-
Previous Article
A note for the global stability of a delay differential equation of hepatitis B virus infection
- MBE Home
- This Issue
-
Next Article
A note on the replicator equation with explicit space and global regulation
A simple analysis of vaccination strategies for rubella
1. | Department of Mathematics and Applications, University of Naples Federico II, via Cintia, I-80126 Naples |
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: 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] | |
[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: http://www.who.int/immunization_monitoring/data/en/ (Select Member State: China) |
show all references
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: 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] | |
[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: http://www.who.int/immunization_monitoring/data/en/ (Select Member State: China) |
[1] |
Shujing Gao, Dehui Xie, Lansun Chen. Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 77-86. doi: 10.3934/dcdsb.2007.7.77 |
[2] |
Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete and Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69 |
[3] |
Chandrani Banerjee, Linda J. S. Allen, Jorge Salazar-Bravo. Models for an arenavirus infection in a rodent population: consequences of horizontal, vertical and sexual transmission. Mathematical Biosciences & Engineering, 2008, 5 (4) : 617-645. doi: 10.3934/mbe.2008.5.617 |
[4] |
Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929 |
[5] |
Liming Cai, Maia Martcheva, Xue-Zhi Li. Epidemic models with age of infection, indirect transmission and incomplete treatment. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2239-2265. doi: 10.3934/dcdsb.2013.18.2239 |
[6] |
Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control and Related Fields, 2022, 12 (1) : 201-223. doi: 10.3934/mcrf.2021007 |
[7] |
Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. Optimal control for an epidemic in populations of varying size. Conference Publications, 2015, 2015 (special) : 549-561. doi: 10.3934/proc.2015.0549 |
[8] |
Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045 |
[9] |
Linhua Zhou, Meng Fan, Qiang Hou, Zhen Jin, Xiangdong Sun. Transmission dynamics and optimal control of brucellosis in Inner Mongolia of China. Mathematical Biosciences & Engineering, 2018, 15 (2) : 543-567. doi: 10.3934/mbe.2018025 |
[10] |
Chunxiao Ding, Zhipeng Qiu, Huaiping Zhu. Multi-host transmission dynamics of schistosomiasis and its optimal control. Mathematical Biosciences & Engineering, 2015, 12 (5) : 983-1006. doi: 10.3934/mbe.2015.12.983 |
[11] |
Moatlhodi Kgosimore, Edward M. Lungu. The Effects of Vertical Transmission on the Spread of HIV/AIDS in the Presence of Treatment. Mathematical Biosciences & Engineering, 2006, 3 (2) : 297-312. doi: 10.3934/mbe.2006.3.297 |
[12] |
Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure and Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97 |
[13] |
Claus Kirchner, Michael Herty, Simone Göttlich, Axel Klar. Optimal control for continuous supply network models. Networks and Heterogeneous Media, 2006, 1 (4) : 675-688. doi: 10.3934/nhm.2006.1.675 |
[14] |
Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006 |
[15] |
Sanjukta Hota, Folashade Agusto, Hem Raj Joshi, Suzanne Lenhart. Optimal control and stability analysis of an epidemic model with education campaign and treatment. Conference Publications, 2015, 2015 (special) : 621-634. doi: 10.3934/proc.2015.0621 |
[16] |
Kazuyuki Yagasaki. Optimal control of the SIR epidemic model based on dynamical systems theory. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2501-2513. doi: 10.3934/dcdsb.2021144 |
[17] |
Colin J. Cotter, Michael John Priestley Cullen. Particle relabelling symmetries and Noether's theorem for vertical slice models. Journal of Geometric Mechanics, 2019, 11 (2) : 139-151. doi: 10.3934/jgm.2019007 |
[18] |
Folashade B. Agusto. Optimal control and cost-effectiveness analysis of a three age-structured transmission dynamics of chikungunya virus. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 687-715. doi: 10.3934/dcdsb.2017034 |
[19] |
Fred Brauer. Some simple epidemic models. Mathematical Biosciences & Engineering, 2006, 3 (1) : 1-15. doi: 10.3934/mbe.2006.3.1 |
[20] |
Fred Brauer, Zhilan Feng, Carlos Castillo-Chávez. Discrete epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (1) : 1-15. doi: 10.3934/mbe.2010.7.1 |
2018 Impact Factor: 1.313
Tools
Metrics
Other articles
by authors
[Back to Top]