2011, 8(1): 199-222. doi: 10.3934/mbe.2011.8.199

Epidemic spread of influenza viruses: The impact of transient populations on disease dynamics

1. 

Department of Mathematical Sciences, University of Puerto Rico-Mayagüez, Mayagüez, PR 00686, United States

2. 

Department of Mathematical Sciences, Montclair State University, 1 Normal Avenue, Montclair, NJ 07043

3. 

Mathematical, Computational and Modeling Science Center, School of Mathematics and Statistic, Arizona State University, Tempe, AZ 85287, United States

Received  June 2010 Revised  September 2010 Published  January 2011

The recent H1N1 ("swine flu") pandemic and recent H5N1 ("avian flu") outbreaks have brought increased attention to the study of the role of animal populations as reservoirs for pathogens that could invade human populations. It is believed that pigs acquired flu strains from birds and humans, acting as a mixing vessel in generating new influenza viruses. Assessing the role of animal reservoirs, particularly reservoirs involving highly mobile populations (like migratory birds), on disease dispersal and persistence is of interests to a wide range of researchers including public health experts and evolutionary biologists. This paper studies the interactions between transient and resident bird populations and their role on dispersal and persistence. A metapopulation framework based on a system of nonlinear ordinary differential equations is used to study the transmission dynamics and control of avian diseases. Simplified versions of mathematical models involving a limited number of migratory and resident bird populations are analyzed. Epidemiological time scales and singular perturbation methods are used to reduce the dimensionality of the model. Our results show that mixing of bird populations (involving residents and migratory birds) play an important role on the patterns of disease spread.
Citation: Karen R. Ríos-Soto, Baojun Song, Carlos Castillo-Chavez. Epidemic spread of influenza viruses: The impact of transient populations on disease dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 199-222. doi: 10.3934/mbe.2011.8.199
References:
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S. Blythe, C. Castillo-Chavez and S. Palmer, Toward a unified theory of sexual mixing and pair formation,, Math. Biosci., 107 (1991), 379. doi: 10.1016/0025-5564(91)90015-B.

[2]

S. Blythe, S. Busenberg and C. Castillo-Chavez, Affinity in paired event probability,, Math. Biosci., 128 (1995), 265. doi: 10.1016/0025-5564(94)00075-B.

[3]

A. C. M. Boon, M. R. Sandbulte, P. Seiler, R. J. Webby, T. Songserm, Y. Guan and R. G. Webster, Role of terrestrial wild birds in ecology of influenza A virus (H5N1),, Emerg. Infect. Dis., 13 (2007), 1720.

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A. Bouma, I. Claassen, K. Naith, D. Klinkenberg, C. A. Donnelly, G. Koch and M. van Boven, Estimation of transmission parameters of H5N1 avian influenza virus in chickens,, PLoS Path., 5 (2009), 1. doi: 10.1371/journal.ppat.1000281.

[5]

S. Busenberg and C. Castillo-Chavez, A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS,, IMA J. Math. Appl. Med. Biol., 8 (1991), 1. doi: 10.1093/imammb/8.1.1.

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D. Butler, Doubts hang over source of bird flu spread,, Nature, 439 (2006).

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Center for Disease Control and Prevention, (2006a)., Key facts about avian influenza (bird flu) and avian influenza A (H5N1) virus,, http://www.cdc.gov/flu/avian/gen-info/facts.htm., ().

[8]

C. Castillo-Chavez, K. Cooke, W. Z. Huang and S. A. Levin, The role of long periods of infectiousness in the dynamics of acquired immunodeficiency syndrome (AIDS),, Mathematical approaches to problems in resource management and epidemiology (Ithaca, (1989), 177.

[9]

C. Castillo-Chavez, A. Capurro, Z. Zellner and J. X. Velasco-Hernandez, El transporte publico y la dinamica de la tuberculosis a nivel poblacional,, Aportaciones Matematicas, 22 (1998), 209.

[10]

C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $R_0$ and its role in global stability,, In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduciton, (2002), 229.

[11]

C. Castillo-Chavez, B. Song and J. Zhang, An epidemic model with virtual mass transportation: The case of smallpox in a large city,, In: Bioterrorism: Mathematical Modeling Applications in Homeland Security, (2003), 173.

[12]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosci. Eng., 1 (2004), 361.

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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. doi: 10.1038/nature03974.

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H. Chen et al., Establishment of multiple sublineages of H5N1 influenza virus in Asia: Implications for pandemic control,, PNAS, 103 (2006), 2845. doi: 10.1073/pnas.0511120103.

[17]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous population,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324.

[18]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosc., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

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M. Friend and C. J. Franson, Avian Influenza,, In: Field Manual of Wildlife Diseases: General Field Procedures and Diseases of Birds., (1999), 181.

[23]

S. Gourley, R. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay,, SIAM Journal on Applied Dynamical Systems, 9 (2010), 589. doi: 10.1137/090767261.

[24]

F. Hoppensteadt, Asymptotic stability in singular perturbation problems. II. Problems having matched asymptotic expansion solutions,, J. Diff. Eqns., 15 (1974), 510. doi: 10.1016/0022-0396(74)90070-9.

[25]

W. Z. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission,, SIAM J. Appl. Math., 52 (1992), 835. doi: 10.1137/0152047.

[26]

I. Iglesias, A. M. Perez., J. M. Sánchez-Vizcaíno, M. J. Muñoz, M. Martínez and A. De La Torre, Reproductive ratio for the local spread of highly pathogenic avian influenza in wild bird populations of Europe, 2005-2008,, Epidemiol. Infect., 14 (2010), 1.

[27]

J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analyzing HIV transmission: The effect of contact patters,, Math. Biosci., 92 (1988), 119. doi: 10.1016/0025-5564(88)90031-4.

[28]

H. K. Leong, C. S. Goh, S. T. Chew et al., Prevention and control of avian influenza in Singapore,, Ann. Acad. Med. Singap., 37 (2008), 504.

[29]

J. Liu et al., Highly pathogenic H5N1 influenza virus infection in migratory birds,, Science, 309 (2005). doi: 10.1126/science.1115273.

[30]

R. Liu, V. R. S. K. Duvvuri and J. Wu, Spread pattern formation of H5N1-avian influenza and its implications for control strategies,, Math. Model. Nat. Phenom., 3 (2008), 161. doi: 10.1051/mmnp:2008048.

[31]

D. Normile, Are wild birds to blame?,, Science, 310 (2005), 426. doi: 10.1126/science.310.5747.426.

[32]

D. Normile, Evidence points to migratory birds in H5N1 spread,, Science, 311 (2006). doi: 10.1126/science.311.5765.1225.

[33]

B. Olsen, V. J. Munster, A. Wallensten, J. Waldenstrom, A. D. M. E. Osterhaus and R. O. M. Fouchier, Global patterns of influenza A virus in wild birds,, Science, 312 (2006), 384. doi: 10.1126/science.1122438.

[34]

K. D. Redd, J. K. Meece, J. S. Henkel and S. K. Shukla, Birds, migration and emerging zoonoses: West Nile virus, lyme diseases, influenza A and enteropathongens,, Clinical Medicine and Research, 1 (2003), 5. doi: 10.3121/cmr.1.1.5.

[35]

K. R. Ríos-Soto, "Dispersal and Disease Dynamics in Populations with and without Demography,'', Ph.D. thesis, (2008).

[36]

C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations,, SIAM J. Appl. Math., 52 (1992), 541. doi: 10.1137/0152030.

[37]

B. Song, C. Castillo-Chavez and J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts,, Math. Biosci., 180 (2002), 187. doi: 10.1016/S0025-5564(02)00112-8.

[38]

World Health Organization (2005), Avian influenza frequently asked questions,, http://www.who.int/csr/disease/avian_influenza/avian_faqs/en/index.html., ().

show all references

References:
[1]

S. Blythe, C. Castillo-Chavez and S. Palmer, Toward a unified theory of sexual mixing and pair formation,, Math. Biosci., 107 (1991), 379. doi: 10.1016/0025-5564(91)90015-B.

[2]

S. Blythe, S. Busenberg and C. Castillo-Chavez, Affinity in paired event probability,, Math. Biosci., 128 (1995), 265. doi: 10.1016/0025-5564(94)00075-B.

[3]

A. C. M. Boon, M. R. Sandbulte, P. Seiler, R. J. Webby, T. Songserm, Y. Guan and R. G. Webster, Role of terrestrial wild birds in ecology of influenza A virus (H5N1),, Emerg. Infect. Dis., 13 (2007), 1720.

[4]

A. Bouma, I. Claassen, K. Naith, D. Klinkenberg, C. A. Donnelly, G. Koch and M. van Boven, Estimation of transmission parameters of H5N1 avian influenza virus in chickens,, PLoS Path., 5 (2009), 1. doi: 10.1371/journal.ppat.1000281.

[5]

S. Busenberg and C. Castillo-Chavez, A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS,, IMA J. Math. Appl. Med. Biol., 8 (1991), 1. doi: 10.1093/imammb/8.1.1.

[6]

D. Butler, Doubts hang over source of bird flu spread,, Nature, 439 (2006).

[7]

Center for Disease Control and Prevention, (2006a)., Key facts about avian influenza (bird flu) and avian influenza A (H5N1) virus,, http://www.cdc.gov/flu/avian/gen-info/facts.htm., ().

[8]

C. Castillo-Chavez, K. Cooke, W. Z. Huang and S. A. Levin, The role of long periods of infectiousness in the dynamics of acquired immunodeficiency syndrome (AIDS),, Mathematical approaches to problems in resource management and epidemiology (Ithaca, (1989), 177.

[9]

C. Castillo-Chavez, A. Capurro, Z. Zellner and J. X. Velasco-Hernandez, El transporte publico y la dinamica de la tuberculosis a nivel poblacional,, Aportaciones Matematicas, 22 (1998), 209.

[10]

C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $R_0$ and its role in global stability,, In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduciton, (2002), 229.

[11]

C. Castillo-Chavez, B. Song and J. Zhang, An epidemic model with virtual mass transportation: The case of smallpox in a large city,, In: Bioterrorism: Mathematical Modeling Applications in Homeland Security, (2003), 173.

[12]

C. Castillo-Chavez and B. Song, Dynamical models of tuberculosis and their applications,, Math. Biosci. Eng., 1 (2004), 361.

[13]

Center for Disease Control (Accessed 2009a), Key facts about Swine flu,, http://www.cdc.gov/h1n1flu/key_facts.htm., ().

[14]

Center for Disease Control (Accessed 2009b), Questions and answers: H1N1 Flu (Swine Flu) and you., http://www.cdc.gov/h1n1flu/qa.htm., ().

[15]

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. doi: 10.1038/nature03974.

[16]

H. Chen et al., Establishment of multiple sublineages of H5N1 influenza virus in Asia: Implications for pandemic control,, PNAS, 103 (2006), 2845. doi: 10.1073/pnas.0511120103.

[17]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous population,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324.

[18]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosc., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6.

[19]

Federation of American Scientists (Accessed 2009), 1918 influenza A (H1N1) fact sheet,, http://www.fas.org/programs/ssp/bio/factsheets/H1N1factsheet.html, ().

[20]

Food and Agricultural Organization of the United Nations (2005), FAO AIDE news special issue. Update on avian influenza situation,, (As of 12/11/2005) - Issue no. 36, (): 1.

[21]

Food and Agricultural Organization of the United Nations (Accessed 2006), Animal health special report, wild birds and avian influenza, 1-5., http://www.fao.org/ag/againfo/subjects/en/health/diseases-cards/avian_HPAIrisk.html., ().

[22]

M. Friend and C. J. Franson, Avian Influenza,, In: Field Manual of Wildlife Diseases: General Field Procedures and Diseases of Birds., (1999), 181.

[23]

S. Gourley, R. Liu and J. Wu, Spatiotemporal distributions of migratory birds: Patchy models with delay,, SIAM Journal on Applied Dynamical Systems, 9 (2010), 589. doi: 10.1137/090767261.

[24]

F. Hoppensteadt, Asymptotic stability in singular perturbation problems. II. Problems having matched asymptotic expansion solutions,, J. Diff. Eqns., 15 (1974), 510. doi: 10.1016/0022-0396(74)90070-9.

[25]

W. Z. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission,, SIAM J. Appl. Math., 52 (1992), 835. doi: 10.1137/0152047.

[26]

I. Iglesias, A. M. Perez., J. M. Sánchez-Vizcaíno, M. J. Muñoz, M. Martínez and A. De La Torre, Reproductive ratio for the local spread of highly pathogenic avian influenza in wild bird populations of Europe, 2005-2008,, Epidemiol. Infect., 14 (2010), 1.

[27]

J. A. Jacquez, C. P. Simon, J. Koopman, L. Sattenspiel and T. Perry, Modeling and analyzing HIV transmission: The effect of contact patters,, Math. Biosci., 92 (1988), 119. doi: 10.1016/0025-5564(88)90031-4.

[28]

H. K. Leong, C. S. Goh, S. T. Chew et al., Prevention and control of avian influenza in Singapore,, Ann. Acad. Med. Singap., 37 (2008), 504.

[29]

J. Liu et al., Highly pathogenic H5N1 influenza virus infection in migratory birds,, Science, 309 (2005). doi: 10.1126/science.1115273.

[30]

R. Liu, V. R. S. K. Duvvuri and J. Wu, Spread pattern formation of H5N1-avian influenza and its implications for control strategies,, Math. Model. Nat. Phenom., 3 (2008), 161. doi: 10.1051/mmnp:2008048.

[31]

D. Normile, Are wild birds to blame?,, Science, 310 (2005), 426. doi: 10.1126/science.310.5747.426.

[32]

D. Normile, Evidence points to migratory birds in H5N1 spread,, Science, 311 (2006). doi: 10.1126/science.311.5765.1225.

[33]

B. Olsen, V. J. Munster, A. Wallensten, J. Waldenstrom, A. D. M. E. Osterhaus and R. O. M. Fouchier, Global patterns of influenza A virus in wild birds,, Science, 312 (2006), 384. doi: 10.1126/science.1122438.

[34]

K. D. Redd, J. K. Meece, J. S. Henkel and S. K. Shukla, Birds, migration and emerging zoonoses: West Nile virus, lyme diseases, influenza A and enteropathongens,, Clinical Medicine and Research, 1 (2003), 5. doi: 10.3121/cmr.1.1.5.

[35]

K. R. Ríos-Soto, "Dispersal and Disease Dynamics in Populations with and without Demography,'', Ph.D. thesis, (2008).

[36]

C. P. Simon and J. A. Jacquez, Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations,, SIAM J. Appl. Math., 52 (1992), 541. doi: 10.1137/0152030.

[37]

B. Song, C. Castillo-Chavez and J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts,, Math. Biosci., 180 (2002), 187. doi: 10.1016/S0025-5564(02)00112-8.

[38]

World Health Organization (2005), Avian influenza frequently asked questions,, http://www.who.int/csr/disease/avian_influenza/avian_faqs/en/index.html., ().

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