An alternative to the constant vaccination strategy could be the administration of a large number of doses on "immunization days" with the aim of maintaining the basic reproduction number to be below one. This strategy, known as pulse vaccination, has been successfully applied for the control of many diseases especially in low-income countries. In this paper, we analytically prove (without being computer-aided) the existence of chaotic dynamics in the classical SIR model with pulse vaccination. To the best of our knowledge, this is the first time in which a theoretical proof of chaotic dynamics is given for an epidemic model subject to pulse vaccination. In a realistic public health context, our analysis suggests that the combination of an insufficient vaccination coverage and high birth rates could produce chaotic dynamics and an increment of the number of infectious individuals.
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Figure 1. Representation of $ \Upsilon(0.3,T) $ as a function of $ T $. Fixed parameters in both panels $ \beta = \gamma = \mu = 1 $. (Left) $ \lambda = 0.25 $. In this case, $ R_{0}<1 $ and $ \Upsilon(0.3,T)<1 $ for all $ T>0 $. (Right) $ \lambda = 3 $. In this case, $ R_{0}>1 $ and $ \Upsilon (0.3,T) $ exhibits a non-monotone behavior
Figure 4. Pictorial description of $ \Gamma_{x_{0}} $, $ \Gamma_{x_{0}+\delta} $ (continuous curves), and $ t_{v}(\Gamma_{x_{0}}) $ $ t_{v}(\Gamma_{x_{0}+\delta}) $ (dashed lines). Geometrically, condition (31) means that the intersection of both annulus is below the line $ z = \widetilde{z}_{max} $, (red line)
[1] |
Z. Agur, L. Cojocaru, G. Mazor, R. M. Anderson and Y. L. Danon, Pulse mass measles vaccination across age cohorts, Proc. Natl. Acad. Sci. U. S. A., 90 (1993), 11698-11702.
doi: 10.1073/pnas.90.24.11698.![]() ![]() |
[2] |
P. G. Barrientos, J. A. Rodriguez and A. Ruiz-Herrera, Chaotic dynamics in the seasonally forced SIR epidemic model, J. Math. Biol., 75 (2017), 1655-1668.
doi: 10.1007/s00285-017-1130-9.![]() ![]() ![]() |
[3] |
N. Bharti, et al., Explaining seasonal fluctuations of measles in Niger using night time lights imagery, Science, 334 (2011), 1424-1427.
doi: 10.1126/science.1210554.![]() ![]() |
[4] |
C. J. Browne, R. J. Smith and L. Bourouiba, From regional pulse vaccination to global disease eradication: insights from a mathematical model of poliomyelitis, J. Math. Biol., 71 (2015), 215-253.
doi: 10.1007/s00285-014-0810-y.![]() ![]() ![]() |
[5] |
S. V. Chincholikar and R. D. Prayag, Evaluation of pulse-polio immunisation in rural area of Maharashtra, Indian J. Pediatr., 67 (2000), 647-649.
doi: 10.1007/BF02762174.![]() ![]() |
[6] |
M. Choisy, J. F. Guegan and P. Rohani, Dynamics of infectious diseases and pulse vaccination: teasing apart the embedded resonance effects, Physica D, 223 (2006), 26-35.
doi: 10.1016/j.physd.2006.08.006.![]() ![]() ![]() |
[7] |
S. N. Chow and D. Wang, On the monotonicity of the period function of some second order equations, $\check{C}$asopis P$\check{e}$st. Mat., 111 (1986), 14–25.
![]() ![]() |
[8] |
A. D'Onofrio, Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures, Math. Comput. Model., 36 (2002), 473-489.
doi: 10.1016/S0895-7177(02)00177-2.![]() ![]() ![]() |
[9] |
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.![]() ![]() ![]() |
[10] |
D. J. Earn, P. Rohani, B. M. Bolker and B. T. Grenfell, A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670.
doi: 10.1126/science.287.5453.667.![]() ![]() |
[11] |
M. J. Ferrari, et al., The dynamics of measles in sub-Saharan Africa, Nature, 451 (2008), 679-684.
doi: 10.1038/nature06509.![]() ![]() |
[12] |
T. C. Germann, K. Kadau, I. M. Longini and C. A. Macken, Mitigation strategies for pandemic influenza in the United States, Proc. Natl. Acad. Sci. U. S. A., 103 (2006), 5935-5940.
doi: 10.1073/pnas.0601266103.![]() ![]() |
[13] |
H. Heesterbeek, et al., Modeling infectious disease dynamics in the complex landscape of global health, Science, 347 (2015), 4339.
doi: 10.1126/science.aaa4339.![]() ![]() |
[14] |
T. J. John, Immunisation against polioviruses in developing countries, Rev. Med. Virol., 3 (1993), 149-160.
![]() |
[15] |
M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2011.
![]() ![]() |
[16] |
A. Korobeinikov and P. K. Maini, A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence, Math. Biosci. Eng., 1 (2004), 57-60.
doi: 10.3934/mbe.2004.1.57.![]() ![]() ![]() |
[17] |
X. Liu, Y. Takeuchi and S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol., 253 (2008), 1-11.
doi: 10.1016/j.jtbi.2007.10.014.![]() ![]() ![]() |
[18] |
L. Mailleret and L. Valerie, A note on semi-discrete modelling in the life sciences, Philos. Trans. R. Soc. A - Math. Phys. Eng., 367 (2009), 4779-4799.
doi: 10.1098/rsta.2009.0153.![]() ![]() ![]() |
[19] |
A. Margheri, C. Rebelo and F. Zanolin, Chaos in periodically perturbed planar Hamiltonian systems using linked twist maps, J. Differ. Equ., 249 (2010), 3233-3257.
doi: 10.1016/j.jde.2010.08.021.![]() ![]() ![]() |
[20] |
A. Medio, M. Pireddu and F. Zanolin, Chaotic dynamics for maps in one and two dimensions: a geometrical method and applications to economics, Int. J. Bifurcation Chaos, 19 (2009), 3283-3309.
doi: 10.1142/S0218127409024761.![]() ![]() ![]() |
[21] |
D. C. Quadros, et al., Eradication of poliomyelitis: progress in the Americas, Pediatr. Infect. Dis. J., 10 (1991), 222-229.
doi: 10.1097/00006454-199103000-00011.![]() ![]() |
[22] |
M. Rey and P. G. Marc, The global eradication of poliomyelitis: Progress and problems, Comp. Immunol. Microbiol. Infect. Dis., 31 (2008), 317-325.
doi: 10.1016/j.cimid.2007.07.013.![]() ![]() |
[23] |
P. Rohani, D. E. J. Earn and B. T. Grenfell, Opposite patterns of synchrony in sympatric disease metapopulations, Science, 286 (1999), 968-971.
doi: 10.1126/science.286.5441.968.![]() ![]() |
[24] |
A. Ruiz-Herrera and F. Zanolin, Horseshoes in 3D equations with applications to Lotka-Volterra systems, NoDea-Nonlinear Differ. Equ. Appl., 22 (2015), 877-897.
doi: 10.1007/s00030-014-0307-9.![]() ![]() ![]() |
[25] |
A. B. Sabin, Measles, killer of millions in developing countries: strategy for rapid elimination and continuing control, Eur. J. Epidemiol., 7 (1993), 1-22.
![]() |
[26] |
B. Shulgin, L. Stone and Z. Agur, Pulse vaccination strategy in the SIR epidemic model, Bull. Math. Biol., 60 (1998), 1123-1148.
doi: 10.1016/S0092-8240(98)90005-2.![]() ![]() |
[27] |
H. L. Smith, Subharmonic bifurcation in an SIR epidemic model, J. Math. Biol., 17 (1983), 163-177.
doi: 10.1007/BF00305757.![]() ![]() ![]() |
[28] |
L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536.
doi: 10.1038/nature05638.![]() ![]() |
[29] |
L. Stone, B. Shulgin and Z. Agur, Theoretical examination of the pulse vaccination policy in the SIR epidemic model, Math. Comput. Model., 31 (2000), 207-216.
doi: 10.1016/S0895-7177(00)00040-6.![]() ![]() ![]() |
[30] |
A. J. Terry, Pulse vaccination strategies in a metapopulation SIR model, Math. Biosci. Eng., 7 (2010), 455-477.
doi: 10.3934/mbe.2010.7.455.![]() ![]() ![]() |
Representation of
Illustration of the energy levels of system (28)
Pictorial description of
Pictorial description of
Pictorial description of the annulus