A model of HIV/AIDS with staged progression and amelioration. (English) Zbl 1008.92032

Summary: An epidemic model with multiple stages of infection is studied. The model allows for infected individuals to move from advanced stages of infection back to less advanced stages of infection. A threshold parameter which determines the local stability of the disease free equilibrium is found and interpreted. Under conditions on the parameters, global stability is demonstrated using techniques involving compound matrices.


92D30 Epidemiology
34D20 Stability of solutions to ordinary differential equations
93C95 Application models in control theory
34C60 Qualitative investigation and simulation of ordinary differential equation models
Full Text: DOI


[1] Hyman, J.M.; Li, J.; Stanley, E.A., The differential infectivity and staged progression models for the transmission of HIV, Math. biosci., 155, 77, (1999) · Zbl 0942.92030
[2] Lin, X.; Hethcote, H.W.; van den Driessche, P., An epidemiological model for HIV/AIDS with proportional recruitment, Math. biosci., 118, 181, (1993) · Zbl 0793.92011
[3] Hale, J.K., Ordinary differential equations, (1969), John Wiley New York · Zbl 0186.40901
[4] Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J., On the definition and the computation of the basic reproduction ratio \(R0\) in models for infectious diseases in heterogeneous populations, J. math. biol., 28, 365, (1990) · Zbl 0726.92018
[5] Simon, C.P.; Jacquez, J.A., Reproduction numbers and the stability of equilibria of SI models for heterogeneous populations, SIAM J. appl. math., 52, 541, (1992) · Zbl 0765.92019
[6] Hofbauer, J.; Sigmund, K., Evolutionary games and population dynamics, (1998), Cambridge University Cambridge · Zbl 0914.90287
[7] Li, M.Y.; Muldowney, J.S., Dynamics of differential equations on invariant manifolds, J. differ. equat., 168, 295, (2000) · Zbl 0983.34033
[8] Muldowney, J.S., Compound matrices and ordinary differential equations, Rocky mount. J. math., 20, 857, (1990) · Zbl 0725.34049
[9] Li, M.Y., Dulac criteria for autonomous systems having an invariant affine manifold, J. math. anal. appl., 199, 374, (1996) · Zbl 0851.34031
[10] Li, M.Y.; Graef, J.R.; Wang, L.; Karsai, J., Global dynamics of a SEIR model with varying total population size, Math. biosci., 160, 191, (1999) · Zbl 0974.92029
[11] Martin, R.H., Logarithmic norms and projections applied to linear differential systems, J. math. anal. appl., 45, 432, (1974) · Zbl 0293.34018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.