Permanence and Almost Periodic Solutions to an Epidemic Model with Delay and Feedback Control
This paper is concerned with an epidemic model with delay. By using the comparison theorem of the differential equation and constructing a suitable Lyapunov functional, Some sufficient conditions which guarantee the permeance and existence of a unique globally attractive positive almost periodic solution of the model are obtain. Finally, an example is employed to illustrate our result.
[1] Z. Mukandavire, W. Garira, C. Chiyaka, Asymptotic properties of an
HIV/AIDS model with a time delay, Journal of Mathematical Analysis
and Applications 330 (2) (2007) 834-852.
[2] Fang Zhang, Xiao-Qiang Zhao, A periodic epidemic model in a patchy
environment, Journal of Mathematical Analysis and Applications 325 (1)
(2007) 496-516.
[3] Alberto d’Onofrio, Stability properties of pulse vaccination strategy in
SEIR epidemic model, Mathematical Biosciences 179 (1) (2002) 57-72.
[4] Xin-An Zhang, Lansun Chen, The periodic solution of a class of
epidemic models, Computers and Mathematics with Applications 38 (3-4)
(1999) 61-71.
[5] A. Lajmanovich, T.A. Yorke, Adeterministic model for gonorrhea in a
nonhomogeneous population, Mathematical Biosciences 28 (1976) 221-
236.
[6] G. Aronsson, I. Mellander, Adeterministic model in biomathematics,
asymptotic behavior and threshold conditions, Mathematical Biosciences
49 (1980) 207-222.
[7] L. Chen, J. Sun, The positive periodic solution for an epidemic model,
Journal of Systems Science and Mathematical Sciences 26 (4) (2006) 456-
466 (in Chinese)
[8] B. Xiao, B. Liu, Exponential convergence of an epidemic model with
continuously distributed delays, Mathematical and Computer Modelling
48 (3-4) (2008) 541-547.
[9] C. Zhao, M. Yang, G. Peng, Exponential convergence of an epidemic
model with time-varying delaysNonlinear Analysis: Real World Applications11
(2010) 117-121.
[10] S. Mohamad, K. Gopalsamy, H. Akca, Exponential stability of artificial
neural networks with distributed delays and large impulses, Nonlinear
Analysis: Real World Applications 9 (3) (2008) 872-888.
[11] A.Y. Mitrophanov, M. Borodovsky, Convergence rate estimation for
the TKF91 model of biological sequence length evolution, Mathematical
Biosciences 209 (2) (2007) 470-485.
[12] K. Gopalsamy, P. Wen, Feedback regulation of logistic growth, Internat.
J. Math. Math. Sci. 16 (1993) 177-192.
[13] Y.K. Li, P. Liu, Positive periodic solutons of a class of functional
differential systems with feedback controls, Nonlinear Anal. 57 (2004)
655-666.
[14] B.S. Lalli, J. Yu, Feedback regulation of a logistic growth, Dynam.
Systems Appl. 5 (1996) 117-124.
[15] Q. Liu, R. Xu, Persisrence and global stablity for a delayed nonautonomous
single-species model with dispersal and feedback control,
Differential Equations Dynam. Systems 11 (3-4) (2003) 353-367.
[16] Y.K. Li, Positive periodic solutions for neutral functional differential
equations with distributed delays and feedback control, Nonlinear Anal.
Real World Appl. 9 (2008) 2214-2221.
[17] F.D. Chen, Z. Li, Note on the permanence of a competitive system with
infinite delay and feedback controls, Nonlinear Anal. Real World Appl. 8
(2007) 680-687.
[18] P.H. Leslie, Some further notes on the use of matrices in population
mathematics, Biometrika 35 (1948) 213-245.
[19] A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture
Notes in Mathematics, Springer, Berlin, Germany, 1974.
[20] C. Y. He, Almost Periodic Differential Equations, Higher Education
Publishing House, Beijing, China, 1992.
[1] Z. Mukandavire, W. Garira, C. Chiyaka, Asymptotic properties of an
HIV/AIDS model with a time delay, Journal of Mathematical Analysis
and Applications 330 (2) (2007) 834-852.
[2] Fang Zhang, Xiao-Qiang Zhao, A periodic epidemic model in a patchy
environment, Journal of Mathematical Analysis and Applications 325 (1)
(2007) 496-516.
[3] Alberto d’Onofrio, Stability properties of pulse vaccination strategy in
SEIR epidemic model, Mathematical Biosciences 179 (1) (2002) 57-72.
[4] Xin-An Zhang, Lansun Chen, The periodic solution of a class of
epidemic models, Computers and Mathematics with Applications 38 (3-4)
(1999) 61-71.
[5] A. Lajmanovich, T.A. Yorke, Adeterministic model for gonorrhea in a
nonhomogeneous population, Mathematical Biosciences 28 (1976) 221-
236.
[6] G. Aronsson, I. Mellander, Adeterministic model in biomathematics,
asymptotic behavior and threshold conditions, Mathematical Biosciences
49 (1980) 207-222.
[7] L. Chen, J. Sun, The positive periodic solution for an epidemic model,
Journal of Systems Science and Mathematical Sciences 26 (4) (2006) 456-
466 (in Chinese)
[8] B. Xiao, B. Liu, Exponential convergence of an epidemic model with
continuously distributed delays, Mathematical and Computer Modelling
48 (3-4) (2008) 541-547.
[9] C. Zhao, M. Yang, G. Peng, Exponential convergence of an epidemic
model with time-varying delaysNonlinear Analysis: Real World Applications11
(2010) 117-121.
[10] S. Mohamad, K. Gopalsamy, H. Akca, Exponential stability of artificial
neural networks with distributed delays and large impulses, Nonlinear
Analysis: Real World Applications 9 (3) (2008) 872-888.
[11] A.Y. Mitrophanov, M. Borodovsky, Convergence rate estimation for
the TKF91 model of biological sequence length evolution, Mathematical
Biosciences 209 (2) (2007) 470-485.
[12] K. Gopalsamy, P. Wen, Feedback regulation of logistic growth, Internat.
J. Math. Math. Sci. 16 (1993) 177-192.
[13] Y.K. Li, P. Liu, Positive periodic solutons of a class of functional
differential systems with feedback controls, Nonlinear Anal. 57 (2004)
655-666.
[14] B.S. Lalli, J. Yu, Feedback regulation of a logistic growth, Dynam.
Systems Appl. 5 (1996) 117-124.
[15] Q. Liu, R. Xu, Persisrence and global stablity for a delayed nonautonomous
single-species model with dispersal and feedback control,
Differential Equations Dynam. Systems 11 (3-4) (2003) 353-367.
[16] Y.K. Li, Positive periodic solutions for neutral functional differential
equations with distributed delays and feedback control, Nonlinear Anal.
Real World Appl. 9 (2008) 2214-2221.
[17] F.D. Chen, Z. Li, Note on the permanence of a competitive system with
infinite delay and feedback controls, Nonlinear Anal. Real World Appl. 8
(2007) 680-687.
[18] P.H. Leslie, Some further notes on the use of matrices in population
mathematics, Biometrika 35 (1948) 213-245.
[19] A. M. Fink, Almost Periodic Differential Equations, vol. 377 of Lecture
Notes in Mathematics, Springer, Berlin, Germany, 1974.
[20] C. Y. He, Almost Periodic Differential Equations, Higher Education
Publishing House, Beijing, China, 1992.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:54483", author = "Chenxi Yang and Zhouhong Li", title = "Permanence and Almost Periodic Solutions to an Epidemic Model with Delay and Feedback Control", abstract = "This paper is concerned with an epidemic model with delay. By using the comparison theorem of the differential equation and constructing a suitable Lyapunov functional, Some sufficient conditions which guarantee the permeance and existence of a unique globally attractive positive almost periodic solution of the model are obtain. Finally, an example is employed to illustrate our result.
", keywords = "Permanence, Almost periodic solution, Epidemic
model, Delay, Feedback control.", volume = "8", number = "1", pages = "1-7", }
