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October  2011, 16(3): 867-881. doi: 10.3934/dcdsb.2011.16.867

Long time behavior of some epidemic models

Fang Li 1, and  Nung Kwan Yip 1,

1. 

Department of Mathematics, Purdue University, 150 N. University St., West Lafayette, IN 47907, United States

Received  August 2010 Revised  March 2011 Published  June 2011

In this paper, we prove two results concerning the long time behavior of two systems of reaction diffusion equations motivated by the S-I-R model in epidemic modeling. The results generalize and simplify previous approaches. In particular, we consider the presence of directed diffusions between the two species. The new system contains an ill-posed region for arbitrary parameters. Our result is established under the assumption of small initial data.
Keywords: epidemic models., Long time behavior.
Mathematics Subject Classification: Primary: 35B40, 35K51; Secondary: 92B05, 92D3.
Citation: Fang Li, Nung Kwan Yip. Long time behavior of some epidemic models. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 867-881. doi: 10.3934/dcdsb.2011.16.867
References:
[1]

S. Busenberg and C. Castillo-Chavez, Interaction, pair formation and force of infection terms in sexually transmitted diseases, Mathematical and Statistical Approaches to AIDS Epidemiology, 289-300, Lecture Notes in Biomath., 83, Springer, Berlin, 1989.

[2]

C. Castillo-Chavez and H. R. Thieme, On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic, Mathematical and Statistical Approaches to AIDS Epidemiology, 157-176, Lecture Notes in Biomath., 83, Springer, Berlin, 1989.

[3]

P. C. Fife, Asymptotic States for Equations of Reaction and Diffusion, Bull. AMS, 84 (1978), 693-726. doi: 10.1090/S0002-9904-1978-14502-9.

[4]

M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1.

[5]

W. S. C. Gurney and R. M. Nisbet, The Regulation of Inhomogeneous Populations, J. Theor. Biol., 52 (1975), 441-457. doi: 10.1016/0022-5193(75)90011-9.

[6]

M. Iannelli, R. Loro, F. A. Milner, A. Pugliese and G. Rabbiolo, An AIDS model with distributed incubation and variable infectiousness: applications to i.v. drug users in Latium, Italy, Eur. J. Epidemiol., 8 (1992), 585-593. doi: 10.1007/BF00146381.

[7]

W. O. Kermack and A. G. McKendrick, A Contribution to the Mathematical Theory of Epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721 doi: 10.1098/rspa.1927.0118.

[8]

T. G. Kurtz and J. Xiong, Particle representation for a class of nonlinear SPDEs, Stoch. Proc. and Their Appl., 83 (1999), 103-126. doi: 10.1016/S0304-4149(99)00024-1.

[9]

F. A. Milner and R. Zhao, S-I-R Model with Directed Spatial Diffusion, Mathematical Population Studies, 15 (2008), 160-181. doi: 10.1080/08898480802221889.

[10]

M. Mimura and M. Yamaguti, Pattern formation in interacting and diffusing systems in population biology, Adv. Biophys., 15 (1982), 19-65. doi: 10.1016/0065-227X(82)90004-1.

[11]

B. K. Oksendal, "Stochastic Differential Equations: An Introduction with Applications," 6th edition, Springer-Verlag, Berlin, 2003.

[12]

G. F. Webb, A reaction-diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl., 84 (1981), 150-161. doi: 10.1016/0022-247X(81)90156-6.

show all references

References:
[1]

S. Busenberg and C. Castillo-Chavez, Interaction, pair formation and force of infection terms in sexually transmitted diseases, Mathematical and Statistical Approaches to AIDS Epidemiology, 289-300, Lecture Notes in Biomath., 83, Springer, Berlin, 1989.

[2]

C. Castillo-Chavez and H. R. Thieme, On the role of variable infectivity in the dynamics of the human immunodeficiency virus epidemic, Mathematical and Statistical Approaches to AIDS Epidemiology, 157-176, Lecture Notes in Biomath., 83, Springer, Berlin, 1989.

[3]

P. C. Fife, Asymptotic States for Equations of Reaction and Diffusion, Bull. AMS, 84 (1978), 693-726. doi: 10.1090/S0002-9904-1978-14502-9.

[4]

M. E. Gurtin and R. C. MacCamy, On the diffusion of biological populations, Math. Biosci., 33 (1977), 35-49. doi: 10.1016/0025-5564(77)90062-1.

[5]

W. S. C. Gurney and R. M. Nisbet, The Regulation of Inhomogeneous Populations, J. Theor. Biol., 52 (1975), 441-457. doi: 10.1016/0022-5193(75)90011-9.

[6]

M. Iannelli, R. Loro, F. A. Milner, A. Pugliese and G. Rabbiolo, An AIDS model with distributed incubation and variable infectiousness: applications to i.v. drug users in Latium, Italy, Eur. J. Epidemiol., 8 (1992), 585-593. doi: 10.1007/BF00146381.

[7]

W. O. Kermack and A. G. McKendrick, A Contribution to the Mathematical Theory of Epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721 doi: 10.1098/rspa.1927.0118.

[8]

T. G. Kurtz and J. Xiong, Particle representation for a class of nonlinear SPDEs, Stoch. Proc. and Their Appl., 83 (1999), 103-126. doi: 10.1016/S0304-4149(99)00024-1.

[9]

F. A. Milner and R. Zhao, S-I-R Model with Directed Spatial Diffusion, Mathematical Population Studies, 15 (2008), 160-181. doi: 10.1080/08898480802221889.

[10]

M. Mimura and M. Yamaguti, Pattern formation in interacting and diffusing systems in population biology, Adv. Biophys., 15 (1982), 19-65. doi: 10.1016/0065-227X(82)90004-1.

[11]

B. K. Oksendal, "Stochastic Differential Equations: An Introduction with Applications," 6th edition, Springer-Verlag, Berlin, 2003.

[12]

G. F. Webb, A reaction-diffusion model for a deterministic diffusive epidemic, J. Math. Anal. Appl., 84 (1981), 150-161. doi: 10.1016/0022-247X(81)90156-6.

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