zbMATH — the first resource for mathematics

Determination of optimal vaccination strategies using an orbital stability threshold from periodically driven systems. (English) Zbl 1282.92020
Summary: We analyse a periodically driven SIR epidemic model for childhood related diseases, where the contact rate and vaccination rate parameters are considered to be periodic. The aim is to define optimal vaccination strategies for the control of childhood related infections. Stability analysis of the uninfected solution is the tool for setting up the control function. The optimal solutions are sought within a set of susceptible population profiles. Our analysis reveals that a periodic vaccination strategy hardly contributes to the stability of the uninfected solution if the human residence time (life span) is much larger than the contact rate period. However, if the human residence time and the contact rate match periods, we observe some positive effects of periodic vaccination. Such a vaccination strategy would be useful in the developing world, where human life spans are shorter, or basically in the case of vaccination of livestock or small animals whose life-spans are relatively shorter.

92C60 Medical epidemiology
49N90 Applications of optimal control and differential games
49J15 Existence theories for optimal control problems involving ordinary differential equations
Full Text: DOI
[1] Agur, Z; Cojocaru, L; Mazor, G; Anderson, R; Danon, Y, Pulse mass measles vaccination across age cohorts, Proc Natl Acad Sci, 90, 11698-11702, (1993)
[2] Alexander, ME; Moghadas, SM; Rohani, P; Summers, AR, Modelling the effect of a booster vaccination on disease epidemiology, J Math Biol, 52, 290-306, (2006) · Zbl 1086.92010
[3] Amann H, Escher J (2001) Analysis III. Birkhäuser, Basel · Zbl 0995.26001
[4] Bacaer, N; Abdurahman, X, Resonance of the epidemic threshold in a periodic environment, J Math Biol, 57, 649-673, (2008) · Zbl 1161.92044
[5] Bacaer, N; Ait Dads, E, Genealogy with seasonality, the basic reproduction number, and the influenza pandemic, J Math Biol, 62, 741-762, (2011) · Zbl 1232.92063
[6] Bacaer N, Ait Dads E (2012) On the biological interpretation of a definition for the parameter R0 in periodic population models. J Math Biol. doi:10.1007/s00285-011-0479-4 · Zbl 1303.92111
[7] Bacaer, N; Gomes, MGM, On the final fize of epidemics with seasonality, Bull Math Biol, 71, 1954-1966, (2009) · Zbl 1180.92074
[8] Bacaer, N; Guernaoui, S, The epidemic threshold of vector-borne diseases with seasonality, J Math Biol, 53, 421-436, (2006) · Zbl 1098.92056
[9] Castillo-Chavez, C; Feng, Z, Global stability of an age structured model for TB and its applications to optimal vaccination strategies, Math Biosci, 151, 135-154, (1998) · Zbl 0981.92029
[10] Chicone C (1999) Ordinary differential equations with applications. Springer, New York · Zbl 0937.34001
[11] Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases. Wiley, Chichester · Zbl 0997.92505
[12] d’Onofrio, A, Stability properties of pulse vaccination strategy in SEIR epidemic model, Math Biosci, 179, 57-72, (2002) · Zbl 0991.92025
[13] Evans LC (1998) Partial differential equations. American Mathematical Society, Providence · Zbl 0902.35002
[14] Gao S, Teng Z, Nieto JJ, Torres A (2007) Analysis of an SIR epidemic model with pulse vaccination and distributed time dely. J Biomed Biotechnol. doi:10.1155/2007/64870 · Zbl 0981.92029
[15] Hadeler KP, Müller J (1996a) Vaccination in age structured populations I: the reproduction number. In: Isham V, Medley G (eds) Models for infectious human diseases: their structure and relation to data. Cambridge University Press, Cambridge, pp 90-101
[16] Hadeler KP, Müller J (1996b) Vaccination in age structured populations II: optimal vaccination strategies. In: Isham V, Medley G (eds) Models for infectious human diseases: their structure and relation to data. Cambridge University Press, Cambridge, pp 102-114
[17] Ioos G, Daniel DD (1990) Elementary stability and bifurcation theory. Springer, New York
[18] Khasin M, Dykman MI, Meerson B (2010) Speeding up disease extinction with a limited amount of vaccine. Phys Rev E 81:5 (no. 051925)
[19] Keeling MJ, Rohani P (2007) Modeling infectious diseases in humans and animals. Princeton University Press, Princeton · Zbl 1279.92038
[20] Müller, J, Optimal vaccination patterns in age-structured populations, SIAM J Appl Math, 59, 222-241, (1998) · Zbl 0919.92033
[21] Müller, J, Optimal vaccination patterns in age-structured populations: endemic case, Math Comput Model, 31, 149-160, (2000) · Zbl 1043.92515
[22] Mossong, J; Müller, CP, Estimation of the basic reproduction number of measles during an outbreak in a partially vaccinated population, Epidemiol Infect, 124, 273-278, (2000)
[23] Neubert, MG, Marine reserves and optima harvesting, Ecol Lett, 6, 843-849, (2003)
[24] Nokes, DJ; Swinton, J, The control of childhood viral infections by pulse vaccination, Math Med Biol, 12, 29-53, (1995) · Zbl 0832.92024
[25] Nokes DJ, Swinton J (1997) Vaccination in pulses: a strategy for global eradication of measles and polio? Trends Microbiol 14:1
[26] Onyango NO, Müller J (2013) Optimal vaccination strategies: orbital stability analysis versus instantaneous stability threshold (unpublished) · Zbl 1232.92063
[27] Shulgin, B; Stone, L; Agur, Z, Pulse vaccination strategy in the SIR epidemic model, Bull Math Biol, 60, 1123-1148, (1998) · Zbl 0941.92026
[28] Shulgin, B; Stone, L; Agur, Z, Theoretical examination of pulse vaccination policy in the SIR epidemic model, Math Comput Model, 31, 207-215, (2000) · Zbl 1043.92527
[29] Thieme HR (2003) Mathematics in population biology. Princeton University Press, Princeton · Zbl 1054.92042
[30] UN Reports (2007) World population prospects: the 2006 revision, highlights, working paper no. ESA/P/WP.202. United Nations Department of economic and social affairs: population division, New York · Zbl 1043.92515
[31] WHO and UNICEF (2010) Immunization summary: a statistical reference containing data through 2008. United Nations Children’s Fund (UNICEF), New York
[32] Wesley, CL; Allen, LJS, The basic reproduction number in epidemic models with periodic demographics, J Biol Dyn, 3, 116-129, (2009) · Zbl 1342.92287
[33] Wickwire, KH, Mathematical models for the control of pests and infectious diseases: a survey, Theor Popul Biol, 11, 182-238, (1977) · Zbl 0356.92001
[34] Yosida K (1980) Functional analysis. Springer, Berlin · Zbl 0435.46002
[35] Zhou, Y; Liu, H, Stability of periodic solutions for an SIS model with pulse vaccination, Math Comput Model, 38, 299-308, (2003) · Zbl 1045.92042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.