Degla, Guy Existence and uniqueness results for a smooth model of periodic infectious diseases. (English) Zbl 1470.45006 Abstr. Appl. Anal. 2016, Article ID 1708527, 4 p. (2016). Summary: We prove the existence of a curve (with respect to the scalar delay) of periodic positive solutions for a smooth model of Cooke-Kaplan’s integral equation by using the implicit function theorem under suitable conditions. We also show a situation in which any bounded solution with a sufficiently small delay is isolated, clearing an asymptotic stability result of Cooke and Kaplan. MSC: 45G10 Other nonlinear integral equations 92D30 Epidemiology × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Cooke, K. L.; Kaplan, J. L., A periodicity threshold theorem for epidemics and population growth, Mathematical Biosciences, 31, 1-2, 87-104, (1976) · Zbl 0341.92012 · doi:10.1016/0025-5564(76)90042-0 [2] Leggett, R. W.; Williams, L. R., A fixed point theorem with application to an infectious disease model, Journal of Mathematical Analysis and Applications, 76, 1, 91-97, (1980) · Zbl 0448.47044 · doi:10.1016/0022-247x(80)90062-1 [3] Nussbaum, R., A periodicity threshold theorem for some nonlinear integral equations, SIAM Journal on Mathematical Analysis, 9, 2, 356-376, (1978) · Zbl 0385.45007 · doi:10.1137/0509024 [4] Agarwal, R. P.; O’Regan, D., Periodic solutions to nonlinear integral equations on the infinite interval modelling infectious disease, Nonlinear Analysis: Theory, Methods & Applications, 40, 1–8, 21-35, (2000) · Zbl 0958.45011 · doi:10.1016/s0362-546x(00)85002-6 [5] Amann, H.; Escher, J., Analysis II, (1999), Birkhäuser · Zbl 0940.26001 · doi:10.1007/978-3-0348-8972-8 [6] Ambrosetti, A.; Prodi, G., A Primer of Nonlinear Analysis. A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics, 34, (1993), Cambridge University Press · Zbl 0781.47046 [7] Ambrosetti, A.; Malchiodi, A., Nonlinear Analysis and Semilinear Elliptic Problems. Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge Studies in Advanced Mathematics, 104, (2007), Cambridge University Press · Zbl 1125.47052 · doi:10.1017/cbo9780511618260 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.