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May  2020, 19(5): 2533-2548. doi: 10.3934/cpaa.2020111

Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination

Alfonso Ruiz Herrera

Departamento de Matemáticas, Universidad de Oviedo, Oviedo, Spain

Received  November 2018 Revised  October 2019 Published  March 2020

Fund Project: The author is supported by the project MTM2017-87697P

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.

Keywords: Pulse vaccination, chaos, SIR model, stretching along the paths.
Mathematics Subject Classification: Primary: 92B05, 34C28; Secondary: 34C60.
Citation: Alfonso Ruiz Herrera. Paradoxical phenomena and chaotic dynamics in epidemic models subject to vaccination. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2533-2548. doi: 10.3934/cpaa.2020111
References:
[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.  Google Scholar

[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.  Google Scholar

[3]

N. Bharti, Explaining seasonal fluctuations of measles in Niger using night time lights imagery, Science, 334 (2011), 1424-1427.  doi: 10.1126/science.1210554.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. J. Ferrari, The dynamics of measles in sub-Saharan Africa, Nature, 451 (2008), 679-684.  doi: 10.1038/nature06509.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

T. J. John, Immunisation against polioviruses in developing countries, Rev. Med. Virol., 3 (1993), 149-160.   Google Scholar

[15] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2011.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. C. Quadros, Eradication of poliomyelitis: progress in the Americas, Pediatr. Infect. Dis. J., 10 (1991), 222-229.  doi: 10.1097/00006454-199103000-00011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[27]

H. L. Smith, Subharmonic bifurcation in an SIR epidemic model, J. Math. Biol., 17 (1983), 163-177.  doi: 10.1007/BF00305757.  Google Scholar

[28]

L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536.  doi: 10.1038/nature05638.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[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.  Google Scholar

[3]

N. Bharti, Explaining seasonal fluctuations of measles in Niger using night time lights imagery, Science, 334 (2011), 1424-1427.  doi: 10.1126/science.1210554.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

M. J. Ferrari, The dynamics of measles in sub-Saharan Africa, Nature, 451 (2008), 679-684.  doi: 10.1038/nature06509.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

T. J. John, Immunisation against polioviruses in developing countries, Rev. Med. Virol., 3 (1993), 149-160.   Google Scholar

[15] M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2011.   Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

D. C. Quadros, Eradication of poliomyelitis: progress in the Americas, Pediatr. Infect. Dis. J., 10 (1991), 222-229.  doi: 10.1097/00006454-199103000-00011.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[27]

H. L. Smith, Subharmonic bifurcation in an SIR epidemic model, J. Math. Biol., 17 (1983), 163-177.  doi: 10.1007/BF00305757.  Google Scholar

[28]

L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536.  doi: 10.1038/nature05638.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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
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Figure 2.  Illustration of the energy levels of system (28)
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Figure 3.  Pictorial description of $ \Gamma_{x_{0}} $ (continuous curve) and $ t_{v}(\Gamma_{x_{0}}) $ (dashed curve)
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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)
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Figure 5.  Pictorial description of the annulus $ \mathcal{A}_{1} $, $ \mathcal{A}_{2} $ and the topological rectangles $ \mathcal{R}_{1} $, $ \mathcal{R}_{2} $. Notice that $ \mathcal{Q}\subset \mathcal{R}_{2} $
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