×

zbMATH — the first resource for mathematics

Role of incidence function in vaccine-induced backward bifurcation in some HIV models. (English) Zbl 1134.92026
Summary: The phenomenon of backward bifurcation in disease models, where a stable endemic equilibrium co-exists with a stable disease-free equilibrium when the associated reproduction number is less than unity, has important implications for disease control. In such a scenario, the classical requirement of the reproduction number being less than unity becomes only a necessary, but not sufficient, condition for disease elimination.
This paper addresses the role of the choice of incidence functions in a vaccine-induced backward bifurcation in HIV models. Several examples are given where backward bifurcations occur using standard incidence, but not with their equivalents that employ mass action incidence. Furthermore, this result is independent of the type of vaccination program adopted. These results emphasize the need for further work on the incidence functions used in HIV models.

MSC:
92C60 Medical epidemiology
34D20 Stability of solutions to ordinary differential equations
34C23 Bifurcation theory for ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arino, J.; McCluskey, C.C.; van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. appl. math., 64, 260, (2003) · Zbl 1034.92025
[2] ()
[3] ()
[4] Blower, S.M.; McLean, A.R., Prophylactic vaccines, risk behaviour change, and the probability of eradicating HIV in San Francisco, Science, 265, 1451, (1994)
[5] Brander, C.; Frahm, N.; Walker, B.D., The challenges of host and viral diversity in HIV vaccine design, Curr. opin. immunol., 18, 4, 430, (2006)
[6] Brauer, F.; van den Driessche, P., Models for transmission of disease with immigration of infectives, Math. biosci., 171, 143, (2001) · Zbl 0995.92041
[7] Brauer, F., Backward bifurcations in simple vaccination models, J. math. anal. appl., 298, 2, 418, (2004) · Zbl 1063.92037
[8] Brauer, F., Some simple epidemic models, Math. biosci. eng., 3, 1, 1, (2006) · Zbl 1089.92042
[9] Castillo-Chavez, C.; Cooke, K.; Huang, W.; Levin, S.A., Results on the dynamics for models for the sexual transmission of the human immunodeficiency virus, Appl. math. lett., 2, 327, (1989) · Zbl 0703.92022
[10] Castillo-Chavez, C.; Cooke, K.; Huang, W.; Levin, S.A., The role of long incubation periods in the dynamics of HIV/AIDS. part 2: multiple group models, (), 200
[11] Castillo-Chavez, C.; Song, B., Dynamical models of tuberculosis and their applications, Math. biosci. eng., 1, 2, 361, (2004) · Zbl 1060.92041
[12] Dietz, K., Transmission and control of arbovirus disease, (), 104 · Zbl 0322.92023
[13] Dushoff, J.; Wenzhang, H.; Castillo-Chavez, C., Backwards bifurcations and catastrophe in simple models of fatal diseases, J. math. biol., 36, 227, (1998) · Zbl 0917.92022
[14] Elbasha, E.H.; Gumel, A.B., Theoretical assessment of public health impact of imperfect prophylactic HIV-1 vaccines with therapeutic benefits, Bull. math. biol., 68, 577, (2006) · Zbl 1334.91060
[15] Feng, Z.; Castillo-Chavez, C.; Capurro, F., A model for tuberculosis with exogenous reinfection, Theor. pop. biol., 57, 235, (2000) · Zbl 0972.92016
[16] Gomez-Acevedo, H.; Li, M.Y., Backward bifurcation in a model for HTLV-I infection of CD4^+ T cells, Bull. math. biol., 67, 1, 101, (2005) · Zbl 1334.92231
[17] Gumel, A.B.; Connell McCluskey, C.; van den Driessche, P., Mathematical study of a staged-progression HIV model with imperfect vaccine, Bull. math. biol., 68, 2105, (2006) · Zbl 1296.92124
[18] Hadeler, K.P.; Castillo-Chavez, C., A core group model for disease transmission, Math. biosci., 128, 41, (1995) · Zbl 0832.92021
[19] Hale, J.K., Ordinary differential equations, (1969), John Wiley and Sons New York · Zbl 0186.40901
[20] Hethcote, H.W., Qualitative analysis of communicable disease models, Math. biosci., 28, 335, (1976) · Zbl 0326.92017
[21] Hethcote, H.W.; van Ark, J.W., Epidemiology models for heterogeneous populations: proportionate mixing, parameter estimation, and immunization programs, Math. biosci., 84, 85, (1987) · Zbl 0619.92006
[22] Hethcote, H.W., A thousand and one epidemic models, (), 504 · Zbl 0819.92020
[23] Hethcote, H.W., The mathematics of infectious diseases, SIAM rev., 42, 4, 599, (2000) · Zbl 0993.92033
[24] Hyman, J.M.; Li, J.; Stanley, E.A., The differential infectivity and staged progression models for the transmission of HIV, Math. biosci., 208, 227, (1999)
[25] Hyman, J.M.; Li, J., The reproductive number for an HIV model with differential infectivity and staged progression, Linear algebra its appl. v.398, 1-3, 101, (2005) · Zbl 1062.92060
[26] IAVI (2006), AIDS vaccine blueprint 2006: actions to strengthen global research and development, International AIDS Vaccine Initiative (IAVI), New York, 2006.
[27] Kermack, W.O.; McKendrick, A.G., A contribution to the mathematical theory of epidemics, Proc. roy. soc. A, 115, 700, (1927) · JFM 53.0517.01
[28] Kribs-Zaleta, C.; Valesco-Hernandez, J., A simple vaccination model with multiple endemic states, Math biosci., 164, 183, (2000) · Zbl 0954.92023
[29] Kribs-Zaleta, C., To switch or taper off: the dynamics of saturation, Math. biosci., 192, 137, (2004) · Zbl 1072.92044
[30] Lajmanovich, A.; Yorke, J.A., A deterministic model for gonorrhea in a non-homogeneous population, Math. biosci., 28, 221, (1976) · Zbl 0344.92016
[31] Lakshmikantham, V.; Leela, S.; Martynyuk, A.A., Stability analysis of nonlinear systems, (1989), Marcel Dekker Inc. New York and Basel · Zbl 0676.34003
[32] Markel, H., The search for effective HIV vaccines, N. engl. J. med., 353, 753, (2005)
[33] McLean, A.R.; Blower, S.M., Imperfect vaccines and herd immunity to HIV, Proc. roy. soc. lond. B, 253, 9, (1993)
[34] O’Brien, T.; Busch, N.; Donegan, E.; Ward, J.; Wong, L., Heterosexual transmission of human immunodeficiency virus type 1 from transfusion recipients to their sex partners, J. acquir. immune. defic. syndr., 7, 705, (1994)
[35] Perelson, A.; Nelson, P., Mathematical analysis of HIV-1 dynamics in vivo, SIAM rev., 41, 3, (1999) · Zbl 1078.92502
[36] Simon, C.P.; Jacquez, J.A., Reproduction numbers and the stability of equilibrium of SI models for heterogeneous populations, SIAM J. appl. math., 52, 541, (1992) · Zbl 0765.92019
[37] Smith, H.L.; Waltman, P., The theory of the chemostat, (1995), Cambridge University · Zbl 0860.92031
[38] UNAIDS (2006), Report on the global AIDS epidemic: a UNAIDS 10th anniversary special edition, Joint United Nations Programme on HIV/AIDS (UNAIDS), May 2006.
[39] van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. biosci., 180, 29, (2002) · Zbl 1015.92036
[40] Wang, W., Backward bifurcation of an epidemic model with treatment, Math. biosci., 201, 1-2, 58, (2006) · Zbl 1093.92054
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.