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Acta Biotheoretica

, Volume 61, Issue 1, pp 141–172 | Cite as

Random Modelling of Contagious Diseases

  • J. DemongeotEmail author
  • O. Hansen
  • H. Hessami
  • A. S. Jannot
  • J. Mintsa
  • M. Rachdi
  • C. Taramasco
Regular Article

Abstract

Modelling contagious diseases needs to include a mechanistic knowledge about contacts between hosts and pathogens as specific as possible, e.g., by incorporating in the model information about social networks through which the disease spreads. The unknown part concerning the contact mechanism can be modelled using a stochastic approach. For that purpose, we revisit SIR models by introducing first a microscopic stochastic version of the contacts between individuals of different populations (namely Susceptible, Infective and Recovering), then by adding a random perturbation in the vicinity of the endemic fixed point of the SIR model and eventually by introducing the definition of various types of random social networks. We propose as example of application to contagious diseases the HIV, and we show that a micro-simulation of individual based modelling (IBM) type can reproduce the current stable incidence of the HIV epidemic in a population of HIV-positive men having sex with men (MSM).

Keywords

Social networks Contagious diseases Stochastic epidemic modelling HIV 

Supplementary material

References

  1. Allen L (2008) An introduction to stochastic epidemic models. Mathematical Epidemiology 1945:81–130CrossRefGoogle Scholar
  2. Amarasekare P (1998) Interactions between local dynamics and dispersal: insights from single species models. Theor Popul Biol 53(44):59Google Scholar
  3. Arino J, van den Driessche P (2003) The basic reproduction number in a multi-city compartmental epidemic model. LNCIS 294:135–142Google Scholar
  4. Artalejo AJR, Economou A, Lopez-Herrero MJ (2010) On the number of recovering individuals in the SIS and SIR stochastic epidemic models. Math Biosci 228:45–55CrossRefGoogle Scholar
  5. Auvert B, Taljaard D, Lagarde E, Sobngwi-Tambekou J, Sitta JR, Puren A (2005) Randomized, controlled intervention trial of male circumcision for reduction of HIV infection risk: the ANRS 1265 Trial. PLoS Med 2:e298CrossRefGoogle Scholar
  6. Bahar A, Mao X (2004) Stochastic delay Lotka–Volterra model. J Math Anal Appl 292:364–380CrossRefGoogle Scholar
  7. Bailey NTJ (1963) The simple stochastic epidemic: a complete solution in terms of known functions. Biometrika 50:235–240Google Scholar
  8. Bailey RC, Moses S, Parker CB, Agot K, Maclean I, Krieger JN, Williams CF, Campbell RT, Ndinya-Achola JO (2007) Male circumcision for HIV prevention in young men in Kisumu, Kenya: a randomised controlled trial. Lancet 369:643–656CrossRefGoogle Scholar
  9. Ball F, Neal P (2002) A general model for stochastic SIR epidemics with two levels of mixing. Math Biosci 180:73–102CrossRefGoogle Scholar
  10. Barreira L, Valls C (2010) Stability of delay equations via Lyapunov functions. J Math Anal Appl 365:797–805CrossRefGoogle Scholar
  11. Barth J (2002) What should we do about the obesity epidemic? Pract Diabetes Int 19:119–122CrossRefGoogle Scholar
  12. Bartholomay AF (1958a) On the linear birth and death processes of biology as Markoff chains. Bull Math Biophys 20:97–118CrossRefGoogle Scholar
  13. Bartholomay AF (1958b) Stochastic models for chemical reactions: I. Theory of the unimolecular reaction process. Bull. Math. Biophys. 20:175–190CrossRefGoogle Scholar
  14. Bartholomay AF (1959) Stochastic models for chemical reactions: II. The unimolecular rate constant. Bull Math Biophys 21:363–373CrossRefGoogle Scholar
  15. Beier JC (1998) Malaria parasite development in mosquitoes. Ann Rev Entomol 43:519–543CrossRefGoogle Scholar
  16. Ben Amor H, Demongeot J, Elena A, Sené S (2008) Structural sensitivity of neural and genetic networks. Lect Notes Comput Sci 5317:973–986CrossRefGoogle Scholar
  17. Beretta E, Capasso V (1986) Global stability results for a multi-group SIR epidemic model. In: Hallam TG, Gross IJ, Levin SA (eds) Mathematical ecology. World Scientific, Singapore, pp 317–342Google Scholar
  18. Beretta E, Hara T, Ma W, Takeuchi Y (2001) Global asymptotic stability of a SIR epidemic model with distributed time delay. Nonlinear Anal 47:4107–4115CrossRefGoogle Scholar
  19. Bernoulli D (1760) Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages de l’inoculation pour la prévenir. Mémoire Acad Roy Sci, ParisGoogle Scholar
  20. Bochner S (1933) Abstrakte fastperiodische Funktionen. Acta Mathematica 61:150–184CrossRefGoogle Scholar
  21. Bouyssou A, Janier M, Dupin N, Alcaraz I, Vernay-Vaïsse C, Basselier B, Spenatto N, Dhotte P, Castano F, Semaille C, Gallay A (2011) La syphilis en France: analyse des données de surveillance sur 10 ans, Bulletin épidémiologique hebdomadaire 26-27-28:295–297Google Scholar
  22. Bricault G (2008) Naissance d’un ordre hospitalier. Publication AFAA, GrenobleGoogle Scholar
  23. Britton T (2010) Stochastic epidemic models: a survey. Math Biosci 225:24–35CrossRefGoogle Scholar
  24. Brownlee J (1915) On the curve of the epidemic. Br Med J 1:799–800CrossRefGoogle Scholar
  25. Caputo JG, Sarels B (2011) Reaction-diffusion front crossing, a local defect. Phys Rev E 84:041108CrossRefGoogle Scholar
  26. Charlebois ED, Das M, Porco TC, Havlir DV (2011) The effect of expanded antiretroviral treatment strategies on the HIV epidemic among men who have sex with men in San Francisco. Clin Infect Dis 52:1046–1049CrossRefGoogle Scholar
  27. Christakis N, Fowler J (2006) The spread of obesity in a large social network over 32 years. N Engl J Med 355:77–82Google Scholar
  28. Clerc M, Gallay A, Imounga L, Le Roy C, Peuchant O, Bébéar C, Goulet V, Barbeyrac B (2011) Évolution du nombre de lymphogranulomatoses vénériennes rectales et d’infections rectales à Chlamydia trachomatis à souches non L en France entre 2002 et 2009. Bul Epidémiol Hebd 26–28:310–313Google Scholar
  29. Cohen-Cole E, Fletcher JM (2008) Is obesity contagious? Social networks vs. environmental factors in the obesity epidemic. J Health Econ 27:1382–1387CrossRefGoogle Scholar
  30. Cori A (2010) Modéliser l’hétérogénéité dans les épidémies: aspects biologiques et comporte-mentaux. PhD Thesis. University Paris VI—Pierre et Marie CurieGoogle Scholar
  31. Dalal N, Greenhalgh D, Mao X (2007) A stochastic model of AIDS and condom use. J Math Anal Appl 325:36–53CrossRefGoogle Scholar
  32. d’Alembert J (1761) Onzième memoire: sur l’application du calcul des probabilités à l’inoculation de la petite vérole; notes sur le mémoire précédent; théorie mathématique de l’inoculation. In: Opuscules mathématiques. David, Paris, t. II, pp 26–95Google Scholar
  33. de Saint-Pol T (2008) Obésité et milieux sociaux en France: les inégalités augmentent. Bull Epidemiol Hebdom 20:175–179Google Scholar
  34. Delbrück M (1940) Statistical fluctuations in autocatalytic reactions. J Chem Phys 8:120–124CrossRefGoogle Scholar
  35. Demongeot J (1977) A stochastic model for the cellular metabolism. In: Recent developments in statistics. North Holland, Amsterdam, pp 655–662Google Scholar
  36. Demongeot J (1981) Existence de solutions périodiques pour une classe de systèmes différentiels gouvernant la cinétique de chaînes enzymatiques oscillantes. Lect Notes Biomath 41:40–62Google Scholar
  37. Demongeot J, Fricot J (1986) Random fields and renewal potentials. NATO ASI Ser F 20:71–84Google Scholar
  38. Demongeot J, Kellershohn N (1983) Glycolytic oscillations: an attempt to an “in vitro” reconstitution of the higher part of glycolysis. Lect Notes Biomath 49:17–31CrossRefGoogle Scholar
  39. Demongeot J, Laurent M (1983) Sigmoidicity in allosteric models. Math Biosci 67:1–17CrossRefGoogle Scholar
  40. Demongeot J, Sené S (2011) The singular power of the environment on nonlinear Hopfield networks. In: CMSB’11. ACM proceedings, New York, pp 55–64Google Scholar
  41. Demongeot J, Waku J (2012) Robustness in genetic regulatory networks, IV. Comptes Rendus Mathématique 350:293–298CrossRefGoogle Scholar
  42. Demongeot J, Elena A, Sené S (2008) Robustness in neural and genetic networks. Acta Biotheor 56:27–49CrossRefGoogle Scholar
  43. Demongeot J, Drouet E, Moreira A, Rechoum Y, Sené S (2009) Micro-RNAs: viral genome and robustness of the genes expression in host. Phil Trans R Soc A 367:4941–4965CrossRefGoogle Scholar
  44. Demongeot J, Elena A, Noual M, Sené S (2011) Random Boolean Networks and Attractors of their Intersecting Circuits. In: AINA’ 11. IEEE proceedings, Piscataway, pp 483–487Google Scholar
  45. Demongeot J, Gaudart J, Mintsa J, Rachdi M (2012a) Demography in epidemics modelling. Commun Pure Appl Anal 11:61–82CrossRefGoogle Scholar
  46. Demongeot J, Gaudart J, Lontos A, Promayon F, Mintsa J, Rachdi M (2012b) Least diffusion zones in morphogenesis and epidemiology. Int J Bifurcat Chaos 22:50028Google Scholar
  47. Dietz K (1967) Epidemics and rumours: a survey. J R Stat Soc Ser A (General) 130:505–528CrossRefGoogle Scholar
  48. Duchon P, Hanusse N, Lebhar E, Schabanel N (2006) Could any graph be turned into a small-world? Theor Comp Sci 355:96–103CrossRefGoogle Scholar
  49. Durrett RT (2010) Some features of the spread of epidemics and information on a random graph. Proc Natl Acad Sci USA 107:4491–4498CrossRefGoogle Scholar
  50. Eisenberg JNS, Desai MA, Levy K, Bates SJ, Liang S, Naumoff K, Scott JC (2007) Environmental determinants of infectious disease: a framework for tracking causal links and guiding public health research. Environ Health Perspect 115:1216–1223CrossRefGoogle Scholar
  51. Elena A (2004) Algorithme pour la simulation de la dynamique des réseaux de régulation génétique. Master Thesis. University J. Fourier, GrenobleGoogle Scholar
  52. Elena A (2009) Robustesse des réseaux d’automates à seuil. University J. Fourier, GrenobleGoogle Scholar
  53. Elena A, Demongeot J (2008) Interaction motifs in regulatory networks and structural robustness. In: IEEE ARES’ 08. IEEE Press, Piscataway, pp 682–686Google Scholar
  54. Elena A, Ben-Amor H, Glade N, Demongeot J (2008) Motifs in regulatory networks and their structural robustness. In: IEEE BIBE’ 08. IEEE Press, Piscataway, pp 234–242Google Scholar
  55. Farr W (1866) Report on the cholera epidemic of 1866 in England. Suppl Ann Rep Reg Gen 29:1867–1868Google Scholar
  56. Filipe JAN, Gibson GJ (2001) Comparing approximations to spatio-temporal models for Epidemics with local Spread. Bull Math Biol 63:603–624CrossRefGoogle Scholar
  57. Fraser C, Hollingsworth TD, Chapman R, De Wolf F, Hanage WP (2007) Variation in HIV-1 set-point viral load: epidemiological analysis and an evolutionary hypothesis. Proc Natl Acad Sci USA 104:17441–17446CrossRefGoogle Scholar
  58. Gaudart J, Giorgi R, Poudiougou B, Ranque S, Doumbo OK, Demongeot J (2007) Spatial cluster detection: principle and application of different general methods. Rev Epid Santé Pub 55:297–306CrossRefGoogle Scholar
  59. Gaudart J, Touré O, Dessay N, Dicko AL, Ranque S, Forest L, Demongeot J, Doumbo OK (2009) Modelling malaria incidence with environmental dependency in a locality of Sudanese savannah area. Mali Malaria J 8:e61CrossRefGoogle Scholar
  60. Gaudart J, Ghassani M, Mintsa J, Rachdi M, Waku J, Demongeot J (2010a) Demography and diffusion in epidemics: Malaria and Black Death spread. Acta Biotheor 58:277–305CrossRefGoogle Scholar
  61. Gaudart J, Ghassani M, Mintsa J, Waku J, Rachdi M, Doumbo OK, Demongeot J (2010b) Demographic and spatial factors as causes of an epidemic spread, the copule approach. Application to the retro-prediction of the Black Death epidemics of 1346. In: IEEE AINA’10. IEEE Press, Piscataway, pp 751–758Google Scholar
  62. Gibson ME (1978) Sir Ronald Ross and his contemporaries. J R Soc Med 71:611–618Google Scholar
  63. Gillespie DT (1970) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22:403–434CrossRefGoogle Scholar
  64. Glade N, Elena A, Fanchon E, Demongeot J, Ben Amor H (2011) Determination, optimization and taxonomy of regulatory networks. The example of Arabidopsis thaliana flower morphogenesis. In: IEEE AINA’ 11. IEEE Press, Piscataway, pp 488–494Google Scholar
  65. Graunt J (1662) Natural and political observations made upon the bills of mortality. In: J. Martin, J. Allestry and T. Dicas (eds) T. Roycroft, LondonGoogle Scholar
  66. Gray RH, Kigozi G, Serwadda D, Makumbi F, Watya S, Nalugoda F et al (2007) Male circumcision for HIV prevention in men in Rakai, Uganda: a randomised trial. Lancet 369:657–666CrossRefGoogle Scholar
  67. Grinsztejn B, Ribaudo H, Cohen MS, HPTN 052 Protocol Team et al (2011) Effects of early versus delayed initiation of antiretroviral therapy (ART) on HIV clinical outcomes: results from the HPTN 052 randomized clinical trial. In: 6th IAS Conference on HIV Pathogenesis, Treatment and Prevention, RomeGoogle Scholar
  68. Guo HB, Li MY, Shuai Z (2006) Global stability of the endemic equilibrium of multi-group SIR epidemic models. Can Appl Math Q 14:259–284Google Scholar
  69. Hallett TB, Smit C, Garnett GP, de Wolf F (2011) Estimating the risk of HIV transmission from homosexual men receiving treatment to their HIV-uninfected partners. Sex Transm Infect 87:17–21CrossRefGoogle Scholar
  70. Hamer WH (1906) Epidemic disease in England. Lancet 1:733–739Google Scholar
  71. Hethcote HW (1978) An immunization model for a heteregenous population. Theor Popul Biol 14:338–349CrossRefGoogle Scholar
  72. Hethcote HW, Levin SA (1995) Periodicity in Epidemiological models. In: Levin SA, Hallam TG, Gross L (eds) Applied mathematical ecology. Biomathematics, vol 18. Springer, Berlin, pp 193–211CrossRefGoogle Scholar
  73. Hethcote HW, Van den Driessche P (1995) An SIS epidemic model with variable population size and a delay. J Math Biol 34:177–194CrossRefGoogle Scholar
  74. Hoare MR (1970) Molecular Markov processes. Nature 226:599–603CrossRefGoogle Scholar
  75. Hollingsworth TD, Anderson RM, Fraser C (2008) HIV-1 transmission, by stage of infection. J Infect Dis 198:687–693CrossRefGoogle Scholar
  76. International Association for the Study of Obesity (2000) Obesity: preventing and managing the global epidemic. International Obesity Task Force Prevalence Data, LondonGoogle Scholar
  77. Ishikawa H, Ishii NagaiAN, Ohmae H, Harada M, Suguri S, Leafasia J (2008) A mathematical model for the transmission of the Plasmodium vivax malaria. Parasitol Int 52:81–93CrossRefGoogle Scholar
  78. Jachimowski JC, McQuarrie DA, Russell ME (1964) A stochastic approach to enzyme-substrate reactions. Biochemistry 3:1732–1736CrossRefGoogle Scholar
  79. Kermack WO, McKendrick AG (1927) A contribution to the mathematical theory of epidemics. Proc R Soc Lond Ser A 115:700–721CrossRefGoogle Scholar
  80. Kermack WO, McKendrick AG (1932) Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc R Soc Lond Ser A 120:138–155Google Scholar
  81. Kermack WO, McKendrick AG (1933) Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity. Proc R Soc Lond Ser A 121:141–194Google Scholar
  82. Koella JC, Antia R (2003) Epidemiological models for the spread of anti-malaria resistance. Malaria J 2:e3CrossRefGoogle Scholar
  83. Koopman JS, Longini IM (1994) The ecological effects of individual exposures and nonlinear disease dynamics in populations. Am J Public Health 84:836–842CrossRefGoogle Scholar
  84. Korobeinikov A, Maini PK (2004) A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence. Math Biosci Eng 1:57–60CrossRefGoogle Scholar
  85. Kretzschmar M (1996) Measures of concurrency in networks and the spread of infectious disease. Math Biosci 133:165–195CrossRefGoogle Scholar
  86. Laitinen J, Power C, Jarvelin MR (2001) Family social class, maternal body mass index, childhood body mass index, and age at menarche as predictors of adult obesity. Am J Clin Nutr 74:287–294Google Scholar
  87. Le Vu S, Le Strat Y, Barin F, Pillonel J, Cazein F, Bousquet V, Brunet S, Thierry D, Semaille C, Meyer L, Desenclos JC (2010) Population-based HIV-1 incidence in France, 2003–08: a modelling analysis. Lancet Infect Dis 10:682–687CrossRefGoogle Scholar
  88. Leclerc PM, Matthews AP, Garenne ML (2009) Fitting the HIV epidemic in Zambia: a two-sex micro-simulation model. PLoS ONE 4:e5439CrossRefGoogle Scholar
  89. Li MY, Wang L (1995) Global stability in some SEIR epidemic models. Math Biosci 125:155–164CrossRefGoogle Scholar
  90. Magal P, Ruan S (2012) SIR models revisited: from the individual level to the population level. Preprint University BordeauxGoogle Scholar
  91. Magal P, McCluskey CC, Webb GF (2010) Lyapunov functional and global asymptotic stability for an infection-age model. Appl Anal 89:1109–1140CrossRefGoogle Scholar
  92. Maillard G, Charles MA, Thibault N, Forhan A, Sermet C, Basdevant A, Eschwege E (1999) Trends in the prevalence of obesity in the French adult population between 1980 and 1991. Int J Obes 23:389–394CrossRefGoogle Scholar
  93. McQuarrie DA (1963) Kinetics of small systems. J Chem Phys 38:433–436CrossRefGoogle Scholar
  94. McQuarrie DA (1967) Stochastic approach to chemical kinetics. J Appl Prob 4:413–478CrossRefGoogle Scholar
  95. McQuarrie DA, Jachimowski CJ, Russell ME (1964) Kinetics of small systems. II. J Chem Phys 40:2914–2921CrossRefGoogle Scholar
  96. Melesse DY, Gumel AB (2010) Global asymptotic properties of an SEIRS model with multiple infectious stages. J Math Anal Appl 366:202–217CrossRefGoogle Scholar
  97. Morris M, Kretzschmar M (2000) A microsimulation study of the effect of concurrent partnerships on the spread of HIV in Uganda. Math Popul Stud 8:109–133CrossRefGoogle Scholar
  98. Murray JM, McDonald AM, Law MG (2009) Rapidly ageing HIV epidemic among men who have sex with men in Australia. Sex Health 6:83–86CrossRefGoogle Scholar
  99. Myers A, Rosen JC (1999) Obesity stigmatization and coping: relation to mental health symptoms, body image, and self-esteem. Int J Obes Relat Metab Disord 23:221–230CrossRefGoogle Scholar
  100. Novi Inverardi PL, Tagliani A (2006) Discrete distributions from moment generating function. Appl Math Comput 182:200–209CrossRefGoogle Scholar
  101. ObEpi-Roche (2009) Enquête épidémiologique nationale sur le surpoids et l’obésité. Enquête INSERM-Roche, ParisGoogle Scholar
  102. Orcutt GH, Greenberger M, Korbel J, Rivlin AM (1961) Microanalysis of socioeconomic systems: a simulation study. Harper, New YorkGoogle Scholar
  103. Orroth KK, Freeman EE, Bakker R, Buvé A, Glynn JR, Boily MC, White RG, Habbema JDF, Hayes RJ (2007) Understanding the differences between contrasting HIV epidemics in east and west Africa: results from a simulation model of the Four Cities Study. Sex Transm Infect 83:i5CrossRefGoogle Scholar
  104. Pilcher CD, Tien HC, Eron JJ Jr, Vernazza PL, Leu SY, Stewart PW, Goh LE, Cohen MS (2004) Brief but efficient: acute HIV infection and the sexual transmission of HIV. J Infect Dis 189:1785–1792CrossRefGoogle Scholar
  105. Pilcher CD, Joaki G, Hoffman IF, Martinson FE, Mapanje C, Stewart PW, Powers KA, Galvin S, Chilongozi D, Gama S, Price MA, Fiscus SA, Cohen MS (2007) Amplified transmission of HIV-1: comparison of HIV-1 concentrations in semen and blood during acute and chronic infection. AIDS 21:1723–1730CrossRefGoogle Scholar
  106. Pinkerton SD, Abramson PR (1997) Effectiveness of condoms in preventing HIV transmission. Soc Sci Med 44:1303–1312CrossRefGoogle Scholar
  107. Porco TC, Martin JN, Page-Shafer KA, Cheng A, Charlebois E, Grant RM, Osmond DH (2004) Decline in HIV infectivity following the introduction of highly active antiretroviral therapy. AIDS 18:81–88CrossRefGoogle Scholar
  108. Quinn TC, Wawer MJ, Sewankambo N, Serwadda D, Li C, Wabwire-Mangen F, Meehan MO, Lutalo T, Gray RH (2000) Viral load and heterosexual transmission of human immunodeficiency virus type 1. Rakai project study group. N Engl J Med 342:921–929CrossRefGoogle Scholar
  109. Reynolds SJ, Makumbi F, Nakigozi G, Kagaayi J, Gray RH, Wawer M, Quinn TC, Serwadda D (2011) HIV-1 transmission among HIV-1 discordant couples before and after the introduction of antiretroviral therapy. AIDS 25:473–477CrossRefGoogle Scholar
  110. Rhodes CJ, Demetrius L (2010) Evolutionary entropy determines invasion success in emergent epidemics. PLoS ONE 5:e12951CrossRefGoogle Scholar
  111. Rogier C, Sallet G (2004) Modélisation du paludisme. Med Trop 64:89–97Google Scholar
  112. Ross R (1910) Prevention of Malaria. John Murray, LondonGoogle Scholar
  113. Ross R (1915) Some a priori pathometric équations. Br Med J 1:546–547CrossRefGoogle Scholar
  114. Ross R (1916) An application of the theory of probabilities to the study of a priori pathometry. Part I. Proc R Soc Lond Ser A 92:204–230CrossRefGoogle Scholar
  115. Ruan S, Xiao D, Beier JC (2008) On the delayed Ross–MacDonald model for Malaria transmission. Bull Math Biol 70:1098–1114CrossRefGoogle Scholar
  116. Sathik MM, Rasheed AA (2011) Social network analysis in an online blogosphere. Int J Eng Sci Technol 3:117–121Google Scholar
  117. Sawers L, Stillwaggon E (2010) Concurrent sexual partnerships do not explain the HIV epidemics in Africa: a systematic review of the evidence. J Int AIDS Soc 13:1–23CrossRefGoogle Scholar
  118. Scharoun-Lee M, Adair LS, Kaufman JS, Gordon-Larsen P (2009) Obesity, race/ethnicity and the multiple dimensions of socioeconomic status during the transition to adulthood: a factor analysis approach. Soc Sci Med 68:708–716CrossRefGoogle Scholar
  119. Seng R, Rolland M, Beck-Wirth G, Souala GF, Deveau C, Delfraissy JF, Goujard C, Meyer L (2011) Trends in unsafe sex and influence of viral load among patients followed since primary HIV infection between 2000 and 2009. AIDS 25:977–988CrossRefGoogle Scholar
  120. Shi R, Chen L (2007) Stage-structured impulsive model for pest management. Discrete Dyn Nat Soc 2007:97608Google Scholar
  121. Shirreff G, Pellis L, Laeyendecker O, Fraser C (2011) Transmission selects for HIV-1 strains of intermediate virulence: a modelling approach. PLoS Comput Biol 7:e1002185CrossRefGoogle Scholar
  122. Taramasco C (2011) Impact de l’obésité sur les structures sociales et impact des structures sociales sur l’obésité? PhD thesis. Ecole Polytechnique, ParisGoogle Scholar
  123. Taramasco C, Demongeot J (2011) Collective intelligence, social networks and propagation of a social disease, the obesity. In: EIDWT’11. IEEE Proceedings, Piscataway, pp 86–90Google Scholar
  124. Tuckwell HC, Williams RJ (2007) Some properties of a simple stochastic epidemic model of SIR type. Math Biosci 208:76–97CrossRefGoogle Scholar
  125. Velter A, Enquête Presse Gay (2004) Maladies Infectieuses. Institut national de veille sanitaire, ParisGoogle Scholar
  126. Velter A, Barin F, Bouyssou A, Le Vu S, Guinard J, Pillonel J, Semaille C (2010) Prévalence du VIH et comportement de dépistage des hommes fréquentant les lieux de convivialité gay parisiens. Prevagay 2009. Bull Epidemiol Hebdom 22:464–467Google Scholar
  127. Wang Y, Wang J, Zhang L (2010) Cross diffusion-induced pattern in an SI model. Math Comput 217:1965–1970Google Scholar
  128. Wilson DP, Law MG, Grulich AE, Cooper DA, Kaldor JM (2008) Relation between HIV viral load and infectiousness: a model-based analysis. Lancet 372:314–320CrossRefGoogle Scholar
  129. Wilson DP, Hoare A, Regan DG, Law MG (2009) Importance of promoting HIV testing for preventing secondary transmissions: modelling the Australian HIV epidemic among men who have sex with men. Sex Health 6:19–33CrossRefGoogle Scholar
  130. World Health Organization (2000) Obesity: preventing and managing the global epidemic. WHO Technical report 894, Geneva Optimal contact process on complex networksGoogle Scholar
  131. Xiridou M, Geskus R, de Wit J, Coutinho R, Kretzschmar M (2003) The contribution of steady and casual partnerships to the incidence of HIV infection among homosexual men in Amsterdam. AIDS 17:1029–1038CrossRefGoogle Scholar
  132. Xiridou M, Geskus R, de Wit J, Coutinho R, Kretzschmar M (2004) Primary HIV infection as source of HIV transmission within steady and casual partnerships among homosexual men. AIDS 18:1311–1320CrossRefGoogle Scholar
  133. Yang R, Zhou T, Xie YB, Lai YC, Wang BH (2008) Optimal contact process on complex networks. Phys Rev E 78:066109CrossRefGoogle Scholar
  134. Yongzhen P, Shaoying L, Changguo L, Lansun C (2009) The dynamics of an impulse delay model with variable coefficients. Appl Math Mod 33:2766–2776CrossRefGoogle Scholar
  135. Yoshida N, Hara T (2007) Global stability of a delayed SIR epidemic model with density dependent birth and death rates. J Comput Appl Math 201:339–347CrossRefGoogle Scholar
  136. Yu J, Jiang D, Shi N (2009) Global stability of two-group SIR model with random perturbation. J Math Anal Appl 360:235–244CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • J. Demongeot
    • 1
    • 4
    Email author
  • O. Hansen
    • 1
  • H. Hessami
    • 1
  • A. S. Jannot
    • 1
  • J. Mintsa
    • 1
  • M. Rachdi
    • 1
    • 5
  • C. Taramasco
    • 1
    • 2
    • 3
    • 4
  1. 1.AGIM, FRE, CNRS 3405, Faculty of Medicine of GrenobleUniversity J. FourierLa TroncheFrance
  2. 2.Escuela de Ingeniería Civil InformáticaUniversidad de ValparaisoValparaisoChile
  3. 3.ISCPIF, Complex Systems Institute of Paris Ile de FranceParisFrance
  4. 4.Universidad de ValparaisoValparaisoChile
  5. 5.University Pierre Mendès-FranceGrenoble Cedex 9France

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