×

zbMATH — the first resource for mathematics

The explicit series solution of SIR and SIS epidemic models. (English) Zbl 1171.92033
Summary: SIR and SIS epidemic models in biology are solved by means of an analytic technique for nonlinear problems, namely the homotopy analysis method (HAM). Both of the SIR and SIS models are described by coupled nonlinear differential equations. A one-parameter family of explicit series solutions is obtained for both models. This parameter has no physical meaning but provides us with a simple way to ensure convergent series solutions to the epidemic models. Our analytic results agree well with the numerical ones. This analytic approach is general and can be applied to get convergent series solutions of some other coupled nonlinear differential equations in biology.

MSC:
92D30 Epidemiology
37N25 Dynamical systems in biology
34A34 Nonlinear ordinary differential equations and systems, general theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kermack, W.O.; McKendrick, A.G., Contribution to the mathematical theory of epidemics, Proc. roy. soc., A115, 700-721, (1927) · JFM 53.0517.01
[2] Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J., On the definition and computation of the basic reproductive ratio in models for infectious diseases in heterogeneous population, J. math. biol., 28, 365-382, (1990) · Zbl 0726.92018
[3] Hethcote, H.W., The mathematics of infectious diseases, SIAM rev., 42, 599-653, (2000) · Zbl 0993.92033
[4] West, R.W.; Thompson, J.R., Models for the simple epidemic, Math. biosci., 141, 29-39, (1997) · Zbl 0881.92030
[5] Nucci, M.C.; Leach, P.G.L., An integrable SIS model, J. math. anal. appl., 290, 506-518, (2004) · Zbl 1067.92054
[6] Zhien, Ma.; Liu, J.; Li, J., Stability analysis for differential infectivity epidemic models, Nonlinear anal.-real., 4, 841-856, (2003) · Zbl 1025.92012
[7] Jing, H.; Zhu, D., Global stability and periodicity on SIS epidemic models with backward bifurcation, Int. J. comput. math., 50, 1271-1290, (2005) · Zbl 1078.92058
[8] Korobeinikov, A.; Wake, G.C., Lyapunov functions and global stability for SIR, SIRS and SIS epidemiological models, Appl. math. lett., 15, 955-960, (2002) · Zbl 1022.34044
[9] Garry, J.S., Fluctuations in a dynamic, intermediate-run IS-LM model applications of the poincar – bendixon theorem, J. econom. theory, 15, 369-375, (1982) · Zbl 0494.90017
[10] Yicang, Z.; Liu, H., Stability of periodic solutions for an SIS model with pulse vaccination, Math. comput. model., 38, 299-308, (2003) · Zbl 1045.92042
[11] Pietro, G.C., How mathematical models have helped to improve understanding the epidemiology of infection, Early hum. dev., 83, 141-148, (2007)
[12] D.B. Meade, Qualitative analysis of an epidemic model with directed disprsion, IMA preprint series, 1992.
[13] N. Singh, Epidemiological models for mutating pathogen with temporary immunity, Ph.D. Dissertation (in English), University of Central Florida, Orlando, FL, 2006.
[14] Murray, J.D., Mathematical biology, (1993), Springer-Verlag · Zbl 0779.92001
[15] S.J. Liao, The proposed homotopy analysis techniques for the solution of nonlinear problems, Ph.D. Dissertation (in English), Shanghai Jiao Tong University, Shanghai, 1992.
[16] Liao, S.J., Beyond perturbation-introduction to the homotopy analysis method, (2003), Chapman & Hall/CRC Boca Raton
[17] Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl. math. comput., 147, 2, 499-513, (2004) · Zbl 1086.35005
[18] Liao, S.J., A kind of approximate solution technique which does not depend upon small parameters: a special example, Int. J. non-linear mech., 30, 371-380, (1995)
[19] Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, stud, Appl. math., 119, 297-354, (2007)
[20] Zhu, S.P., An exact and explicit solution for the valuation of American put option, Quant. finan., 3, 229-242, (2006) · Zbl 1136.91468
[21] M. Sajid, T. Hayat, S. Asghar, Comparison between the HAM and HPM solutions of tin film flows of non-Newtonian fluids on a moving belt, Nonlinear Dyn., doi: 10.1007/S11071-006-91400-y. · Zbl 1181.76031
[22] Abbasbandy, S., The application of the homotopy analysis method to solve a generalized hirota – satsuma coupled KdV equation, Phys. lett. A, 361, 478-483, (2007) · Zbl 1273.65156
[23] Khan, H.; Xu, H., Series solution to thomas – fermi equation, Phys. lett. A, 365, 111-115, (2007) · Zbl 1203.81060
[24] Abbasbandy, S., Homotopy analysis method for heat radiation equations, Int. commun. heat mass transfer, 34, 380-387, (2007)
[25] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Int. J. eng. sci., 42, 123-135, (2004) · Zbl 1211.76009
[26] Abbas, Z.; Sajid, M.; Hayat, T., MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel, Theor. comput. fluid dyn., 20, 229-238, (2006) · Zbl 1109.76065
[27] Shi, Y.R.; Xu, X.J.; Wu, Z.X., Application of the homotopy analysis method to solving nonlinear evolution equations, Acta phys. sin., 55, 1555-1560, (2006) · Zbl 1202.65130
[28] Hayat, T.; Sajid, M., On analytic solution for thin film flow of a forth grade fluid down a vertical cylinder, Phys. lett. A, 361, 316-322, (2007) · Zbl 1170.76307
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.