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Instituto Nacional de Salud Pública

2009, Number 5

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salud publica mex 2009; 51 (5)

Stochastic model of infectious diseases transmission.

Ruiz-Ramírez J, Hernández-Rodríguez GE

Language: Spanish
References: 89
Page: 390-396
PDF size: 142.51 Kb.


ABSTRACT

Objective. Propose a mathematic model that shows how population structure affects the size of infectious disease epidemics. Material and Methods. This study was conducted during 2004 at the University of Colima. It used generalized small-world network topology to represent contacts that occurred within and between families. To that end, two programs in MATLAB were conducted to calculate the efficiency of the network. The development of a program in the C programming language was also required, that represents the stochastic susceptible-infectious-removed model, and simultaneous results were obtained for the number of infected people. Results. An increased number of families connected by meeting sites impacted the size of the infectious diseases by roughly 400%. Discussion. Population structure influences the rapid spread of infectious diseases, reaching epidemic effects.


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  41. Lago F, Huerta R, Corbacho F, Sigüenza J. Fast response and temporal coherent oscillation in small-world networks. Phys Rev Lett 2000;84(12):2758-2761.

  42. Mathias N,Gopal V. Small worlds: How and why. Physical Review. E, Statistical, Nonlinear, And Soft Matter Physics [serie en internet]. 63(2 Pt 1):021117. [Consultado 2009 junio 5]. Disponible en: http: //eprints.lisc. ernet.in/454/1/Small_worlds.pdf.

  43. Sun K, Ouyang Q. Microscopic self-organization in networks. Physical Review. E, Statistical, Nonlinear, And Soft Matter Physics [serie en internet]. ;64(2 Pt 2):026111. [Consultado 2009 junio 5]. Disponible en: http: //www.soporteuv.mx:2126/ehost/pdf?vid=8&hid=9&sid=8bcl511007- 68d2-4c10-82f5-b193d746d30%40sessionmgr11

  44. Hernández-Suárez C, Castillo-Chavez C. A basic result on the integral for birth-death Markov processes. Mathem Biosc 1999;161:95-104.

  45. Shonkwiller R, Thompson M. A validation study of a simulation model for common source epidemics. Inter J Bio-medical Comp 1986;19(3- 4):175-194.

  46. Cohen M. Changing patterns of infectious disease. Nature 2000;406 (6797):762.

  47. Binder S, Levitt A, Sacks J, Hughes J. Emerging Infectious Diseases: Public Health Issues for the 21st Century. Science 284(5418):1311.

  48. Kay KM. Global defense again the infectious disease threat. Communicable diseases 2002. Geneva, Switzerland: WHO, 2003.

  49. Nåsell I. Stochastic models of some endemic infections. Mathematical Biosciences 2002;179(1):1-19.

  50. Ball F, Lyne O. Optimal vaccination policies for stochastic epidemics among a population of households. Mathematical Biosciences [serie en internet]. (2002, May), [Consultado 2009 junio 7];177/178:333.

  51. Becker N. The uses of epidemic models. Biometrics 1979;35:295-305.

  52. Altmann M. Susceptible-infected-removed epidemic models with dynamic partnerships. J Mathem Biol 1995;33:661-675.

  53. Heesterbeek JAP, Dietz K. The concept of R0 in epidemic theory. Stat Neerl 1996;50:89-110.

  54. Wallinga J, Edmunds WJ, Kretzschmar S. Perspective: human contact patterns and the spread of airborne infectious diseases. Trends Microbiol 1999;7(9):372-377.

  55. Longini I, Koopman J. Household and community transmission parameters from final distributions of infections in households. Biometrics 1982;38:115-126.

  56. Keeling M, Grenfell B. Disease extinction and community size: Modeling the persistence of measles. Science 1997;275:65-67.

  57. Ball F. Stochastic and deterministic models for sis epidemics a population partitioned into households. Mathem Biosc 1999;156:41-67.

  58. Allen L, Burgin A. Comparison of deterministic and stochastic SIS and SIR models in discrete time. Mathem Biosc 2000;163:1-33.

  59. Chick S, Adams A, Koopman J. Analysis and simulation of a stochastic, discrete-individual model of STD transmission with partnership concurrency. Mathem Biosc 2000;166:45-68.

  60. Duerr HP, Dietz K. Stochastic models for aggregation processes. Mathem Biosc 2000;165:135-145.

  61. Hyman J, Li J. An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations. Mathem Biosc 2000;167:65-86.

  62. Müller J, Kretzschmar M, Dietz K. Contact tracing and deterministic epidemic models. Mathem Biosc 2000;164:39-64.

  63. Ball F, Neal P. A general model for stochastic sir epidemics with two levels of mixing. Mathem Biosc 2002;180:73-102.

  64. Kuperman M, Abramson G. Small world effect in an epidemiological model. Phys Rev Lett [serie en internet]. 86(13):2909-2912. [Consultado 2009 junio5]. Disponible en: http://guillermoabramson.110mb.com/paper/smallworld/swepi.pdf

  65. Lefèbre C, Picard P. Collective Epidemic Models. Mathem Biosc 1996;134:51-70.

  66. Watts D. Small worlds: The dynamic of networks between order and randomness. EUA: Princeton University Press, 1999.

  67. Boots M, Sasaki A. ‘Small worlds’ and the evolution of virulence: Infection occurs locally and at a distance. Proc Roy Soc London B 1999;266:1933-1938.

  68. Kretzschmar M, Morris M. Measures of concurrency in networks and the spread of infectious disease. Mathem Biosc 1996;133:165-195.

  69. Tu Y. How robust is the Internet? (Cover story). Nature 2000;406:353-382.

  70. Aguda B, Goryachev A. From Pathways Databases to Network Models of Switching Behavior. PLoS Computational Biology [serie en internet] 3(9):1674-1678. [Consultado 2009 junio 7]. Disponible en: http://people. mbi.ohid-state.edu/baguda/Agudalab/myPAPERS/2007-PLOS-CB.

  71. Jasch F, Blumen A. Target problem on small-world networks. Physical Review. E, Statistical, Nonlinear, And Soft Matter Physics [serie en internet]. 63(4 Pt 1):041108. [Consultado 2009 junio 5]. Disponible en: http://link.aps.org/doi/10.1103/PhysRevE.63.041108.

  72. Latora V, Marchiori M. Efficient behavior of small-world networks. Phys Rev Lett [serie en internet]. 87(19):198701. [Consultado 2009 junio 5]. Disponible en: http://www.ct.infn.it/˜labra/efficiency_pri_PRL87.pdf

  73. Newman M. The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences of the United States of America 2001;98(2):404-409.

  74. Lloyd A. How viruses spread among computers and people. Science 2001;292:1316-1317.

  75. Koopman J, Chick S, Simon C, Riolo C, Jacquez G. Stochastic effects on endemic infection levels of disseminating versus local contacts. Mathem Biosc 2002;180:49-71.

  76. Fell D, Wagner A. The small world of metabolism. Nat Biotechnol 2000;18:1121-1122.

  77. Watts D, Strogatz S. Collective dynamics of “small-world” networks. Nature 1998;393:440-442.

  78. Ball P. Science is about networking. Nature News [serie en internet] 2001. [Consultado 2008 diciembre 02]. Disponible en: http://www.nature. com/news/2001/010118/full/news010118-2.html.

  79. Collins J, Chow C. It’s a small world. Nature 1998;393:409-410.

  80. Kleinberg J. Navigation in a small world. Nature 2000;406:845.

  81. Hong H, Choi MY, Kim B. Synchronization on small-world networks. Physical Review. E, Statistical, Nonlinear, And Soft Matter Physics [serial on the Internet]. (2002, Feb 24), [cited June 5, 2009];65(2 Pt 2):026139.

  82. Montoya J, Solé R. Small world patterns in food webs. J Theoret Biol 2002;214:405-412.

  83. Xin-She Y. Chaos in small-world networks. Physical Review. E, Statistical, Nonlinear, And Soft Matter Physics [serie en internet]. 63(4 Pt 2):046206. [Consultado 2009 junio 5]. Disponible en: http://link.aps.org/doi/ro.no3/PhysRevE.63.046206.

  84. Davidsen J, Ebel H, Bornholdt S. Emergence of a small world from local interactions: Modeling acquaintance networks. Phys Rev Lett [serie en internet]. 88(12):128701. [Consultado 2009 junio 5]. Disponible en: http://www.//p.uni-bremen.de/complex/pr128701.pdf.

  85. Lago F, Huerta R, Corbacho F, Sigüenza J. Fast response and temporal coherent oscillation in small-world networks. Phys Rev Lett 2000;84(12):2758-2761.

  86. Mathias N,Gopal V. Small worlds: How and why. Physical Review. E, Statistical, Nonlinear, And Soft Matter Physics [serie en internet]. 63(2 Pt 1):021117. [Consultado 2009 junio 5]. Disponible en: http://eprints.lisc.ernet.in/454/1/Small_worlds.pdf.

  87. Sun K, Ouyang Q. Microscopic self-organization in networks. Physical Review. E, Statistical, Nonlinear, And Soft Matter Physics [serie en internet].;64(2 Pt 2):026111. [Consultado 2009 junio 5]. Disponible en: http://www.soporteuv.mx:2126/ehost/pdf?vid=8&hid=9&sid=8bcl511007-68d2-4c10-82f5-b193d746d30%40sessionmgr11

  88. Hernández-Suárez C, Castillo-Chavez C. A basic result on the integral for birth-death Markov processes. Mathem Biosc 1999;161:95-104.

  89. Shonkwiller R, Thompson M. A validation study of a simulation model for common source epidemics. Inter J Bio-medical Comp 1986;19(3-4):175-194.




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