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Memory Effects in Population Dynamics : Spread of InfectiousDisease as a Case Study

Published online by Cambridge University Press:  06 June 2012

A. Pimenov ,
T.C. Kelly ,
A. Korobeinikov ,
M.J.A. O’Callaghan ,
A.V. Pokrovskii  and
D. Rachinskii
A. Pimenov
Affiliation:
Weierstrass Institute, Mohrenstrasse 39, D-10117 Berlin, Germany
T.C. Kelly
Affiliation:
Department of Biology, Earth and Environmental Science, University College Cork, Ireland
A. Korobeinikov*
Affiliation:
MACSI, Department of Mathematics and Statistics, University of Limerick, Ireland
M.J.A. O’Callaghan
Affiliation:
Department of Applied Mathematics, University College Cork, Ireland
A.V. Pokrovskii
Affiliation:
Department of Applied Mathematics, University College Cork, Ireland
D. Rachinskii
Affiliation:
Department of Applied Mathematics, University College Cork, Ireland
*
⋆ Corresponding author. E-mail: [email protected]
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Abstract

Modification of behaviour in response to changes in the environment or ambientconditions, based on memory, is typical of the human and, possibly, many animalspecies.One obvious example of such adaptivity is, for instance, switching to a saferbehaviour when in danger, from either a predator or an infectious disease. In humansociety such switching to safe behaviour is particularly apparent during epidemics.Mathematically, such changes of behaviour in response to changes in the ambient conditionscan be described by models involving switching. In most cases, this switching is assumedto depend on the system state, and thus it disregards the history and, therefore, memory.Memory can be introduced into a mathematical model using a phenomenon known as hysteresis.We illustrate this idea using a simple SIR compartmental model that is applicable inepidemiology. Our goal is to show why and how hysteresis can arise in such a model, andhow it may be applied to describe a variety of memory effects. Our other objective is tointroduce a unified paradigm for mathematical modelling with memory effects inepidemiology and ecology. Our approach treats changing behaviour as an irreversible flowrelated to large ensembles of elementary exchange operations that recently has beensuccessfully applied in a number of other areas, such as terrestrial hydrology, andmacroeconomics. For the purposes of illustrating these ideas in an application to biology,we consider a rather simple case study and develop a model from first principles. Weaccompany the model with extensive numerical simulations which exhibit interestingqualitative effects.

Keywords

mathematical epidemiologySIR modelhysteresisPETSadaptationmemory effectsequilibriuminfectious diseasePreisach operatoroperator-differential equationsdynamicspublic informationolfactory
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 3: Epidemiology , 2012 , pp. 204 - 226
Copyright
© EDP Sciences, 2012

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References

Amo, L., Galván, I., Tomás, G., Sanz, J.J., Predator odour recognition and avoidance in a songbird. Functional Ecology, 22 (2008), 289-293. doi :10.1111/j.1365-2435.2007.01361.x CrossRefGoogle Scholar
R.M. Anderson, R.M. May, Infectious Diseases of Humans : Dynamics and Control. Oxford University Press, Oxford, 1992.
Antia, R., Ganusov, V.V., Ahmed, R., The role of models in understanding CD8+ T - Cell memory, Nature Reviews Immunology, 5 (2005), 101–111. CrossRefGoogle ScholarPubMed
Appelbe, B., Flynn, D., McNamara, H., O’Kane, J.P., Pimenov, A., Pokrovskii, A., Rachinskii, D., Zhezherun, A., Rate-independent hysteresis in terrestrial hydrology, A Vegetated Soil Model with Preisach Hysteresis. IEEE Control Systems Magazine, 1 (2009), 44–69. CrossRefGoogle Scholar
Appelbe, B., Rachinskii, D., Zhezherun, A., Hopf bifurcation in a van der Pol type oscillator with magnetic hysteresis. Physica B, 403 (2008), No. 2–3, 301–304. CrossRefGoogle Scholar
N.T. Bailey, The Mathematical Theory of Infectious Diseases (2-nd edition). Charles Griffin and Co. Ltd., 1975.
Banks, P.B., Dickman, C.R., Alien predation and the effects of multiple levels of prey naivete. Trends Ecol. Evol., 22 (2007), No. 5, 229–230. CrossRefGoogle ScholarPubMed
C. Barnard, Animal Behaviour - Mechanism, Development, Function and Evolution. Pearson/ Prentice Hall, London, 2004.
Bolhuis, J.J., Wynne, C.D.L., Can evolution explain how minds work ?. Nature, 458 (2009), 832–833. CrossRefGoogle ScholarPubMed
Bolhuis, J.J., Macphail, E.M., A critique of the neuroecology of learning and memory. Trends Cogn. Sci., 5 (2001), 426–433. CrossRefGoogle ScholarPubMed
Brokate, M., Pokrovskii, A., Rachinskii, D., Asymptotic stability of continuum sets of periodic solutions to systems with hysteresis. J. Math. Anal. Appl., 319 (2006), No. 1, 94–109. CrossRefGoogle Scholar
M. Brokate, A. Pokrovskii, D. Rachinskii, O. Rasskazov, Differential equations with hysteresis via a canonical example. in The Science of Hysteresis (Bertotti and Mayergoyz, editors). Vol. 1, pp. 125–291, Elsevier Science, 2005. ISBN : 978-0-12-480874-4
M. Brokate, J. Sprekels, Hysteresis and Phase Transitions. Springer, New York, 1996.
V. Capasso, Mathematical Structures of Epidemic Systems. Lecture Notes in Biomathematics, Vol. 97. Springer-Verlag, Heidelberg, 1993.
Clancy, C.F., O’Callaghan, M.J.A., Kelly, T.C., A multi-scale problem arising in a model of avian flu virus in a seabird colony. J. Phys. Conf. Ser., 55 (2006), 45-54. CrossRefGoogle Scholar
Clayton, D., The influence of parasites on host-sexual selection, Parasitology Today, 7 (1991), 329-334 CrossRefGoogle ScholarPubMed
Clayton, N.S., Griffiths, D.P., Emery, N.J., Dickinson, A., Elements of episodic memory in animals, Phil. Trans. R. Soc. Lond B, 356 (2001), 1483–1491. CrossRefGoogle ScholarPubMed
Collett, T.S., Collett, M., Memory use in insect visual navigation, Nature Reviews Neuroscience, 3 (2002), 542–552. CrossRefGoogle ScholarPubMed
Cornelius, J.M., Breuner, C.W., Hahn, T.P., Under a neighbour’s influence : public information affects stress hormones and behaviour of a songbird. Proc. R. Soc. B, 277 (2010), 2399–2404. doi : 10.1098/rspb.2010.0164 CrossRefGoogle Scholar
Cox, J.G. and Lima, S.L., Naivete and an aquatic — terrestrial dichotomy in the effects of introduced predators. Trends Ecol. Evol., 21 (2006), No. 12, 674–680. CrossRefGoogle ScholarPubMed
Cross, R., McNamara, H., Pokrovskii, A., Modelling macroeconomic flows related to large ensembles of elementary exchange operations. Physica B, 403 (2008), 451–455. CrossRefGoogle Scholar
Cross, R., McNamara, H., Pokrovskii, A.V., Rachinskii, D., A new paradigm for modelling hysteresis in macroeconomic flows. Physica B, 403 (2007), 2–3, 231–236. CrossRefGoogle Scholar
Danchin, E., Giraldeau, L-A., Valone, T.J., Wagner, R.H., Public Information : From Nosy Neighbours to Cultural Evolution. Science, 305 (2006), 487–491. CrossRefGoogle Scholar
Mark Davis, Invasion Biology. Oxford University Press, 2009.
De Waal, F.B.M., Darwin’s last laugh. Nature, 460 (2009), 175. CrossRefGoogle ScholarPubMed
Deng, W., Aimone, J.B., Gage, F.H., New neurons and new memories : how does adult hippocampal neurogenesis affect learning and memory. Nature Reviews Neuroscience, 11 (2010), 229–233. CrossRefGoogle ScholarPubMed
Emery, N.J., Clayton, N.S., Effects of experience and social context of prospective caching strategies in scrub jays. Nature, 414 (2004), 443–446. CrossRefGoogle ScholarPubMed
Emery, N.J., Dally, J., Clayton, N.S., Western scrub jays (Aphelocoma californica) use cognitive strategies to protect their caches from thieving conspecifics. Animal Cognition, 7 (2004), 37–43. CrossRefGoogle ScholarPubMed
Everett, D.H., Whitton, W.I., A general approach to hysteresis. Transactions of the Faraday Society, 48 (1952), 749–757. CrossRefGoogle Scholar
J.A. Ewing, Experimental research in magnetism. Trans. R. Soc. Lond., 176 (1895), II.
Ferguson, N.M., Donnelly, C.A., Anderson, R.M., The Foot-and-Mouth Epidemic in Great Britain : Pattern of Spread and Impact of Interventions. Science, 292 (2001), No. 5519, 1155–1160. CrossRefGoogle ScholarPubMed
Flynn, D., Rasskazov, O., On the integration of an ODE involving the derivative of a Preisach nonlinearity. J. Phys. Conf. Ser., 22 (2005), 43–55. doi :10.1088/1742-6596/22/1/003 CrossRefGoogle Scholar
Forterre, P., The origin of viruses and their possible role in major evolutionary transitions. Virus Research, 117 (2006), 5–16. CrossRefGoogle ScholarPubMed
Fraser, C., Donnelly, C.A., Cauchemez, S., Hanage, W.P., Van Kerkhove, M.D.T., Hollingsworth, D., Griffin, J., Baggaley, R.F., Jenkins, H.E., Lyons, E.J., Jombart, T., Hinsley, W.R., Grassly, N.C., Balloux, F., Ghani, A.C., Ferguson, N.M., Rambaut, A., Pybus, O.G., Lopez-Gatell, H., Alpuche-Aranda, C.M., Bojorquez Chapela, Ietza, Zavala, E.P., Espejo Guevara, Dulce Ma., Checchi, F., Garcia, E., Hugonnet, S., Roth, C. : The WHO Rapid Pandemic Assessment Collaboration. Pandemic Potential of a Strain of Influenza A (H1N1) : Early Findings. Science, 324 (2009), 1557–1561. CrossRefGoogle ScholarPubMed
Goldenfeld, N., Woese, C., Biology’s next revolution. Nature, 445 (2007), 369. CrossRefGoogle ScholarPubMed
Hamilton, W.D., Zuk, M., Heritable true fitness and bright birds : a role for parasites ?. Science, 218 (1982), 384-387. CrossRefGoogle Scholar
Hampton, R., Healy, S.D., Shettlewort, S.J., Kamil, A., Neurecologists are not made of straw. Trends Cogn. Sci., 6 (2002), 1–2. CrossRefGoogle Scholar
Hawkins, R.D., Kandel, E., Bailey, C.B., Molecular mechanisms of memory storage in Aplysia, Biological Bulletin, 210 (2006), 174–191. CrossRefGoogle ScholarPubMed
Healy, S.D., de Kort, S.R., Clayton, N.S., The Hippocampus, spatial memory and food hoarding : a puzzle revisited. Trends Ecol. Evol., 20 (2005), 17–22. CrossRefGoogle ScholarPubMed
Heffernan, J.M., Smith, R.J., Wahl, L.M., Perspectives on the basic reproductive ratio. J. R. Soc. Interface, 2 (2005), No. 4, 281-93. doi :10.1098/rsif.2005.0042 CrossRefGoogle ScholarPubMed
Hyman, J.M., Li, J., Differential susceptibility epidemic models. J. Math. Biol., 50 (2005), No. 62, 626–644. CrossRefGoogle ScholarPubMed
Kalachev, L.V., Kelly, T.C., O’Callaghan, M.J., Pokrovskii, A.V., Pokrovskiy, A.A., Analysis of threshold-type behaviour in mathematical models of the intrusion of a novel macroparasite in a host colony. Math. Med. Biol., 28 (2011), No. 4, 287-333. doi : 10.1093/imammb/dqq013CrossRefGoogle Scholar
Kandel, E., The Molecular Biology of Memory Storage : A Dialogue between Genes and Synapses. Science, 294 (2001), 1030–1038. CrossRefGoogle ScholarPubMed
M.J. Keeling, P. Rohani, Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton, 2008.
Kermack, W.O., McKendrick, A.G., A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A, 115 (1927), 700–721. CrossRefGoogle Scholar
Koonin, E.V., Darwinian Evolution in the Light of Genomics. Nucleic Acids Research, 37 (2009), 1011–1034. CrossRefGoogle ScholarPubMed
Koonin, E.V., Wolf, Y.I., Genomics of Bacteria and Archaea; the Emerging Dynamic View of the Prokaryotic World. Nucleic Acids Research, 36 (2008), 6688–6719. CrossRefGoogle ScholarPubMed
Korobeinikov, A., Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol., 68 (2006), 615–626. CrossRefGoogle ScholarPubMed
Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol., 69 (2007), 1871–1886. CrossRefGoogle ScholarPubMed
Korobeinikov, A., Global asymptotic properties of virus dynamics models with dose dependent parasite reproduction and virulence, and nonlinear incidence rate. Math. Med. Biol., 26 (2009), No. 3, 225–239. CrossRefGoogle Scholar
Korobeinikov, A., Stability of ecosystem : global properties of a general predator-prey model. Math. Med. Biol., 26 (2009), 309–321. CrossRefGoogle ScholarPubMed
Korobeinikov, A., Global properties of a general predator-prey model with non-symmetric attack and consumption rate. Discrete Cont. Dyn.-B, 14 (2010), No. 3, 1095–1103. CrossRefGoogle Scholar
Korobeinikov, A., Maini, P.K., Nonlinear incidence and stability of infectious disease models. Math. Med. Biol., 22 (2005), 113–128. CrossRefGoogle ScholarPubMed
M.A. Krasnosel’skii, A.V. Pokrovskii, Systems with Hysteresis. Nauka, Moscow, 1983 (English edition : Springer, 1989).
P. Krejci, P. O’Kane, A. Pokrovskii, D. Rachinskii, Properties of solutions to a class of differential models incorporating Preisach hysteresis operator. Physica D, in press,(2011). doi :10.1016/j.physd.2011.05.005
P. Krejci, P. O’Kane, A. Pokrovskii, D. Rachinskii, Stability results for a soil model with singular hysteretic hydrology. 5th International Workshop on Multi-Rate Processes and Hysteresis (MURPHYS 2010) IOP Publishing, J. Phys. Conf. Ser. 268 (2011) 012016 doi :10.1088/1742-6596/268/1/012016 CrossRef
Levenson, J., Sweatt, J.D., Epigenetic mechanisms in memory formation. Nature Reviews Neuroscience, 6 (2005), 105–117. CrossRefGoogle ScholarPubMed
Lipsitch, M., Cohen, T., Cooper, B., Robins, J.M., Ma, S., James, L., Gopalakrishna, G., Chew, S.K., Tan, C.C., Samore, M.H., Fisman, D., Murray, M., Transmission Dynamics and Control of Severe Acute Respiratory Syndrome. Science, 300 (2003), No. 5627, 1966-1970. CrossRefGoogle ScholarPubMed
Macphail, E.M., Bolhuis, J., The evolution of intelligence : adaptive specialisations versus general process. Biological Reviews, 76 (2001), 341–364. CrossRefGoogle Scholar
Mattick, J.S., Mehler, M.F., RNA editing, DNA recoding and the evolution of human cognition. Trends Neurosci., 31 (2008), 227–233. CrossRefGoogle ScholarPubMed
I.D. Mayergoyz, Mathematical Models for Hysteresis. Springer, New York, 1991.
I.D. Mayergoyz, Mathematical Models of Hysteresis And Their Applications. Elsevier, 2003.
Menzel, R., Greggers, U., Smith, A., Berger, S., Brandt, S., Bundrock, G., Plumpe, T., Schaupp, F., Silke, S., Stindt, J., Stollhoff, N., Watzl, S., Honey bees navigate according to a map-like memory. Proceedings of the National Academy of Sciences, 102 (2006), 3040–3045. CrossRefGoogle ScholarPubMed
Neel, L., Theories des lois d’aimantation de Lord Rayleigh 1, 2. Cahiers de Physique, 12 (1942), 1–20 ; 13 (1943), 19–30. Google Scholar
T. Piersma, J.A. van Gils, The Flexible Phenotype A Body - Centered Integration of Ecology, Physiology and Behaviour. Oxford University Press, 2011.
A. Pimenov, Stability and bifurcations of systems with hysteresis and multistable systems. Ph.D. dissertation, University College Cork, Ireland, 2009.
Pimenov, A., Rachinskii, D., Linear Stability Analysis of Systems with Preisach Memory. Discrete Cont. Dyn.-B, 4 (2009), 997–1018. CrossRefGoogle Scholar
Preisach, P., Uber die magnetische Nachwirkung. Zeitschrift für Physik, 94 (1935), 277–302. CrossRefGoogle Scholar
Rodriguez, A., Hausberger, M., Clergeau, P., Flexibility in European starlings’ use of social information : experiments with decoys in different populations. Animal Behaviour, 80 (2010), 965-97. doi :10.1016/ j.anbehav. 2010.08.010 CrossRefGoogle Scholar
The Science of Hysteresis (Bertotti and Mayergoyz, editors). Vol. 1-3, 125–291, Elsevier Science, 2005. ISBN : 978-0-12-480874-4.
Tulving, E., Episodic Memory : From Mind to Brain. Ann. Rev. Psychology, 53 (2002), 1–25. CrossRefGoogle Scholar
Templeton, J.J., Giraldea, Luc-Alain, Vicarious sampling : the use of personal and public information by starlings foraging in a simple patchy environment. Behav Ecol Sociobiol, 38 (1996), 105-114. CrossRefGoogle Scholar
A. Visintin, Differential Models of Hysteresis. Springer, Berlin, 1994.
Wagner, R.H., Danchin, E., A taxonomy of biological information. Oikos, 119 (2010), 203–209. doi : 10.1111/j.1600-0706.2009.17315.x CrossRefGoogle Scholar
Ward, M.P., Habitat selection by dispersing yellow-headed blackbirds : evidence of prospecting and the use of public information. Oecologia, 145 (2005), 650–657. doi : 10.1007/s00442-005-0179-0 CrossRefGoogle Scholar
Wonham, M., Lewis, M., Renclawowicz, J., van den Driessche, P., Transmission assumptions generate conflicting predictions in host-vector disease models : a case study in West Nile virus. Ecology Letters, 9 (2006), 706–725. CrossRefGoogle ScholarPubMed
Yu, X., Tsibane, T., McGraw, P.A., House, F.S., Keefer, C.J., Hicar, M.D., Tumpey, T.M., Pappas, C., Perrone, L.A., Martinez, O., Stevens, J., Wilson, I.A., Aguilar, P.V., Altschuler, E.L., Basler, C.F., Crowe, J.E. Jr, Neutralizing antibodies derived from the B cells of 1918 influenza pandemic survivors. Nature, 445 (2008), 532–536. CrossRefGoogle Scholar
Zanetti, M., Franchini, G., T cell memory and protective immunity by vaccination : is more better ?. Trends Immunol., 27 (2005), 511–517. CrossRefGoogle ScholarPubMed
Zhao, C., Deng, W., Gage, F.W., Mechanisms and Functional Implications of Adult Neurogenesis. Cell, 132 (2008), 645–660. CrossRefGoogle ScholarPubMed