Progression age enhanced backward bifurcation in an epidemic model with super-infection

Journal of Mathematical Biology

Volumn 46, Issue 5, 2003, Pages 385-424

  Martcheva, Maia ^a,b   Thieme, Horst R ^c  

^a POLYTECHNIC UNIVERSITY   (United States)

^b Cornell University ^*  (United States)

^c ARIZONA STATE UNIVERSITY   (United States)

Author keywords

Alternating stability; Backward bifurcation; Break point density; Dose dependent latent period; Multiple endemic equilibria; Progression age structure; Progressive and quiescent latent stages; Super infection; Threshold type disease activation

Indexed keywords

ARTICLE; BIOLOGICAL MODEL; HUMAN; HUMAN IMMUNODEFICIENCY VIRUS INFECTION; MALARIA; MICROBIOLOGY; PARASITOLOGY; SCHISTOSOMIASIS; SUPERINFECTION; TUBERCULOSIS;

HIV INFECTIONS; HUMANS; MALARIA; MODELS, BIOLOGICAL; SCHISTOSOMIASIS; SUPERINFECTION; TUBERCULOSIS;

EID: 1642284439     PISSN: 03036812     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00285-002-0181-7     Document Type: Article

References (48)

1. Anderson, R.M., May, R.M.: Infectious Diseases of Humans, Oxford University Press, 1991
2. Andreasen, V., Lin, J., Levin, S.A.: The dynamics of cocirculating influenza strains conferring partial cross-immunity. J. Math. Biol. 35, 825-842 (1997)
3. Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H.P., Moustakas, U., Nagel, R., Neubrander, F., Schlotterbeck, U.: One-parameter Semigroups of Positive Operators, Lect. Notes Math. 1184, Berlin, New York, Tokyo, Springer-Verlag, 1986
4. Bailey, N.T.J.: The Mathematical Theory of Infectious Diseases and its Applications. Sec. Ed., Griffin, London, 1975
5. Blower, S.M., Porco, T., Lietman, T.: Tuberculosis: the evolution of antibiotic resistence and the design of epidemic control strategies, in Mathematical Models in Medical and Health Sciences, Eds. M. A. Horn, G. Simonett, G. Webb, 51-72, Vanderbilt Univ. Press, Nashville, TN, 1998
6. Blower, S.M., McLean, A.R., Porco, T., Sanches, M., Small, P., Hopewell, P., Moss, A.: The intrinsic transmission dynamics of tuberculosis epidemics. Nature Medicine 1, 815-821 (1995)
7. Blower, S.M., Small, P., Hopewell, P.: Control strategies for tuberculosis epidemics: new models for old problems. Science 273, 497-500 (1996)
8. Castillo-Chavez, C., Feng, Z.: Mathematical models for the disease dynamics of tuberculosis, in Advances in Mathematical Population Dynamics - Molecules, Cells and Man, Eds. O. Arino, D. Axelrod, M. Kimmel, Ser. Math. Biol. Med., Vol.6: 629-656, World Scientific Publishing, Singapore, New Jersey, London, Hong Kong, 1997
9. Castillo-Chavez, C., Feng, Z.: To treat or not to treat: the case of tuberculosis. J. Math. Biol. 35, 629-659 (1997)
10. Castillo-Chavez, C., Feng, Z.: Global stability of an age-structure model for TB and its application to optimal vaccination strategies. Math. Biosci. 151, 135-154 (1998)
11. Castillo-Chavez, C., Huang, W.: Age-structured core group model and its impact on STD dynamics. In: Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, Eds. C. Castillo-Chavez with S. Blower, P. van den Driessche, D. Kirschner, and A.-A. Yakubu: 261-273, Springer-Verlag, New York, 2002
12. Chow, S.-N., Hale, J.K.: Methods of Bifurcation Theory, Berlin-Heidelberg New York: Springer Verlag, 1982
13. Clark, C.W.: Mathematical Bioeconomics: The Optimal Management of Renewable Resourses, John Wiley & Sons, New York, London, Sydney, Toronto, 1976
14. Cooke, K.: Functional-differential equations: some models and perturbation problems, Differential Equations and Dynamical Systems, Eds. J. Hale, J. LaSalle, eds, 167-183, New York: Academic Press, 1967
15. Desch, W., Schappacher, W.: Linearized stability for nonlinear semigroups, in Differential Equations in Banach Spaces, Eds. A. Favini, E. Obrecht, Lec. Notes Math. 1223, 61-73, Springer-Verlag, Berlin, Heidelberg, 1986
16. Dietz, K.: Mathematical models for transmission and control of malaria, In: Malaria: Principles and Practice of Malariology. Eds. W. Wernsdorfer, I. McGregor, 1091-1133 Churchill Livingstone, Edinburgh, 1988
17. van den Driessche, P., Watmough, J.: A Simple SIS epidemic model with a backward bifurcation. J. Math. Biol. 40(6), 525-540 (2000)
18. Dushoff, J., Huang, W., Castillo-Chavez, C.: Backwards bifurcations and catastrophe in simple models of fatal diseases. J. Math. Biol. 36(3), 227-248 (1998)
19. Feng, Z., Castillo-Chavez, C., Capurro, A.: A model for tuberculosis with exogeneous reinfection. Theor. Pop. Biol. 57, 235-247 (2000)
20. Z. Feng, W. Huang & C. Castillo-Chavez, On the role of variable latent periods in mathmatical models for tuberculosis. J. Dynam. Diff. Eq. 13(2), 425-452 (2001)
21. Feng, Z., Iannelli, M., Milner, F.A.: A two-strain TB model with age of infection. SIAM J. App. Math. 62(5), 1634-1656 (2002)
22. Greenhalgh, D., Diekmann, O., de Jong, M.C.M.: Subcritical endemic steady states in mathematical models for animal infections with incomplete immunity. Math. Biosci. 165(1), 1-25 (2000)
23. Greenhalgh, D., Doyle, M., Lewis, F.: A mathematical treatment of AIDS and condom use. IMA J. Math. Applied in Medecine and Biology 18, 225-262 (2001)
24. Hadeler, K.P.: Hysteresis in a model for parasitic infection, Numerical methods for bifurcation problems (Dortmund, 1983). Internat. Schriftenreihe Numer. Math., 70, 171-180, Birkhäuser, Basel, 1984
25. Hadeler, K.P., Castillo-Chavez, C.: A core group model for disease transmission. Math. Biosci. 128, 41-55 (1995)
26. Hadeler, K.P., van den Driessche, P.: Backward bifurcation in epidemic control. Math. Biosci. 146, 15-35 (1997)
27. Hethcote, H.W.: A thousand and one epidemic model. in Frontiers in Mathematical Biology, Ed. S.A. Levin, Lecture Notes in Biomathematics, Vol. 100: 504-515, Springer-Verlag, Berlin, Heidelberg, 1994
28. Hoppensteadt, F., Waltman, P.: A problem in theory of epidemics. Math. Biosci. 9, 71-91 (1970)
29. Hoppensteadt, F., Waltman, P.: A problem in theory of epidemics II. Math. Biosci. 12, 133-145 (1971)
30. Hale, J.K.: Asymptotic Behavior of Dissipative Systems, AMS, Providence, 1988
31. Huang, W., Cooke, K., Castilo-Chavez, C.: Stability and bifurcation for a multiple-group model for the dynamics of HIV transmission, SIAM J. Appl. Math. 52(3), 835-854 (1992)
32. Iannelli, M.: Mathematical Theory of Age-structured Population Dynamics, Applied Mathematics Monographs 7, Giardini Editori, Pisa, 1995
33. Kribs-Zaleta, C.M., Velasco-Hernández, J.X.: A simple vaccination model with multiple endemic states. Math. Biosci. 164, 183-201 (2000)
34. Kribs-Zaleta, C.M., Martcheva, M.: Vaccination strategies and backward bifurcation in age-since-infection structured model. Math. Biosci. 177/178, 317-332 (2002)
35. Liu, W.-m., Hethcote, H.W., Levin, S.A.: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25, 359-380 (1987)
36. Liu, W.-m.: Dose-dependent latent period and periodicity of infectious diseases. J. Math. Biol. 31, 487-494 (1993)
37. Liu, W.-m., van den Driessche, P.: Epidemiological models with varying population size and dose-dependent latent period. Math. Biosci. 128, 57-69 (1995)
38. Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. New York: Springer-Verlag, 1983
39. Porco, T., Blower, S.M.: Quantifying the intrinsic transmission dynamics of tuberculosis epidemics. Theor. Pop. Biol. 54(2), 117-132 (1998)
40. Sanches, M., Blower, S.M.: Uncertainty and sensitivity analysis of the basic reproduction number: Tuberculosis as an example. Amer. J. Epidem. 145(12), 1127-1138 (1997)
41. Smith, H.L.: Hopf bifurcation in a system of functional equations modeling the spread of an infectious disease. SIAM J. Appl. Math. 43, 370-385 (1983)
42. Tu, S., Ross, E.: Minimum transition values and the dynamics of subcritical bifurcation. SIAM J. Appl. Math. 46(2), 189-198 (1986)
43. Thieme, H.R.: Semiflows generated by Lipschitz perturbations of non-densely defined operators, Differential Integral Equations 3(6): 1035-1066 (1990)
44. Thieme, H.R.: Uniform persistence and permanence for non-autonomous semiflows in population biology, Math. Biosci. 166, 173-201 (2000)
45. Thieme, H.R.: Quasi-compact semigroups via bounded perturbation, in Advances in Mathematical Population Dynamics - Molecules, Cells and Man, Eds. O. Arino, D. Axelrod, M. Kimmel, Ser. Math. Biol. Med., Vol.6: 691-711, World Scientific Publishing, Singapore, New Jersey, London, Hong Kong, 1997
46. Thieme, H.R.: The transition through stages with arbitrary length distributions, and applications in epidemics, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods and Theory, Eds. C. Castillo-Chavez with S. Blower, P. van den Driessche, D. Kirschner, and A.-A. Yakubu: 45-84, New York: Springer-Verlag, 2002
47. Waltman, P.: Deterministic Threshold Models in the Theory of Epidemics, Lecture Notes in Biomathematics 1, Berlin-Heidelberg, New York: Springer-Verlag, 1974
48. Yosida, K.: Functional Analysis, second edition, Berlin-Heidelberg, New York: Springer-Verlag, 1968