Herzog, Gerd; Redheffer, Ray Nonautonomous SEIRS and Thron models for epidemiology and cell biology. (English) Zbl 1067.92053 Nonlinear Anal., Real World Appl. 5, No. 1, 33-44 (2004). Summary: Remarks on positivity of certain vector-valued functions are applied to the SEIRS equations for infectious diseases and to a nonlinear system recently introduced in connection with cell biology by C. D. Thron [Biophys. Chem. 79, 95–106 (1999)]. In both cases the formerly constant coefficients are replaced by functions of time, and in both cases we give conditions under which the solutions exist globally and are positive. For the nonautonomous SEIRS equations conditions are also given which ensure that the limiting population as \(t\to \infty\) contains susceptible individuals only, hence is disease-free. Cited in 25 Documents MSC: 92D30 Epidemiology 92C37 Cell biology 34C60 Qualitative investigation and simulation of ordinary differential equation models 34C11 Growth and boundedness of solutions to ordinary differential equations Keywords:SEIRS equation; Positive solutions; Quasimonotonicity; Limiting behavior PDFBibTeX XMLCite \textit{G. Herzog} and \textit{R. Redheffer}, Nonlinear Anal., Real World Appl. 5, No. 1, 33--44 (2004; Zbl 1067.92053) Full Text: DOI References: [1] Herzog, G., On quasimonotone increasing systems of ordinary differential equations, Portugal. Math., 57, 1, 35-43 (2000) · Zbl 0946.34006 [2] Hethcote, H. W., The mathematics of infectious diseases, SIAM Rev., 42, 599-653 (2000) · Zbl 0993.92033 [3] Kuznetsov, Yu. A.; Piccardi, C., Bifurcation analysis of periodic SEIR and SIR epidemic models, J. Math. Biol., 32, 109-121 (1994) · Zbl 0786.92022 [4] Lasota, A., Sur l’effet épidermique extérieur et intérieur pour les inégalités difféerentielles ordinaires, Ann. Polon. Math., VI, 259-264 (1959) · Zbl 0090.29904 [5] Li, M. Y.; Graef, J. R.; Wang, L.; Karsai, J., Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160, 191-213 (1999) · Zbl 0974.92029 [6] Li, M. Y.; Muldowney, J. S.; van den Driessche, P., Global stability of SEIRS models in epidemiology, Canad. Appl. Math. Quart., 7, 409-425 (1999) · Zbl 0976.92020 [7] Liu, W.; Hethcote, H. W.; Levin, S. A., Dynamical behavior of epidemiological models with nonlinear rates, J. Math. Biol., 25, 359-380 (1987) · Zbl 0621.92014 [8] Mukherjee, D.; Chattopadhyay, J.; Tapaswi, P. K., Global stability results of epidemiological models with nonlinear incidence rates, Math. Comput. Modelling, 18, 89-92 (1993) · Zbl 0791.92020 [9] Redheffer, R. M., Stability by Freshman Calculus, Amer. Math. Monthly, 71, 6, 656-659 (1964) · Zbl 0125.32602 [10] Redheffer, R., Gewöhnliche Differentialungleichungen mit quasimonotonen Funktionen in normierten Räumen, Arch. Rational Mech. Anal., 52, 2, 121-133 (1973) · Zbl 0289.34094 [11] Thron, C. D., Mathematical analysis of binary activation of a cell cycle kinase which down-regulates its own inhibitor, Biophys. Chem., 79, 95-106 (1999) [12] Walter, W., Gewöhnliche Differential-Ungleichungen im Banachraum, Arch. Math., 20, 36-47 (1969) · Zbl 0195.38402 [13] Walter, W., Differential and Integral Inequalities (1970), Springer: Springer Berlin, Heidelberg, New York [14] Ważewski, T., Certaines propositions de caractère “épidermique” relatives aux inégalités différentielles, Ann. Soc. Polon. Math., 24, fasc. 2, 1-12 (1953) · Zbl 0046.31301 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.