×

zbMATH — the first resource for mathematics

Global dynamics of delay epidemic models with nonlinear incidence rate and relapse. (English) Zbl 1208.34125
Summary: A mathematical model for a disease with a general exposed distribution, the possibility of relapse and nonlinear incidence rate is proposed. By the method of Lyapunov functionals, it is shown that the disease dies out if \(\operatorname{Re}_{0}\leq 1\) and that the disease becomes endemic if \(\operatorname{Re}_{0}>1\). Applications are also made to the special case with a discrete delay and the result confirms that the endemic equilibrium is globally asymptotically stable.

MSC:
34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K20 Stability theory of functional-differential equations
92D30 Epidemiology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Chin, J., Control of communicable diseases manual, (1999), American Public Health Association Washington
[2] Martin, S.W., Livestock disease eradication: evaluation of the cooperative state federal bovine tuberculosis eradication program, (1994), National Academy Press Washington
[3] VanLandingham, K.E.; Marsteller, H.B.; Ross, G.W.; Hayden, F.G., Relapse of herpes simplex encephalitis after conventional acyclovir therapy, Jama, 259, 1051-1053, (1988)
[4] van den Driessche, P.; Wang, L.; Zou, X., Modeling diseases with latency and relapse, Math. biosci. eng., 4, 205-219, (2007) · Zbl 1123.92018
[5] Brown, G.C.; Hasibuan, R., Conidial discharge and transmission efficiency of neozygites floridana, an entomopathogenic fungus infecting two-spotted spider mites under laboratory conditions, J. invertebr. pathol., 65, 10-16, (1995)
[6] Capasso, V.; Serio, G., A generalisation of the kermack – mckendrick deterministic epidemic model, Math. biosci., 42, 43-61, (1978) · Zbl 0398.92026
[7] Gao, S.; Chen, L.; Nieto, J.J.; Torres, A., Analysis of a delayed epidemic model with pulse vaccination and saturation incidence, Vaccine, 24, 6037-6045, (2006)
[8] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. math. biol., 25, 359-380, (1987) · Zbl 0621.92014
[9] Liu, W.M.; Levin, S.A.; Iwasa, Y., Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. math. biol., 23, 187-204, (1986) · Zbl 0582.92023
[10] Yuan, Z.; Wang, L., Global stability of epidemiological models with group mixing and nonlinear incidence rates, Nonlinear anal. RWA, 11, 995-1004, (2010) · Zbl 1254.34075
[11] G. Huang, Y. Takeuchi, W. Ma, D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence Rate, Bull. Math. Biol., doi:10.1007/s11538-009-9487-6. · Zbl 1197.92040
[12] Li, M.Y.; Shuai, Z., Global-stability problem for coupled systems of differential equations on networks, J. differential equations, 248, 1-20, (2010) · Zbl 1190.34063
[13] Li, M.Y.; Shuai, Z.; Wang, C., Global stability of multi-group epidemic models with distributed delays, J. math. anal. appl., 361, 38-47, (2010) · Zbl 1175.92046
[14] Liu, S.; Wang, L., Global stability of an HIV-1 model with distributed intracellular delays and a combination therapy, Math. biosci. eng., 7, 677-687, (2010)
[15] McCluskey, C.C., Global stability for an SEIR epidemiological model with varying infectivity and infinite delay, Math. biosci. and eng., 6, 603-610, (2009) · Zbl 1190.34108
[16] McCluskey, C.C., Complete global stability for an SIR epidemic model with delay-distributed or discrete, Nonlinear anal. RWA, 11, 55-59, (2010) · Zbl 1185.37209
[17] Miller, R.K., Nonlinear Volterra integral equations, (1971), W.A. Benjamin Inc. New York · Zbl 0448.45004
[18] Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A.J., On the definition and the computation of the basic reproduction ratio \(\operatorname{Re}_0\) in models for infectious diseases in heterogeneous populations, J. math. biol., 28, 365-382, (1990) · Zbl 0726.92018
[19] Hale, J.K.; Kato, J., Phase space for retarded equation with infinite delay, Funk. ekv., 21, 11-41, (1978) · Zbl 0383.34055
[20] Hino, Y.; Murakami, S.; Naito, T., Functional differential equations with infinite delay, (1991), Springer-Verlag Germany · Zbl 0732.34051
[21] Feng, Z.; Huang, W.; Castillo-Chavez, C., On the role of variable latent periods in mathematical models for tuberculosis, J. dynam. differential equations, 13, 425-452, (2001) · Zbl 1012.34045
[22] Hale, J.; Waltman, P., Persistence in infinite-dimensional systems, SIAM J. math. anal., 20, 388-395, (1989) · Zbl 0692.34053
[23] Röst, G.; Wu, J., SEIR epidemiological model with varying infectivity and infinite delay, Math. biosci. eng., 5, 389-402, (2008) · Zbl 1165.34421
[24] Hale, J.K.; Verduyn Lunel, S.M., ()
[25] Liu, S.; Beretta, E., Stage-structured predator – prey model with the beddington – deangelis functional response, SIAM J. appl. math., 66, 1101-1129, (2006) · Zbl 1110.34059
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.