# American Institute of Mathematical Sciences

2011, 8(3): 677-687. doi: 10.3934/mbe.2011.8.677

## A simple analysis of vaccination strategies for rubella

 1 Department of Mathematics and Applications, University of Naples Federico II, via Cintia, I-80126 Naples

Received  October 2010 Revised  October 2010 Published  June 2011

We consider an SEIR epidemic model with vertical transmission introduced by Li, Smith and Wang, [23], and apply optimal control theory to assess the effects of vaccination strategies on the model dynamics. The strategy is chosen to minimize the total number of infectious individuals and the cost associated with vaccination. We derive the optimality system and solve it numerically. The theoretical findings are then used to simulate a vaccination campaign for rubella in China.
Citation: Bruno Buonomo. A simple analysis of vaccination strategies for rubella. Mathematical Biosciences & Engineering, 2011, 8 (3) : 677-687. doi: 10.3934/mbe.2011.8.677
##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases in Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. [2] E. Asano, L. J. Gross, S. Lenhart and L. A. Real, Optimal control of vaccine distribution in a rabies metapopulation model, Math. Biosci. Engineering, 5 (2008), 219-238. [3] H. Behncke, Optimal control of deterministic epidemics, Optim. Control Appl. Meth., 21 (2000), 269-285. doi: 10.1002/oca.678. [4] K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile virus, Bull. Math. Biol., 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0. [5] C. Bowman and A. B. Gumel, Optimal vaccination strategies for an influenza-like illness in a heterogeneous population, in "Mathematical Studies on Human Disease Dynamics," 31-49, Contemp. Math., 410, Amer. Math. Soc., Providence, RI, 2006. [6] F. Brauer, P. van den Driessche and J. Wu, editors, "Mathematical Epidemiology," Lecture Notes in Mathematics, 1945, Mathematical Biosciences Subseries, Springer-Verlag, Berlin, 2008. [7] B. Buonomo and D. Lacitignola, On the use of the geometric approach to global stability for three-dimensional ODE systems: A bilinear case, J. Math. Anal. Appl., 348 (2008), 255-266. doi: 10.1016/j.jmaa.2008.07.021. [8] S. Busenberg and K. Cooke, "Vertically Transmitted Diseases. Models and Dynamics," Biomathematics, 23, Springer-Verlag, Berlin, 1993. [9] V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomath., 97, Springer-Verlag, Berlin, 1993. [10] World Health Organization, Western Pacific Region. Countries and Areas: China 2007., Available from:, \url{http://www.wpro.who.int/countries/2007/chn/}., (). [11] F. T. Cutts and E. Vynnycky, Modelling the incidence of congenital rubella syndrome in developing countries, Int. J. Epidemiol., 28 (1999), 1176-1184. doi: 10.1093/ije/28.6.1176. [12] A. d'Onofrio, Globally stable vaccine-induced eradication of horizontally and vertically transmitted infectious disease with periodic contact rates and disease-dependent demographic factors in the population, Appl. Math. Comput., 140 (2003), 537-547. doi: 10.1016/S0096-3003(02)00251-5. [13] A. d'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Math. Lett., 18 (2005), 729-732. doi: 10.1016/j.aml.2004.05.012. [14] L. Dontigny, M. Y. Arsenault, M. J. Martel, et al., Rubella in pregnancy, Society of Obstetricians and Gyneacologists of Canada clinical practice guidelines, J. Obstet. Gynaecol. Can., 30 (2008), 152-68. [15] H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6 (2009), 469-492. doi: 10.3934/mbe.2009.6.469. [16] L. Gao and H. Hethcote, Simulations of rubella vaccination strategies in China, Math. Biosci., 202 (2006), 371-385. doi: 10.1016/j.mbs.2006.02.005. [17] K. Hattaf, M. Rachik, S. Saadi, Y. Tabit and N. Yousfi, Optimal control of tuberculosis with exogenous reinfection, Appl. Math. Sci. (Ruse), 3 (2009), 231-240. [18] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [19] H. R. Joshi, S. Lenhart, M. Y. Li and L. Wang, Optimal control methods applied to disease models, in "Mathematical Studies on Human Disease Dynamics," 187-207, Contemp. Math., 410, Amer. Math. Soc., Providence, RI, 2006. [20] E. Jung, S. Iwami, Y. Takeuchi and Tae-Chang Jo, Optimal control strategy for prevention of avian influenza pandemic, J. Theor. Biol., 260 (2009), 220-229. doi: 10.1016/j.jtbi.2009.05.031. [21] E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473. [22] S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007. [23] M. Y. Li, H. L. Smith and L. Wang, Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62 (2001), 58-69. doi: 10.1137/S0036139999359860. [24] M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449. [25] M. Y. Li and L. Wang, Global stability in some SEIR epidemic models, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory," 295-311, IMA Volumes in Mathematics and its Applications, 126 (2002), Springer-Verlag, New York. [26] X.-Z. Li and L.-L. Zhou, Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate, Chaos Solitons Fractals, 40 (2009), 874-884. doi: 10.1016/j.chaos.2007.08.035. [27] Matlab. Release 13, The mathworks Inc., Natich, MA, 2002. [28] E. Miller, J. Cradock-Watson and T. Pollock, Consequences of confirmed maternal rubella at successive stages of pregnancy, Lancet, 320 (1982), 781-784. doi: 10.1016/S0140-6736(82)92677-0. [29] R. Morton and K. H. Wickwire, On the optimal control of a deterministic epidemic, Advances in Appl. Probability, 6 (1974), 622-635. doi: 10.2307/1426183. [30] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962. [31] S. P. Sethi and P. W. Staats, Optimal control of some simple deterministic epidemic models, J. Opl. Res. Soc., 29 (1978), 129-136. [32] L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638. [33] X. Yan and Y. Zou, Control of epidemics by quarantine and isolation strategies in highly mobile populations, Int. J. Inform. Sys. Science, 5 (2009), 271-286. [34] K. H. Wickwire, Optimal isolation policies for deterministic and stochastic epidemics, Math. Biosci., 26 (1975), 325-346. doi: 10.1016/0025-5564(75)90020-6. [35] World Health Organization, "Immunization Surveillance, Assessment and Monitoring. Data Statistics and Graphics,", Available from: \url{http://www.who.int/immunization_monitoring/data/en/} (Select Member State: China), ().

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##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases in Humans: Dynamics and Control," Oxford University Press, Oxford, 1991. [2] E. Asano, L. J. Gross, S. Lenhart and L. A. Real, Optimal control of vaccine distribution in a rabies metapopulation model, Math. Biosci. Engineering, 5 (2008), 219-238. [3] H. Behncke, Optimal control of deterministic epidemics, Optim. Control Appl. Meth., 21 (2000), 269-285. doi: 10.1002/oca.678. [4] K. W. Blayneh, A. B. Gumel, S. Lenhart and T. Clayton, Backward bifurcation and optimal control in transmission dynamics of West Nile virus, Bull. Math. Biol., 72 (2010), 1006-1028. doi: 10.1007/s11538-009-9480-0. [5] C. Bowman and A. B. Gumel, Optimal vaccination strategies for an influenza-like illness in a heterogeneous population, in "Mathematical Studies on Human Disease Dynamics," 31-49, Contemp. Math., 410, Amer. Math. Soc., Providence, RI, 2006. [6] F. Brauer, P. van den Driessche and J. Wu, editors, "Mathematical Epidemiology," Lecture Notes in Mathematics, 1945, Mathematical Biosciences Subseries, Springer-Verlag, Berlin, 2008. [7] B. Buonomo and D. Lacitignola, On the use of the geometric approach to global stability for three-dimensional ODE systems: A bilinear case, J. Math. Anal. Appl., 348 (2008), 255-266. doi: 10.1016/j.jmaa.2008.07.021. [8] S. Busenberg and K. Cooke, "Vertically Transmitted Diseases. Models and Dynamics," Biomathematics, 23, Springer-Verlag, Berlin, 1993. [9] V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomath., 97, Springer-Verlag, Berlin, 1993. [10] World Health Organization, Western Pacific Region. Countries and Areas: China 2007., Available from:, \url{http://www.wpro.who.int/countries/2007/chn/}., (). [11] F. T. Cutts and E. Vynnycky, Modelling the incidence of congenital rubella syndrome in developing countries, Int. J. Epidemiol., 28 (1999), 1176-1184. doi: 10.1093/ije/28.6.1176. [12] A. d'Onofrio, Globally stable vaccine-induced eradication of horizontally and vertically transmitted infectious disease with periodic contact rates and disease-dependent demographic factors in the population, Appl. Math. Comput., 140 (2003), 537-547. doi: 10.1016/S0096-3003(02)00251-5. [13] A. d'Onofrio, On pulse vaccination strategy in the SIR epidemic model with vertical transmission, Appl. Math. Lett., 18 (2005), 729-732. doi: 10.1016/j.aml.2004.05.012. [14] L. Dontigny, M. Y. Arsenault, M. J. Martel, et al., Rubella in pregnancy, Society of Obstetricians and Gyneacologists of Canada clinical practice guidelines, J. Obstet. Gynaecol. Can., 30 (2008), 152-68. [15] H. Gaff and E. Schaefer, Optimal control applied to vaccination and treatment strategies for various epidemiological models, Math. Biosci. Eng., 6 (2009), 469-492. doi: 10.3934/mbe.2009.6.469. [16] L. Gao and H. Hethcote, Simulations of rubella vaccination strategies in China, Math. Biosci., 202 (2006), 371-385. doi: 10.1016/j.mbs.2006.02.005. [17] K. Hattaf, M. Rachik, S. Saadi, Y. Tabit and N. Yousfi, Optimal control of tuberculosis with exogenous reinfection, Appl. Math. Sci. (Ruse), 3 (2009), 231-240. [18] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [19] H. R. Joshi, S. Lenhart, M. Y. Li and L. Wang, Optimal control methods applied to disease models, in "Mathematical Studies on Human Disease Dynamics," 187-207, Contemp. Math., 410, Amer. Math. Soc., Providence, RI, 2006. [20] E. Jung, S. Iwami, Y. Takeuchi and Tae-Chang Jo, Optimal control strategy for prevention of avian influenza pandemic, J. Theor. Biol., 260 (2009), 220-229. doi: 10.1016/j.jtbi.2009.05.031. [21] E. Jung, S. Lenhart and Z. Feng, Optimal control of treatments in a two-strain tuberculosis model, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 473-482. doi: 10.3934/dcdsb.2002.2.473. [22] S. Lenhart and J. T. Workman, "Optimal Control Applied to Biological Models," Chapman & Hall/CRC Mathematical and Computational Biology Series, Chapman & Hall/CRC, Boca Raton, FL, 2007. [23] M. Y. Li, H. L. Smith and L. Wang, Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62 (2001), 58-69. doi: 10.1137/S0036139999359860. [24] M. Y. Li and J. S. Muldowney, A geometric approach to global-stability problems, SIAM J. Math. Anal., 27 (1996), 1070-1083. doi: 10.1137/S0036141094266449. [25] M. Y. Li and L. Wang, Global stability in some SEIR epidemic models, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory," 295-311, IMA Volumes in Mathematics and its Applications, 126 (2002), Springer-Verlag, New York. [26] X.-Z. Li and L.-L. Zhou, Global stability of an SEIR epidemic model with vertical transmission and saturating contact rate, Chaos Solitons Fractals, 40 (2009), 874-884. doi: 10.1016/j.chaos.2007.08.035. [27] Matlab. Release 13, The mathworks Inc., Natich, MA, 2002. [28] E. Miller, J. Cradock-Watson and T. Pollock, Consequences of confirmed maternal rubella at successive stages of pregnancy, Lancet, 320 (1982), 781-784. doi: 10.1016/S0140-6736(82)92677-0. [29] R. Morton and K. H. Wickwire, On the optimal control of a deterministic epidemic, Advances in Appl. Probability, 6 (1974), 622-635. doi: 10.2307/1426183. [30] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Interscience Publishers John Wiley & Sons, Inc., New York-London, 1962. [31] S. P. Sethi and P. W. Staats, Optimal control of some simple deterministic epidemic models, J. Opl. Res. Soc., 29 (1978), 129-136. [32] L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638. [33] X. Yan and Y. Zou, Control of epidemics by quarantine and isolation strategies in highly mobile populations, Int. J. Inform. Sys. Science, 5 (2009), 271-286. [34] K. H. Wickwire, Optimal isolation policies for deterministic and stochastic epidemics, Math. Biosci., 26 (1975), 325-346. doi: 10.1016/0025-5564(75)90020-6. [35] World Health Organization, "Immunization Surveillance, Assessment and Monitoring. Data Statistics and Graphics,", Available from: \url{http://www.who.int/immunization_monitoring/data/en/} (Select Member State: China), ().
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