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2012, 11(1): 61-82. doi: 10.3934/cpaa.2012.11.61

Demography in epidemics modelling

1. 

AGIM Laboratory, FRE 3405 UJF-CNRS, TIMB Team, University J. Fourier of Grenoble (UJF), Faculty of Medicine, 38700 La Tronche, France

2. 

LERTIM, EA 3283, Aix-Marseille University, Faculty of Medicine, 27 Bd Jean Moulin, 13385 Marseille Cedex 5, France

3. 

AGIM Laboratory, FRE 3405 CNRS, TIMB Team, University P. Mendès-France of Grenoble (UPMF), Faculty of Medicine, 38700 La Tronche, France, France

Received  February 2010 Revised  January 2011 Published  September 2011

Classical models of epidemics by Ross and McKendrick have to be revisited in order to take into account the demography (fecundity, mortality and migration) both of host and vector populations and also the diffusion and mutation of infectious agents. The classical models are supposing the populations involved in the infectious disease to be constant during the epidemic wave, but the presently observed pandemics show that the duration of their spread during months or years imposes to take into account the host and vector population changes, and also the transient or permanent migration and diffusion of hosts (susceptible or infected), as well as those of vectors and infectious agents. One example is presented concerning the malaria in Mali.
Citation: Jacques Demongeot, Jean Gaudart, Julie Mintsa, Mustapha Rachdi. Demography in epidemics modelling. Communications on Pure & Applied Analysis, 2012, 11 (1) : 61-82. doi: 10.3934/cpaa.2012.11.61
References:
[1]

L. Abbas, J. Demongeot and N. Glade, Synchrony in reaction-diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves,, Phil. Trans. Royal Soc. A., 367 (2009), 4829.

[2]

J. d'Alembert, Opuscules Mathématiques,, David, 2 (1761), 35.

[3]

T. Balenghien, F. Fouque, P. Sabatier and D. J. Bicout, Horse-, bird-, and human-seeking behavior and seasonal abundance of mosquitoes in a West Nile virus focus of southern France,, J. Med. Entomol., 43 (2006), 936.

[4]

S. Barry and N. Gualde, The biggest epidemics of history,, L'Histoire, 310 (2006), 38.

[5]

T. P. Baum, N. Pasqual, F. Thuderoz, V. Hierle, D. Chaume, M. P. Lefranc, E. Jouvin-Marche, P. Marche and J. Demongeot, IMGT/GeneInfo: enhancing V(D)J recombination database accessibility,, Nucleic Acids Res., 32 (2004), 51.

[6]

O. J. Benedictow, "The Black Death 1346-1353: The Complete History,", Boydell Press, (2004).

[7]

D. Bernoulli, it Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des advantages de l'inoculation pour la prévenir,, M\'em. Acad. Roy. Sci., (1760).

[8]

D. J. Bicout, K. Chalvet-Monfray and P. Sabatier, Infection persistence time of Aedes breeding habitats,, Physica A: Statistical Mechanics and its Applications, 305 (2002), 597.

[9]

D. J. Bicout and P. Sabatier, Mapping rift valley fever vectors and prevalence using rainfall variations,, Vector-Borne and Zoonotic Diseases, 4 (2004), 33.

[10]

N. Brouhns and M. Denuit, Risque de longévité et rente viagère,, Institut de Statistique Universit\'e Catholique, (0137).

[11]

L. Demetrius, Relations between demographic parameters,, Demography, 16 (1979), 329.

[12]

L. Demetrius and J. Demongeot, A thermodynamic approach in the modelling of the cellular cycle,, Biometrics, 40 (1984).

[13]

J. Demongeot, Coupling of Markov processes and Holley's inequalities for Gibbs measures,, in Proc. IXth Prague Conference on Information Theory, (1983), 183.

[14]

J. Demongeot and J. Fricot, Random fields and renewal potentials,, NATO ASI Serie F, 20 (1986), 71.

[15]

J. Demongeot and L. Demetrius, La dérive démographique et la sélection naturelle: Etude empirique de la France (1850-1965),, Population, 2 (1989), 231.

[16]

J. Demongeot, D. Benaouda and C. Jezequel, Dynamical confinement in neural networks and cell cycle,, Chaos, 5 (1995), 167.

[17]

J. Demongeot and J. Waku, Counter-examples for the population size growth in demography,, Math. Pop. Studies, 12 (2005), 199.

[18]

J. Demongeot, N. Glade and L. Forest, Liénard systems and potential-Hamiltonian decomposition. I Methodology,, Comptes Rendus Math\'ematique, 344 (2007), 121.

[19]

J. Demongeot, N. Glade and L. Forest, Liénard systems and potential-Hamiltonian decomposition. II Algorithm,, Comptes Rendus Math\'ematique, 344 (2007), 191.

[20]

J. Demongeot, N. Glade, A. Moreira and L. Vial, RNA relics and origin of life,, Int. J. Molecular Sciences, 10 (2009), 3420.

[21]

J. Demongeot, Biological boundaries and biological age,, Acta Biotheoretica, 57 (): 397.

[22]

J. Demongeot, J. Gaudart, A. Lontos, J. Mintsa, E. Promayon and M. Rachdi, Least diffusion zones in morphogenesis and epidemiology,, Int. J. Bifurcation and Chaos, ().

[23]

J. M. O. Depinay, C. M. Mbogo, G. Killeen, B. Knols, J. Beier, J., Carlson, J. Dusho, P. Billingsley, H. Mwambi, J. Githure, A. M. Toure and F. E. McKenzie, A simulation model of African Anopheles ecology and population dynamics for the analysis of malaria transmission,, Malaria J., 3 (2004).

[24]

K. Dietz and J. A. P. Heesterbeek, Daniel Bernoulli抯 epidemiological model revisited,, Math. Biosci., 180 (2002).

[25]

K. Dietz and J. A. P. Heesterbeek, Bernoulli was ahead of modern epidemiology,, Nature, 408 (2000).

[26]

O. K. Doumbo, It takes a village: medical research and ethics in Mali,, Science, 307 (2005), 679.

[27]

J Dutertre, étude d'un modèle épidémiologique appliqué au paludisme,, Ann. Soc. Belge. M\'ed. Trop., 56 (1976), 127.

[28]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 353.

[29]

L. Forest, N. Glade and J. Demongeot, Liénard systems and potential-Hamiltonian decomposition. Applications,, C. R. Acad. Sci. Biologies, 330 (2007), 97.

[30]

J. Gaudart, R. Giorgi, B. Poudiougou, S. Ranque, O. Doumbo and J. Demongeot, Spatial cluster detection: principle and application of different general methods,, Rev. Epidemiol. Sant\'e Publique, 55 (2007), 297.

[31]

J. Gaudart, O. Touré, N. Dessay, A. L. Dicko, S. Ranque, L. Forest, J. Demongeot and O. K. Doumbo, Modelling malaria incidence with environmental dependency in a locality of Sudanese savannah area, Mali,, Malar. J., 8 (2009).

[32]

J. Gaudart, M. Ghassani, J. Mintsa, M. Rachdi, J. Waku and J. Demongeot, Demography and diffusion in epidemics: malaria and black death spread, , Acta Biotheoretica, 58 (2010), 2.

[33]

J. Gaudart, M. J. Ghassani, J. Mintsa, J. Waku, M. Rachdi, O. K. Doumbo and J. Demongeot, 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,, IEEE AINA' 10 & BLSMC' 10, (2010).

[34]

N. Glade, L. Forest and J. Demongeot, Liénard systems and potential-Hamiltonian decomposition. III Applications in biology,, Comptes Rendus Math\'ematique, 344 (2007), 253.

[35]

M. Horie, T. Honda, Y. Suzuki, Y. Kobayashi, T. Daito, T. Oshida, K. Ikuta, P. Jern, T. Gojobori, J. M. Coffin and K. Tomonaga, Endogenous non-retroviral RNA virus elements in mammalian genomes,, Nature, 463 (2010), 84.

[36]

J. Huang, E. D. Walker, P. E. Otienoburu, F. Amimo, J. Vulule and J. R. Miller, 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 (2006).

[37]

A. Kaddar, Stability analysis in a delayed SIR epidemic model with a saturated incidence rate,, Nonlinear Analysis: Modelling and Control, 15 (2010), 299.

[38]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity,, Proceedings of the Royal Society of London Series A, 138 (1932), 55.

[39]

W. O. Kermack and A. G. McKendrick, Contributions to the Mathematical Theory of Epidemics. III. Further studies of the problem of endemicity,, Proceedings of the Royal Society of London Series A, 141 (1933), 94.

[40]

K. Khan, J. Arino, W. Hu, J. Raposo, J. Sears, F. Calderon, C. Heidebrecht, M. Macdonald, J. Liauw, A. Chan and M. Gardam, Spread of a novel Influenza A (H1N1) virus via global airline transportation,, New England Journal of Medicine, 361 (2009), 212.

[41]

J. H. Lambert, "Beitrge zum Gebrauche der Mathematik und deren Anwendung,", Dritter Theil, (1772).

[42]

G. J. Lépine, Rapport de six des douze commissaires (contre linoculation),, Paris, (1764), 40.

[43]

P. H. Leslie, On the use of matrices in certain population mathematics,, Biometrika, 33 (1945), 183.

[44]

G. Mac Donald, "The Epidemiology and Control of Malaria,", Oxford University Press, (1957).

[45]

G. C. de Magny, C. Paroissin, B. Cazelles, M. de Lara, J. F. Delmas and J. F. Guégan, Modeling environmental impacts of plankton reservoirs on cholera population dynamics,, ESAIM, 14 (2005), 156.

[46]

P. L. (Moreau de) Maupertuis, "Vénus physique,", Oeuvres, (1745).

[47]

N. May, "Impartial Remarks on the Suttonian Method of Inoculation,", London, (1770).

[48]

R. M. May and R. M. Anderson, Spatial heterogeneity and the design of immunization programs,, Math. Biosci., 72 (1984).

[49]

V. Mendez, J. Fort, H. G. Rotstein and S. Fedotov, Speed of reaction-diffusion fronts in spatially heterogeneous media,, Phys. Rev. E, 68 (2003).

[50]

A. G. McKendrick, Applications of mathematics to medical problems,, Proc. Edinburgh Mathematical Society, 44 (1925), 1.

[51]

J. A. Murray, "Fata Variolarum Insitionis in Suecia,", G\, (1763).

[52]

J. D. Murray, "Mathematical Biology I & II,", Springer, (2002).

[53]

P. I. Ndiaye, D. J. Bicout, B. Mondet and P. Sabatier, Rainfall triggered dynamics of Aedes mosquito aggressiveness,, J. Theor. Biol., 243 (2006), 222.

[54]

T. Porphyre, D. J. Bicout and P Sabatier, Modelling the abundance of mosquito vectors versus flooding dynamics,, Ecological modelling, 183 (2004), 173.

[55]

M. Porte, "Passion des formes. A René Thom,", ENS Editions, (1994).

[56]

V. Rialle, F. Duchêne, N. Noury, L. Bajolle and J. Demongeot, Health 'smart' home: information technology for patients at home,, Telemedicine Journal and E-Health, 8 (2002), 395.

[57]

R. Ross, An application of the theory of probabilities to the study of a priori pathometry. Part I,, Proceedings of the Royal Society of London Series A, 92 (1916), 204.

[58]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,, Dynamics of Continuous, 10 (2003), 863.

[59]

F. Teymoori, O. Hansen, A. Franco and J. Demongeot, Dynamic projection of old aged disability in Iran: DOPAMID microsimulation,, IEEE ARES-CISIS' 10, (2010).

[60]

R. Thom, "Stabilité structurelle et Morphogenèse,", Benjamin, (1972).

[61]

F. Thuderoz, M. A. Simonet, O. Hansen, A. Dariz, T. P. Baum, V. Hierle, J. Demongeot, P. N., Marche and E. Jouvin-Marche, From the TCRAD rearrangement quantification to the computational simulation of the locus behavior,, PloS Comp. Biol., 6 (2010).

[62]

J. Trembley, "Recherches sur la mortalité de la petite vérole,", M\'em. Acad. Roy. Sci., (1796).

[63]

M. B. Usher, A matrix model for forest management,, Biometrics, 25 (1969), 309.

[64]

E. C. Zeeman, Controversy in science: on the ideas of Daniel Bernoulli and René Thom,, Nieuw Arch. Wisk., 1 (1993), 257.

show all references

References:
[1]

L. Abbas, J. Demongeot and N. Glade, Synchrony in reaction-diffusion models of morphogenesis: applications to curvature-dependent proliferation and zero-diffusion front waves,, Phil. Trans. Royal Soc. A., 367 (2009), 4829.

[2]

J. d'Alembert, Opuscules Mathématiques,, David, 2 (1761), 35.

[3]

T. Balenghien, F. Fouque, P. Sabatier and D. J. Bicout, Horse-, bird-, and human-seeking behavior and seasonal abundance of mosquitoes in a West Nile virus focus of southern France,, J. Med. Entomol., 43 (2006), 936.

[4]

S. Barry and N. Gualde, The biggest epidemics of history,, L'Histoire, 310 (2006), 38.

[5]

T. P. Baum, N. Pasqual, F. Thuderoz, V. Hierle, D. Chaume, M. P. Lefranc, E. Jouvin-Marche, P. Marche and J. Demongeot, IMGT/GeneInfo: enhancing V(D)J recombination database accessibility,, Nucleic Acids Res., 32 (2004), 51.

[6]

O. J. Benedictow, "The Black Death 1346-1353: The Complete History,", Boydell Press, (2004).

[7]

D. Bernoulli, it Essai d'une nouvelle analyse de la mortalité causée par la petite vérole, et des advantages de l'inoculation pour la prévenir,, M\'em. Acad. Roy. Sci., (1760).

[8]

D. J. Bicout, K. Chalvet-Monfray and P. Sabatier, Infection persistence time of Aedes breeding habitats,, Physica A: Statistical Mechanics and its Applications, 305 (2002), 597.

[9]

D. J. Bicout and P. Sabatier, Mapping rift valley fever vectors and prevalence using rainfall variations,, Vector-Borne and Zoonotic Diseases, 4 (2004), 33.

[10]

N. Brouhns and M. Denuit, Risque de longévité et rente viagère,, Institut de Statistique Universit\'e Catholique, (0137).

[11]

L. Demetrius, Relations between demographic parameters,, Demography, 16 (1979), 329.

[12]

L. Demetrius and J. Demongeot, A thermodynamic approach in the modelling of the cellular cycle,, Biometrics, 40 (1984).

[13]

J. Demongeot, Coupling of Markov processes and Holley's inequalities for Gibbs measures,, in Proc. IXth Prague Conference on Information Theory, (1983), 183.

[14]

J. Demongeot and J. Fricot, Random fields and renewal potentials,, NATO ASI Serie F, 20 (1986), 71.

[15]

J. Demongeot and L. Demetrius, La dérive démographique et la sélection naturelle: Etude empirique de la France (1850-1965),, Population, 2 (1989), 231.

[16]

J. Demongeot, D. Benaouda and C. Jezequel, Dynamical confinement in neural networks and cell cycle,, Chaos, 5 (1995), 167.

[17]

J. Demongeot and J. Waku, Counter-examples for the population size growth in demography,, Math. Pop. Studies, 12 (2005), 199.

[18]

J. Demongeot, N. Glade and L. Forest, Liénard systems and potential-Hamiltonian decomposition. I Methodology,, Comptes Rendus Math\'ematique, 344 (2007), 121.

[19]

J. Demongeot, N. Glade and L. Forest, Liénard systems and potential-Hamiltonian decomposition. II Algorithm,, Comptes Rendus Math\'ematique, 344 (2007), 191.

[20]

J. Demongeot, N. Glade, A. Moreira and L. Vial, RNA relics and origin of life,, Int. J. Molecular Sciences, 10 (2009), 3420.

[21]

J. Demongeot, Biological boundaries and biological age,, Acta Biotheoretica, 57 (): 397.

[22]

J. Demongeot, J. Gaudart, A. Lontos, J. Mintsa, E. Promayon and M. Rachdi, Least diffusion zones in morphogenesis and epidemiology,, Int. J. Bifurcation and Chaos, ().

[23]

J. M. O. Depinay, C. M. Mbogo, G. Killeen, B. Knols, J. Beier, J., Carlson, J. Dusho, P. Billingsley, H. Mwambi, J. Githure, A. M. Toure and F. E. McKenzie, A simulation model of African Anopheles ecology and population dynamics for the analysis of malaria transmission,, Malaria J., 3 (2004).

[24]

K. Dietz and J. A. P. Heesterbeek, Daniel Bernoulli抯 epidemiological model revisited,, Math. Biosci., 180 (2002).

[25]

K. Dietz and J. A. P. Heesterbeek, Bernoulli was ahead of modern epidemiology,, Nature, 408 (2000).

[26]

O. K. Doumbo, It takes a village: medical research and ethics in Mali,, Science, 307 (2005), 679.

[27]

J Dutertre, étude d'un modèle épidémiologique appliqué au paludisme,, Ann. Soc. Belge. M\'ed. Trop., 56 (1976), 127.

[28]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. Eugenics, 7 (1937), 353.

[29]

L. Forest, N. Glade and J. Demongeot, Liénard systems and potential-Hamiltonian decomposition. Applications,, C. R. Acad. Sci. Biologies, 330 (2007), 97.

[30]

J. Gaudart, R. Giorgi, B. Poudiougou, S. Ranque, O. Doumbo and J. Demongeot, Spatial cluster detection: principle and application of different general methods,, Rev. Epidemiol. Sant\'e Publique, 55 (2007), 297.

[31]

J. Gaudart, O. Touré, N. Dessay, A. L. Dicko, S. Ranque, L. Forest, J. Demongeot and O. K. Doumbo, Modelling malaria incidence with environmental dependency in a locality of Sudanese savannah area, Mali,, Malar. J., 8 (2009).

[32]

J. Gaudart, M. Ghassani, J. Mintsa, M. Rachdi, J. Waku and J. Demongeot, Demography and diffusion in epidemics: malaria and black death spread, , Acta Biotheoretica, 58 (2010), 2.

[33]

J. Gaudart, M. J. Ghassani, J. Mintsa, J. Waku, M. Rachdi, O. K. Doumbo and J. Demongeot, 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,, IEEE AINA' 10 & BLSMC' 10, (2010).

[34]

N. Glade, L. Forest and J. Demongeot, Liénard systems and potential-Hamiltonian decomposition. III Applications in biology,, Comptes Rendus Math\'ematique, 344 (2007), 253.

[35]

M. Horie, T. Honda, Y. Suzuki, Y. Kobayashi, T. Daito, T. Oshida, K. Ikuta, P. Jern, T. Gojobori, J. M. Coffin and K. Tomonaga, Endogenous non-retroviral RNA virus elements in mammalian genomes,, Nature, 463 (2010), 84.

[36]

J. Huang, E. D. Walker, P. E. Otienoburu, F. Amimo, J. Vulule and J. R. Miller, 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 (2006).

[37]

A. Kaddar, Stability analysis in a delayed SIR epidemic model with a saturated incidence rate,, Nonlinear Analysis: Modelling and Control, 15 (2010), 299.

[38]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics. II. The problem of endemicity,, Proceedings of the Royal Society of London Series A, 138 (1932), 55.

[39]

W. O. Kermack and A. G. McKendrick, Contributions to the Mathematical Theory of Epidemics. III. Further studies of the problem of endemicity,, Proceedings of the Royal Society of London Series A, 141 (1933), 94.

[40]

K. Khan, J. Arino, W. Hu, J. Raposo, J. Sears, F. Calderon, C. Heidebrecht, M. Macdonald, J. Liauw, A. Chan and M. Gardam, Spread of a novel Influenza A (H1N1) virus via global airline transportation,, New England Journal of Medicine, 361 (2009), 212.

[41]

J. H. Lambert, "Beitrge zum Gebrauche der Mathematik und deren Anwendung,", Dritter Theil, (1772).

[42]

G. J. Lépine, Rapport de six des douze commissaires (contre linoculation),, Paris, (1764), 40.

[43]

P. H. Leslie, On the use of matrices in certain population mathematics,, Biometrika, 33 (1945), 183.

[44]

G. Mac Donald, "The Epidemiology and Control of Malaria,", Oxford University Press, (1957).

[45]

G. C. de Magny, C. Paroissin, B. Cazelles, M. de Lara, J. F. Delmas and J. F. Guégan, Modeling environmental impacts of plankton reservoirs on cholera population dynamics,, ESAIM, 14 (2005), 156.

[46]

P. L. (Moreau de) Maupertuis, "Vénus physique,", Oeuvres, (1745).

[47]

N. May, "Impartial Remarks on the Suttonian Method of Inoculation,", London, (1770).

[48]

R. M. May and R. M. Anderson, Spatial heterogeneity and the design of immunization programs,, Math. Biosci., 72 (1984).

[49]

V. Mendez, J. Fort, H. G. Rotstein and S. Fedotov, Speed of reaction-diffusion fronts in spatially heterogeneous media,, Phys. Rev. E, 68 (2003).

[50]

A. G. McKendrick, Applications of mathematics to medical problems,, Proc. Edinburgh Mathematical Society, 44 (1925), 1.

[51]

J. A. Murray, "Fata Variolarum Insitionis in Suecia,", G\, (1763).

[52]

J. D. Murray, "Mathematical Biology I & II,", Springer, (2002).

[53]

P. I. Ndiaye, D. J. Bicout, B. Mondet and P. Sabatier, Rainfall triggered dynamics of Aedes mosquito aggressiveness,, J. Theor. Biol., 243 (2006), 222.

[54]

T. Porphyre, D. J. Bicout and P Sabatier, Modelling the abundance of mosquito vectors versus flooding dynamics,, Ecological modelling, 183 (2004), 173.

[55]

M. Porte, "Passion des formes. A René Thom,", ENS Editions, (1994).

[56]

V. Rialle, F. Duchêne, N. Noury, L. Bajolle and J. Demongeot, Health 'smart' home: information technology for patients at home,, Telemedicine Journal and E-Health, 8 (2002), 395.

[57]

R. Ross, An application of the theory of probabilities to the study of a priori pathometry. Part I,, Proceedings of the Royal Society of London Series A, 92 (1916), 204.

[58]

S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays,, Dynamics of Continuous, 10 (2003), 863.

[59]

F. Teymoori, O. Hansen, A. Franco and J. Demongeot, Dynamic projection of old aged disability in Iran: DOPAMID microsimulation,, IEEE ARES-CISIS' 10, (2010).

[60]

R. Thom, "Stabilité structurelle et Morphogenèse,", Benjamin, (1972).

[61]

F. Thuderoz, M. A. Simonet, O. Hansen, A. Dariz, T. P. Baum, V. Hierle, J. Demongeot, P. N., Marche and E. Jouvin-Marche, From the TCRAD rearrangement quantification to the computational simulation of the locus behavior,, PloS Comp. Biol., 6 (2010).

[62]

J. Trembley, "Recherches sur la mortalité de la petite vérole,", M\'em. Acad. Roy. Sci., (1796).

[63]

M. B. Usher, A matrix model for forest management,, Biometrics, 25 (1969), 309.

[64]

E. C. Zeeman, Controversy in science: on the ideas of Daniel Bernoulli and René Thom,, Nieuw Arch. Wisk., 1 (1993), 257.

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