Optimal Vaccination Strategies in an SIR Epidemic Model with Time Scales

Affiliation(s)

School of Mathematics, University of Nairobi, Nairobi, Kenya.

TU Munich, Centre of Mathematical Science, Munich, Germany.

School of Mathematics, University of Nairobi, Nairobi, Kenya.

TU Munich, Centre of Mathematical Science, Munich, Germany.

ABSTRACT

Childhood related diseases such as measles are
characterised by short periodic outbreaks lasting about 2
weeks. This means therefore that the timescale at which such diseases operate
is much shorter than the time scale of the human population dynamics. We
analyse a compartmental model of the SIR type with periodic coefficients and
different time scales for 1) disease dynamics and 2) human
population dynamics. Interest is to determine the optimal vaccination strategy
for such diseases. In a model with time scales, Singular Perturbation theory is used to
determine stability condition for the disease

Cite this paper

O. Owuor, M. Johannes and M. Kibet, "Optimal Vaccination Strategies in an SIR Epidemic Model with Time Scales,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 1-14. doi: 10.4236/am.2013.410A2001.

O. Owuor, M. Johannes and M. Kibet, "Optimal Vaccination Strategies in an SIR Epidemic Model with Time Scales,"

References

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s00285-013-0648-8

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

[2] M. J. Ferrari, F. G. Rebecca, B. Nita, J. K. Conlan, O. N. Bjornstad, L. J. Wolfson, P. J. Guerin, A. Djibo and B. T. Grenfell, “The Dynamics of Measles in Sub-Saharan Africa,” Nature-Articles, Vol. 451, No. 7, 2008, pp. 679684.

[3] M. E. Alexander, S. M. Moghadas, P. Rohani and A. R. Summers, “Modelling the Effect of a Booster Vaccination on Disease Epidemiology,” Journal of Mathematical Biology, Vol. 52, No. 3, 2006, pp. 290-306. http://dx.doi.org/10.1007/s00285-005-0356-0

[4] M. Eichner and K. P. Hadeler, “Deterministic Models for the Eradication of Poliomyelitis: Vaccination with the Inactivated (IPV) and Attenuated (OPV) Polio Virus Vaccine,” Mathematical Biosciences, Vol. 127, No. 2, 1995, pp. 149-166. http://dx.doi.org/10.1016/0025-5564(94)00046-3

[5] P. Rohani, D. J. D. Earn, B. Finkenstaedt and B. T. Grenfell, “Population Dynamics Interference among Childhood Diseases,” Proceedings of the Royal Society of London, Vol. 265, No. 1410, 1998, pp. 2033-2041. http://dx.doi.org/10.1098/rspb.1998.0537

[6] P. Rohani, M. J. Kelling and B. T. Grenfell, “The Interplay between Determinism and Stochasticity in Childhood Diseases,” The American Naturalist, Vol. 159, No. 5, 2002, pp. 469-481. http://dx.doi.org/

10.1086/339467

[7] Y. Zhou and H. Liu, “Stability of Periodic Solutions for an SIS Model with Pulse Vaccination,” Mathematical and Computer Modelling, Vol. 38, No. 3-4, 2003, pp. 299-308. http://dx.doi.org/

10.1016/S0895-7177(03)90088-4

[8] B. Shulgin, L. Stone and Z. Agur, “Theoretical Examination of Pulse Vaccination Policy in the SIR Epidemic Model,” Mathematical and Computer Modelling, Vol. 31, No. 4-5, 2000, pp. 207-215. http://dx.doi.org/10.1016/S0895-7177(00)00040-6

[9] A. d’Onofrio, “Stability Properties of Pulse Vaccination Strategy in SEIR Epidemic Model,” Mathematical Biosciences, Vol. 179, No. 1, 2002, pp. 57-72. http://dx.doi.org/10.1016/S0025-5564(02)00095-0

[10] K. P. Hadeler and J. Mueller, “Vaccination in Age Structured Populations I: The Reproduction Number,” In: V. Isham and G. Medley, Ed., Models for Infectious Human Diseases: Their Structure and Relation to Data, Cambridge University Press, Cambridge, 1996, pp. 90-101. http://dx.doi.org/10.1017/

CBO9780511662935.013

[11] K. P. Hadeler and J. Mueller, “Vaccination in Age Structured Populations II: Optimal Vaccination Strategies,” In: V. Isham and G. Medley, Eds., Models for Infectious Human Diseases: Their Structure and Relation to Data, Cambridge University Press, Cambridge, 1996, pp. 102114. http://dx.doi.org/

10.1017/CBO9780511662935.014

[12] R. E. O’Malley, “Introduction to Singular Perturbations,” Academic Press, New York, 1974.

[13] R. E. O’Malley, “Singular Perturbation Methods for Ordinary Differential Equations,” Springer-Verlag, New York, 1991. http://dx.doi.org/10.1007/978-1-4612-0977-5

[14] N. Fenichel, “Geometric Singular Perturbation Theory for Ordinary Differential Equations,” Journal of Differential Equations, Vol. 31, No. 1, 1979, pp. 53-98. http://dx.doi.org/10.1016/0022-0396(79)90152-9

[15] N. Onyango and J. Müller, “Determination of Optimal Vaccination Strategies Using an Orbital Stability Threshold from Periodically Driven Systems,” Journal of Mathematical Biology, in Press. http://dx.doi.org/10.1007/

s00285-013-0648-8

[16] H. R. Thieme, “Mathematics in Population Biology,” Princeton University Press, Princeton, 2003.

[17] J. Müller, “Optimal Vaccination Patterns in Age-Structured Populations,” SIAM Journal on Applied Mathematics, Vol. 59, No. 1, 1998, pp. 222-241. http://dx.doi.org/10.1137/S0036139995293270

[18] J. Müller, “Optimal Vaccination Patterns in Age-Structured Populations: Endemic Case,” Mathematical and Computer Modelling, Vol. 31, No. 4-5, 2000, pp. 149-160. http://dx.doi.org/10.1016/S0895-7177(00)00033-9

[19] B. Shulgin, L. Stone and Z. Agur, “Pulse Vaccination Strategy in the SIR Epidemic Model,” Bulletin of Mathematical Biology, Vol. 60, No. 6, 1998, pp. 1123-1148.

[20] N. Bacaer, M. Gabriel and M. Gomes, “On the Final Size of Epidemics with Seasonality,” Bulletin of Mathematical Biology, Vol. 71, No. 8, 2009, pp. 1954-1966. http://dx.doi.org/10.1007/s11538-009-9433-7

[21] L. C. Evans, “Partial Differential Equations,” American Mathematical Society, Providence and Rhodes Island, 1998.

[22] K. Yosida, “Functional Analysis,” Springer-Verlag, Berlin, 1980.