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We study the well-known SIR (susceptible, infected, recoverd) model with nonlinear complex incidence rates. Firstly,a series of coordinate transformations are carried out to change the equations as the amenable Hamiltonian systems. Secondly the Melnikov's method is used to establish the conditions of existence of chaotic motion and find the analytically critical values of homoclinic bifurcation. Good agreement can be found between numerical results and analytical results.
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Keywords:
- SIR(susceptible /
- infected /
- recoverd) model /
- chaotic motion /
- Melnikovs method /
- homoclinic bifurcation
[1] Kermack W O, McKendrick A G 1927 Proc. Roy. Soc. A 115 700
[2] Anderson R M, May R M 1991 Infectious Diseases of Humans (Oxford: Oxford University Press)
[3] Aron J L, Schwartz I B 1984 Theor. Biol. 110 665
[4] Kuznetsov, Yu A, Piccardi C 1994 Math. Biol. 32 109
[5] Olsen L F, Shaffer W M 1990 Science 249 499
[6] Hethcote H W, Driessche P V D 1991 Math. Biol. 29 271
[7] Liu W M, Hethcote H W, Levin S A 1987 Math. Biol. 25 359
[8] Liu W M, Levin S A, Iwasa Y 1986 Math. Biol. 23 187
[9] Yi H T 2009 MS dissertation (Hunan: Hunan university) (in Chinese) [尹海涛 2009 硕士学位论文(湖南:湖南大学)]
[10] Li J 2005 MS dissertation (Nanjing: Nanjing university of science and technology) (in Chinese) [李 静 2005 硕士学位论文 (南京:南京理工大学)] 〖11] Ni S J, Weng W G, Fan W C 2009 Acta Phys. Sin. 58 3707 (in Chinese) [倪顺江、翁文国、范维澄 2009 物理学报 58 3707]
[11] Liu M X, Ruan J 2009 Chin. Phys. B 18 2115
[12] Liu M X, Ruan J 2009 Chin. Phys. B 18 5111
[13] Shi H J, Duan Z S, Chen G R, Li R 2009 Chin. Phys. B 18 3309
[14] Pei W D, Chen Z Q, Yuan Z Z 2008 Chin. Phys. B 17 0373
[15] Zhong L, Weng J Q 2000 Acta Phys. Sin. 49 0626 (in Chinese) [钟 玲、翁甲强 2000 物理学报 49 0626]
[16] Xu D, Li X, Wang X F 2007 Acta Phys. Sin. 56 1313 (in Chinese) [许 丹、李 翔、汪小帆 2007 物理学报 56 1313]
[17] Glendinning P, Perry L P 1997 Math. Biol. 34 359
[18] Chow S N, Li C Z, Wang D 1994 Normal forms and bifurcation of plannar vector fields (Cambridge: Cambridge University Press)
[19] Yang X L, Xu W, Sun Z K 2006 Acta Phys. Sin. 55 1678 (in Chinese) [杨晓丽、徐 伟、孙中奎 2006 物理学报 55 1678]
[20] Liu Z R 2002 The analytical method of chaos (Shanghai:Shanghai university press) p56 (in Chinese) [刘增荣 2002 混沌研究中的解析方法(上海:上海大学出版社)第56页]
[21] Lei Y M, Xu W 2007 Acta Phys. Sin. 56 5103 (in Chinese) [雷佑銘、徐 伟 2007 物理学报 56 5103]
[22] Guckenheimer J, Holmes P 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vectors Fields (New York: New York Springer)
[23] Yang X L, Xu W 2009 Acta Phys Sin. 58 3722 (in Chinese) [杨晓丽、徐 伟 2009 物理学报 58 3722]
[24] Baesens N C, Nicolis G 1983 Z. Phys. B 52 345
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[1] Kermack W O, McKendrick A G 1927 Proc. Roy. Soc. A 115 700
[2] Anderson R M, May R M 1991 Infectious Diseases of Humans (Oxford: Oxford University Press)
[3] Aron J L, Schwartz I B 1984 Theor. Biol. 110 665
[4] Kuznetsov, Yu A, Piccardi C 1994 Math. Biol. 32 109
[5] Olsen L F, Shaffer W M 1990 Science 249 499
[6] Hethcote H W, Driessche P V D 1991 Math. Biol. 29 271
[7] Liu W M, Hethcote H W, Levin S A 1987 Math. Biol. 25 359
[8] Liu W M, Levin S A, Iwasa Y 1986 Math. Biol. 23 187
[9] Yi H T 2009 MS dissertation (Hunan: Hunan university) (in Chinese) [尹海涛 2009 硕士学位论文(湖南:湖南大学)]
[10] Li J 2005 MS dissertation (Nanjing: Nanjing university of science and technology) (in Chinese) [李 静 2005 硕士学位论文 (南京:南京理工大学)] 〖11] Ni S J, Weng W G, Fan W C 2009 Acta Phys. Sin. 58 3707 (in Chinese) [倪顺江、翁文国、范维澄 2009 物理学报 58 3707]
[11] Liu M X, Ruan J 2009 Chin. Phys. B 18 2115
[12] Liu M X, Ruan J 2009 Chin. Phys. B 18 5111
[13] Shi H J, Duan Z S, Chen G R, Li R 2009 Chin. Phys. B 18 3309
[14] Pei W D, Chen Z Q, Yuan Z Z 2008 Chin. Phys. B 17 0373
[15] Zhong L, Weng J Q 2000 Acta Phys. Sin. 49 0626 (in Chinese) [钟 玲、翁甲强 2000 物理学报 49 0626]
[16] Xu D, Li X, Wang X F 2007 Acta Phys. Sin. 56 1313 (in Chinese) [许 丹、李 翔、汪小帆 2007 物理学报 56 1313]
[17] Glendinning P, Perry L P 1997 Math. Biol. 34 359
[18] Chow S N, Li C Z, Wang D 1994 Normal forms and bifurcation of plannar vector fields (Cambridge: Cambridge University Press)
[19] Yang X L, Xu W, Sun Z K 2006 Acta Phys. Sin. 55 1678 (in Chinese) [杨晓丽、徐 伟、孙中奎 2006 物理学报 55 1678]
[20] Liu Z R 2002 The analytical method of chaos (Shanghai:Shanghai university press) p56 (in Chinese) [刘增荣 2002 混沌研究中的解析方法(上海:上海大学出版社)第56页]
[21] Lei Y M, Xu W 2007 Acta Phys. Sin. 56 5103 (in Chinese) [雷佑銘、徐 伟 2007 物理学报 56 5103]
[22] Guckenheimer J, Holmes P 1983 Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vectors Fields (New York: New York Springer)
[23] Yang X L, Xu W 2009 Acta Phys Sin. 58 3722 (in Chinese) [杨晓丽、徐 伟 2009 物理学报 58 3722]
[24] Baesens N C, Nicolis G 1983 Z. Phys. B 52 345
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