Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-28T03:34:18.606Z Has data issue: false hasContentIssue false
  • English
  • Français

SPATIAL HETEROGENEITY IN SIMPLE DETERMINISTIC SIR MODELS ASSESSED ECOLOGICALLY

Part of: Mathematical biology in general

Published online by Cambridge University Press:  09 April 2013

E. K. WATERS ,
H. S. SIDHU  and
G. N. MERCER
E. K. WATERS*
Affiliation:
School of Physical, Environmental and Mathematical Sciences, The University of New South Wales Canberra, Australian Defence Force Academy, PO Box 7916, Canberra BC 2610, Australia University of Notre Dame Australia, PO Box 944, Broadway, NSW 2007, Australia
H. S. SIDHU*
Affiliation:
School of Physical, Environmental and Mathematical Sciences, The University of New South Wales Canberra, Australian Defence Force Academy, PO Box 7916, Canberra BC 2610, Australia
G. N. MERCER*
Affiliation:
National Centre for Epidemiology and Population Health, The Australian National University, Canberra, ACT 0200, Australia
*
✉edward.waters.nsw@gmail.com
✉h.sidhu@adfa.edu.au
✉geoff.mercer@anu.edu.au
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Patchy or divided populations can be important to infectious disease transmission. We first show that Lloyd’s mean crowding index, an index of patchiness from ecology, appears as a term in simple deterministic epidemic models of the SIR type. Using these models, we demonstrate that the rate of movement between patches is crucial for epidemic dynamics. In particular, there is a relationship between epidemic final size and epidemic duration in patchy habitats: controlling inter-patch movement will reduce epidemic duration, but also final size. This suggests that a strategy of quarantining infected areas during the initial phases of a virulent epidemic might reduce epidemic duration, but leave the population vulnerable to future epidemics by inhibiting the development of herd immunity.

Keywords

SIR modelecologyinfectious diseasesspatial dynamics

MSC classification

Secondary: 92B05: General biology and biomathematics
Type
Research Article
Information
The ANZIAM Journal , Volume 54 , Issue 1-2 , October 2012 , pp. 23 - 36
Copyright
Copyright ©2013 Australian Mathematical Society 

References

Anderson, R. M. and May, R. M., Infectious diseases of humans: dynamics and control (Oxford University Press, Oxford, 1991).CrossRefGoogle Scholar
Anonymous, “Influenza in a boarding school”, Br. Med. J. 1 (1978) 586; doi:10.1136/bmj.1.6112.586.CrossRefGoogle Scholar
Arino, J., Davis, J. R., Hartley, D., Jordan, R., Miller, J. M. and van den Driessche, P., “A multi-species epidemic model with spatial dynamics”, Math. Med. Biol. 22 (2005) 129–142; doi:10.1093/imammb/dqi003.CrossRefGoogle ScholarPubMed
Arino, J. and van den Driessche, P., “Metapopulations epidemic models. a survey”, in: Nonlinear dynamics and evolution equations, Volume 48 of Fields Inst. Commun. (eds Brunner, H. et al. ), (American Mathematical Society, Providence, RI, 2006), 1–12.Google Scholar
Barlow, N. D., “A spatially aggregated disease/host model for bovine Tb in New Zealand possum populations”, J. Appl. Ecol. 28 (1991) 777–793; doi:10.2307/2404207.CrossRefGoogle Scholar
Bartlett, M. S., “Measles periodicity and community size”, J. R. Statist. Soc. Ser. A 120 (1957) 48–70; doi:10.2307/2342553.CrossRefGoogle Scholar
Begg, K. et al. , “Australia’s notifiable diseases status, 2006: annual report of the national notifiable diseases surveillance system—results: sexually transmissible diseases”, Commun. Dis. Intell. 32 (2008) 139–208.Google Scholar
Bez, N., “On the use of Lloyd’s index of patchiness”, Fisheries Oceanography 9 (2000) 372–376; doi:10.1046/j.1365-2419.2000.00148.x.CrossRefGoogle Scholar
Downing, J. A., “Spatial heterogeneity: evolved behaviour or mathematical artefact?”, Nature 323 (1986) 255–257; doi:10.1038/323255a0.CrossRefGoogle Scholar
Fujita, K., “A new method for describing predator–prey relations in discontinuous environments”, Res. Popul. Ecol. 26 (1984) 379–388; doi:10.1007/BF02515501.CrossRefGoogle Scholar
Fulford, G. R., Roberts, M. G. and Heesterbeek, J. A. P., “The metapopulation dynamics of an infectious disease: tuberculosis in possums”, Theor. Popul. Biol. 61 (2002) 15–29; doi:10.1006/tpbi.2001.1553.CrossRefGoogle ScholarPubMed
Garnett, G. P., Swinton, J., Brunham, R. C. and Anderson, R. M., “Gonococcal infection, infertility, and population growth: II. The influence of heterogeneity in sexual behaviour”, Math. Med. Biol. 9 (1992) 127–144; doi:10.1093/imammb/9.2.127.CrossRefGoogle ScholarPubMed
Iwao, S., “A new regression method for analyzing the aggregation pattern of animal populations”, Res. Popul. Ecol. 10 (1968) 1–20; doi:10.1007/BF02514729.CrossRefGoogle Scholar
Iwao, S., “A note on the related concepts ‘mean crowding’ and ‘mean concentration’”, Res. Popul. Ecol. 17 (1976) 240–242; doi:10.1007/BF02530775.CrossRefGoogle Scholar
Iwao, S., “Analysis of spatial association between two species based on the interspecies mean crowding”, Res. Popul. Ecol. 18 (1977) 243–260; doi:10.1007/BF02510851.CrossRefGoogle Scholar
Jacobs, J. H. et al. , “Searching for sharp drops in the incidence of pandemic A/H1N1 influenza by single year of age”, PLoS ONE 7 (2012) e42328; doi:10.1371/journal.pone.0042328.CrossRefGoogle ScholarPubMed
Kimura, M. and Weiss, G. H., “The stepping stone model of population structure and the decrease of genetic correlation with distance”, Genetics 49 (1964) 561–576.CrossRefGoogle ScholarPubMed
Lloyd, M., “Mean crowding”, J. Animal Ecol. 36 (1967) 1–30; doi:10.2307/3012.CrossRefGoogle Scholar
Ma, J. and Earn, D. J. D., “Generality of the final size formula for an epidemic of a newly invading infectious disease”, Bull. Math. Biol. 68 (2006) 679–702; doi:10.1007/s11538-005-9047-7.CrossRefGoogle ScholarPubMed
Maruyama, T., “On the rate of decrease of heterozygosity in circular stepping stone models of populations”, Theor. Popul. Biol. 1 (1970) 101–119; doi:10.1016/0040-5809(70)90044-4.CrossRefGoogle ScholarPubMed
Matthiopoulos, J., Harwood, J. and Thomas, L., “Metapopulation consequences of site fidelity for colonially breeding mammals and birds”, J. Animal Ecol. 74 (2005) 716–727; doi:10.1111/j.1365-2656.2005.00970.x.CrossRefGoogle Scholar
Parham, P. E. and Ferguson, N. M., “Space and contact networks: capturing the locality of disease transmission”, J. R. Soc. Interface 3 (2006) 483–493; doi:10.1098/rsif.2005.0105.CrossRefGoogle ScholarPubMed
R Core Team. R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, 2012.Google Scholar
Svendsen, G. E., “Behavioral and environmental factors in the spatial distribution and population dynamics of a yellow-bellied marmot population”, Ecology 55 (1974) 760–771; doi:10.2307/1934412.CrossRefGoogle Scholar
Turner, K. M. E., Adams, E. J., Gay, N., Ghani, A. C., Mercer, C. and Edmunds, W. J., “Developing a realistic sexual network model of chlamydia transmission in Britain”, Theor. Biol. Med. Model. 3 (2006); doi:10.1186/1742-4682-3-3.CrossRefGoogle ScholarPubMed
Waters, E. K., “Aggregation and competitive exclusion: explaining the coexistence of Human Papillomavirus types and the effectiveness of limited vaccine conferred cross-immunity”, Acta Biotheoretica 60 (2012) 333–356; doi:10.1007/s10441-012-9161-5.CrossRefGoogle ScholarPubMed