The pluses and minuses of ℛ0

Journal of the Royal Society Interface

Volumn 4, Issue 16, 2007, Pages 949-961

  Roberts, M G ^a  

^a MASSEY UNIVERSITY   (New Zealand)

Author keywords

Backward bifurcation; Basic reproduction number; Infectious diseases; Mathematical epidemiology; Type reproduction number

Indexed keywords

EPIDEMIC; INFECTION; MATHEMATICAL ANALYSIS; REVIEW; ANIMAL; ANIMAL DISEASE; ARTICLE; BIOLOGICAL MODEL; COMMUNICABLE DISEASE; DISEASE TRANSMISSION; STATISTICAL MODEL; STATISTICS; WILD ANIMAL;

ANIMAL DISEASES; ANIMALS; ANIMALS, WILD; COMMUNICABLE DISEASES; DISEASE TRANSMISSION, HORIZONTAL; MODELS, BIOLOGICAL; MODELS, STATISTICAL;

EID: 34948821143     PISSN: 17425689     EISSN: 17425662     Source Type: Journal    
DOI: 10.1098/rsif.2007.1031     Document Type: Review

References (41)

1. Anderson, R. M. & May, R. M. 1991 Infectious diseases of humans: dynamics and control. Oxford, UK: Oxford University Press.
2. Andreasen, V. & Pugliese, A. 1995 Pathogen coexistence induced by density-dependent host mortality. J. Theor. Biol. 177, 159-165.
3. Ball, F. & Becker, N. G. 2006 Control of transmission with two types of infection. Math. Biosci. 200, 170-187. (doi:10.1016/j.mbs.2005. 12.024)
4. Blower, S. M., McLean, A. R., Porco, T. C., Small, P. M., Hopewell, P. C., Sanchez, M. A. & Moss, A. R. 1995 The intrinsic transmission dynamics of tuberculosis epidemics. Nat. Med. 1, 815-821. (doi:10.1038/nm0895-815)
5. Dieckmann, U. & Metz, J. A. J. 2006 Surprising evolutionary predictions from enhanced ecological realism. Theor. Popul. Biol. 69, 263-281. (doi:10.1016/j.tpb.2005.12.001)
6. Dieckmann, U., Metz, J. A. J., Sabelis, M. W. & Sigmund, K. 2002 Adaptive dynamics of infectious diseases. Cambridge, UK: Cambridge University Press.
7. Diekmann, O. & Heesterbeek, J. A. P. 2000 Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Wiley series in mathematical and computational biology. Chichester, UK: Wiley.
8. 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28, 365-382. (doi:10.1007/BF00178324)
9. Diekmann, O., Heesterbeek, J. A. P. & Metz, J. A. J. 1995 The legacy of Kermack and McKendrick. In Epidemic models: their structure and relation to data (ed. D. Mollison), pp. 95-115. Cambridge, UK: Cambridge University Press.
10. Edelstein-Keshet, L. 2005 Mathematical models in biology. Philadelphia, PA: Society for Industrial and Applied Mathematics.
11. Esteva, L. & Vargas, C. 1998 Analysis of a dengue disease transmission model. Math. Biosci. 150, 131-151. (doi:10.1016/S0025-5564(98)10003-2)
12. Esteva, L. & Vargas, C. 2000 Influence of vertical and mechanical transmission on the dynamics of dengue disease. Math. Biosci. 167, 51-64. (doi:10.1016/S0025-5564(00)00024-9)
13. Feng, Z. & Velasco-Hernandez, J. X. 1997 Competitive exclusion in a vector-host model for the dengue fever. J. Math. Biol. 35, 523-544. (doi:10.1007/s002850050064)
14. Feng, Z., Castillo-Chavez, C. & Capurro, A. F. 2000 A model for tuberculosis with exogenous reinfection. Theor. Popul. Biol. 57, 235-247. (doi:10.1006/tpbi.2000.1451)
15. Gomes, M. G. M., Franco, A. O., Gomes, M. C. & Medley, G. F. 2004 The reinfection threshold promotes variability in tuberculosis epidemiology and vaccine efficacy. Proc. R. Soc. B 271, 617-623. (doi:10.1098/rspb.2003.2606)
16. Gomes, M. G. M., Margheri, A., Medley, G. F. & Rebelo, C. 2005 Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence. J. Math. Biol. 51, 414-430. (doi:10.1007/s00285-005-0331-9)
17. Crassly, N. C. & Fraser, C. 2006 Seasonal infectious disease epidemiology. Proc. R. Soc. B 273, 2541-2550. (doi:10.1098/rspb. 2006.3604)
18. Haydon, D., Chase-Topping, M., Shaw, D. J., Matthews, L., Friar, J., Wilesmith, J. & Woolhouse, M. E. J. 2003 The construction and analysis of epidemic trees with reference to the 2001 UK foot-and-mouth outbreak. Proc. R. Soc. B 270, 121-127. (doi:10.1098/rspb.2002.2191)
19. 0 and a recipe for its calculation. Acta Biotheor. 50, 189-204. (doi:10.1023/A:1016599411804)
20. Heesterbeek, J. A. P. & Roberts, M. G. 1995a Threshold quantities for helminth infections. J. Math. Biol. 33, 415-434. (doi:10.1007/BF00176380)
21. Heesterbeek, J. A. P. & Roberts, M. G. 1995b Threshold quantities for infectious diseases in periodic environments. J. Biol. Syst. 3, 779-787. (doi:10.1142/S021833909500071X)
22. Heesterbeek, J. A. P. & Roberts, M. G. 2007 The type-reproduction number T in models for infectious disease control. Math. Biosci. 206, 3-10. (doi:10.1016/j.mbs.2004.10.013)
23. Heffernan, J. M., Smith, R. J. & Wahl, L. M. 2005 Perspectives on the basic reproductive ratio. J. R. Soc. Interface 2, 281-293. (doi:10.1098/rsif.2005.0042)
24. Jordan, D. W. & Smith, P. 1999 Nonlinear ordinary differential equations, 3rd edn. Oxford, UK: Oxford University Press.
25. 0 in heteregeneous populations. J. Theor. Biol. 242, 634-642. (doi:10.1016/j.jtbi.2006.04.003)
26. Kermack, W. O. & McKendrick, A. G. 1927 A Contribution to the mathematical theory of epidemics. I. Proc. R. Soc. A 115, 700-721.
27. (Reprinted with parts II. and III. in Bulletin of Mathematical Biology, 53 (1991) 33-118.) (doi:10.1098/rspa.1927.0118)
28. Kiss, I. Z., Green, D. M. & Kao, R. R. 2006 Infectious disease control using contact tracing in random and scale-free networks. J. R. Soc. Interface 3, 55-62. (doi:10.1098/rsif.2005.0079)
29. May, R. M. 2006 Network structure and the biology of populations. Trends Ecol. Evol. 21, 394-399. (doi:10.1016/j.tree.2006.03.013)
30. Medley, G. F., Lindop, N. A., Edmunds, W. J. & Nokes, D. J. 2001 Hepatitis-B virus endemicity: heterogeneity, catastrophic dynamics and control. Nat. Med. 5, 619-624. (doi:10.1038/87953)
31. Roberts, M. G. 1996 Bovine tuberculosis in possum populations, and its eradication or control by culling or vaccination. J. Anim. Ecol. 65, 451-464. (doi:10.2307/5780)
32. Roberts, M. G. & Dobson, A. P. 1995 The population dynamics of communities of parasitic helminths. Math. Biosci. 126, 191-214. (doi:10.1016/0025-5564(94)00036-Y)
33. Roberts, M. G. & Heesterbeek, J. A. P. 2003 A new method for estimating the effort required to control infectious diseases. Proc. R. Soc. B 270, 1359-1364. (doi:10.1098/rspb.2003.2339)
34. Roberts, M. G. & Jowett, J. 1996 An SEI model with density dependent demographics and epidemiology. IMA J. Math. Appl. Med. Biol. 13, 245-257.
35. Roberts, M. G. & Kao, R. R. 1998 The dynamics of an infectious disease in a population with birth pulses. Math. Biosci. 149, 23-36. (doi:10.1016/S0025-5564(97)10016-5)
36. Roberts, M. G. & Tobias, M. I. 2000 Predicting and preventing measles epidemics in New Zealand: application of a mathematical model. Epidemiol. Infect. 124, 279-287. (doi:10.1017/S0950268899003556)
37. Roberts, M. G., Baker, M., Jennings, L. C., Sertsou, G. & Wilson, N. 2007 A model for the spread and control of pandemic influenza in an isolated geographical region. J. R. Soc. Interface 4, 325-330. (doi:10.1098/rsif.2006.0176)
38. Safan, M., Heesterbeek, J. A. P. & Dietz, K. 2006 The minimum effort required to eradicate infections in models with backward bifurcation. J. Math. Biol. 53, 703-718. (doi:10.1007/s00285-006-0028-8)
39. Soewono, E. & Supriatna, A. K. 2001 A two-dimensional model for the transmission of dengue fever disease. Bull. Malaysian Math. Sci. Soc. 24, 49-57.
40. Strogatz, S. H. 1994 Nonlinear dynamics and chaos. Cambridge, MA: Westview Press.
41. van den Driessche, P. & Watmough, J. 2002 Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29-48. (doi:10.1016/S0025- 5564(02)00108-6)