American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Stability of the spectral gap for the Boltzmann multi-species operator linearized around non-equilibrium maxwell distributions
  • CPAA Home
  • This Issue
  • Next Article
    A stochastic threshold for an epidemic model with isolation and a non linear incidence
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 and 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.

[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, 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, 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, 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.

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.

[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, 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, 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, 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.

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 Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Illustration of the energy levels of system (28)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  Pictorial description of $ \Gamma_{x_{0}} $ (continuous curve) and $ t_{v}(\Gamma_{x_{0}}) $ (dashed curve)
Figure Options

Download full-size image

Download as PowerPoint slide

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)
Figure Options

Download full-size image

Download as PowerPoint slide

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} $
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Alan J. Terry. Pulse vaccination strategies in a metapopulation SIR model. Mathematical Biosciences & Engineering, 2010, 7 (2) : 455-477. doi: 10.3934/mbe.2010.7.455
[2]

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
[3]

Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1141-1157. doi: 10.3934/mbe.2017059
[4]

Urszula Ledzewicz, Heinz Schättler. On optimal singular controls for a general SIR-model with vaccination and treatment. Conference Publications, 2011, 2011 (Special) : 981-990. doi: 10.3934/proc.2011.2011.981
[5]

Jing Hui, Lansun Chen. Impulsive vaccination of sir epidemic models with nonlinear incidence rates. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 595-605. doi: 10.3934/dcdsb.2004.4.595
[6]

Hiroshi Ito, Michael Malisoff, Frédéric Mazenc. Strict Lyapunov functions and feedback controls for SIR models with quarantine and vaccination. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022029
[7]

Jinyan Wang, Yanni Xiao, Robert A. Cheke. Modelling the effects of contaminated environments in mainland China on seasonal HFMD infections and the potential benefit of a pulse vaccination strategy. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5849-5870. doi: 10.3934/dcdsb.2019109
[8]

Shuang-Lin Jing, Hai-Feng Huo, Hong Xiang. Modelling the effects of ozone concentration and pulse vaccination on seasonal influenza outbreaks in Gansu Province, China. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 1877-1911. doi: 10.3934/dcdsb.2021113
[9]

Kaijen Cheng, Kenneth Palmer. Chaos in a model for masting. Discrete and Continuous Dynamical Systems - B, 2015, 20 (7) : 1917-1932. doi: 10.3934/dcdsb.2015.20.1917
[10]

Eric A. Carlen, Maria C. Carvalho, Jonathan Le Roux, Michael Loss, Cédric Villani. Entropy and chaos in the Kac model. Kinetic and Related Models, 2010, 3 (1) : 85-122. doi: 10.3934/krm.2010.3.85
[11]

Panagiotes A. Voltairas, Antonios Charalambopoulos, Dimitrios I. Fotiadis, Lambros K. Michalis. A quasi-lumped model for the peripheral distortion of the arterial pulse. Mathematical Biosciences & Engineering, 2012, 9 (1) : 175-198. doi: 10.3934/mbe.2012.9.175
[12]

Najwa Najib, Norfifah Bachok, Norihan Md Arifin, Fadzilah Md Ali. Stability analysis of stagnation point flow in nanofluid over stretching/shrinking sheet with slip effect using buongiorno's model. Numerical Algebra, Control and Optimization, 2019, 9 (4) : 423-431. doi: 10.3934/naco.2019041
[13]

Tao Feng, Zhipeng Qiu. Global analysis of a stochastic TB model with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2923-2939. doi: 10.3934/dcdsb.2018292
[14]

IvÁn Area, FaÏÇal NdaÏrou, Juan J. Nieto, Cristiana J. Silva, Delfim F. M. Torres. Ebola model and optimal control with vaccination constraints. Journal of Industrial and Management Optimization, 2018, 14 (2) : 427-446. doi: 10.3934/jimo.2017054
[15]

Wisdom S. Avusuglo, Kenzu Abdella, Wenying Feng. Stability analysis on an economic epidemiological model with vaccination. Mathematical Biosciences & Engineering, 2017, 14 (4) : 975-999. doi: 10.3934/mbe.2017051
[16]

Sun-Ho Choi, Hyowon Seo, Minha Yoo. A multi-stage SIR model for rumor spreading. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2351-2372. doi: 10.3934/dcdsb.2020124
[17]

Bum Il Hong, Nahmwoo Hahm, Sun-Ho Choi. SIR Rumor spreading model with trust rate distribution. Networks and Heterogeneous Media, 2018, 13 (3) : 515-530. doi: 10.3934/nhm.2018023
[18]

Tomás Caraballo, Renato Colucci. A comparison between random and stochastic modeling for a SIR model. Communications on Pure and Applied Analysis, 2017, 16 (1) : 151-162. doi: 10.3934/cpaa.2017007
[19]

Zhen Jin, Zhien Ma. The stability of an SIR epidemic model with time delays. Mathematical Biosciences & Engineering, 2006, 3 (1) : 101-109. doi: 10.3934/mbe.2006.3.101
[20]

Yan Li, Wan-Tong Li, Guo Lin. Traveling waves of a delayed diffusive SIR epidemic model. Communications on Pure and Applied Analysis, 2015, 14 (3) : 1001-1022. doi: 10.3934/cpaa.2015.14.1001

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (287)
  • HTML views (104)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy