Globally stable vaccine-induced eradication of horizontally and vertically transmitted infectious diseases with periodic contact rates and disease-dependent demographic factors in the population. (English) Zbl 1018.92030

Summary: Within the framework of SEIR-like epidemic models, we studied the conditions for stable eradication of some families of vertically and horizontally transmitted infectious diseases in the case of periodically varying contact rates. By means of Floquet’s theory, we found a condition for the eradication solution to be locally asymptotically stable. We then demonstrated that the same condition guarantees also that this vaccine-induced disease-free solution is globally asymptotically stable.
A model with interacting populations is also considered. In the final part of this work, we extended the model by taking into account the variation of population size, the impact of disease-related deaths and reduction of fertility.


92D30 Epidemiology
34D05 Asymptotic properties of solutions to ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
Full Text: DOI


[1] Busenberg, S.; Cooke, K., Vertically transmitted diseases, (1992), Springer-Verlag Berlin-Heidelberg-New York · Zbl 0512.92017
[2] Chang, M.H., Universal hepatitis B vaccination in Taiwan and the incidence of hepatocarcinoma in children, New engl. J. med., 336, 1855-1859, (1997)
[3] Duncan, C.J.; Duncan, S.R.; Scott, S., Oscillatory dynamics of small-pox and the impact of vaccination, J. th. biol., 183, 447-454, (1996)
[4] Coppel, W., Asymptotic behaviour of dynamical systems, (1965), Heat Boston, MA · Zbl 0154.09301
[5] ()
[6] Greenhalgh, D.; Das, R., Modelling epidemics with variable contact rates, Th. pop. biol., 47, 129-179, (1995) · Zbl 0833.92018
[7] Kuznetsov, Yu.A.; Piccardi, C., Bifurcation analysis of periodic SEIR ans SIR epidemic models, J. math. biol., 32, 109-121, (1994) · Zbl 0786.92022
[8] Li, M.Y.; Muldowney, J.S., A geometric approach to global stability problem, SIAM J. math. an., 27, 1070-1083, (1996) · Zbl 0873.34041
[9] Li, M.Y.; Smith, H.L.; Wang, L., Global dynamics of a SEIR epidemic model with vertical transmission, SIAM J. appl. math., 62, 1, 58-69, (2002) · Zbl 0991.92029
[10] Li, M.Y.; Wang, L., Global stability in some SEIR models, () · Zbl 1022.92035
[11] London, W.P.; Yorke, J.A., Recurrent outbreaks of measles, chickenpox and mumps, Am. J. epid., 98, 453-468, (1973)
[12] Pourabbas, E.; d’Onofrio, A.; Rafanelli, M., A method to estimate the incidence of communicable diseases under seasonal fluctuations with application to cholera, Appl. math. comput., 118, 161-174, (2001) · Zbl 1017.92032
[13] Smith, H.L., Subharmonic bifurcation in a S-I-R epidemic model, J. math. biol., 17, 163-177, (1983) · Zbl 0578.92023
[14] Smith, H.L., Monotone dynamical systems, (1995), American Mathematical Society Providence RI
[15] Schwartz, I.B.; Smith, H.L., Infinite subharmonics bifurcation in an SEIR epidemic model, J. math. biol., 18, 233-253, (1983) · Zbl 0523.92020
[16] Schwartz, I.B., Small amplitude, long period outbreaks in seasonally driven epidemic, J. math. biol., 30, 473-491, (1992) · Zbl 0745.92026
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