016/j.cpc.2010.03.014

[22] S. J. Liao and Y. Tan, “A General Approach to Obtain Series Solutions of Nonlinear Differential Equations,” Studies in Applied Mathematics, Vol. 119, No. 4, 2007, pp. 297-355. doi:10.1111/j.1467-9590.2007.00387.x

[23] S. Busenberg, K. Cooke and M. Iannelli, “Endemic Thresholds and Stability in a Class of Age-Structured Epidemics,” SIAM Journal Applied Mathematics, Vol. 48, No. 6, December 1988, pp. 1379-1395. doi:10.1137/0148085

[24] S. Busenberg, M. Iannelli and H. Thieme, “Global Behaviour of an Age Structured Epidemic Model,” SIAM Journal on Mathematical Analysis, Vol. 22, No. 4, July 1991, pp. 1065-1080. doi:10.1137/0522069

[25] M. Iannelli, F. Milner and A. Pugliese, “Analytical and Numerical Results for the Age Structured SIS Epidemic Model with Mixed Inter-Intracohort Transmission,” SIAM Journal on Mathematical Analysis, Vol. 23, No. 3, May 1992, pp. 662-688. doi:10.1137/0523034