×

zbMATH — the first resource for mathematics

Global stability of a delayed SIRS model with temporary immunity. (English) Zbl 1142.34354
Summary: This paper addresses a time-delayed SIRS model with a linear incidence rate. Immunity gained by experiencing the disease is temporary; whenever infected, the disease individuals will return to the susceptible class after a fixed period of time. First, the local and global stabilities of the infection-free equilibrium are analyzed, respectively. Second, the endemic equilibrium is formulated in terms of the incidence rate, and two sufficient conditions for its locally asymptotic stability are found, one being proved theoretically, while the other being shown by introducing an auxiliary optimization problem and solving this problem with the help of Matlab toolbox. Finally, by using a Lyapunov functional, a sufficient criterion for the global stability of the endemic equilibrium is established.

MSC:
34D23 Global stability of solutions to ordinary differential equations
92D30 Epidemiology
Software:
Matlab
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aron, J.L., Acquired immunity dependent upon exposure in an SIRS epidemic model, Math biosci, 88, 37-47, (1988) · Zbl 0637.92007
[2] Brauer, F.; van den Driessche, P., Models for transmission of disease with immigration of infectives, Math biosci, 171, 143-171, (2001) · Zbl 0995.92041
[3] Beretta, E.; Kuang, Y., Modeling and analysis of a marine bacteriophage infection with latency period, Nonlinear anal real world appl, 2, 35-74, (2001) · Zbl 1015.92049
[4] Beretta, E.; Takeuchi, Y., Convergence results in SIR epidemic models with varying population sizes, Nonlinear anal theory method appl, 28, 1909-1921, (1997) · Zbl 0879.34054
[5] Busenberg, S.; Cooke, K.L., Periodic solutions of s periodic nonlinear delay differential equation, SIAM J appl math, 35, 704-721, (1978) · Zbl 0391.34022
[6] Li, G.; Zhen, J., Global stability of an SEI epidemic model with general contact rate, Chaos, solitons & fractals, 23, 997-1004, (2005) · Zbl 1062.92062
[7] Thieme, H.R.; van den Driessche, P., Global stability in cyclic epidemic models with disease fatalities, Fields inst commun, 21, 459-472, (1999) · Zbl 0924.92018
[8] Zeng, G.; Chen, L.; Sun, L., Complexity of an SIR epidemic dynamics model with impulsive vaccination control, Chaos, solitons & fractals, 26, 495-505, (2005) · Zbl 1065.92050
[9] Cooke, K.L.; van den Driessche, P., Analysis of an SEIRS epidemic model with two delays, J math biol, 35, 240-260, (1993) · Zbl 0865.92019
[10] Wang, K.; Wang, W.; Liu, X., Viral infection model with periodic lytic immune response, Chaos, solitons & fractals, 28, 90-99, (2006) · Zbl 1079.92048
[11] Wang, W., Global behavior of an SEIRS epidemic model with time delays, Appl math lett, 15, 423-428, (2002) · Zbl 1015.92033
[12] Greenhalgh, D.; Khan, Q.J.A.; Lewis, F.I., Recurrent epidemic cycles in an infectious disease model with a time delay in loss of vaccine immunity, Nonlinear anal, 63, 779-788, (2005) · Zbl 1222.92055
[13] Li, G.; Zhen, J., Global stability of a SEIR epidemic model with infectious force in latent. infected and immune period, Chaos, solitons & fractals, 25, 1177-1184, (2005) · Zbl 1065.92046
[14] Kyrychko, Y.N.; Nlyuss, K.B., Global properties of a delayed SIR model with temporary immunity and nonlinear incidence rate, Nonlinear anal real world appl, 6, 495-507, (2005) · Zbl 1144.34374
[15] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press San Diego · Zbl 0777.34002
[16] Hale, J.K.; Waltman, P., Persistence in infinite-dimensional systems, SIAM J math anal, 20, 335-356, (1976)
[17] Hale, J.K., Theory of functional differential equations, (1977), Springer New York · Zbl 0425.34048
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.