Pulse Vaccination Strategy in an Epidemic Model with Two Susceptible Subclasses and Time Delay

ABSTRACT

In this paper, an impulsive epidemic model with time delay is proposed, which susceptible population is divided into two groups: high risk susceptibles and non-high risk susceptibles. We introduce two thresholds R1, R2 and demonstrate that the disease will be extinct if R1<1 and persistent if R2 >1 . Our results show that larger pulse vaccination rates or a shorter the period of pulsing will lead to the eradication of the disease. The conclusions are confirmed by numerical simulations.

In this paper, an impulsive epidemic model with time delay is proposed, which susceptible population is divided into two groups: high risk susceptibles and non-high risk susceptibles. We introduce two thresholds R1, R2 and demonstrate that the disease will be extinct if R1<1 and persistent if R2 >1 . Our results show that larger pulse vaccination rates or a shorter the period of pulsing will lead to the eradication of the disease. The conclusions are confirmed by numerical simulations.

Cite this paper

nullY. Luo, S. Gao and S. Yan, "Pulse Vaccination Strategy in an Epidemic Model with Two Susceptible Subclasses and Time Delay,"*Applied Mathematics*, Vol. 2 No. 1, 2011, pp. 57-63. doi: 10.4236/am.2011.21007.

nullY. Luo, S. Gao and S. Yan, "Pulse Vaccination Strategy in an Epidemic Model with Two Susceptible Subclasses and Time Delay,"

References

[1] X. B. Zhang, H. F. Huo, X. K. Sun and Q. Fu, “The Differential Susceptibility SIR Epidemic Model with Time Delay and Pulse Vaccination,” Journal of Applied Mathematics and Computing, Vol. 34, No. 1-2, 2009, pp. 287-298.

[2] Z. Agur, L. Cojocaru, R. Anderson and Y. Danon, “Pulse Mass Measles Vaccination across Age Cohorts,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 90, No. 24, 1993, pp. 11698-11702. doi:10.1073/pnas.90.24.11698

[3] W. O. Kermack and A. G. McKendrick, “Contributions to the Mathematical Theory of Epidemics—II: The Problem of Endemicity,” Proceedings of the Royal Society Series A, Vol. 138, No. 834, 1932, pp. 55-83. doi:10.1098/rspa. 1932.0171

[4] W. O. Kermack and A. G. McKendrick, “Contributions to the Mathematical Theory of Epidemics—III: Further Studies of the Problem of Endemicity,” Proceedings of the Royal Society Series A, Vol. 141, No. 843, 1933, pp. 94-122. doi:10.1098/rspa.1933.0106

[5] R. M. Anderson and R. M. May, “Population Biology of Infectious Disease: Part I,” Nature, Vol. 280, 1979, pp. 361-367. doi:10.1038/280361a0

[6] S. Gao, L. Chen and J. J. Nieto, “Angela Torres, Analysis of a Delayed Epidemic Model with Pulse Vaccination and Saturation Incidence,” Vaccine, Vol. 24, No. 35-36, 2006, pp. 6037-6045. doi:10.1016/j.vaccine.2006.05.018

[7] C. McCluskey, “Global Stability for a Class of Mass Action Systems Allowing for Latency in Tuberculosis,” Journal of Mathematical Analysis and Applications, Vol. 338, No. 1, 2008, pp. 518-535. doi:10.1016/j.jmaa.2007. 05.012

[8] K. L. Cooke and P. van den Driessche, “Analysis of an SEIRS Epidemic Model with Two Delays,” Journal of Mathematical Biology, Vol. 35, No. 2, 1996, pp. 240-260. doi:10.1007/s002850050051

[9] R. M. Anderson and R. M. May, “Infectious Diseases of Humans, Dynamics and Control,” Oxford University Press, Oxford, 1992.

[10] O. Diekmann and J. A. P. Heesterbeek, “Mathematical Epidemiology of Infectious Diseases,” John Wiley & Sons, Chisteter, 2000.

[11] F. Brauer and C. C. Castillo, “Mathematical Models in Population Biology and Epidemiology,” Springer, New York, 2000.

[12] W. O. Kermack and A. G. McKendrick, “A Contribution to the Mathematical Theory of Epidemics I,” Proceedings of the Royal Society Series A, Vol. 115, No. 772, 1927, pp. 700-721. doi:10.1098/rspa.1927.0118

[13] S. Gao, Z. Teng and D. Xie, “Analysis of a Delayed SIR Epidemic Model with Pulse Vaccination,” Chaos, Solitons & Fractals, Vol. 40, No. 2, 2009, pp. 1004-1011. doi:10.1016/j.chaos.2007.08.056

[14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific, Singapore, 1989.

[1] X. B. Zhang, H. F. Huo, X. K. Sun and Q. Fu, “The Differential Susceptibility SIR Epidemic Model with Time Delay and Pulse Vaccination,” Journal of Applied Mathematics and Computing, Vol. 34, No. 1-2, 2009, pp. 287-298.

[2] Z. Agur, L. Cojocaru, R. Anderson and Y. Danon, “Pulse Mass Measles Vaccination across Age Cohorts,” Proceedings of the National Academy of Sciences of the United States of America, Vol. 90, No. 24, 1993, pp. 11698-11702. doi:10.1073/pnas.90.24.11698

[3] W. O. Kermack and A. G. McKendrick, “Contributions to the Mathematical Theory of Epidemics—II: The Problem of Endemicity,” Proceedings of the Royal Society Series A, Vol. 138, No. 834, 1932, pp. 55-83. doi:10.1098/rspa. 1932.0171

[4] W. O. Kermack and A. G. McKendrick, “Contributions to the Mathematical Theory of Epidemics—III: Further Studies of the Problem of Endemicity,” Proceedings of the Royal Society Series A, Vol. 141, No. 843, 1933, pp. 94-122. doi:10.1098/rspa.1933.0106

[5] R. M. Anderson and R. M. May, “Population Biology of Infectious Disease: Part I,” Nature, Vol. 280, 1979, pp. 361-367. doi:10.1038/280361a0

[6] S. Gao, L. Chen and J. J. Nieto, “Angela Torres, Analysis of a Delayed Epidemic Model with Pulse Vaccination and Saturation Incidence,” Vaccine, Vol. 24, No. 35-36, 2006, pp. 6037-6045. doi:10.1016/j.vaccine.2006.05.018

[7] C. McCluskey, “Global Stability for a Class of Mass Action Systems Allowing for Latency in Tuberculosis,” Journal of Mathematical Analysis and Applications, Vol. 338, No. 1, 2008, pp. 518-535. doi:10.1016/j.jmaa.2007. 05.012

[8] K. L. Cooke and P. van den Driessche, “Analysis of an SEIRS Epidemic Model with Two Delays,” Journal of Mathematical Biology, Vol. 35, No. 2, 1996, pp. 240-260. doi:10.1007/s002850050051

[9] R. M. Anderson and R. M. May, “Infectious Diseases of Humans, Dynamics and Control,” Oxford University Press, Oxford, 1992.

[10] O. Diekmann and J. A. P. Heesterbeek, “Mathematical Epidemiology of Infectious Diseases,” John Wiley & Sons, Chisteter, 2000.

[11] F. Brauer and C. C. Castillo, “Mathematical Models in Population Biology and Epidemiology,” Springer, New York, 2000.

[12] W. O. Kermack and A. G. McKendrick, “A Contribution to the Mathematical Theory of Epidemics I,” Proceedings of the Royal Society Series A, Vol. 115, No. 772, 1927, pp. 700-721. doi:10.1098/rspa.1927.0118

[13] S. Gao, Z. Teng and D. Xie, “Analysis of a Delayed SIR Epidemic Model with Pulse Vaccination,” Chaos, Solitons & Fractals, Vol. 40, No. 2, 2009, pp. 1004-1011. doi:10.1016/j.chaos.2007.08.056

[14] V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, “Theory of Impulsive Differential Equations,” World Scientific, Singapore, 1989.