# American Institute of Mathematical Sciences

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

## Modelling seasonal influenza in Israel

 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.
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. [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. [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. [4] O. Diekmann and J. Hesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," 1st edition, Wiley, New York, 2000. [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. [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. [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. [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. [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. [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. [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. [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. [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., Preprint. doi: 10.1098/rsif.2010.0515. [14] E. D. Kilbourne and J. L. Schulman, Airborne transmission of influenza virus infection in mice, Nature, 195 (1961), 1129-1130. [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. [16] Anne Moscona, Neuraminidase inhibitors for influenza, N. Engl. J. Med., 353 (2005), 1363-1373. doi: 10.1056/NEJMra050740. [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. [18] K. G. Nicholson, J. M. Wood and M. Zambon, Influenza, Lancet, 362 (2003), 1733-1745. doi: 10.1016/S0140-6736(03)14854-4. [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. [20] Christopher W. Potter, A history of influenza, J. Appl. Microbiol., 91 (2001), 572-579. [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. [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. [23] H. E. Soper, The interpretation of periodicity in disease prevalence, J. R. Stat. Soc., 92 (1929), 34-73. doi: 10.2307/2341437. [24] L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638. [25] R. J. Webby and R. G Webster, Are We Ready for Pandemic Influenza?, Science, 302 (2003), 1519-1522. doi: 10.1126/science.1090350.

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##### 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. [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. [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. [4] O. Diekmann and J. Hesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," 1st edition, Wiley, New York, 2000. [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. [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. [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. [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. [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. [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. [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. [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. [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., Preprint. doi: 10.1098/rsif.2010.0515. [14] E. D. Kilbourne and J. L. Schulman, Airborne transmission of influenza virus infection in mice, Nature, 195 (1961), 1129-1130. [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. [16] Anne Moscona, Neuraminidase inhibitors for influenza, N. Engl. J. Med., 353 (2005), 1363-1373. doi: 10.1056/NEJMra050740. [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. [18] K. G. Nicholson, J. M. Wood and M. Zambon, Influenza, Lancet, 362 (2003), 1733-1745. doi: 10.1016/S0140-6736(03)14854-4. [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. [20] Christopher W. Potter, A history of influenza, J. Appl. Microbiol., 91 (2001), 572-579. [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. [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. [23] H. E. Soper, The interpretation of periodicity in disease prevalence, J. R. Stat. Soc., 92 (1929), 34-73. doi: 10.2307/2341437. [24] L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533-536. doi: 10.1038/nature05638. [25] R. J. Webby and R. G Webster, Are We Ready for Pandemic Influenza?, Science, 302 (2003), 1519-1522. doi: 10.1126/science.1090350.
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