Abstract
Seasonality is a complex force in nature that affects multiple processes in wild animal populations. In particular, seasonal variations in demographic processes may considerably affect the persistence of a pathogen in these populations. Furthermore, it has been long observed in computer simulations that under seasonal perturbations, a host–pathogen system can exhibit complex dynamics, including the transition to chaos, as the magnitude of the seasonal perturbation increases. In this paper, we develop a seasonally perturbed Susceptible-Infected-Recovered model of avian influenza in a seabird colony. Numerical simulations of the model give rise to chaotic recurrent epidemics for parameters that reflect the ecology of avian influenza in a seabird population, thereby providing a case study for chaos in a host– pathogen system. We give a computer-assisted exposition of the existence of chaos in the model using methods that are based on the concept of topological hyperbolicity. Our approach elucidates the geometry of the chaos in the phase space of the model, thereby offering a mechanism for the persistence of the infection. Finally, the methods described in this paper may be immediately extended to other infections and hosts, including humans.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Advanced Numerical Differential Equation Solving in Mathematica (2011) Wolfram Research. http://reference.wolfram.com/mathematica/tutorial/NDSolveOverview.html. Accessed 8 October 2011
Altizer S, Dobson A, Hosseini P, Hudson P, Pascual M, Rohani P (2006) Seasonality and the dynamics of infectious diseases. Ecol Lett 9: 467–484
Anderson, RM, May, RM (eds) (1982) Population biology of infectious diseases. Springer, Berlin
Anishchenko VS, Vadivasova TE, Strelkova GI, Kopeikin AS (1998) Chaotic attractors of two-dimensional invertible maps. Discr Dyn Nat Soc 2: 249–256
Aron JL, Schwartz IB (1984) Seasonality and period-doubling bifurcations in an epidemic model. J Theor Biol 110: 665–679
Barbosa A, Palacios MJ (2009) Health of Antarctic birds: a review of their parasites, pathogens and diseases. Polar Biol 32: 1095–1115
Billings L, Schwartz IB (2002) Exciting chaos with noise: unexpected dynamics in epidemic outbreaks. J Math Biol 44: 31–48
Bogdanova MI, Daunt F, Newell M, Phillips RA, Harris MP, Wanless S (2011) Seasonal interactions in the black-legged kittiwake, Rissa tridactyla: links between breeding performance and winter distribution. Proc R Soc B 278: 2412–2418
Bolker BM, Grenfell BT (1993) Chaos and biological complexity in measles dynamics. Proc R Soc B 251: 75–81
Brown CR, Brown MB (1996) Coloniality in the Cliff Swallow—the effect of group size on social behaviour. The University of Chicago Press, Chicago
Chen H, Smith GJD, Zhang SY, Qin K, Wang J, Li KS, Webster RG, Peiris JSM, Guan Y (2005) H5N1 virus outbreak in migratory waterfowl. Nature 436: 191–192
Clayton, DH, Moore, J (eds) (1997) Host–parasite evolution: general principles and avian models. Oxford University Press, Oxford
Cox EA, Mortell MP, Pokrovskii AV, Rasskazov O (2005) On chaotic wave patterns in periodically forced steady state KdVB and extended KdVB equations. Proc R Soc A 461: 2857–2885
Crespin L, Harris MP, Lebreton JD, Frederiksen M, Wanless S (2006) Recruitment to a seabird population depends on environmental factors and on population size. J Anim Ecol 75: 228–238
Croxall, JP (eds) (1987) Seabirds: feeding ecology and role in marine ecosystems. Cambridge University Press, Cambridge
Dane DS, Miles JAR, Stoker MGP (1953) A disease of Manx Shearwaters: further observations in the field. J Anim Ecol 22: 123–133
Daszak P, Cunningham AA, Hyatt AD (2000) Emerging infectious diseases of wildlife—threats to biodiversity and human health. Science 287: 483–489
Day S, Junge O, Mischaikow K (2004) A rigorous numerical method for the global analysis of infinite-dimensional discrete dynamical systems. SIAM J Appl Dyn Syst 3: 117–160
Deimling K (1985) Nonlinear functional analysis. Springer, Berlin
Descamps S, Gilchrist HG, Bety J, Buttler IE, Forbes MR (2009) Costs of reproduction in a long-lived bird: large clutch size is associated with low survival in the presence of a highly virulent disease. Biol Lett 5: 278–281
Diallo O, Koné Y (2007) Melnikov analysis of chaos in a general epidemiological model. Nonlinear Anal Real World Appl 8: 20–26
Dobson A (2004) Population dynamics of pathogens with multiple host species. Am Nat 164: S64–S78
Dushoff J, Plotkin JB, Levin SA, Earn DJB (2004) Dynamical resonance can account for seasonality of influenza epidemics. Proc Natl Acad Sci USA 101: 16915–16916
Earn DJD, Rohani P, Grenfell BT (1998) Persistence, chaos and synchrony in ecology and epidemiology. Proc R Soc B 265: 7–10
Earn DJD, Rohani P, Bolker BM, Grenfell BT (2000) A simple model for complex dynamical transitions in epidemics. Science 287: 667–670
Galias Z, Zgliczynski P (1998) Computer assisted proof of chaos in the Lorenz equations. Phys D 115: 165–188
Gameiro M, Gedeon T, Kalies W, Kokubu H, Mischaikow K, Oka H (2007) Topological horseshoes of travelling waves for a fast-slow predator-prey system. J Differ Dyn Equ 19: 623–654
Gani R, Hughes H, Fleming D, Griffin T, Medlock J, Leach S (2005) Potential impact of antiviral drug use during influenza pandemic. Emerg Infect Dis 11: 1355–1362
Glendinning P, Perry LP (1997) Melnikov analysis of chaos in a simple epidemiological model. J Math Biol 35: 359–373
Greenman J, Kamo M, Boots M (2004) External forcing of ecological and epidemiological systems: a resonance approach. Phys D 190: 136–151
Grenfell BT, Bolker BM, Kleczkowski A (1995) Seasonality and extinction in chaotic metapopulations. Proc R Soc B 259: 97–103
Guckenheimer J, Holmes P (1983) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer, New York
Hamer KC, Schreiber EA, Burger J (2001) Breeding biology, life histories, and life history—environment interactions in seabirds. In: Schreiber EA, Burger J (eds) Biology of marine birds. CRC Press, Florida, pp 217–263
Hénon M (1976) A two-dimensional mapping with a strange attractor. Commun Math Phys 50: 69–77
Hirsch MW, Smale S, Devaney RL (2004) Differential equations, dynamical systems and an introduction to chaos. Academic Press, San Diego
Ireland JM, Norman RA, Greenman JV (2004) The effect of seasonal host birth rates on population dynamics: the importance of resonance. J Theor Biol 231: 229–238
Jones KE, Patel NG, Levy MA, Storeygard A, Balk D, Gittleman JL, Daszak P (2008) Global trends in emerging infectious diseases. Nature 451: 990–994
Katok A, Hasselblatt B (1995) Introduction to the modern theory of dynamical systems. Cambridge University Press, New York
Keeling MJ, Rohani P, Grenfell BT (2001) Seasonally forced disease dynamics explored as switching between attractors. Phys D 148: 317–335
Keesing F, Holt RD, Ostfeld RS (2006) Effects of species diversity on disease risk. Ecol Lett 9: 485–498
Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc A 115: 700–721
Kilpatrick AM, Chmura AA, Gibbons DW, Fleischer RC, Marra PP, Daszak P (2006) Predicting the global spread of H5N1 avian influenza. Proc Natl Acad Sci USA 103: 19368–19373
Kitchens BP (1998) Symbolic dynamics: one-sided, two-sided and countable state Markov shifts. Springer, Berlin
Krasnosel’skii MA, Zabreiko PP (1984) Geometrical methods of nonlinear analysis. Springer, Berlin
Krauss S, Stallknecht DE, Negovetich NJ, Niles LJ, Webby RJ, Webster RG (2010) Coincident ruddy turnstone migration and horseshoe crab spawning creates an ecological ‘hotspot’ for influenza viruses. Proc R Soc B 277: 3373–3379
Li TY, Yorke JA (1975) Period three implies chaos. Am Math Mon 82: 985–992
Liu WM, Hethcote HW, Levin SA (1987) Dynamical behavior of epidemiological models with nonlinear incidence rates. J Math Biol 25: 359–380
Mallory ML, Robinson SA, Hebert CE, Forbes MR (2010) Seabirds as indicators of aquatic ecosystem conditions: a case for gathering multiple proxies of seabird health. Mar Pollut Bull 60: 7–12
McNamara HA, Pokrovskii AV (2006) Hysteresis in the trade cycle. Phys B 372: 202–206
Mischaikow K (2002) Topological techniques for efficient rigorous computation in dynamics. Acta Numer 11: 435–477
Mischaikow K, Mrozek M (1998) Chaos in the Lorenz equations: a computer assisted proof. Part II: details. Math Comput 67: 1023–1046
Muzaffar SB, Jones IL (2004) Parasites and diseases of the auks (Alcidae) of the world and their ecology—a review. Mar Ornithol 32: 121–146
Nelson JB (1978) The Sulidae: gannets and boobies. Aberdeen University, Study Series, vol 154, Oxford
Nettleship, DN, Birkhead, TR (eds) (1985) The Atlantic Alcidae. The evolution, distribution and biology of the Auks inhabiting the Atlantic ocean and adjacent water areas. Academic Press, London
Nuttall PA (1984) Tick-borne viruses in seabird colonies. Seabird 7: 31–42
Nuttall PA, Harrap KA (1982) Isolation of a Coronavirus during studies on Puffinosis, a disease of the Manx Shearwater (Puffinus puffinus). Arch Virol 73: 1–13
Nuttall PA, Kelly TC, Carey D, Moss SR, Harrap KA (1984) Mixed infections with tick-borne viruses in a seabird colony in Eire. Arch Virol 79: 35–44
O’Brien VA, Moore AT, Young GR, Komar N, Reisen WK, Brown CR (2011) An enzootic vector-borne virus is amplified at epizootic levels by an invasive avian host. Proc R Soc B 278: 239–246
Olsen LF, Schaffer WM (1990) Chaos versus noisy periodicity: alternative hypotheses for childhood epidemics. Science 249: 499–504
O’Regan SM, Kelly TC, Korobeinikov A, O’Callaghan MJA, Pokrovskii AV (2008) Qualitative and numerical investigations of the impact of a novel pathogen on a seabird colony. J Phys Conf Ser 138. doi:10.1088/1742-6596/138/1/012018
O’Regan SM, Kelly TC, Korobeinikov A, O’Callaghan MJA, Pokrovskii AV (2010) Lyapunov functions for SIR and SIRS epidemic models. Appl Math Lett 23: 446–448
Österblom H, Van Der Jeugd HP, Olsson O (2004) Adult survival and avian cholera in Common Guillemots Uria aalge in the Baltic Sea. Ibis 146: 531–534
Pavlovsky EN (1966) Natural nidality of transmissible diseases. University of Illinois Press, Urbana
Peterson AT, Benz BW, Papes M (2007) Highly pathogenic H5N1 avian influenza: entry pathways into North America via bird migration. PLoS ONE 2, e261. doi:10.1371/journal.pone.0000261
Pireddu M (2009) Fixed points and chaotic dynamics for expansive-contractive maps in Euclidean spaces, with some applications. PhD thesis, Università à di Udine, Italy, arXiv:0910.3832v1
Pireddu M, Zanolin F (2008) Chaotic dynamics in the Volterra predator–prey model via linked twist maps. Opusc Math 28: 567–592
Pokrovskii A, Zhezherun A (2008) Topological degree in analysis of chaotic behavior in singularly perturbed systems. Chaos 18: 023130
Pokrovskii A, Rasskazov O, Studdert R (2005) Split-hyperbolicity, hysteresis and Lang–Kobayashi equations. In: Mortell MP, O’Malley RE, Pokrovskii A, Sobolev V (eds) Singular perturbations and hysteresis. Society for Industrial and Applied Mathematics, Philadelphia, pp 299–338
Pokrovskii A, Rasskazov O, Visetti D (2007) Homoclinic trajectories and chaotic behaviour in a piecewise linear oscillator. Discr Cont Dyn B 8: 943–970
Pokrovskii AV (1997) Topological shadowing and split-hyperbolicity. Funct Differ Equ 4: 335–360
Pokrovskii AV, Rasskazov OA (2004) On the use of the topological degree theory in broken orbits analysis. Proc Am Math Soc 132: 567–577
Pokrovskii AV, Szybka SJ, McInerney JG (2001) Topological degree in locating homoclinic structures for discrete dynamical systems. Proc Russ Acad Nat Sci Nonlinear Dyn Laser React Syst 5: 152–183
Rand DA, Wilson HB (1991) Chaotic stochasticity: a ubiquitous source of unpredictability in epidemics. Proc R Soc B 246: 179–184
Rasskazov O (2003) Methods of geometrical analysis of complicated dynamics, with applications to models of semi-conductor lasers. PhD thesis, University College Cork, Ireland
Rohani P, Breban R, Stallknecht DE, Drake JM (2009) Environmental transmission of low pathogenicity avian influenza viruses and its implications for pathogen invasion. Proc Natl Acad Sci USA 106: 10365–10369
Rolland V, Barbraud C, Weimerskirch H (2009) Assessing the impact of fisheries, climate and disease on the dynamics of the Indian yellow-nosed albatross. Biol Conserv 142: 1084–1095
Ruelle D (1989) Elements of differentiable dynamics and bifurcation theory. Academic Press, Boston
Schwartz IB, Shaw LB, Cummings DAT, Billings L, McCrary M, Burke DS (2005) Chaotic desynchronization of multi-strain diseases. Phys Rev E 72: 066201–066206
Shulgin B, Stone L, Agur Z (1998) Pulse vaccination strategy in the SIR epidemic model. Bull Math Biol 60: 1123–1148
Sovada MA, Pietz PJ, Converse KA, King DT, Hofmeister EK, Scherr P, Ip HS (2008) Impact of West Nile virus and other mortality factors on American white pelicans at breeding colonies in the northern plains of North America. Biol Conserv 141: 1021–1031
Stone L, Olinky R, Huppert A (2007) Seasonal dynamics of recurrent epidemics. Nature 446: 533–536
Strogatz SH (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview Press, Cambridge
Swinton J, Woolhouse MEJ, Begon ME, Dobson AP, Ferroglio E, Grenfell BT, Guberti V, Hails RS, Heesterbeek JAP, Lavazza A, Roberts MG, White PJ, Wilson K (2002) Microparasite transmission and persistence. In: Hudson PJ, Rizzoli A, Grenfell BT, Heesterbeek H, Dobson AP (eds) The ecology of wildlife diseases. Oxford University Press, Oxford, pp 83–101
The Birds of the Western Palearctic Interactive Version 2 (2007) On DVD-ROM for Windows and Mac OSX. Accessed 8 October 2011
Tel T, Lai YC, Gruiz M (2008) Noise-induced chaos: a consequence of long deterministic transients. Int J Bifurcation Chaos 18: 509–520
Tiensin T, Nielen M, Vernooij H, Songserm T, Kalpravidh W, Chotiprasatintara S, Chaisingh A, Wongkasemjit S, Chanachai K, Thanapongtham W, Srisuvan T, Stegeman A (2007) Transmission of the highly pathogenic avian influenza virus H5N1 within flocks during the 2004 epidemic in Thailand. J Infect Dis 196: 1679–1684
Waller LJ, Underhill LG (2007) Management of avian cholera Pasteurella multocida on Dyer Island, South Africa, 2002–2005. Afr J Mar Sci 29: 105–111
Wiggins S (2003) Introduction to applied nonlinear dynamical systems and chaos. Springer, New York
Wittenburger JF, Hunt GL Jr (1985) The adaptive significance of coloniality in birds. In: Farner DS, King JR, Parkes KC (eds) Avian biology, vol 8. Academic Press, New York, pp 1–78
Young LC, VanderWerf EA (2008) Prevalence of avian pox virus and effect on the fledging success of Laysan Albatross. J Field Ornithol 79: 93–98
Zgliczynski P (1996) Fixed point index for iterations of maps, topological horseshoe and chaos. Topol Methods Nonlinear Anal 8: 169–177
Zgliczynski P (1997) Computer assisted proof of chaos in the Rössler equations and the Hénon map. Nonlinearity 10: 243–252
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
O’Regan, S.M., Kelly, T.C., Korobeinikov, A. et al. Chaos in a seasonally perturbed SIR model: avian influenza in a seabird colony as a paradigm. J. Math. Biol. 67, 293–327 (2013). https://doi.org/10.1007/s00285-012-0550-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00285-012-0550-9