Abstract
The classical models of epidemics dynamics by Ross and McKendrick have to be revisited in order to incorporate elements coming from the demography (fecundity, mortality and migration) both of host and vector populations and from the diffusion and mutation of infectious agents. The classical approach is indeed dealing with populations supposed to be constant during the epidemic wave, but the presently observed pandemics show duration of their spread during years imposing to take into account the host and vector population changes as well as the transient or permanent migration and diffusion of hosts (susceptible or infected), as well as vectors and infectious agents. Two examples are presented, one concerning the malaria in Mali and the other the plague at the middle-age.
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Abbas L, Demongeot J, Glade N (2009) Synchrony in reaction-diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves. Philos Trans R Soc A 367:4829–4862
Balenghien T, Fouque F, Sabatier P, Bicout DJ (2006) Horse-, bird-, and human-seeking behavior and seasonal abundance of mosquitoes in a West West Nile virus focus of southern France. J Med Entomol 43:936–946
Barry S, Gualde N (2006) The biggest epidemics of history. L’Histoire 310:38–49
Baum TP, Pasqual N, Thuderoz F, Hierle V, Chaume D, Lefranc MP, Jouvin-Marche E, Marche P, Demongeot J (2004) IMGT/GeneInfo: enhancing V(D)J recombination database accessibility. Nucleic Acids Res 32:51–54
Beaudoin D, Lakhal-Chaieb L (2008) Archimedean copula model selection under dependent truncation. Stat Med 27:4440–4454
Benedictow OJ (2004) The black death 1346–1353: the complete history. Boydell Press, Woodbridge
Bernoulli D (1760) Essai d’une nouvelle analyse de la mortalité causée par la petite vérole, et des avantages del’inoculation pour la prévenir. Mém Acad Roy Sci, Paris
Bicout DJ, Sabatier P (2004) Mapping rift valley fever vectors and prevalence using rainfall variations. Vector-Borne Zoonotic Dis 4:33–42
Bicout DJ, Chalvet-Monfray K, Sabatier P (2002) Infection persistence time of Aedes breeding habitats. Physica A Stat Mech Appl 305:597–603
Brandenburg A, Multamaki T (2004) How long can left and right handed life forms coexist? Int J Astrobiol 3:209–219
Brossollet J, Mollaret H (1994) Pourquoi la peste ? Le rat, la puce et le bubon. Gallimard, Paris
Brouhns N, Denuit M (2001) Risque de longévité et rente viagère. Institut de Statistique Université Catholique, Louvain, Discussion Paper 0137
Cantor NF (2001) In the wake of the plague: the black death and the world it made. Free Press, New York
Carpentier E (1993) Une ville devant la peste, Orvieto et la peste noire de 1348. De Boeck Université, Bruxelles
Christakos G, Olea RA, Serre ML, Yu HL, Wang LL (2005) Interdisciplinary public health reasoning and epidemic modelling: the case of black death. Springer, Berlin
Cohn SK Jr (2002) The black death: end of a paradigm. Am Hist Rev 107:1–54
Cooke RM, Morales-Napoles O (2006) Competing risk and the cox proportional hazard model. J Stat Plan Inference 136:1621–1637
D’Alembert J (1761) Opuscules Mathématiques. David, Paris
de Magny GC, Paroissin C, Cazelles B, de Lara M, Delmas JF, Guégan JF (2005) Modeling environmental impacts of plankton reservoirs on cholera population dynamics. Esaim 14:156–173
Demetrius L (1979) Relations between demographic parameters. Demography 16:329–338
Demongeot J (1983) Thesis. Université J. Fourier, Grenoble
Demongeot J (2009) Biological boundaries and biological age. Acta Biotheor 57:397–419
Demongeot J, Demetrius L (1989) La dérive démographique et la sélection naturelle: Etude empirique de la France (1850–1965). Population 2:231–248
Demongeot J, Fricot J (1986) Random fields and renewal potentials. NATO ASI Serie F 20:71–84
Demongeot J, Glade N, Forest L (2007a) Liénard systems and potential-Hamiltonian decomposition. I Methodology. Comptes Rendus Mathématique 344:121–126
Demongeot J, Glade N, Forest L (2007b) Liénard systems and potential-Hamiltonian decomposition. II Algorithm. Comptes Rendus Mathématique 344:191–194
Demongeot J, Glade N, Moreira A, Vial L (2009) RNA relics and origin of life. Int J Mol Sci 10:3420–3441
Depinay JMO, Mbogo CM, Killeen G, Knols B, Beier J, Carlson J, Dusho J, Billingsley P, Mwambi H, Githure J, Toure AM, McKenzie FE (2004) A simulation model of African Anopheles ecology and population dynamics for the analysis of malaria transmission. Malaria J 3:29
Dietz K, Heesterbeek JAP (2000) Bernoulli was ahead of modern epidemiology. Nature 408:513
Dietz K, Heesterbeek JAP (2002) Daniel Bernoulli’s epidemiological model revisited. Math Biosci 180:1–21
Doliger C (2006) Démographie, fécondité et croissance économique en France. Thesis Université Louis Pasteur, Strasbourg
Doumbo OK (2005) It takes a village: medical research and ethics in Mali. Science 307:679–681
Drancourt M, Aboudharam G, Signoli M, Dutour O, Raoult D (1998) Detection of 400-year-old Yersinia pestis DNA in human dental pulp: an approach to the diagnosis of ancient septicemia. Proc Natl Acad Sci USA 95:2637–12640
Dutertre J (1976) Étude d’un modèle épidémiologique appliqué au paludisme. Ann Soc Belge Méd Trop 56:127–141
Eckert EA (2000) The black death and the transformation of the West. Bull Hist Med 74:356–357
Fisher RA (1937) The wave of advance of advantageous genes. Ann Eugenics 7:353–369
Forest L, Glade N, Demongeot J (2007) Liénard systems and potential-Hamiltonian decomposition. Applications. C R Acad Sci Biol 330:97–106
Gani R, Leach S (2004) Epidemiologic determinants for modelling pneumonic plague outbreaks. Emerg Infect Dis 10:608–614
Gaudart J, Giorgi R, Poudiougou B, Ranque S, Doumbo OK, Demongeot J (2007) Spatial cluster detection: principle and application of different general methods. Rev Epidemiol Santé Publique 55:297–306
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. Malar J 8:61
Gaudart J, Ghassani M, Mintsa J, Waku J, Rachdi M, Doumbo OK, Demongeot J (2010) Demographic and spatial factors as causes of an epidemic spread, the copule approach. Application to the retro-prediction of the black death epidemy of 1346. In: IEEE AINA’ 10 & BLSMC’ 10. IEEE Press, Piscataway, pp 751–758
Glade N, Forest L, Demongeot J (2007) Liénard systems and potential-Hamiltonian decomposition. III Applications in biology. Comptes Rendus Mathématique 344:253–258
Horie M, Honda T, Suzuki Y, Kobayashi Y, Daito T, Oshida T, Ikuta K, Jern P, Gojobori T, Coffin JM, Tomonaga K (2010) Endogenous non-retroviral RNA virus elements in mammalian genomes. Nature 463:84–88
Horrox R (1994) The black death. Manchester University Press, Manchester
Huang J, Walker ED, Otienoburu PE, Amimo F, Vulule J, Miller JR (2006) Laboratory tests of oviposition by the african malaria mosquito, Anopheles gambiae, on dark soil as influenced by presence or absence of vegetation. Malaria J 5:88
Kelly J (2005) The great mortality: an intimate history of the black death, the most devastating plague of all time. Harper Collins, New York
Kermack WO, McKendrick AG (1932) Contributions to the mathematical theory of epidemics. II. The problem of endemicity. Proc R Soc Lond A 138:55–83
Kermack WO, McKendrick AG (1933) Contributions to the mathematical theory of epidemics. III. Further studies of the problem of endemicity. Proc R Soc Lond A 141:94–122
Khan K, Arino J, Hu W, Raposo J, Sears J, Calderon F, Heidebrecht C, Macdonald M, Liauw J, Chan A, Gardam M (2009) Spread of a novel Influenza A (H1N1) virus via global airline transportation. N Engl J Med 361:212–214
L’Épine GJ (1764) Rapport de six des douze commissaires (contre l’inoculation). Quillau, Paris, pp 40–43
La peste humaine en 1997 (1999) Relevé Epidémiologique Hebdomadaire OMS 74, 340–344
Lambert JH (1772) Beiträge zum Gebrauche der Mathematik und deren Anwendung”. Dritter Theil, Berlin
Leslie PH (1945) On the use of matrices in certain population mathematics. Biometrika 33:183–212
Mac Donald G (1957) The epidemiology and control of malaria. Oxford University Press, London
Maupertuis PL Moreau de (1745, reed. 1965) Vénus physique, in Oeuvres. Georg Olms, Hildesheim
May N (1770) Impartial remarks on the Suttonian method of inoculation. Tilley, Wheble & Brown, London
May RM, Anderson RM (1984) Spatial heterogeneity and the design of immunization programs. Math Biosci 72:83–111
McKendrick AG (1925) Applications of mathematics to medical problems. Proc Edinburgh Math Soc 44:1–34
Mendez V, Fort J, Rotstein HG, Fedotov S (2003) Speed of reaction-diffusion fronts in spatially heterogeneous media. Phys Rev E 68:041105
Mischlewski A (1995) Un ordre hospitalier au Moyen Âge. Les chanoines réguliers de Saint-Antoine en Viennois. PUG, Grenoble
Mocellin G, Experton I (1992) Saint Antoine et l’Ordre des Antonins aux XVIIème et XVIIIème siècles. Ed. Musée, St-Antoine l’Abbaye
Mocellin-Spicuzza G (2002) Chroniques d’une abbaye au Moyen-Age. Guérir l’âme et le corps. Ed. Musée, Saint-Antoine l’Abbaye
Murray JA (1763) Fata variolarum insitionis in Suecia. Doctoral dissertation University Göttingen, Göttingen
Murray JD (2002) Mathematical biology I & II. Springer, Berlin
Ndiaye PI, Bicout DJ, Mondet B, Sabatier P (2006) Rainfall triggered dynamics of Aedes mosquito aggressiveness. J Theor Biol 243:222–229
Neerinckx SB, Peterson AT, Gulinck H, Deckers J, Leirs H (2008) Geographic distribution and ecological niche of plague in sub-Saharan Africa. Int J Health Geogr 7:54
Orent W (2001) Will the black death return? Discover 22:1–10
Orent W (2004) Plague: the mysterious past and mystifying furture of the world’s most dangerous disease. Free Press, New York
Porphyre T, Bicout DJ, Sabatier P (2004) Modelling the abundance of mosquito vectors versus flooding dynamics. Ecol Modell 183:173–181
Porte M (1994) Passion des formes. A René Thom. ENS Editions, Paris
Prentice MB, Rahalison L (2007) Plague. Lancet 369:1196–1207
Raoult D, Aboudharam G, Crubézy E, Larrouy G, Ludes B, Drancourt M (2000) Molecular identification of ‘Suicide PCR’ of Yersinia pestis as the agent of medieval black death. Proc Natl Acad Sci USA 97:12800–12803
Renouard Y (1948) Conséquences et intérêt démographiques de la Peste noire de 1348. Population 3:459–466
Ross R (1916) An application of the theory of probabilities to the study of a priori pathometry. Part I. Proc R Soc Lond A 92:204–230
Russell JC (1948) British medieval population. Univ. of New Mexico, Albuquerque
Russell JC (1972) Population in Europe. In: Cipolla CM (ed) The Fontana economic history of Europe, vol. I, the middle ages. Collins/Fontana, Glasgow, pp 25–71
Scott S, Duncan C (2004) Return of the black death: the world’s greatest serial killer West Sussex. Wiley, New York
Stenseth NC, Atshabar BB, Begon M, Belmain SR, Bertherat E, Carniel E, Gage KL, Leirs H, Rahalison L (2008) Plague: past, present, and future. PLOS Med 5:e3
Thom R (1972) Stabilité structurelle et Morphogenèse. Benjamin, New York
Thuderoz F, Simonet MA, Hansen O, Dariz A, Baum TP, Hierle V, Demongeot J, Marche PN, Jouvin-Marche E (2010) From the TCRAD rearrangement quantification to the computational simulation of the locus behavior. PloS Comp Biol 6:e1000682
Trembley J (1796) Recherches sur la mortalité de la petite vérole. Mém Acad Roy Sci, Paris
Usher MB (1969) A matrix model for forest management. Biometrics 25:309–315
Villani G (2001) Florence, 1336–38, the chronicle of Giovanni Villani. In: Medieval trade in the mediterranean world, Lopez, R.S. & I.W. Raymond Trads., Columbia Un. Press, New York
Wheelis M (2002) Biological warfare at the 1346 Siege of Caffa. Emerg Infect Dis 8:971–975
WHO (1999) La peste humaine en 1997. Relevé Epidémiologique Hebdomadaire OMS 74:340–344
Wood JW (2003) The temporal dynamics of the fourteenth-century black death: new evidence from english ecclesiastical records. Hum Biol 75:427–448
Zeeman EC (1993) Controversy in science: on the ideas of Daniel Bernoulli and René Thom. Nieuw Arch Wisk 11:257
Zhang SY, Yu L, Daszak P (2008) EcoHealth and the black death in the year of the rat. Ecohealth 5:99–100
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We are indebted to L. Forest for his invaluable contribution to models, simulations and for his unforgettable friendship.
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Gaudart, J., Ghassani, M., Mintsa, J. et al. Demography and Diffusion in Epidemics: Malaria and Black Death Spread. Acta Biotheor 58, 277–305 (2010). https://doi.org/10.1007/s10441-010-9103-z
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DOI: https://doi.org/10.1007/s10441-010-9103-z