American Institute of Mathematical Sciences

Advanced
2011, 8(2): 561-573. doi: 10.3934/mbe.2011.8.561

Modelling seasonal influenza in Israel

Oren Barnea 1, , Rami Yaari 1, , Guy Katriel 1, and  Lewi Stone 1,

1. 

Biomathematics Unit, Department of Zoology, Faculty of Life Sciences, Tel Aviv University, Tel Aviv 69978, Israel, Israel, Israel, Israel

Received  March 2010 Revised  August 2010 Published  April 2011

Mathematical modeling approaches are used to study the epidemic dynamics of seasonal influenza in Israel. The recent availability of highly resolved ten year timeseries of influenza cases provides an opportunity for modeling and estimating important epidemiological parameters in the Israeli population. A simple but well known SIR discrete-time deterministic model was fitted to consecutive epidemics allowing estimation of the initial number of susceptibles in the population $S_0$, as well as the reproductive number $R_0$ each year. The results were corroborated by implementing a stochastic model and using a maximum likelihood approach. The paper discusses the difficulties in estimating these important parameters especially when the reporting rate of influenza cases might only be known with limited accuracy, as is generally the case. In such situations invariant parameters such as the percentage of susceptibles infected, and the effective reproductive rate might be preferred, as they do not depend on reporting rate. Results are given based on the Israeli timeseries.
Keywords: seasonality., basic reproductive ratio, model fitting, Epidemiology, $R_0$, influenza.
Mathematics Subject Classification: Primary: 92D3.
Citation: Oren Barnea, Rami Yaari, Guy Katriel, Lewi Stone. Modelling seasonal influenza in Israel. Mathematical Biosciences & Engineering, 2011, 8 (2) : 561-573. doi: 10.3934/mbe.2011.8.561
References:
[1]

V. Andreasen, J. Lin and S. A. Levin, The dynamics of cocirculating influenza strains conferring partial cross-immunity, J. Math. Biol., 35 (1997), 825-842. doi: 10.1007/s002850050079.  Google Scholar

[2]

J. J, Cannell, R. Vieth, J. C. Umhau, M. F. Holick, W. B. Grant, S. Madronich, C. F. Garland and E Giovannucci, Epidemic influenza and vitamin D, Epidemiol. Infect., 134 (2006), 1129-1140. Google Scholar

[3]

N. J Cox and K. Subbarao, Global epidemiology of influenza: Past and present, Annu. Rev. Med., 51 (2000), 407-421. doi: 10.1146/annurev.med.51.1.407.  Google Scholar

[4]

O. Diekmann and J. Hesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," 1st edition, Wiley, New York, 2000. Google Scholar

[5]

N. M. Ferguson, D. A. T. Cummings, C. Fraser, J. C. Cajka, P. C. Cooley and D. S. Burke, Strategies for mitigating an influenza pandemic, Nature, 442 (2006), 448-452. doi: 10.1038/nature04795.  Google Scholar

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M. J. Ferrari, O. N. Bjórnstad and A. P. Dobson, Estimation and inference of $R_0$ of an infectious pathogen by a removal method, Math. Biosci., 198 (2005), 14-26. doi: 10.1016/j.mbs.2005.08.002.  Google Scholar

[7]

P. E. M Fine and J. A. Clarkson, Measles in England and Wales-I: An Analysis of Factors Underlying Seasonal Patterns, Int. J. Epidemiol., 11 (1982), 5-14. doi: 10.1093/ije/11.1.5.  Google Scholar

[8]

B. S Finkelman, C. Viboud, K. Koelle, M. J. Ferrari, N. Bharti and B. T. Grenfell, Global Patterns in Seasonal Activity of Influenza A/H3N2, A/H1N1, and B from 1997 to 2005: Viral Coexistence and Latitudinal Gradients, PLoS ONE, 2 (2007), e1296. doi: 10.1371/journal.pone.0001296.  Google Scholar

[9]

T. C. Germann, K. Kadau, I. M. Longini, Jr. and C. A. Macken, Mitigation strategies for pandemic influenza in the United States, PNAS, 103 (2006), 5935-5940. doi: i:10.1073/pnas.0601266103.  Google Scholar

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H. Heesterbeek, The law of mass-action in epidemiology: A historical perspective, in "Ecological Paradigms Lost - Routes of Theory Change" (eds. K. Cuddington and B. Beisner), Academic Press, 2005, 81-105. Google Scholar

[11]

A. D. Heymann, I. Hoch, L. Valinsky, E. Kokia and D. M. Steinberg, School Closure May Be Effective In Reducing Transmission Of Respiratory Viruses In The Community, Epidemiol. Infect., 137 (2009), 1369-1376. doi: 10.1017/S0950268809002556.  Google Scholar

[12]

G. Katriel and L. Stone, Pandemic dynamics and the breakdown of herd immunity, PLoS ONE, 5 (2010), e9565. doi: 10.1371/journal.pone.0009565.  Google Scholar

[13]

G. Katriel, R. Yaari, A. Huppert, U. Roll and L. Stone, Modelling the initial phase of an epidemic using incidence and infection network data: 2009 H1N1 pandemic in Israel as a case study,, J. R. Soc., ().  doi: 10.1098/rsif.2010.0515.  Google Scholar

[14]

E. D. Kilbourne and J. L. Schulman, Airborne transmission of influenza virus infection in mice, Nature, 195 (1961), 1129-1130. Google Scholar

[15]

J. D. Mathews, C. T. McCaw, J. McVernon, E. S. McBryde and J. M. McCaw, A biological model for influenza transmission: Pandemic planning implications of asymptomatic infection and immunity, PLoS ONE, 2 (2007), e1220. doi: 10.1371/journal.pone.0001220.  Google Scholar

[16]

Anne Moscona, Neuraminidase inhibitors for influenza, N. Engl. J. Med., 353 (2005), 1363-1373. doi: 10.1056/NEJMra050740.  Google Scholar

[17]

J.S. Nguyen-Van-Tam, Epidemiology of influenza, in "Textbook of Influenza" (eds. K.G Nicholson, R.G. Webster and A.J. Hay), Malden: Blackwell Science, 1998, 181-206. Google Scholar

[18]

K. G. Nicholson, J. M. Wood and M. Zambon, Influenza, Lancet, 362 (2003), 1733-1745. doi: 10.1016/S0140-6736(03)14854-4.  Google Scholar

[19]

R. Olinky, A. Huppert and L. Stone, Seasonal dynamics and thresholds governing recurrent epidemics, J. Math. Biol., 56 (2008), 827-839. doi: 10.1007/s00285-007-0140-4.  Google Scholar

[20]

Christopher W. Potter, A history of influenza, J. Appl. Microbiol., 91 (2001), 572-579. Google Scholar

[21]

C. A. Russell, T. C. Jones, I. G. Barr, N. J. Cox, R. J. Garten, V. Gregory, I. D. Gust, A. W. Hampson, A. J. Hay, A. C. Hurt, J. C. de Jong, A. Kelso, A. I. Klimov, T. Kageyama, N. Komadina, A. S. Lapedes, Y. P. Lin, A. Mosterin, M. Obuchi, T. Odagiri, A. D. M. E. Osterhaus, G. F. Rimmelzwaan, M. W. Shaw, E. Skepner, K. Stohr, M. Tashiro, R. A. M. Fouchier and D. J. Smith, The global circulation of of seasonal Influenza A (H3N2) viruses, Science, 320 (2008), 340-346. doi: 10.1126/science.1154137.  Google Scholar

[22]

D. J. Smith, A. S. Lapedes, J. C. de Jong, T. M. Bestebroer, G. F. Rimmelzwaan, A. D. M. E. Osterhaus and R. A. M. Fouchier, Mapping the Antigenic and Genetic Evolution of Influenza Virus, Science, 305 (2004), 371-376. doi: 10.1126/science.1097211.  Google Scholar

[23]

H. E. Soper, The interpretation of periodicity in disease prevalence, J. R. Stat. Soc., 92 (1929), 34-73. doi: 10.2307/2341437.  Google Scholar

[24]

L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638.  Google Scholar

[25]

R. J. Webby and R. G Webster, Are We Ready for Pandemic Influenza?, Science, 302 (2003), 1519-1522. doi: 10.1126/science.1090350.  Google Scholar

show all references

References:
[1]

V. Andreasen, J. Lin and S. A. Levin, The dynamics of cocirculating influenza strains conferring partial cross-immunity, J. Math. Biol., 35 (1997), 825-842. doi: 10.1007/s002850050079.  Google Scholar

[2]

J. J, Cannell, R. Vieth, J. C. Umhau, M. F. Holick, W. B. Grant, S. Madronich, C. F. Garland and E Giovannucci, Epidemic influenza and vitamin D, Epidemiol. Infect., 134 (2006), 1129-1140. Google Scholar

[3]

N. J Cox and K. Subbarao, Global epidemiology of influenza: Past and present, Annu. Rev. Med., 51 (2000), 407-421. doi: 10.1146/annurev.med.51.1.407.  Google Scholar

[4]

O. Diekmann and J. Hesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," 1st edition, Wiley, New York, 2000. Google Scholar

[5]

N. M. Ferguson, D. A. T. Cummings, C. Fraser, J. C. Cajka, P. C. Cooley and D. S. Burke, Strategies for mitigating an influenza pandemic, Nature, 442 (2006), 448-452. doi: 10.1038/nature04795.  Google Scholar

[6]

M. J. Ferrari, O. N. Bjórnstad and A. P. Dobson, Estimation and inference of $R_0$ of an infectious pathogen by a removal method, Math. Biosci., 198 (2005), 14-26. doi: 10.1016/j.mbs.2005.08.002.  Google Scholar

[7]

P. E. M Fine and J. A. Clarkson, Measles in England and Wales-I: An Analysis of Factors Underlying Seasonal Patterns, Int. J. Epidemiol., 11 (1982), 5-14. doi: 10.1093/ije/11.1.5.  Google Scholar

[8]

B. S Finkelman, C. Viboud, K. Koelle, M. J. Ferrari, N. Bharti and B. T. Grenfell, Global Patterns in Seasonal Activity of Influenza A/H3N2, A/H1N1, and B from 1997 to 2005: Viral Coexistence and Latitudinal Gradients, PLoS ONE, 2 (2007), e1296. doi: 10.1371/journal.pone.0001296.  Google Scholar

[9]

T. C. Germann, K. Kadau, I. M. Longini, Jr. and C. A. Macken, Mitigation strategies for pandemic influenza in the United States, PNAS, 103 (2006), 5935-5940. doi: i:10.1073/pnas.0601266103.  Google Scholar

[10]

H. Heesterbeek, The law of mass-action in epidemiology: A historical perspective, in "Ecological Paradigms Lost - Routes of Theory Change" (eds. K. Cuddington and B. Beisner), Academic Press, 2005, 81-105. Google Scholar

[11]

A. D. Heymann, I. Hoch, L. Valinsky, E. Kokia and D. M. Steinberg, School Closure May Be Effective In Reducing Transmission Of Respiratory Viruses In The Community, Epidemiol. Infect., 137 (2009), 1369-1376. doi: 10.1017/S0950268809002556.  Google Scholar

[12]

G. Katriel and L. Stone, Pandemic dynamics and the breakdown of herd immunity, PLoS ONE, 5 (2010), e9565. doi: 10.1371/journal.pone.0009565.  Google Scholar

[13]

G. Katriel, R. Yaari, A. Huppert, U. Roll and L. Stone, Modelling the initial phase of an epidemic using incidence and infection network data: 2009 H1N1 pandemic in Israel as a case study,, J. R. Soc., ().  doi: 10.1098/rsif.2010.0515.  Google Scholar

[14]

E. D. Kilbourne and J. L. Schulman, Airborne transmission of influenza virus infection in mice, Nature, 195 (1961), 1129-1130. Google Scholar

[15]

J. D. Mathews, C. T. McCaw, J. McVernon, E. S. McBryde and J. M. McCaw, A biological model for influenza transmission: Pandemic planning implications of asymptomatic infection and immunity, PLoS ONE, 2 (2007), e1220. doi: 10.1371/journal.pone.0001220.  Google Scholar

[16]

Anne Moscona, Neuraminidase inhibitors for influenza, N. Engl. J. Med., 353 (2005), 1363-1373. doi: 10.1056/NEJMra050740.  Google Scholar

[17]

J.S. Nguyen-Van-Tam, Epidemiology of influenza, in "Textbook of Influenza" (eds. K.G Nicholson, R.G. Webster and A.J. Hay), Malden: Blackwell Science, 1998, 181-206. Google Scholar

[18]

K. G. Nicholson, J. M. Wood and M. Zambon, Influenza, Lancet, 362 (2003), 1733-1745. doi: 10.1016/S0140-6736(03)14854-4.  Google Scholar

[19]

R. Olinky, A. Huppert and L. Stone, Seasonal dynamics and thresholds governing recurrent epidemics, J. Math. Biol., 56 (2008), 827-839. doi: 10.1007/s00285-007-0140-4.  Google Scholar

[20]

Christopher W. Potter, A history of influenza, J. Appl. Microbiol., 91 (2001), 572-579. Google Scholar

[21]

C. A. Russell, T. C. Jones, I. G. Barr, N. J. Cox, R. J. Garten, V. Gregory, I. D. Gust, A. W. Hampson, A. J. Hay, A. C. Hurt, J. C. de Jong, A. Kelso, A. I. Klimov, T. Kageyama, N. Komadina, A. S. Lapedes, Y. P. Lin, A. Mosterin, M. Obuchi, T. Odagiri, A. D. M. E. Osterhaus, G. F. Rimmelzwaan, M. W. Shaw, E. Skepner, K. Stohr, M. Tashiro, R. A. M. Fouchier and D. J. Smith, The global circulation of of seasonal Influenza A (H3N2) viruses, Science, 320 (2008), 340-346. doi: 10.1126/science.1154137.  Google Scholar

[22]

D. J. Smith, A. S. Lapedes, J. C. de Jong, T. M. Bestebroer, G. F. Rimmelzwaan, A. D. M. E. Osterhaus and R. A. M. Fouchier, Mapping the Antigenic and Genetic Evolution of Influenza Virus, Science, 305 (2004), 371-376. doi: 10.1126/science.1097211.  Google Scholar

[23]

H. E. Soper, The interpretation of periodicity in disease prevalence, J. R. Stat. Soc., 92 (1929), 34-73. doi: 10.2307/2341437.  Google Scholar

[24]

L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638.  Google Scholar

[25]

R. J. Webby and R. G Webster, Are We Ready for Pandemic Influenza?, Science, 302 (2003), 1519-1522. doi: 10.1126/science.1090350.  Google Scholar

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