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

  • A. Pimenov (a1), T.C. Kelly (a2), A. Korobeinikov (a3), M.J.A. O’Callaghan (a4), A.V. Pokrovskii (a4) and D. Rachinskii (a4)...


Modification of behaviour in response to changes in the environment or ambient conditions, based on memory, is typical of the human and, possibly, many animal species.One obvious example of such adaptivity is, for instance, switching to a safer behaviour when in danger, from either a predator or an infectious disease. In human society such switching to safe behaviour is particularly apparent during epidemics. Mathematically, such changes of behaviour in response to changes in the ambient conditions can be described by models involving switching. In most cases, this switching is assumed to 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 in epidemiology. Our goal is to show why and how hysteresis can arise in such a model, and how it may be applied to describe a variety of memory effects. Our other objective is to introduce a unified paradigm for mathematical modelling with memory effects in epidemiology and ecology. Our approach treats changing behaviour as an irreversible flow related to large ensembles of elementary exchange operations that recently has been successfully applied in a number of other areas, such as terrestrial hydrology, and macroeconomics. 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. We accompany the model with extensive numerical simulations which exhibit interesting qualitative effects.


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[1] 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
[2] R.M. Anderson, R.M. May, Infectious Diseases of Humans : Dynamics and Control. Oxford University Press, Oxford, 1992.
[3] Antia, R., Ganusov, V.V., Ahmed, R., The role of models in understanding CD8+ T - Cell memory, Nature Reviews Immunology, 5 (2005), 101111.
[4] 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), 4469.
[5] 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, 301304.
[6] N.T. Bailey, The Mathematical Theory of Infectious Diseases (2-nd edition). Charles Griffin and Co. Ltd., 1975.
[7] Banks, P.B., Dickman, C.R., Alien predation and the effects of multiple levels of prey naivete. Trends Ecol. Evol., 22 (2007), No. 5, 229230.
[8] C. Barnard, Animal Behaviour - Mechanism, Development, Function and Evolution. Pearson/ Prentice Hall, London, 2004.
[9] Bolhuis, J.J., Wynne, C.D.L., Can evolution explain how minds work ?. Nature, 458 (2009), 832833.
[10] Bolhuis, J.J., Macphail, E.M., A critique of the neuroecology of learning and memory. Trends Cogn. Sci., 5 (2001), 426433.
[11] 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, 94109.
[12] 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
[13] M. Brokate, J. Sprekels, Hysteresis and Phase Transitions. Springer, New York, 1996.
[14] V. Capasso, Mathematical Structures of Epidemic Systems. Lecture Notes in Biomathematics, Vol. 97. Springer-Verlag, Heidelberg, 1993.
[15] 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.
[16] Clayton, D., The influence of parasites on host-sexual selection, Parasitology Today, 7 (1991), 329-334
[17] 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), 14831491.
[18] Collett, T.S., Collett, M., Memory use in insect visual navigation, Nature Reviews Neuroscience, 3 (2002), 542552.
[19] 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), 23992404. doi : 10.1098/rspb.2010.0164
[20] 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, 674680.
[21] Cross, R., McNamara, H., Pokrovskii, A., Modelling macroeconomic flows related to large ensembles of elementary exchange operations. Physica B, 403 (2008), 451455.
[22] Cross, R., McNamara, H., Pokrovskii, A.V., Rachinskii, D., A new paradigm for modelling hysteresis in macroeconomic flows. Physica B, 403 (2007), 2–3, 231236.
[23] Danchin, E., Giraldeau, L-A., Valone, T.J., Wagner, R.H., Public Information : From Nosy Neighbours to Cultural Evolution. Science, 305 (2006), 487491.
[24] Mark Davis, Invasion Biology. Oxford University Press, 2009.
[25] De Waal, F.B.M., Darwin’s last laugh. Nature, 460 (2009), 175.
[26] 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), 229233.
[27] Emery, N.J., Clayton, N.S., Effects of experience and social context of prospective caching strategies in scrub jays. Nature, 414 (2004), 443446.
[28] 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), 3743.
[29] Everett, D.H., Whitton, W.I., A general approach to hysteresis. Transactions of the Faraday Society, 48 (1952), 749757.
[30] J.A. Ewing, Experimental research in magnetism. Trans. R. Soc. Lond., 176 (1895), II.
[31] 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, 11551160.
[32] Flynn, D., Rasskazov, O., On the integration of an ODE involving the derivative of a Preisach nonlinearity. J. Phys. Conf. Ser., 22 (2005), 4355. doi :10.1088/1742-6596/22/1/003
[33] Forterre, P., The origin of viruses and their possible role in major evolutionary transitions. Virus Research, 117 (2006), 516.
[34] 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), 15571561.
[35] Goldenfeld, N., Woese, C., Biology’s next revolution. Nature, 445 (2007), 369.
[36] Hamilton, W.D., Zuk, M., Heritable true fitness and bright birds : a role for parasites ?. Science, 218 (1982), 384-387.
[37] Hampton, R., Healy, S.D., Shettlewort, S.J., Kamil, A., Neurecologists are not made of straw. Trends Cogn. Sci., 6 (2002), 12.
[38] Hawkins, R.D., Kandel, E., Bailey, C.B., Molecular mechanisms of memory storage in Aplysia, Biological Bulletin, 210 (2006), 174191.
[39] 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), 1722.
[40] 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
[41] Hyman, J.M., Li, J., Differential susceptibility epidemic models. J. Math. Biol., 50 (2005), No. 62, 626644.
[42] 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/dqq013
[43] Kandel, E., The Molecular Biology of Memory Storage : A Dialogue between Genes and Synapses. Science, 294 (2001), 10301038.
[44] M.J. Keeling, P. Rohani, Modeling Infectious Diseases in Humans and Animals. Princeton University Press, Princeton, 2008.
[45] Kermack, W.O., McKendrick, A.G., A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. A, 115 (1927), 700721.
[46] Koonin, E.V., Darwinian Evolution in the Light of Genomics. Nucleic Acids Research, 37 (2009), 10111034.
[47] Koonin, E.V., Wolf, Y.I., Genomics of Bacteria and Archaea; the Emerging Dynamic View of the Prokaryotic World. Nucleic Acids Research, 36 (2008), 66886719.
[48] Korobeinikov, A., Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission. Bull. Math. Biol., 68 (2006), 615626.
[49] Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol., 69 (2007), 18711886.
[50] 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, 225239.
[51] Korobeinikov, A., Stability of ecosystem : global properties of a general predator-prey model. Math. Med. Biol., 26 (2009), 309321.
[52] 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, 10951103.
[53] Korobeinikov, A., Maini, P.K., Nonlinear incidence and stability of infectious disease models. Math. Med. Biol., 22 (2005), 113128.
[54] M.A. Krasnosel’skii, A.V. Pokrovskii, Systems with Hysteresis. Nauka, Moscow, 1983 (English edition : Springer, 1989).
[55] 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
[56] 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
[57] Levenson, J., Sweatt, J.D., Epigenetic mechanisms in memory formation. Nature Reviews Neuroscience, 6 (2005), 105117.
[58] 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.
[59] Macphail, E.M., Bolhuis, J., The evolution of intelligence : adaptive specialisations versus general process. Biological Reviews, 76 (2001), 341364.
[60] Mattick, J.S., Mehler, M.F., RNA editing, DNA recoding and the evolution of human cognition. Trends Neurosci., 31 (2008), 227233.
[61] I.D. Mayergoyz, Mathematical Models for Hysteresis. Springer, New York, 1991.
[62] I.D. Mayergoyz, Mathematical Models of Hysteresis And Their Applications. Elsevier, 2003.
[63] 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), 30403045.
[64] Neel, L., Theories des lois d’aimantation de Lord Rayleigh 1, 2. Cahiers de Physique, 12 (1942), 120 ; 13 (1943), 19–30.
[65] T. Piersma, J.A. van Gils, The Flexible Phenotype A Body - Centered Integration of Ecology, Physiology and Behaviour. Oxford University Press, 2011.
[66] A. Pimenov, Stability and bifurcations of systems with hysteresis and multistable systems. Ph.D. dissertation, University College Cork, Ireland, 2009.
[67] Pimenov, A., Rachinskii, D., Linear Stability Analysis of Systems with Preisach Memory. Discrete Cont. Dyn.-B, 4 (2009), 9971018.
[68] Preisach, P., Uber die magnetische Nachwirkung. Zeitschrift für Physik, 94 (1935), 277302.
[69] 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
[70] The Science of Hysteresis (Bertotti and Mayergoyz, editors). Vol. 1-3, 125–291, Elsevier Science, 2005. ISBN : 978-0-12-480874-4.
[71] Tulving, E., Episodic Memory : From Mind to Brain. Ann. Rev. Psychology, 53 (2002), 125.
[72] 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.
[73] A. Visintin, Differential Models of Hysteresis. Springer, Berlin, 1994.
[74] Wagner, R.H., Danchin, E., A taxonomy of biological information. Oikos, 119 (2010), 203209. doi : 10.1111/j.1600-0706.2009.17315.x
[75] Ward, M.P., Habitat selection by dispersing yellow-headed blackbirds : evidence of prospecting and the use of public information. Oecologia, 145 (2005), 650657. doi : 10.1007/s00442-005-0179-0
[76] 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), 706725.
[77] 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), 532536.
[78] Zanetti, M., Franchini, G., T cell memory and protective immunity by vaccination : is more better ?. Trends Immunol., 27 (2005), 511517.
[79] Zhao, C., Deng, W., Gage, F.W., Mechanisms and Functional Implications of Adult Neurogenesis. Cell, 132 (2008), 645660.


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

  • A. Pimenov (a1), T.C. Kelly (a2), A. Korobeinikov (a3), M.J.A. O’Callaghan (a4), A.V. Pokrovskii (a4) and D. Rachinskii (a4)...


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