Global stability analysis of an SIR epidemic model with demographics and time delay on networks. (English) Zbl 1395.92169
Summary: In this paper, a susceptible-infected-recovery (SIR) epidemic model is governed with demographics and time delay on networks. Firstly, the basic reproduction number \(R_0\) is derived dependent on birth rate, death rate, recovery rate and transmission rate. The disease-free equilibrium of the model is stable when \(R_0 \leq 1\) and unstable when \(R_0 > 1\). Secondly, based on a Jacobian matrix calculated along with the disease-free equilibrium, we find that the system does not occur Hopf branch under the disease-free equilibrium. Thirdly, the global asymptotic stability of a disease-free equilibrium and a unique endemic equilibrium are proved by structuring two Lyapunov functions. Finally, numerical simulations are performed to illustrate the analysis results.
References:
| [1] | Beretta, E.; Takeuchi, Y., Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33, 250-260, (1995) · Zbl 0811.92019 |
| [2] | Cooke, K. L.; Van Den Driessche, P., Analysis of an SEIRS epidemic mode1 with two delays, J. Math. Biol., 35, 240-260, (1996) · Zbl 0865.92019 |
| [3] | Hethcotew, Herbert W.; van den Driessche, P., Two SIS epidemiologic models with delays, J. Math. Biol., 40, 3-26, (2000) · Zbl 0959.92025 |
| [4] | Wei, H. M.; Li, X. Z.; Martcheva, M., An epidemic model of a vector-borne disease with direct transmission and time delay, J. Math. Anal. Appl., 342, 895-908, (2008) · Zbl 1146.34059 |
| [5] | Naresha, R.; Tripathi, A.; Tchuenche, J. M.; Sharma, D., Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate, Comput. Math. Appl., 58, 348-359, (2009) · Zbl 1189.34098 |
| [6] | Ma, W. B.; Song, M., Global stability of an SIR epidemic model with time delay, Appl. Math. Lett., 17, 1141-1145, (2004) · Zbl 1071.34082 |
| [7] | M. Song, G.C. Liu, Z.H. Li, H.Y. Wu, Delay-independent stability of an SIR epidemic model with time delay, in: Proceedings of the Conference of Biomathematics Tai’an, PR China 25-29, 2008, pp. 389-392. |
| [8] | Wang, W. D., Global behavior of an SEIRS epidemic model with time delays, Appl. Math. Lett., 15, 423-428, (2002) · Zbl 1015.92033 |
| [9] | Kaddar, A., Stability analysis in a delayed SIR epidemic model with a saturated incidence rate, Nonlinear Anal. Model. Control, 15, 299-306, (2010) · Zbl 1227.34084 |
| [10] | Xu, R.; Ma, Z. E., Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. RWA, 10, 3175-3189, (2009) · Zbl 1183.34131 |
| [11] | McCluskey, C. C., Global stability for an SIR epidemic model with delay and nonlinear incidence, Nonlinear Anal. RWA, 11, 3106-3109, (2010) · Zbl 1197.34166 |
| [12] | Zhang, J. Z.; Jin, Z.; Yan, J. R.; Sun, G. Q., Stability and Hopf bifurcation in a delayed competition system, Nonlinear Anal., 70, 658-670, (2009) · Zbl 1166.34049 |
| [13] | Jiang, Z. C.; Wei, J. J., Stability and bifurcation analysis in a delayed SIR model, Chaos Solitons Fractals, 38, 609-619, (2008) · Zbl 1131.92055 |
| [14] | Jin, Z.; Ma, Z. E.; Han, M. A., Global stability of an SIRS epidemic model with delays, Acta Math. Sci., 26B, 2, 291-306, (2006) · Zbl 1090.92044 |
| [15] | Zhang, T. L.; Teng, Z. D., Global behavior and permanence of SIRS epidemic model with time delay, Nonlinear Anal. RWA, 9, 1409-1424, (2008) · Zbl 1154.34390 |
| [16] | Li, C. H.; Tsai, C. C.; Yang, S. Y., Permanence of an SIR epidemic model with density dependent birth rate and distributed time delay, Appl. Math. Comput., 218, 1682-1693, (2011) · Zbl 1228.92066 |
| [17] | Guo, H. B.; Li, Michael Y.; Shuai, Z. S., Global stability of the endemic equilibrium of multigroup SIR epidemic models, Appl. Math. Q., 14, 3, (2006) · Zbl 1148.34039 |
| [18] | Li, Michael Y.; Shuai, Z. S., Global stability problem for coupled systems of differential equations on networks, J. Differential Equations, 248, 1-20, (2010) · Zbl 1190.34063 |
| [19] | Li, Michael Y.; Shuai, Z. S.; Wang, C. C., Global stability of multi-group epidemic models with distributed delays, J. Math. Anal. Appl., 361, 38-47, (2010) · Zbl 1175.92046 |
| [20] | Korobeinikov, A., Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull. Math. Biol., 71, 75-83, (2009) · Zbl 1169.92041 |
| [21] | Huanga, G.; Takeuchi, Y.; Ma, W. B.; Wei, D. J., Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72, 1192-1207, (2010) · Zbl 1197.92040 |
| [22] | Zhang, J. P.; Jin, Z., The analysis of an epidemic model on networks, Appl. Math. Comput., 217, 7053-7064, (2011) · Zbl 1211.92053 |
| [23] | Liu, J. Z.; Tang, Y. F.; Yang, Z. R., The spread of disease with birth and death on networks, vol. 9, 1-8, (2008), Elsevier Science |
| [24] | Xua, X. J.; Peng, H. O.; Wang, X. M.; Wang, Y. H., Epidemic spreading with time delay in complex networks, Physica A, 367, 525-530, (2006) |
| [25] | Zou, S. F.; Wu, J. H.; Chen, Y. M., Multiple epidemic waves in delayed susceptible-infected-recovered models on complex networks, Phys. Rev. E, 83, 056121, (2011) |
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