Stability analysis of within-host parasite models with delays

Mathematical Biosciences

Volumn 209, Issue 1, 2007, Pages 51-75

  Iggidr, Abderrahman ^a   Mbang, Joseph ^a,b   Sallet, Gauthier ^a  

^a UNIVERSITÉ DE LORRAINE   (France)

^b UNIVERSITY OF YAOUNDÉ I   (Cameroon)

Author keywords

Delays; Global stability; Nonlinear dynamical systems; Parasite models

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; DYNAMICAL SYSTEMS; GLOBAL OPTIMIZATION; MEDICAL PROBLEMS;

GLOBAL STABILITY; NONLINEAR DYNAMICAL SYSTEMS; PARASITE MODELS;

MATHEMATICAL MODELS;

ANALYTICAL METHOD; HOST-PARASITE INTERACTION; NUMERICAL MODEL; PARASITE INFESTATION;

ARTICLE; CELL PROLIFERATION; HEPATITIS B VIRUS; HOST PARASITE INTERACTION; HUMAN IMMUNODEFICIENCY VIRUS; MATHEMATICAL MODEL; NONHUMAN; NONLINEAR SYSTEM; PARASITE; PLASMODIUM MALARIAE; T LYMPHOCYTE; TARGET CELL;

HIV INFECTIONS; HIV-1; HUMANS; INFECTION; MODELS, BIOLOGICAL;

EID: 34547744392     PISSN: 00255564     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.mbs.2007.01.008     Document Type: Article

References (40)

1. Perelson A. Modeling the interaction of the immune system with HIV. In: Castillo-Chavez C. (Ed). Mathematical and Statistical Approaches to AIDS Epidemiology with HIV (1989), Springer-Verlag 350-370
2. Perelson A.S., Kirschner D.E., and De Boer R. Dynamics of HIV infection of CD4+ T cells. Math. Biosci. 114 (1993) 81-125
3. Perelson A.S., and Nelson P.W. Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41 1 (1999) 3-44 electronic
4. Anderson R.M., May R.M., and Gupta S. Non-linear phenomena in host-parasite interactions. Parasitology 99 Suppl. (1989) S59-S79
5. Anderson R.M. Complex dynamic behaviours in the interaction between parasite population and the host's immune system. Int. J. Parasitol. 28 (1998) 551-566
6. Hetzel C., and Anderson R.M. The within-host cellular dynamics of bloodstage malaria: theoretical and experimental studies. Parasitology 113 Pt 1 (1996) 25-38
7. Nowak M.A., Bonhoeffer S., Hill A., Boehme R., Thomas H., and McDade H. Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. USA 93 (1996) 4398-4402
8. Nowak M.A., and Bangham C. Population dynamics of immune responses to persistent viruses. Science 272 (1996) 74-79
9. R.J. Payne, M.A. Nowak, B.S. Blumberg, The dynamics of hepatitis B virus infection, Proc. Natl. Acad. Sci. USA 93.
10. Nowak M.A., and May R.M. Virus Dynamics. Mathematical Principles of Immunology and Virology (2000), Oxford University Press
11. Molineaux L., and Dietz K. Review of intra-host models of malaria. Parassitologia 41 1-3 (2000) 221-231
12. Jacquez J.A., and Simon C.P. Qualitative theory of compartmental systems with lags. Math. Biosci. 180 (2002) 329-362
13. Nelson P.W., Murray J., and Perelson A.S. A model of HIV-1 pathogenesis that includes an intracellular delay. Math. Biosci. 163 (2000) 201-215
14. Mittler J.E., Sulzer B., Neumann A.U., and Perelson A.S. Influence of delayed viral production on viral dynamics in HIV-1 infected patients. Math. Biosci. 152 2 (1998) 143-163
15. Nelson P.W., and Perelson A.S. Mathematical analysis of delay differential equation models of HIV-1 infection. Math. Biosci. 179 1 (2002) 73-94
16. Jacquez J.A. Modeling with compartments. BioMedware (1999)
17. Jacquez J.A. Compartmental analysis in biology and medicine. BioMedware (1996)
18. MacDonald N. Time Lags in Biological Models, Lecture Notes in Biomathematics No 27 (1978), Springer-Verlag, Berlin
19. Cox D., and Miller H. The Theory of Stochastic Processes (1965), Chapman and Hall
20. Grossman Z., Feinberg M., Kuznetsov K., Dimitrov D., and Paul W. HIV infection: how effective is drug combination treatment. Immunol. Today 19 (1998) 528-532
21. Lloyd A.L. Destabilization of epidemic models with the inclusion of realistic distributions of infectious periods. Proc. R. Soc. London. B. Biol. Sci. 268 1470 (2001) 985-993
22. Lloyd A.L. The dependance of viral parameter estimates on the assumed viral life cycle: limitations of studies of viral load data. Proc. R. Soc. London. B. Biol. Sci. 268 (2001) 847-854
23. Lloyd A.L. Realistic distributions of infectious periods in epidemic models: changing patterns of persistence and dynamics. Theor. Popul. Biol. 60 (2001) 59-71
24. van den Driessche P., and Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180 (2002) 29-48
25. 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol. 28 4 (1990) 365-382
26. Sontag E.D. Mathematical Control Theory, Deterministic Finite Dimensional Systems, No. 6 in Texts in Applied Mathematics (1990), Springer-Verlag
27. Luenberger D.G. Introduction to Dynamic Systems. Theory, Models, and Applications (1979), John Wiley & Sons, Ltd.
28. Lee E., and Markus L. Foundations of Optimal Control Theory (1967), John Wiley & Sons. Ltd.
29. Jacquez J.A., and Simon C.P. Qualitative theory of compartmental systems. SIAM Rev. 35 1 (1993) 43-79
30. Beretta E., and Takeuchi Y. Global stability of lotka-volterra diffusion models with continuous time delay. SIAM J. Appl. Math. 48 (1988) 627-651
31. Hethcote H.W. The mathematics of infectious diseases. SIAM Rev. 42 4 (2000) 599-653 electronic
32. Diekmann O., and Heesterbeek J.A.P. Mathematical Epidemiology of Infectious Diseases, model building, analysis and interpretation, Wiley Series in Mathematical and Computational Biology (2000), John Wiley & Sons Ltd., Chichester
33. Butler G.J., and Waltman P. Persistence in dynamical systems. J. Differ. Equations 63 (1986) 255-263
34. Hale J.K. Asymptotic Behavior of Dissipative Systems (1988), A.M.S., Providence
35. Nagumo M. Über die lage der integralkurven gewohnlicher differentialgleichungen. Proc. Phys.-Math. Soc. Japan 24 (1942) 551-559
36. Bhatia N.P., and Szegö G.P. Dynamical Systems: Stability Theory and Applications (1967), Springer-Verlag
37. LaSalle J. Stability theory for ordinary differential equations. J. Differ. Equations 41 (1968) 57-65
38. A.S. Perelson, P.W. Nelson, Modeling viral infections, in: An Introduction To Mathematical Modeling in Physiology, Cell biology, and Immunology (New Orleans, LA, 2001), vol. 59, Proc. Sympos. Appl. Math., Amer. Math. Soc., Providence, RI, 2002, pp. 139-172.
39. Schwartz L. Methods of Mathematical Physics (1964), Addison-Wesley, Reading Mass.
40. De Leenheer P., and Smith H.L. Virus dynamics: a global analysis. SIAM J. Appl. Math. 63 4 (2003) 1313-1327