×

zbMATH — the first resource for mathematics

On the global stability of a delayed epidemic model with transport-related infection. (English) Zbl 1231.34128
Summary: We study the global dynamics of a time delayed epidemic model proposed by J. Liu, J. Wu and Y. Zhou [Rocky Mt. J. Math. 38, No. 5, 1525–1540 (2008; Zbl 1194.34111)] describing disease transmission dynamics among two regions due to transport-related infection. We prove that if an endemic equilibrium exists then it is globally asymptotically stable for any length of time delay by constructing a Lyapunov functional. This suggests that the endemic steady state for both regions is globally asymptotically stable regardless of the length of the travel time when the disease is transferred between two regions by human transport.

MSC:
34K20 Stability theory of functional-differential equations
37N25 Dynamical systems in biology
34D23 Global stability of solutions to ordinary differential equations
92D30 Epidemiology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] WHO, Severe acute respiratory syndrome (SARS): status of the outbreak and lessons for the immediate future, Geneva, May 20, 2003.
[2] Khan, K.; Arino, J.; Hu, W.; Raposo, P.; Sears, J.; Calderon, F.; Heidebrecht, C.; Macdonald, M.; Liauw, J.; Chan, A.; Gardam, M., Spread of a novel influenza a (H1N1) virus via global airline transportation, N. engl. J. med., 361, 2, 212-214, (2009)
[3] Ruan, S.; Wang, W.; Levin, S.A., The effect of global travel on the spread of SARS, Math. biosci. eng., 3, 1, 205-218, (2006) · Zbl 1089.92049
[4] Wang, W.; Zhao, X.-Q., An epidemic model in a patchy environment, Math. biosci., 190, 1, 97-112, (2004) · Zbl 1048.92030
[5] Arino, J.; van den Driessche, P., A multi-city epidemic model, Math. popul. stud., 10, 3, 175-193, (2003) · Zbl 1028.92021
[6] Arino, J., Diseases in metapopulations, (), 64-122
[7] Cui, J.; Takeuchi, Y.; Saito, Y., Spreading disease with transport-related infection, J. theoret. biol., 239, 3, 376-390, (2006)
[8] Takeuchi, Y.; Liu, X.; Cui, J., Global dynamics of SIS models with transport-related infection, J. math. anal. appl., 329, 2, 1460-1471, (2007) · Zbl 1154.34353
[9] Liu, J.; Wu, J.; Zhou, Y., Modeling disease spread via transport-related infection by a delay differential equation, Rocky mountain J. math., 38, 5, 1525-1540, (2008) · Zbl 1194.34111
[10] Liu, X.; Takeuchi, Y., Spread of disease with transport-related infection and entry screening, J. theoret. biol., 242, 2, 517-528, (2006)
[11] Hale, J.K.; Waltman, P., Persistence in infinite-dimensional systems, SIAM J. math. anal., 20, 2, 388-395, (1989) · Zbl 0692.34053
[12] Saito, Y.; Hara, T.; Ma, W., Necessary and sufficient conditions for permanence and global stability of a lotka – volterra system with two delays, J. math. anal. appl., 236, 2, 534-556, (1999) · Zbl 0944.34059
[13] Hale, J.K.; Verduyn Lunel, S.M., ()
[14] LaSalle, J.P., The stability of dynamical systems, (1976), Society for Industrial and Applied Mathematics Philadelphia, Pa., with an appendix: “Limiting equations and stability of nonautonomous ordinary differential equations” by Z. Artstein, Regional Conference Series in Applied Mathematics · Zbl 0364.93002
[15] Suzuki, M.; Matsunaga, H., Stability criteria for a class of linear differential equations with off-diagonal delays, Discrete contin. dyn. syst., 24, 4, 1381-1391, (2009) · Zbl 1180.34082
[16] Mischaikow, K.; Smith, H.; Thieme, H.R., Asymptotically autonomous semiflows: chain recurrence and Lyapunov functions, Trans. amer. math. soc., 347, 5, 1669-1685, (1995) · Zbl 0829.34037
[17] Carlos, C.-C.; Thieme, H.R., Asymptotically autonomous epidemic models, (), 33-50
[18] Enatsu, Y.; Nakata, Y.; Muroya, Y., Global stability of SIR epidemic models with a wide class of nonlinear incidence rates and distributed delays, Discrete contin. dyn. syst. ser. B, 15, 1, 61-74, (2011) · Zbl 1217.34127
[19] Nakata, Y., Global dynamics of a viral infection model with a latent period and beddington – deangelis response, Nonlinear anal. TMA, 74, 9, 2929-2940, (2011) · Zbl 1227.34086
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.