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Mathematical analysis and dynamic active subspaces for a long term model of HIV
Effect of the epidemiological heterogeneity on the outbreak outcomes
Biomathematics and Epidemiology, EPSP -TIMC, UMR 5525 CNRS, Grenoble Alpes University, VetAgro Sup Lyon, 1 avenue Bourgelat -69280 Marcy l'Etoile, France |
Multi-host pathogens infect and are transmitted by different kinds of hosts and, therefore, the host heterogeneity may have a great impact on the outbreak outcome of the system. This paper deals with the following problem: consider the system of interacting and mixed populations of hosts epidemiologically different, what would be the outbreak outcome for each host population composing the system as a result of mixing in comparison to the situation with zero mixing? To address this issue we have characterized the epidemic response function for a single-host population and defined a heterogeneity index measuring how host systems are epidemiologically different in terms of generation time, basic reproduction number $R_0$ and, therefore, epidemic response function. Based on the individual epidemiological characteristics of populations, with heterogeneities and mixing affinities, the response of subpopulations in a multi-host system is compared to that of a single-host system. The case of a two-host system, in which the infection transmission depends solely on the infection susceptibility of the receiver, is analyzed in detail. Three types of responses are observed: dilution, amplification or no effect, corresponding to lower, higher or equal attack rates, respectively, for a host population in an interacting multi-host system compared to the zero-mixing situation. We find that no effect is generally observed for zero heterogeneity. A dilution effect is always observed for all the host populations when their individual $R_{0,i} <1$. Whereas, when at least one of the individual $R_{0,i}>1$, then the hosts "$i$" with $R_{0,i}>R_{0,j}$ undergo a dilution effect while the hosts "$j$" undergo an amplification effect.
References:
[1] |
F. R. Adler,
The effects of averaging on the basic reproduction ratio, Mathematical Biosciences, 111 (1992), 89-98.
doi: 10.1016/0025-5564(92)90080-G. |
[2] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans/ Dynamics and Control, Oxford Science Publications, Oxford, 1991. Google Scholar |
[3] |
D. J. Bicout, Modélisation des Maladies Vectorielles, Habilitation á Diriger des Recherches -Université Joseph Fourier -Grenoble I, 2006. Google Scholar |
[4] |
J. D. Brown, D. E. Stallknecht and D. E. Swayne, Experimental infection of swans and geese with highly pathogenic avian influenza virus (H5N1) of asian lineage, Emerging Infectious Diseases, 14 (2008), 136-142. Google Scholar |
[5] |
J. D. Brown, D. E. Stallknecht, J. R. Beck, D. L. Suarez and D. E. Swayne, Susceptibility of North american ducks and gulls to (H5N1) highly pathogenic avian influenza viruses, Emerging Infectious Diseases, 12 (2006), 1663-1670. Google Scholar |
[6] |
H. Chen, Y. Li, Z. Li, J. Shi, K. Shinya, G. Deng, Q. Qi, G. Tian, S. Fan, H. Zhao, Y. Sun and Y. Kawaoka, Properties and Dissemination of H5N1 Viruses Isolated during an Influenza Outbreak in Migratory Waterfowl in Western China, Journal of Virology, 80 (2006), 5976-5983. Google Scholar |
[7] |
H. Chen, G. J. D. Smith, S. Y. Zhang, K. Qin, J. Wang, K. S. Li, R. G. Webster, J. S. M. Peiris and Y. Guan, H5N1 virus outbreak in migratory waterfowl, Nature, 436 (2005), 191-192. Google Scholar |
[8] |
M. de Jong, O. Diekmann and H. Heesterbeek, How does transmission of infection depend on population size, In Epidemic models: their structure and relation to data (eds. D. Mollison) Cambridge: Press Syndicate of the University of Cambridge, (1995), 84–94. Google Scholar |
[9] |
M. C. M. de Jong, O. Diekmann and J. A. P. Heesterbeek, The computation of R0 for discrete-time epidemic models with dynamic heterogeneity, Mathematical Biosciences, 119 (1994), 97-114. Google Scholar |
[10] |
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz,
On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[11] |
O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts,
The construction of next-generation matrices for compartmental epidemic models, Journal of the Royal Society Interface, 7 (2010), 873-885.
doi: 10.1098/rsif.2009.0386. |
[12] |
A. P. Dobson,
Population dynamics of pathogens with multiple host species, Am. Nat., 164 (2004), S64-S78.
doi: 10.1086/424681. |
[13] |
D. Doctrinal, S. Ruette, J. Hars, M. Artois and D. J. Bicout, Spatial and temporal analysis of the highly pathogenic avian influenza (H5N1) outbreak in the Dombes Area, France in 2006, Wildfowl, 2 (2009), 202-214. Google Scholar |
[14] |
J. Dushoff and S. Levin,
The effects of population heterogeneity on disease invasion, Mathematical Biosciences, 128 (1995), 25-40.
doi: 10.1016/0025-5564(94)00065-8. |
[15] |
P. L. Flint, Applying the scientific method when assessing the influence of migratory birds on the dispersal of H5N1, Virology Journal, 4 (2007), 132 (1-3). Google Scholar |
[16] |
L. Gall-Reculé, F. X. Briand, A. Schmitz, O. Guionie, P. Massin and V. Jestin, Double introduction of highly pathogenic H5N1 avian influenza virus into France in early 2006, Avian Pathology, 37 (2008), 15-23. Google Scholar |
[17] |
M. Gauthier-Clerc, C. Lebarbenchon and F. Thomas, Recent expansion of highly pathogenic avian influenza H5N1: a critical review, Ibis, 149 (2007), 202-214. Google Scholar |
[18] |
V. Guberti and S. H. Newman, Guidelines on Wild Bird Surveillance for Highly Pathogenic Avian Influenza H5N1 Virus, Journal of Wildlife Diseases, 43 (2007), S29-S34. Google Scholar |
[19] |
J. Hars, S. Ruette, M. Benmergui, C. Fouque, J. Y. Fournier, A. Legouge, M. Cherbonnel, D. Baroux, C. Dupuy and V. Jestin, The epidemiology of the highly pathogenic H5N1 avian influenza in Mute Swan (Cygnus olor) and other Anatidae in the Dombes region (France), 2006, J Wildlife Dis, 44 (2008), 811-823. Google Scholar |
[20] |
J. Hars, S. Ruette, M. Benmergui, C. Fouque, J. Y. Fournier, A. Legouge, M. Cherbonnel, D. Baroux, C. Dupuy and V. Jestin, Rôle Epidémiologique du Cygne Tuberculé et des Autres Anatidés Dans L'épisode D'influenza Aviaire H5N1 HP Dans la Dombes en 2006, ONCFS Rapport Scientifique, 2006. Google Scholar |
[21] |
J. A. P. Heesterbeek, Abrief history of R0 and a recipe for its calculation, Acta Biotheoretica, 50 (2002), 189-204. Google Scholar |
[22] |
D. Kalthoff, A. Breithaupt, J. P. Teifke, A. Globig, T. Harder, T. C. Mettenleiter and M. Beer, Highly pathogenic avian influenza virus (H5N1) in experimentally infected adult mute swans, Emerging Infectious Diseases, 14 (2008), 1267-1270. Google Scholar |
[23] |
J. Keawcharoen, D. van Riel, G. van Amerongen, T. Bestebroer, W. E. Beyer, R. van Lavieren, A. D. M. E. Osterhaus, R. A. M. Fouchier and T. Kuiken, Wild ducks as long-distance vectors of highly pathogenic avian influenza virus (H5N1), Emerging Infectious Diseases, 14 (2008), 600-607. Google Scholar |
[24] |
F. Keesing, R. D. Holt and R. S. Ostfeld,
Effects of species diversity on disease risk, Ecology letters, 9 (2006), 485-498.
doi: 10.1111/j.1461-0248.2006.00885.x. |
[25] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721. Google Scholar |
[26] |
H. Kida, R. Yanagawa and Y. Matsuoka, Duck influenza lacking evidence of disease signs and immune response, Infect. Immun, 30 (1980), 547-553. Google Scholar |
[27] |
A. M. Kilpatrick, A. A. Chmura, D.W. Gibbons, R. C. Fleischer, P. P. Marra and P. Daszak, Predicting the global spread of H5N1 avian influenza, Proc Natl Acad Sci USA, 103 (2006), 19368-19373. Google Scholar |
[28] |
J. Liu, H. Xiao, F. Lei, Q. Zhu, K. Qin, X. -w Zhang, X. -l. Zhang, D. Zhao, G. Wang, Y. Feng, J. Ma, W. Liu, J. Wang and G. F. Gao, Highly pathogenic H5N1 influenza virus infection in migratory birds, Science, 309 (2005), 1206. Google Scholar |
[29] |
H. Nishiura, B. Hoye, M. Klaassen, S. Bauer and H. Heesterbeek,
How to find natural reservoir hosts from endemic prevalence in a multi-host population: A case study of influenza in waterfowl, Epidemics, 1 (2009), 118-128.
doi: 10.1016/j.epidem.2009.04.002. |
[30] |
B. Olsen, V. J. Munster, A. Wallensten, J. Waldenström, A. D. M. E. Osterhaus and R. A. M. Fouchier,
Global Patterns of Influenza A Virus in Wild Birds, Science, 312 (2006), 384-388.
doi: 10.1126/science.1122438. |
[31] |
M. René and D. J. Bicout, Influenza aviaire: Modélisation du risque d'infection des oiseaux á partir d'étangs contaminés, Epidémiologie et santé animale, 51 (2007), 95-109. Google Scholar |
[32] |
A. Satelli, S. Tarantola and K. P.-S. Chan, Quantitative model-independent method for global sensitivity analysis of model output, Technometrics, 41 (1999), 39-56. Google Scholar |
[33] |
M. E. J. Woolhouse, L. H. Taylor and D. T. Haydon, Population biology of multi-host pathogens, Science, 292 (2001), 1109-1112. Google Scholar |
[34] |
G. Zhang, D. Shoham, S. Davydof, J. D. Castello, S. O. Rogers and D. Gilichinsky, Evidence of influenza A virus RNA in Siberian lake ice, Journal of Virology, 80 (2006), 12229-12235. Google Scholar |
show all references
References:
[1] |
F. R. Adler,
The effects of averaging on the basic reproduction ratio, Mathematical Biosciences, 111 (1992), 89-98.
doi: 10.1016/0025-5564(92)90080-G. |
[2] |
R. M. Anderson and R. M. May, Infectious Diseases of Humans/ Dynamics and Control, Oxford Science Publications, Oxford, 1991. Google Scholar |
[3] |
D. J. Bicout, Modélisation des Maladies Vectorielles, Habilitation á Diriger des Recherches -Université Joseph Fourier -Grenoble I, 2006. Google Scholar |
[4] |
J. D. Brown, D. E. Stallknecht and D. E. Swayne, Experimental infection of swans and geese with highly pathogenic avian influenza virus (H5N1) of asian lineage, Emerging Infectious Diseases, 14 (2008), 136-142. Google Scholar |
[5] |
J. D. Brown, D. E. Stallknecht, J. R. Beck, D. L. Suarez and D. E. Swayne, Susceptibility of North american ducks and gulls to (H5N1) highly pathogenic avian influenza viruses, Emerging Infectious Diseases, 12 (2006), 1663-1670. Google Scholar |
[6] |
H. Chen, Y. Li, Z. Li, J. Shi, K. Shinya, G. Deng, Q. Qi, G. Tian, S. Fan, H. Zhao, Y. Sun and Y. Kawaoka, Properties and Dissemination of H5N1 Viruses Isolated during an Influenza Outbreak in Migratory Waterfowl in Western China, Journal of Virology, 80 (2006), 5976-5983. Google Scholar |
[7] |
H. Chen, G. J. D. Smith, S. Y. Zhang, K. Qin, J. Wang, K. S. Li, R. G. Webster, J. S. M. Peiris and Y. Guan, H5N1 virus outbreak in migratory waterfowl, Nature, 436 (2005), 191-192. Google Scholar |
[8] |
M. de Jong, O. Diekmann and H. Heesterbeek, How does transmission of infection depend on population size, In Epidemic models: their structure and relation to data (eds. D. Mollison) Cambridge: Press Syndicate of the University of Cambridge, (1995), 84–94. Google Scholar |
[9] |
M. C. M. de Jong, O. Diekmann and J. A. P. Heesterbeek, The computation of R0 for discrete-time epidemic models with dynamic heterogeneity, Mathematical Biosciences, 119 (1994), 97-114. Google Scholar |
[10] |
O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz,
On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[11] |
O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts,
The construction of next-generation matrices for compartmental epidemic models, Journal of the Royal Society Interface, 7 (2010), 873-885.
doi: 10.1098/rsif.2009.0386. |
[12] |
A. P. Dobson,
Population dynamics of pathogens with multiple host species, Am. Nat., 164 (2004), S64-S78.
doi: 10.1086/424681. |
[13] |
D. Doctrinal, S. Ruette, J. Hars, M. Artois and D. J. Bicout, Spatial and temporal analysis of the highly pathogenic avian influenza (H5N1) outbreak in the Dombes Area, France in 2006, Wildfowl, 2 (2009), 202-214. Google Scholar |
[14] |
J. Dushoff and S. Levin,
The effects of population heterogeneity on disease invasion, Mathematical Biosciences, 128 (1995), 25-40.
doi: 10.1016/0025-5564(94)00065-8. |
[15] |
P. L. Flint, Applying the scientific method when assessing the influence of migratory birds on the dispersal of H5N1, Virology Journal, 4 (2007), 132 (1-3). Google Scholar |
[16] |
L. Gall-Reculé, F. X. Briand, A. Schmitz, O. Guionie, P. Massin and V. Jestin, Double introduction of highly pathogenic H5N1 avian influenza virus into France in early 2006, Avian Pathology, 37 (2008), 15-23. Google Scholar |
[17] |
M. Gauthier-Clerc, C. Lebarbenchon and F. Thomas, Recent expansion of highly pathogenic avian influenza H5N1: a critical review, Ibis, 149 (2007), 202-214. Google Scholar |
[18] |
V. Guberti and S. H. Newman, Guidelines on Wild Bird Surveillance for Highly Pathogenic Avian Influenza H5N1 Virus, Journal of Wildlife Diseases, 43 (2007), S29-S34. Google Scholar |
[19] |
J. Hars, S. Ruette, M. Benmergui, C. Fouque, J. Y. Fournier, A. Legouge, M. Cherbonnel, D. Baroux, C. Dupuy and V. Jestin, The epidemiology of the highly pathogenic H5N1 avian influenza in Mute Swan (Cygnus olor) and other Anatidae in the Dombes region (France), 2006, J Wildlife Dis, 44 (2008), 811-823. Google Scholar |
[20] |
J. Hars, S. Ruette, M. Benmergui, C. Fouque, J. Y. Fournier, A. Legouge, M. Cherbonnel, D. Baroux, C. Dupuy and V. Jestin, Rôle Epidémiologique du Cygne Tuberculé et des Autres Anatidés Dans L'épisode D'influenza Aviaire H5N1 HP Dans la Dombes en 2006, ONCFS Rapport Scientifique, 2006. Google Scholar |
[21] |
J. A. P. Heesterbeek, Abrief history of R0 and a recipe for its calculation, Acta Biotheoretica, 50 (2002), 189-204. Google Scholar |
[22] |
D. Kalthoff, A. Breithaupt, J. P. Teifke, A. Globig, T. Harder, T. C. Mettenleiter and M. Beer, Highly pathogenic avian influenza virus (H5N1) in experimentally infected adult mute swans, Emerging Infectious Diseases, 14 (2008), 1267-1270. Google Scholar |
[23] |
J. Keawcharoen, D. van Riel, G. van Amerongen, T. Bestebroer, W. E. Beyer, R. van Lavieren, A. D. M. E. Osterhaus, R. A. M. Fouchier and T. Kuiken, Wild ducks as long-distance vectors of highly pathogenic avian influenza virus (H5N1), Emerging Infectious Diseases, 14 (2008), 600-607. Google Scholar |
[24] |
F. Keesing, R. D. Holt and R. S. Ostfeld,
Effects of species diversity on disease risk, Ecology letters, 9 (2006), 485-498.
doi: 10.1111/j.1461-0248.2006.00885.x. |
[25] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. Roy. Soc. Lond. A, 115 (1927), 700-721. Google Scholar |
[26] |
H. Kida, R. Yanagawa and Y. Matsuoka, Duck influenza lacking evidence of disease signs and immune response, Infect. Immun, 30 (1980), 547-553. Google Scholar |
[27] |
A. M. Kilpatrick, A. A. Chmura, D.W. Gibbons, R. C. Fleischer, P. P. Marra and P. Daszak, Predicting the global spread of H5N1 avian influenza, Proc Natl Acad Sci USA, 103 (2006), 19368-19373. Google Scholar |
[28] |
J. Liu, H. Xiao, F. Lei, Q. Zhu, K. Qin, X. -w Zhang, X. -l. Zhang, D. Zhao, G. Wang, Y. Feng, J. Ma, W. Liu, J. Wang and G. F. Gao, Highly pathogenic H5N1 influenza virus infection in migratory birds, Science, 309 (2005), 1206. Google Scholar |
[29] |
H. Nishiura, B. Hoye, M. Klaassen, S. Bauer and H. Heesterbeek,
How to find natural reservoir hosts from endemic prevalence in a multi-host population: A case study of influenza in waterfowl, Epidemics, 1 (2009), 118-128.
doi: 10.1016/j.epidem.2009.04.002. |
[30] |
B. Olsen, V. J. Munster, A. Wallensten, J. Waldenström, A. D. M. E. Osterhaus and R. A. M. Fouchier,
Global Patterns of Influenza A Virus in Wild Birds, Science, 312 (2006), 384-388.
doi: 10.1126/science.1122438. |
[31] |
M. René and D. J. Bicout, Influenza aviaire: Modélisation du risque d'infection des oiseaux á partir d'étangs contaminés, Epidémiologie et santé animale, 51 (2007), 95-109. Google Scholar |
[32] |
A. Satelli, S. Tarantola and K. P.-S. Chan, Quantitative model-independent method for global sensitivity analysis of model output, Technometrics, 41 (1999), 39-56. Google Scholar |
[33] |
M. E. J. Woolhouse, L. H. Taylor and D. T. Haydon, Population biology of multi-host pathogens, Science, 292 (2001), 1109-1112. Google Scholar |
[34] |
G. Zhang, D. Shoham, S. Davydof, J. D. Castello, S. O. Rogers and D. Gilichinsky, Evidence of influenza A virus RNA in Siberian lake ice, Journal of Virology, 80 (2006), 12229-12235. Google Scholar |









The reduced equivalent reproduction number, η1, for host 1, and global reproductive number

Reduced equivalent reproduction numbers ηi (i = 1, 2) and global reproductive number

heterogeneity | outbreak response | |
host 1 | host 2 | |
dilution | dilution | |
amplification | dilution | |
no effect | no effect | |
dilution | dilution | |
no effect | amplification |
heterogeneity | outbreak response | |
host 1 | host 2 | |
dilution | dilution | |
amplification | dilution | |
no effect | no effect | |
dilution | dilution | |
no effect | amplification |
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