Skip to main content

Advertisement

SpringerLink
Log in

Seasonality in epidemic models: a literature review

  • Published:
Ricerche di Matematica Aims and scope Submit manuscript

    We’re sorry, something doesn't seem to be working properly.

    Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Abstract

We provide a review of some key literature results on the influence of seasonality and other time heterogeneities of contact rates, and other parameters, such as vaccination rates, on the spread of infectious diseases. This is a classical topic where highly theoretical methodologies have provided new insight on the seemingly random behavior observed in epidemic time-series. We follow the line of providing a highly personal non-systematic review of this topic, mainly based on the history of mathematical epidemiology and on the impact of reviewed articles. Our aim is to stress some issues of increasing interest, such as the public health implications of the biomathematical literature and the impact of seasonality on epidemic extinction or elimination.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. Incidentally, the term ‘chaos’ was first introduced in the literature by Yorke himself 2 years later [81].

References

  1. Abdelrazec, A., Lenhart, S., Zhu, H.: Dynamics and optimal control of a West Nile virus model with seasonality. Can. Appl. Math. Q. 23(4), 12–33 (2015)

    Google Scholar 

  2. Altizer, S., Dobson, A., Hosseini, P., Hudson, P., Pascual, M., Rohani, P.: Seasonality and the dynamics of infectious diseases. Ecol. Lett. 9, 467–484 (2006)

    Google Scholar 

  3. Anita, S., Arnautu, V., Capasso, V.: An Introduction to Optimal Control Problems in Life Sciences and Economics. Birkhäuser, Boston (2010)

    MATH  Google Scholar 

  4. Aron, J.L.: Multiple attractors in the response to a vaccination program. Theor. Popul. Biol. 38, 58–67 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Aron, J.L., Schwartz, I.B.: Seasonality and period-doubling bifurcations in an epidemic model. J. Theor. Biol. 110, 665–679 (1984)

    MathSciNet  Google Scholar 

  6. Aronsson, G., Mellander, I.: A deterministic model in biomathematics. Asymptotic behavior and threshold conditions. Math. Biosci. 49, 207–222 (1980)

    MathSciNet  MATH  Google Scholar 

  7. Bacaër, N.: Approximation of the basic reproduction number \(R_0\) for vector-borne diseases with a periodic vector population. Bull. Math. Biol. 69, 1067–1091 (2007)

    MathSciNet  MATH  Google Scholar 

  8. Bacaër, N., Guernaoui, S.: The epidemic threshold of vector-borne diseases with seasonality: the case of cutaneous leishmaniasis in Chichaoua, Morocco. J. Math. Biol. 53, 421–436 (2006)

    MathSciNet  MATH  Google Scholar 

  9. Barrientos, P.G., Angel Rodriguez, J., Ruiz-Herrera, A.: Chaotic dynamics in the seasonally forced SIR epidemic model. J. Math. Biol. 75, 1655–1668 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Bartlett, M.S.: Measles periodicity and community size. J. R. Stat. Soc. A 120, 48–70 (1957)

    Google Scholar 

  11. Ben-Mizrachi, A., Procaccia, I., Grassberger, P.: Characterization of experimental (noisy) strange attractors. Phys. Rev. A 29, 975 (1984)

    Google Scholar 

  12. Billings, L., Schwartz, I.B.: Exciting chaos with noise: unexpected dynamics in epidemic outbreaks. J. Math. Biol. 44, 31–48 (2002)

    MathSciNet  MATH  Google Scholar 

  13. Bjornstad, O.N., Finkenstadt, B.F., Grenfell, B.T.: Dynamics of measles epidemics: estimating scaling of transmission rates using a time series SIR model. Ecol. Monogr. 72(2), 169–184 (2002)

    Google Scholar 

  14. Breen, G.E., Benjamin, B.: Measles in London. The Lancet 254(6579), 620–625 (1949)

    Google Scholar 

  15. Brownlee, J.: An investigation into the periodicity of measles epidemics in London from 1703 to the present day by the method of the periodogram. Phil. Trans. R. Soc. Lond. B 208, 225–250 (1918)

    Google Scholar 

  16. Buonomo, B., Carbone, G., d’Onofrio, A.: Effect of seasonality on the dynamics of an imitation-based vaccination model with public health intervention. Math. Biosci. Eng. 15, 299–321 (2018)

    MathSciNet  MATH  Google Scholar 

  17. Buonomo, B., Della Marca, R.: Optimal bed-net usage for a dengue disease model with mosquito seasonal pattern. Math. Meth. Appl. Sci. https://doi.org/10.1002/mma.4629

  18. Buonomo, B., d’Onofrio, A., Lacitignola, D.: Modeling of pseudo-rational exemption to vaccination for SEIR diseases. J. Math. Anal. Appl. 404, 385–398 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Caminade, C., Kovats, S., Rocklov, J., Tompkins, A.M., Morse, A.P., Colón-González, F.J., Stenlund, H., Martens, P., Lloyd, S.J.: Impact of climate change on global malaria distribution. Proc. Nat. Acad. Sci. U S A 111, 3286–3291 (2014)

    Google Scholar 

  20. Capasso, V.: Mathematical Structures of Epidemic Systems. Springer, Berlin (1993)

    MATH  Google Scholar 

  21. Chaves, L.F., Pascual, M.: Climate cycles and forecasts of cutaneous leishmaniasis, a nonstationary vector-borne disease. PLOS Med. 3, e295 (2006)

    Google Scholar 

  22. Childs, D.Z., Boots, M.: The interaction of seasonal forcing and immunity and the resonance dynamics of malaria. J. R. Soc. Interface 7, 309–319 (2010)

    Google Scholar 

  23. Chitnis, N., Hardy, D., Smith, T.: A periodically-forced mathematical model for the seasonal dynamics of malaria in mosquitoes. Bull. Math. Biol. 74, 1098–1124 (2012)

    MathSciNet  MATH  Google Scholar 

  24. Chow, S.N., Hale, J.K., Mallet-Paret, J.: An example of bifurcation to homoclinic orbits. J. Differ. Equ. 37, 351–373 (1980)

    MathSciNet  MATH  Google Scholar 

  25. Cintron-Arias, A., Banks, H.T., Capaldi, A., Lloyd, A.L.: A sensitivity matrix based methodology for inverse problem formulation. J. Inverse Ill-Posed Probl. 17, 1–20 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Deguen, A., Thomas, G., Chau, N.P.: Estimation of the contact rate in a seasonal SEIR model: application to chickenpox incidence in France. Stat. Med. 19, 1207–1216 (2000)

    Google Scholar 

  27. De Oliveira, J.C.F., Hale, J.K.: Dynamic behavior from bifurcation equations. Tohoku Math. J. 32, 577–592 (1980)

    MathSciNet  MATH  Google Scholar 

  28. Dietz, K.: The incidence of infectious diseases under the influence of seasonal fluctuations. In: Mathematical Models in Medicine, pp. 1–15. Springer, Berlin (1976)

  29. Dietz, K., Molineaux, L., Thomas, A.: A malaria model tested in the African savannah. Bull. World Health Organ. 50, 347–357 (1974)

    Google Scholar 

  30. d’Onofrio, A.: Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Biosci. 179, 57–72 (2002)

    MathSciNet  MATH  Google Scholar 

  31. d’Onofrio, A.: Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures. Math. Comput. Model. 36, 473–489 (2002)

    MathSciNet  MATH  Google Scholar 

  32. d’Onofrio, A.: Vaccination policies and nonlinear force of infection: generalization of an observation by Alexander and Moghadas (2004). Appl. Math. Comput. 168, 613–622 (2005)

    MathSciNet  MATH  Google Scholar 

  33. d’Onofrio, A.: Biomathematical analysis and extension of the new class of epidemic models proposed by Satsuma et al. (2004). Appl. Math. Comput. 170, 125–134 (2005)

    MathSciNet  MATH  Google Scholar 

  34. d’Onofrio, A.: A note on the global behaviour of the network-based SIS epidemic model. Nonlinear Anal. RWA 9, 1567–1572 (2008)

    MathSciNet  MATH  Google Scholar 

  35. d’Onofrio, A., Manfredi, P.: Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases. J. Theor. Biol. 256, 473–478 (2009)

    MathSciNet  Google Scholar 

  36. d’Onofrio, A., Manfredi, P., Salinelli, E.: Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases. Theor. Popul. Biol. 71, 301–317 (2007)

    MATH  Google Scholar 

  37. d’Onofrio, A., Manfredi, P., Salinelli, E.: Fatal SIR diseases and rational exemption to vaccination. Math. Med. Biol. 25, 337–357 (2008)

    MATH  Google Scholar 

  38. Doutor, P., Rodrigues, P., Soares, M., Chalub, F.A.: Optimal vaccination strategies and rational behaviour in seasonal epidemics. J. Math. Biol. 73, 1437–1465 (2016)

    MathSciNet  MATH  Google Scholar 

  39. Duncan, S.R.: Estimating the disease parameters for smallpox in London over the period 1708 to 1748. In: IFAC Proceedings Volumes 38, pp. 1047–1052 (2005)

  40. Duncan, C.J., Duncan, S.R., Scott, S.: Oscillatory dynamics of smallpox and the impact of vaccination. J. Theor. Biol. 183, 447–454 (1996)

    Google Scholar 

  41. Duncan, C.J., Duncan, S.R., Scott, S.: The dynamics of scarlet fever epidemics in England and Wales in the 19th century. Epidemiol. Infect. 117, 493–499 (1996)

    Google Scholar 

  42. Duncan, C.J., Duncan, S.R., Scott, S.: Whooping cough epidemics in London, 1701–1812: infection dynamics, seasonal forcing and the effects of malnutrition. Proc. R. Soc. Lond. B Biol. Sci. 263, 445–450 (1996)

    Google Scholar 

  43. Duncan, C.J., Duncan, S.R., Scott, S.: The dynamics of measles epidemics. Theor. Popul. Biol. 52, 155–163 (1997)

    MATH  Google Scholar 

  44. Duncan, C.J., Duncan, S.R., Scott, S.: The effects of population density and malnutrition on the dynamics of whooping cough. Epidemiol. Infect. 121, 325–334 (1998)

    Google Scholar 

  45. Duncan, S.R., Scott, S., Duncan, C.J.: Modelling the dynamics of scarlet fever epidemics in the 19th century. Eur. J. Epidemiol. 16, 619–626 (2000)

    Google Scholar 

  46. Earn, D.J., Rohani, P., Bolker, B.M., Grenfell, B.T.: A simple model for complex dynamical transitions in epidemics. Science 287(5453), 667–670 (2000)

    Google Scholar 

  47. Eckhoff, P.A., Wenger, E.A.: Spatial agent-based simulation modeling in public health: design, implementation, and applications for malaria epidemiology. In: Arifin, S.M.N., Madey, G.R., Collins, F.H. (eds.) The EMOD Individual-Based Model, 1st edn, pp. 185–208. Wiley, Hoboken (2016). chap.11

    Google Scholar 

  48. Ellner, S.P., Bailey, B.A., Bobashev, G.V., Gallant, A.R., Grenfell, B.T., Nychka, D.W.: Noise and nonlinearity in measles epidemics: combining mechanistic and statistical approaches to population modeling. Am. Nat. 151, 425–440 (1998)

    Google Scholar 

  49. Farmer, J.D., Ott, E., Yorke, J.A.: The dimension of chaotic attractors. Physica D 7, 153–180 (1983)

    MathSciNet  MATH  Google Scholar 

  50. Feigenbaum, M.J.: Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19, 25–52 (1978)

    MathSciNet  MATH  Google Scholar 

  51. Feigenbaum, M.J.: The onset spectrum of turbulence. Phys. Lett. A 74, 375–378 (1979)

    MathSciNet  Google Scholar 

  52. Ferrari, M.J., Djibo, A., Grais, R.F., Bharti, N., Grenfell, B.T., Bjornstad, O.N.: Rural-urban gradient in seasonal forcing of measles transmission in Niger. Proc. R. Soc. Lond. B Biol. Sci. 277, 2775–2782 (2010)

    Google Scholar 

  53. Finkenstadt, B.F., Grenfell, B.T.: Time series modelling of childhood diseases: a dynamical systems approach. J. R. Stat. Soc. Ser. C Appl. Stat. 49(2), 187–205 (2000)

    MathSciNet  MATH  Google Scholar 

  54. Fonda, A.: Uniformly persistent semidynamical systems. Proc. Am. Math. Soc. 104, 111–116 (1988)

    MathSciNet  MATH  Google Scholar 

  55. Glendinning, P., Perry, L.P.: Melnikov analysis of chaos in a simple epidemiological model. J. Math. Biol. 35, 359–373 (1997)

    MathSciNet  MATH  Google Scholar 

  56. Grassberger, P., Procaccia, I.: Characterization of strange attractors. Phys. Rev. Lett. 50, 346 (1983)

    MathSciNet  MATH  Google Scholar 

  57. Grassly, N.C., Fraser, C.: Seasonal infectious disease epidemiology. Proc. R. Soc. Lond. B Biol. Sci. 273(1600), 2541–50 (2006)

    Google Scholar 

  58. Greenman, J., Kamo, M., Boots, M.: External forcing of ecological and epidemiological systems: a resonance approach. Physica D 190, 136–151 (2004)

    MATH  Google Scholar 

  59. Griffin, J.T., Hollingsworth, T.D., Okell, L.C., Churcher, T.S., White, M., Hinsley, W., Bousema, T., Drakeley, C.J., Ferguson, N.M., Basáñez, M.G., Ghani, A.C.: Reducing Plasmodium falciparum malaria transmission in Africa: a model-based evaluation of intervention strategies. PLoS Med. 7, e1000324 (2010)

    Google Scholar 

  60. Grossman, Z.: Oscillatory phenomena in a model of infectious diseases. Theor. Popul. Biol. 18, 204–243 (1980)

    MathSciNet  MATH  Google Scholar 

  61. Guckenheimer, J., Holmes, P.J.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, Berlin (2013)

    MATH  Google Scholar 

  62. Hale, J.K.: Stability from the bifurcation function. In: Differential Equations, pp. 23–30. Academic Press, New York (1980)

  63. Hale, J.K.: Ordinary Differential Equations. Dover Publications, New York (1997)

    MATH  Google Scholar 

  64. Hale, J.K., Taboas, P.: Interaction of damping and forcing in a second order equation. Nonlinear Anal. TMA 2, 77–84 (1978)

    MathSciNet  MATH  Google Scholar 

  65. Heesterbeek, J.A.P., Roberts, M.G.: Threshold quantities for infectious diseases in periodic environments. J. Biol. Syst. 3, 779–787 (1995)

    Google Scholar 

  66. Hethcote, H.W.: Asymptotic behavior in a deterministic epidemic model. Bull. Math. Biol. 35, 607–614 (1973)

    MATH  Google Scholar 

  67. Hoshen, M.B., Morse, A.P.: A weather-driven model of malaria transmission. Malar. J. 3, 32 (2004)

    Google Scholar 

  68. Inoue, M., Kamifukumoto, H.: Scenarios leading to chaos in a forced Lotka–Volterra model. Prog. Theor. Phys. 71, 930–937 (1984)

    MathSciNet  MATH  Google Scholar 

  69. Imam, Z.E., Hosny, A., Alfy, L.B., El Rai, F.: Medical importance of measles in UAR (Egypt). Archiv für die gesamte Virusforschung 16, 49–52 (1965)

    Google Scholar 

  70. Jazwinski, A.H.: Stochastic Processes and Statistical Filtering. Academic Press, New York (1970)

    MATH  Google Scholar 

  71. Kalivianakis, M., Mous, S.L.J., Grasman, J.: Reconstruction of the seasonally varying contact rate for measles. Math. Biosci. 124, 225–234 (1994)

    MATH  Google Scholar 

  72. Keeling, M.J., Rohani, P., Grenfell, B.T.: Seasonally forced disease dynamics explored as switching between attractors. Physica D 148, 317–335 (2001)

    MATH  Google Scholar 

  73. Kernighan, B.W., Pike, R.: The Unix Programming Environment. Englewood Cliffs, Prentice-Hall (1984)

    Google Scholar 

  74. Kernighan, B.W., Ritchie, D.M.: The C Programming Language. Prentice-Hall, Englewood Cliffs (1988)

    MATH  Google Scholar 

  75. Kim, J.E., Lee, H., Lee, C.H., Lee, S.: Assessment of optimal strategies in a two-patch dengue transmission model with seasonality. PLoS ONE 12(3), e0173673 (2017)

    Google Scholar 

  76. Kot, M., Schaffer, W.M., Truty, G.L., Graser, D.J., Olsen, L.F.: Changing criteria for imposing order. Ecol. Model. 43, 75–110 (1988)

    Google Scholar 

  77. Kuznetsov, Y.A., Piccardi, C.: Bifurcation analysis of periodic SEIR and SIR epidemic models. J. Math. Biol. 32, 109–121 (1994)

    MathSciNet  MATH  Google Scholar 

  78. Lajmanovich, A., Yorke, J.A.: A deterministic model for gonorrhea in a nonhomogeneous population. Math. Biosci. 28, 221–236 (1976)

    MathSciNet  MATH  Google Scholar 

  79. Landau, L.D., Lifshitz, E.M.: Mechanics. Pergamon Press, Oxford (1976)

    MATH  Google Scholar 

  80. Lenhart, S., Workman, J.T.: Optimal Control Applied to Biological Models. Chapman & Hall, Boca Raton (2007)

    MATH  Google Scholar 

  81. Li, T.-Y., Yorke, J.A.: Period three implies chaos. Am. Math. Mon. 82, 985–992 (1975)

    MathSciNet  MATH  Google Scholar 

  82. Lloyd, A.L.: Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R. Soc. Lond. B Biol. Sci. 268, 985–993 (2001)

    Google Scholar 

  83. London, W.P., Yorke, J.A.: Recurrent outbreaks of measles, chickenpox and mumps I. Seasonal variation in contact rates. Am. J. Epidemiol. 98, 453–468 (1973)

    Google Scholar 

  84. Magori, K., Legros, M., Puente, M.E., Focks, D.A., Scott, T.W., Lloyd, A.L., Gould, F.: Skeeter Buster: a stochastic, spatially explicit modeling tool for studying Aedes aegypti population replacement and population suppression strategies. PLoS Negl. Trop. Dis. 3, e508 (2009)

    Google Scholar 

  85. Metcalf, C.J.E., Bjornstad, O.N., Grenfell, B.T., Andreasen, V.: Seasonality and comparative dynamics of six childhood infections in pre-vaccination Copenhagen. Proc. R. Soc. Lond. B Biol. Sci. 276, 4111–4118 (2009)

    Google Scholar 

  86. Nakata, Y., Kuniya, T.: Global dynamics of a class of SEIRS epidemic models in a periodic environment. J. Math. Anal. Appl. 325, 230–237 (2010)

    MathSciNet  MATH  Google Scholar 

  87. Olsen, L.F., Schaffer, W.M.: Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics. Science 249, 499–504 (1990)

    Google Scholar 

  88. Olsen, L.F., Truty, G.L., Schaffer, W.M.: Oscillations and chaos in epidemics: a nonlinear dynamic study of six childhood diseases in Copenhagen, Denmark. Theor. Popul. Biol. 33, 344–370 (1988)

    MathSciNet  MATH  Google Scholar 

  89. Plaisier, A.P., van Oortmarssen, G.J., Habbema, J.D.F., Remme, J., Alley, E.S.: ONCHOSIM: a model and computer simulation program for the transmission and control of onchocerciasis. Comput. Methods Programs Biomed. 31, 43–56 (1990)

    Google Scholar 

  90. Picken, R.M.F.: The administrative control of measles. Br. Med. J. 1915–2, 429–430 (1915)

    Google Scholar 

  91. Pourabbas, E., d’Onofrio, A., Rafanelli, M.: A method to estimate the incidence of communicable diseases under seasonal fluctuations with application to cholera. Appl. Math. Comput. 118, 161–174 (2001)

    MathSciNet  MATH  Google Scholar 

  92. Rebelo, C., Margheri, A., Bacaër, N.: Persistence in seasonally forced epidemiological models. J. Math. Biol. 64, 933–949 (2012)

    MathSciNet  MATH  Google Scholar 

  93. Rohani, P., Earn, D.J., Grenfell, B.T.: Opposite patterns of synchrony in sympatric disease metapopulations. Science 286(5441), 968–971 (1999)

    Google Scholar 

  94. Rohani, P., Keeling, M.J., Grenfell, B.T.: The interplay between determinism and stochasticity in childhood diseases. Am. Nat. 159, 469–481 (2002)

    Google Scholar 

  95. Ross, R.: Some quantitative studies in epidemiology. Nature 87(2188), 466–467 (1911)

    MATH  Google Scholar 

  96. Schaffer, W.M., Kot, M.: Nearly one dimensional dynamics in an epidemic. J. Theor. Biol. 112, 403–427 (1985)

    MathSciNet  Google Scholar 

  97. Schenzle, D.: An age-structured model of pre-and post-vaccination measles transmission. Math. Med. Biol. 1, 169–191 (1984)

    MathSciNet  MATH  Google Scholar 

  98. Schwartz, I.B.: Multiple stable recurrent outbreaks and predictability in seasonally forced nonlinear epidemic models. J. Math. Biol. 21, 347–361 (1985)

    MathSciNet  MATH  Google Scholar 

  99. Schwartz, I.B., Smith, H.L.: Infinite subharmonic bifurcation in an SEIR epidemic model. J. Math. Biol. 18, 233–253 (1983)

    MathSciNet  MATH  Google Scholar 

  100. Smith, H.L.: Subharmonic bifurcation in an S-I-R epidemic model. J. Math. Biol. 17, 163–177 (1983)

    MathSciNet  MATH  Google Scholar 

  101. Smith, T., Maire, N., Ross, A., Penny, M., Chitnis, N., Schapira, A., Studer, A., Genton, B., Lengeler, C., Tediosi, F., de Savigny, D., Tanner, M.: Towards a comprehensive simulation model of malaria epidemiology and control. Parasitology 135, 1507–1516 (2008)

    Google Scholar 

  102. Soper, H.E.: The interpretation of periodicity in disease prevalence. J. R. Stat. Soc. 92, 34–73 (1929)

    MATH  Google Scholar 

  103. Stirzaker, D.R.: A perturbation method for the stochastic recurrent epidemic. J. Inst. Math. Appl. 15, 135–160 (1975)

    MathSciNet  MATH  Google Scholar 

  104. Stone, L., Olinky, R., Huppert, A.: Seasonal dynamics of recurrent epidemics. Nature 446, 533–536 (2007)

    MATH  Google Scholar 

  105. Sutherland, I., Fayers, P.M.: Effect of measles vaccination on incidence of measles in the community. Br. Med. J. 1971–1, 698–702 (1971)

    Google Scholar 

  106. Takens, F.: Detecting strange attractors in turbulence. In: Dynamical Systems and Turbulence, Warwick 1980, pp. 366–381. Springer, Berlin (1981)

  107. Tanaka, G., Aihara, K.: Effects of seasonal variation patterns on recurrent outbreaks in epidemic models. J. Theor. Biol. 317, 87–95 (2013)

    MathSciNet  MATH  Google Scholar 

  108. Thieme, H.: Uniform weak implies uniform strong persistence for non-autonomous semiflows. Proc. Am. Math. Soc. 127(8), 2395–2403 (1999)

    MathSciNet  MATH  Google Scholar 

  109. Thieme, H.R.: Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. Biosci. 166(2), 173–201 (2000)

    MathSciNet  MATH  Google Scholar 

  110. Tompkins, A.M., Volker, E.: A regional-scale, high resolution dynamical malaria model that accounts for population density, climate and surface hydrology. Malar. J. 12, 65 (2013)

    Google Scholar 

  111. Wang, Z., Bauch, C.T., Bhattacharyya, S., d’Onofrio, A., Manfredi, P., Perc, M., Perra, N., Salathe, M., Zhao, D.: Statistical physics of vaccination. Phys. Rep. 664, 1–113 (2016)

    MathSciNet  MATH  Google Scholar 

  112. Wang, W., Zhao, X.Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20, 699–717 (2008)

    MathSciNet  MATH  Google Scholar 

  113. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Springer, New York (2003)

    MATH  Google Scholar 

  114. Wolf, A., Swift, J.B., Swinney, L., Vastano, J.A.: Determining Lyapunov exponents from a time series. Physica D 16, 285–317 (1985)

    MathSciNet  MATH  Google Scholar 

  115. Zhang, F., Zhao, X.Q.: A periodic epidemic model in a patchy environment. J. Math. Anal. Appl. 325, 496–516 (2007)

    MathSciNet  MATH  Google Scholar 

  116. Zhao, X.Q.: Dynamical Systems in Population Biology, CMS Books Mathematics, vol. 16. Springer, New York (2003)

    Google Scholar 

Download references

Acknowledgements

The work of BB has been performed under the auspices of the Italian National Group for the Mathematical Physics (GNFM) of the National Institute for Advanced Mathematics (INdAM).

Author information

Authors and Affiliations

  1. Department of Mathematics and Applications, University of Naples Federico II, via Cintia, 80126, Naples, Italy

    B. Buonomo

  2. Department of Public Health and Epidemiology, Swiss Tropical and Public Health Institute, Socinstrasse 57, Postfach, 4002, Basel, Switzerland

    N. Chitnis

  3. International Prevention Research Institute, 95 cours Lafayette, 69006, Lyon, France

    A. d’Onofrio

  4. University of Basel, Basel, Switzerland

    N. Chitnis

Authors
  1. B. Buonomo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. Chitnis
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. d’Onofrio
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. d’Onofrio.

Additional information

This article belongs to the Special Issue: Demographic and temporal heterogeneity in infectious disease epidemiology.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Buonomo, B., Chitnis, N. & d’Onofrio, A. Seasonality in epidemic models: a literature review. Ricerche mat 67, 7–25 (2018). https://doi.org/10.1007/s11587-017-0348-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11587-017-0348-6

Keywords

Mathematics Subject Classification

Search

Navigation