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January  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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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, (). Google Scholar

[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). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[37]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[46]

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

[47]

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

[48]

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

[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). Google Scholar

[50]

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

[51]

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

[52]

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

[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. Google Scholar

[54]

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

[55]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[60]

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

[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). Google Scholar

[62]

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

[63]

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

[64]

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

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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[6]

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

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[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, (). Google Scholar

[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). Google Scholar

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[37]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[41]

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

[42]

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

[43]

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

[44]

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

[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. Google Scholar

[46]

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

[47]

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

[48]

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

[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). Google Scholar

[50]

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

[51]

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

[52]

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

[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. Google Scholar

[54]

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

[55]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[60]

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

[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). Google Scholar

[62]

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

[63]

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

[64]

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

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