AASCIT - Journal - All Issues

International Journal of Bioinformatics and Computational Biology

The Journal

Indexing

Manuscript Information

Connectivity, Continuity and Distance Norm in Mathematical Models for Community Ecology, Epidemiology and Multicellular Pathway Prediction

International Journal of Bioinformatics and Computational Biology
Vol.4 , No. 1, Publication Date: Jun. 20, 2019, Page: 1-10

892 Views Since June 20, 2019, 210 Downloads Since Jun. 20, 2019

Paper in PDF (293K)

Follow on

Authors

[1]	Wilfried Allaerts, Biological Publishing A&O and Immunology Department, Erasmus MC, Rotterdam, The Netherlands.

Abstract

In this paper, we embark on a historical review of the mathematical models developed in the previous century, that were devoted to the study of the geographical spread of biological infections. The basic notions of connectivity, continuity and distance norm as applied by successive bio-mathematicians, starting with the names of Volterra, Turing and Kendall, are highlighted in order to demonstrate their usefulness in several new areas of bio-mathematical research. These new areas include the well-known fields of community ecology and epidemiology, but also the less well-known field of multicellular pathway prediction. The biological interpretation of these abstract mathematical notions, as well as the methodological criteria for these interpretative schemes and their corroboration with empirical evidence are discussed. In particular, we will focus on the boundedness norm in polynomial Lyapunov functions and its application in Markovian models for community assembly and in models for cellular pathways in multicellular systems.

Keywords

Connectivity, Boundedness Norm, Community Ecology, Geographical Spread of Biological Infections, Markovian Models for Multicellular Systems

Reference

[01]	Kermack W O and McKendrick A G 1927. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society (London), series A, 115 700-21.
[02]	Kendall D G 1957. Discussion of ‘Measles periodicity and community size’ by M. S. Bartlett. Journal of the Royal Statistical Society, series A, 120 64-7.
[03]	Thieme H R 1977. A model for the spatial spread of an epidemic. Journal of Mathematical Biology 4 337-51.
[04]	Diekmann O 1978. Thresholds and travelling waves for the geographical spread of infection. Journal of Mathematical Biology 6 109-130.
[05]	Volterra V 1926. Variazione fluttuazioni del numero d’individui in specie animali conviventi. Mem. Acad. Lincei 2: 3-113, translation in: ed R N Chapman (1931), Animal Ecology pp 409-48 (New York: McGraw Hill).
[06]	Turing A M 1952. The chemical basis of morphogenesis. Phil. Trans. R. Soc. London (B) 237 37-72.
[07]	Allaerts W 2003. Fifty years after Alan M. Turing. An extraordinary theory of morphogenesis. Belgian Journal of Zoology 133 3-14.
[08]	Aronson D G and Weinberger H F 1975. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. In: ed J A Goldstein, Partial differential equations and related topics, Lecture notes in Math. vol 446 pp 5-49 (Berlin: Springer).
[09]	Van den Bosch F, Metz J A J and Diekmann O 1990. The velocity of spatial population expansion. Journal of Mathematical Biology 28 529-65.
[10]	Van den Bosch F, Hengeveld R and Metz J A J 1992. Analysing the velocity of animal range expansion. Journal of Biogeography 19 135-50.
[11]	Hengeveld R and Van den Bosch F 1997. Invading into an ecologically non-uniform area. In: B Huntley, et al. (eds.), Past and future rapid environmental changes: the spatial and evolutionary responses of terrestrial biota. NATO AS1 Series, vol 147 pp 217-25 (Berlin, Heidelberg: Springer).
[12]	Diekmann O 1979. Run for your life. A note on the asymptotic speed of propagation of an epidemic. Journal of Differential Equations 33 58-73.
[13]	Cliff A D, Haggett P, Ord J D and Versey G R 1981. Spatial Diffusion (Cambridge: Cambridge University Press).
[14]	Murray J D 1989. Geographical Spread of Epidemics, chapter 20. In: ed J D Murray, Mathematical Biology (Berlin, Heidelberg: Springer).
[15]	Rand D A 1999. Correlation equations and pair approximations for spatial ecologies. In: ed J McGlade, Advanced Theoretical Ecology: Principles and Applications pp. 100-42 (London: Blackwell Science).
[16]	Jeong H, Tombor B, Albert R, Ottval Z N and Barabási A-L 2000. The large-scale organization of metabolic networks. Nature 407 651-54.
[17]	Albert R and Barabási A-L 2000. Topology of Evolving Networks: Local Events and Universality. Physical Review Letters 85 5234-37.
[18]	Diekmann O and Heesterbeek J A P 2000. Mathematical Epidemiology of Infectious Diseases. Model building, analysis and interpretation (New York: John Wiley and Son).
[19]	Nåsell I 1985. Hybrid Models of Tropical Infections (Berlin: Springer).
[20]	Nåsell I 1995. The threshold concept in stochastic epidemic and endemic models. In: ed D Mollison, Epidemic models: their structure and relation to data (Cambridge: Cambridge University Press).
[21]	Bartlett M S 1957. Measles periodicity and community size. Journal of the Royal Statistical Society, series A, 120 48-60.
[22]	Cliff AD, Haggett P and Smallman-Raynor M 1993. Measles: an historical geography of a major human viral disease from global expansion to local retreat, 1840-1990 (London: Blackwell).
[23]	Melzak Z A 1983. Bypasses. A simple approach to complexity (New York: John Wiley and Sons).
[24]	Brinkmann H W and Klotz E A 1971. Linear algebra and analytic geometry (London, Reading, Menlo Park: Addison-Wesley Publishing Company).
[25]	Schönemann P H and Carroll R M 1970. Fitting one matrix to another under choice of a central dilation and a rigid motion. Psychometrika 35 (2) 245-55.
[26]	Gower J C 1975. Generalized Procrustes analysis. Psychometrika 40 (1) 33-51.
[27]	Piskounov N 1980. Calcul différentiel et intégral, tome II (Moscow: Éditions Mir).
[28]	Dedekind R 1872. Stetigkeit und irrationale Zahlen (Braunschweig, 1872); Eg. translation by W W Beman, Continuity and irrational numbers, in: R Dedekind (1963), Essays on the theory of numbers (Dover).
[29]	Fauvel J and Gray J (1987)(eds.). The History of Mathematics (London, Milton Keynes: Mac Millan Press & The Open University).
[30]	Lipschitz R 1877. Lehrbuch der Analyse (Bonn).
[31]	Aksoy A G and Khamsi M A (1990). Nonstandard methods in Fixed Point Theory (Berlin, New York: Springer).
[32]	Kirk W A 1965. A fixed point theorem for mappings which do not increase distances. American Mathematics Monthly 72: 1004-6.
[33]	Allaerts W 1999. Local and global patterns during morphogenesis of the retinotectal topographical mapping in the vertebrate brain. Acta Biotheoretica 47 99-122.
[34]	Armstrong M A 1979. Basic Topology (Berlin: Springer).
[35]	Wilson R J 1975 (2^nd ed.). Introduction to graph theory (London: Longman).
[36]	Keeling M J and Grenfell B T 1997. Disease extinction and community size: modeling the persistance of measles. Science 275 65-7.
[37]	Keeling M J, Rand D A and Morris A J 1997. Correlation models for childhood epidemics. Proceedings of the Royal Society of London, series B 264 1149-56.
[38]	Bezuidenhout C and Grimmett G 1990. The critical contact process dies out. Annals of probability 18 1462-82.
[39]	Stewart I and Golubitsky M 1992. Fearful Symmetry. Is God a Geometer? (London: Blackwell Publishers, Penguin Books).
[40]	Presnov E V and Isaeva V V 1990. Local and global aspects of biological morphogenesis. Speculations in Science and Technology 13 68-75.
[41]	Allaerts W and Roelants H 1993. Positional information limits the self-explaining endeavour in morphogenetic theory (in the sense of Turing). Towards the understanding of the functioning of biological forms. Belgian Journal of Zoology 123 263-82.
[42]	Hengeveld R 1989. Dynamics of biological invasions (London, New York: Chapman & Hall).
[43]	Allaerts W 1999. The biological function paradigm applied to the immunological self-non-self discrimination: critique of Tauber’s phenomenological analysis. Journal for General Philosophy of Science 30 155-71.
[44]	Saunders P T 1992. Collected works of A. M. Turing. vol. 3: Morphogenesis (Amsterdam, London: Elsevier Science Publishers).
[45]	Turing A M 1992 (posthum.). Morphogen theory of phyllotaxis. In: ed P T Saunders, Collected works of A. M. Turing. vol. 3: Morphogenesis, pp. 49-123 (Amsterdam, London: Elsevier Science Publishers).
[46]	Swinton J 2013. Turing, Morphogenesis, and Fibonacci Phyllotaxis: Life in pictures. In: eds S B Cooper and J van Leeuwen, Alan Turing: His Work and Impact, pp. 834-849 (Amsterdam, Boston, London: Elsevier).
[47]	Allaerts W 2018. Why is biodiversity of cardinal importance for public health? International Journal of Environment & Agricultural Science 2 (1): 013.
[48]	Allaerts W 2018. Annotation and predictability of cellular pathways: III. Computability and potential use of parallel Ant Colony Optimization algorithms. Functional and Structural Genomics and Medicine 1 (1): 102-9.
[49]	Law R 1999. Theoretical aspects of community assembly. In: ed J McGlade, Advanced Theoretical Ecology: Principles and Applications pp. 143-171 (London: Blackwell Science).
[50]	Allaerts W and Chang T W 2017. Skewed exposure to environmental antigens complements hygiene hypothesis in explaining the rise of allergy. Acta Biotheoretica 65 (2) 117-34.
[51]	Goh B S 1977. Global stability in many-species systems. American Naturalist 111 135-42.
[52]	Parrilo P A and Jadbabaie A 2008. Approximation of the joint spectral radius using sum of squares. Linear Algebra and its Applications 428 2385-402.
[53]	Lü Q, Xia X Y, Chen R, Miao D J, Chen S S, Quan L J, et al. 2012. When the lowest energy does not induce native structures: Parallel minimization of multi-energy values by hybridizing searching intelligences. PLOS One 7 (9): e44967.