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Global dynamics of an SEI model with acute and chronic stages. (English) Zbl 1131.92058

Summary: A model with acute and chronic stages in a population with exponentially varying size is proposed. An equivalent system is obtained, which has two equilibriums: a disease-free equilibrium and an endemic equilibrium. The stability of these two equilibriums is controlled by the basic reproduction number \(R_{0}\). When \(R_{0}<1\), the disease-free equilibrium is globally stable. When \(R_{0}>1\), the disease-free equilibrium is unstable and the unique endemic equilibrium is locally stable. When \(R_{0}>1\) and \(\gamma =0,\alpha =0\), the endemic equilibrium is globally stable in \(\varGamma ^{0}\).

MSC:

92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
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[1] Bandy, U., Hepatitis C virus (HCV): a silent epidemic, Med. Health R.I. Jun, 82, 223 (1999)
[2] Busenberg, S.; Hadeler, K., Demography and epidemics, Math. Biosci., 101, 63 (1990) · Zbl 0751.92012
[3] Busenerg, S.; van den Driessche, P., Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28, 257 (1990) · Zbl 0725.92021
[4] Colina, R.; Azambuja, C.; Uriarte, R.; Mogdasy, C.; Cristina, J., Evidence of increasing diversification of Hepatitis C viruses, J. Gen. Virol., 80, 1377 (1999)
[5] Coppel, W. A., Stability and Asymptotic Behavior of Differential Equations (1956), Heath: Heath Boston · Zbl 0154.09301
[6] Derrick, W.; van den Driessche, P., A disease transmission model in a nonconstant population, J. Math. Biol., 31, 495 (1993) · Zbl 0772.92015
[7] Esteva, L.; Vargas, C., A model for dengue disease with variable human population, J. Math. Biol., 38, 22 (1999) · Zbl 0981.92016
[8] Feng, Z.; Iannelli, M.; Milnew, F. A., A two-strain tuberculosis model with age of infection, SIAM J. Appl. Math., 62, 1634-1656 (2002) · Zbl 1017.35066
[9] Fiedler, M., Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czechoslovak Math. J., 99, 392 (1974) · Zbl 0345.15013
[10] Hadeler, K.; Castillo-Chavez, C., A core group model for disease transmission, Math. Biosci., 128, 41 (1995) · Zbl 0832.92021
[11] Li, M. Y.; Muldowney, J. S., A geometric approach to the global-stability problems, SIAM J. Math. Anal., 27, 1070 (1996) · Zbl 0873.34041
[12] Liu, W.-M.; van den Driessche, P., Epidemiological models with varying population size and dose-dependent latent period, Math. Biosci., 128, 57 (1995) · Zbl 0832.92023
[13] Ma, Z.; Zhou, Y., Qualitative and Stable Methods of Ordinary Differential Equation (2003), Science Press: Science Press Beijing
[14] Ma, Z.; Zhou, Y.; Wang, W.; Jin, Z., Mathematical Model and Research of Epidemic Dynamics (2004), Science Press: Science Press Beijing
[15] Martcheva, M.; Castillo-Chavez, C., Diseases with chronic stage in a population with varying size, Math. Biosci., 182, 1-25 (2003) · Zbl 1012.92024
[16] Reade, B.; Bowers, R.; Begon, M.; Gaskell, R., A model of disease and vaccination for infections with acute and chronic phases, J. Theoret. Biol., 190, 355 (1998)
[17] Thieme, H., Epidemic and demographic interaction in the spread of potentially fatal diseases in a growing population, Math. Biosci., 111, 99 (1992) · Zbl 0782.92018
[18] Thieme, H.; Castillo-Chavez, C., How may infection-age-dependent infectivity affect the dynamics of HIV/AIDS?, SIAM J. Appl. Math., 53, 1447 (1993) · Zbl 0811.92021
[19] H. Thieme, C. Castillo-Chavez, On the role of variable infectivity in the dynamics of HIV epidemic, Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomathematics, vol. 83, Springer, Berlin, 1993, p. 157.; H. Thieme, C. Castillo-Chavez, On the role of variable infectivity in the dynamics of HIV epidemic, Mathematical and Statistical Approaches to AIDS Epidemiology, Lecture Notes in Biomathematics, vol. 83, Springer, Berlin, 1993, p. 157.
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