The stage-structured epidemic: Linking disease and demography with a multi-state matrix approach model

Theoretical Ecology

Volumn 4, Issue 3, 2011, Pages 301-319

  Klepac, Petra ^a   Caswell, Hal ^b,c  

^a PRINCETON UNIVERSITY   (United States)

^b WOODS HOLE OCEANOGRAPHIC INSTITUTION   (United States)

^c MAX PLANCK INSTITUTE FOR DEMOGRAPHIC RESEARCH   (Germany)

Author keywords

Basic reproductive number; Disease dynamics; Sensitivity; Stage structure

Indexed keywords

DEMOGRAPHIC METHOD; DEMOGRAPHIC SURVEY; DISEASE SPREAD; DISEASE TRANSMISSION; EPIDEMIC; EPIDEMIOLOGY; FERTILITY; HEALTH RISK; HOST-PATHOGEN INTERACTION; INFECTIVITY; MORTALITY; SENSITIVITY ANALYSIS;

EID: 79960597874     PISSN: 18741738     EISSN: 18741746     Source Type: Journal    
DOI: 10.1007/s12080-010-0079-8     Document Type: Article

References (46)

1. Allen LJS, van den Driessche P (2008) The basic reproduction number in some discrete-time epidemic models. J Differ Equ Appl 14: 1127-1147.
2. Altizer S, Dobson A, Hosseini P, Hudson P, Pascual M, Rohani, P (2006) Seasonality and the dynamics of infectious diseases. Ecol Lett 9: 467-484. doi: 10. 1111/j. 1461-0248. 2005. 00879. x.
3. Anderson RM, May RM (1985) Age-related-changes in the rate of disease transmissions-implications for the design of vaccination programs. J Hyg-Cambridge 94: 365-436.
4. Anderson RM, May RM (1991) Infectious diseases of humans. Oxford University Press, Oxford.
5. Begon M, Bennett M, Bowers RG, French NP, Hazel SM, Truner J (2002) A clarification of transmission terms in host-microparasite models: numbers, densities and areas. Epidemiol Infect 129: 127-153.
6. Brauer F (2006) Some simple epidemic models. Math Biosci Eng 3: 1-15.
7. Busenberg SN, Hadeler KP (1990) Demography and epidemics. Math Biosci 101: 63-74.
8. Castillo-Chavez C, Hethcote H, Andreasen V, Levin S, Liu W (1989) Epidemiological models with age structure, proportionate mixing, and cross-immunity. J Math Biol 27: 233-258.
9. Caswell H (2001) Matrix Population Models. Sunauer Associates, Inc. Second edition.
10. Caswell H (2006) Applications of Markov chains in demography In: MAM2006: Markov anniversary meeting. Boson Books, Releigh, pp 319-334.
11. Caswell H (2007) Evolutionary demography: the invasion exponent and the effective population density in nonlinear matrix models In: From energetics to ecosystems: the dynamics and structure of ecological systems. Springer, Dortrecht, pp 237-256.
12. Caswell H (2008) Perturbation analysis of nonlinear matrix population models. Demographic Research 18: 59-116.
13. Caswell H (2009a) Sensitivity and elasticity of density-dependent population models. J Differ Equ Appl 15: 349-369. doi: 10. 1080/10236190802282669.
14. Caswell H (2009b) Stage, age and individual stochasticity in demography. Oikos 118: 1763-1782. doi: 10. 1111/j. 1600-0706. 2009. 17620. x.
15. Cushing JM, Yicang Z (1994) The net reproductive value and stability in matrix population models. Natural Resource Modeling 8: 297-333.
16. Diekmann O, Heesterbeek JAP (2000) Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley, New York.
17. 0 in models for infectious diseases in heterogeneous populations. J Math Biol 28: 365-382.
18. Dietz K (1975) Transmission and control of arboviruses. In: Epidemiology. SIAM, Philadelphia, pp 104-121.
19. Dietz K, Heesterbeek JAP (2002) Daniel Bernoulli's epidemiological model revisited. Math Biosci 180: 1-21.
20. Dobson A (2004) Population dynamics of pathogens with multiple host species. Am Nat 164: S64-S78.
21. Ferrari MJ, Grais RF, Bharti N, Conlan AJK, Bjørnstad ON, Wolfson LJ, Guerin PJ, Djibo A, Grenfell BT (2008) The dynamics of measles in sub-Saharan Africa. Nature 451: 679-684. doi: 10. 1038/nature06509.
22. Grenfell BT, Bjørnstad ON, Kappey J (2001) Travelling waves and spatial hierarchies in measles epidemics. Nature 414: 716-723.
23. Härkönen T, Harding KC, Heide-Jørgensen MP (2002) Rates of increase in age-structured populations: a lesson from the European harbour seals. Can J Zool 80: 1498-1510.
24. Henderson HV, Searle SR (1979) Vec and vech operators for matrices, with some uses in Jacobians and multivariate statistics. Can J Stat 7: 65-81.
25. Henderson H, Searle S (1981) The vec-permutation, the vec operator and Kronecker products: a review. Linear Multilinear Algebra 9: 271-288.
26. Hethcote, H. (2000) The mathematics of infectious diseases. SIAM Rev 42: 599-653.
27. Hunter CM, Caswell H (2005) The use of the vec-permutation matrix in spatial matrix population models. Ecol Model 188: 15-21.
28. Jenouvrier S, Caswell H, Barbraud C, Holland M, Stroeve J, Weimerskirch H (2009) Demographic models and IPCC climate projections predict the decline of an emperor penguin population. PNAS 106: 1844-1847. doi: 10. 1073/pnas. 0806638106.
29. Jenouvrier S, Caswell H, Barbraud C, Weimerskirch H (2010) Mating behavior, population growth and the operational sex ratio: a periodic two-sex model approach. Am Nat 175: 739-752.
30. Klepac P, Pomeroy LW, Bjørnstad ON, Kuiken T, Osterhaus AD, Rijks JM (2009) Stage-structured transmission of phocine distemper virus in the Dutch 2002 outbreak. Proc R Soc B 276: 2469-2476. doi: 10. 1098/rspb. 2009. 0175.
31. Lloyd AL, May RM (1996) Spatial heterogeneity in epidemic models. J Theor Biol 179: 1-11.
32. Lloyd-Smith J, Greig D, Hietala S, Ghneim G, Palmer L, St Leger J, Grenfell B, Gulland F (2007) Cyclical changes in seroprevalence of leptospirosis in California sea lions: endemic and epidemic disease in one host species? BMC Infect Dis 7: 125. doi: 10. 1186/1471-2334-7-125.
33. Magnus J, Neudecker H (1985) Matrix differential calculus with applications to simple, Hadamard, and Kronecker products. J Math Psychol 29: 474-492.
34. Magnus J, Neudecker H (1988) Matrix differential calculus with applications in statistics and econometrics. Wiley, New York.
35. Matser A, Hartemink N, Heesterbeek H, Galvani A, Davis S (2009) Elasticity analysis in epidemiology: an application to tick-borne infections. Ecol Lett 12: 1298-1305. doi: 10. 1111/j. 1461-0248. 2009. 01378. x.
36. Metcalf CJE, Klepac P, Ferrari M, Grais RF, Grenfell BT (2010) Modeling the first dose of measles vaccination: the role of maternal immunity, demographic factors, and delivery systems. Epidemiol Infect (in press).
37. Pitzer VE, Viboud C, Simonsen L, Steiner C, Panozzo CA, Alonso, WJ, Miller MA, Glass RI, Glasser JW et al (2009) Demographic variability, vaccination, and the spatiotemporal dynamics of rotavirus epidemics. Science 325: 290-294. doi: 10. 1126/science. 1172330.
38. Roth WE (1934) On direct product matrices. Bull Am Math Soc 40: 461-468.
39. Schenzle D (1984) An age-structured model of pre- and post-vaccination measles tranmission. Math Med Biol 1: 169-191.
40. Thieme HR (2003) Mathematics in population biology. Princeton series in theoretical and computational biology. Princeton Univeristy Press, Princeton.
41. van den Driessche P, Watmough J (2002) Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission. Math Biosci 180: 32-34.
42. Verdy A, Caswell H (2008) Sensitivity analysis of reactive ecological dynamics. Bull Math Biol 70: 1634-1659.
43. Viboud C, Bjørnstad ON, Smith DL, Simonsen L, Miller MA, Grenfell BT (2006) Synchrony, waves and spatial hierarchy in the spread of influenza. Science 312: 447-451.
44. Wallinga J, Lipsitch M (2007) How generation intervals shape the relationship between growth rates and reproductive numbers. Proc R Soc B 274: 599-604.
45. Wasserberg G, Osnas EE, Rolley RE, Samuel MD (2009) Host culling as an adaptive management tool for chronic wasting disease in white-tailed deer: a modelling study. J Appl Ecol 46: 457-466. doi: 10. 1111/j. 1365-2664. 2008. 01576. x.
46. Xia Y, Bjørnstad ON, Grenfell BT (2004) Measles metapopulation dynamics: A gravity model for epidemiological coupling and dynamics. Am Nat 164: 267-281.