Analysis of a nonlinear integral equation modelling infection diseases

3 years ago

Abstract

To describe the spread of virus diseases with contact rate that varies seasonally, the following delay integral equation has been proposed by K.L. Cooke and J.L. Kaplan

$x\left( t\right) =\int_{t-\tau}^{t}f\left( s,x\left( s\right) \right) ds$

This model can also be interpreted as an evolution equation of a single species population. The purpose of this paper is to describe and improve recent results on this equation, obtained by the authors in the last decade. Our analysis is concerned with the existence, uniqueness, approximation and continuous dependence on data of the positive solutions of the initial value problem, and of the periodic solutions. We use topological methods (fixed point theorems, continuation principle) and monotone iterative techniques.

Authors

Radu Precup
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania

Eduard Kirr
Faculty of Mathematics and Informatics University “Babes-Bolyai”, Cluj, Romania

Keywords

nonlinear integral equation; positive solutions; periodic solutions; fixed point; continuation principle; monotone iterations; continuous dependence;population dynamics.

Paper coordinates

R. Precup, E. Kirr, Analysis of a nonlinear integral equation modelling infection diseases, In: “Proceedings of the International Conference on Analysis and Numerical Computation, Timişoara, May 19-21, 1997”, Şt. Balint ed., West Univ. of Timişoara, 1997, 178-195.

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About this paper

Journal

“Proceedings of the International Conference on Analysis and Numerical Computation, Timişoara

Publisher Name

West Univ. of Timişoara

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[1] K.L. Cooke, J.L. Kaplan, A periodicity threshold theorem for epidemics and population growth, Math. Biosci. 31 (1976), 87-104.
[2] H.L. Smith, On periodic solutions of a delay integral equation modelling epidemics, J. Math. Biol. 4 (1977), 69-80.
[3] R.D. Nussbaum, A periodicity threshold theorem for some nonlinear integral equations, SIAM J. Math. Anal. 9 (1978), 356-367.
[4] J. Kaplan, M. Sorg, J. Yorke, Asymptotic behavior for epidemic equations, Nonlinear Analysis (1978).
[5] S. Busenberg, K. Cooke, Periodic solutions of delay differential equations arising in some models of epidemics, in “Applied Nonlinear Analysis” (Proc. Third Internat. Conf., Univ. Texas, Arlington, Texas, 1978), Academic
Press, New York, 1979, 67-78.
[6] I.A. Rus, Principii §i aplicatii ale teoriei punctului fix, Dacia, Cluj, 1979.
[7] R.W. Leggett, L.R. Williams, A fixed point theorem with application to an infectious disease model, J. Math. Anal. Appl. 76 (1980), 91-97.
[8] H.L. Smith, An abstract threshold theorem for one parameter families of positive noncompact operators, Funkcial Ekvac. 24 (1981), 141-152.
[9] L.R. Williams, R.W. Leggett, Nonzero solutions of nonlinear integral equations modeling infectious disease, SIAM J. Math. Anal. 13 (1982), 112-121.
[10] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
[11] A. Canada, Method of upper and lower solutions for nonlinear integral equations and an application to an infectious disease model, in “Dynamics of Infinite Dimentional Systems”, S.N. Chow and J.K. Hale eds., Springer, Berlin, 1987, 39-44.
[12] D. Guo, V. Lakshmikantham, Positive solutions of nonlinear integral equations arising in infectious diseases, J. Math. Anal. Appl. 134 (1988), 1-8.
[13] I.A. Rus, A delay integral equation from biomathematics, in “Seminar on Differential Equations: Preprint Nr. 3, 1989”, I.A. Rus ed., University “Babe§- Bolyai”, Cluj, 1989, 87-90.
[14] S.G. Hristova, D.D. Bainov, Method for solving the periodic problem for integro-differential equations, Matematiche (Catania) 44 (1989), 149-157.
[15] N.G. Kazakova, D.D. Bainov, An approximate solution of the initial value problem for integro-differential equations with a deviating argument, Math. J. Toyama Univ. 13 (1990), 9-27.
[16] A.M. Fink, J.A. Gatica, Positive almost periodic solutions of some delay integral equations, J. Diff. Eqns. 83 (1990), 166-178.
[17] R. Precup, Positive solutions of the initial value problem for an integral equation modeling infectious disease, in “Seminar on Differential Equations: Preprint Nr. 3, 1991”, I.A. Rus ed., University “Babe§-Bolyai”, Cluj, 1991, 25-30.
[18] R. Precup, Ecuatii integrale neliniare, Univ. “Babes,-Bolyai”, Cluj, 1993.
[19] R. Torrejön, Positive almost periodic solutions of a state-dependent delay nonlinear integral equation, Nonlinear Analysis 20 (1993), 1383-1416.
[20] R. Precup, Periodic solutions for an integral equation from biomathematics via Leray-Schauder principle, Studia Univ. Babe§-Bolyai, Mathematica 39, No. 1 (1994), 47-58.
[21] E. Kirr, Existence of periodic solutions for some integral equations arising in infectious diseases, Studia Univ. Babe§-Bolyai, Mathematica 39, No. 2 (1994), 107-119.
[22] A. Canada, A. Zertiti, Method of upper and lower solutions for nonlinear delay integral equations modelling epidemics and population growth, Math. Models and Methods in Applied Sciences 4 (1994), 107-120.
[23] A. Canada, A. Zertiti, Systems of nonlinear delay integral equations modelling population growth in a periodic environment, Comment. Math. Univ. Carolinae 35, No. 4 (1994), 633-644.
[24] R. Precup, Monotone technique to the initial values problem for a delay integral equation from biomathematics, Studia Univ. Babe§-Bolyai, Mathematica 40, No. 2 (1995), 63-73.
[25] E. Kirr, Existence and continuous dependence on data of the positive solutions of an integral equation from biomathematics, Studia Univ. Babe§- Bolyai, Mathematica 41, No. 2 (1996).
[26] Aid Dads, K. Ezzinbi, 0. Arino, Positive almost periodic solution for some nonlinear delay integral equation, Nonlinear Studies 3 (1996), 85-101.
[27] A. Granas, R. Guenther, J. Lee, Nonlinear Boundary Value Problems for Ordinary Differential Equations, Dissertationes Mathematicae, Vol. 244, Warszawa, 1985.
[28] V. Kolmanovskii, A. Myshkis, Applied Theory of Functional Differential Equations, Kluwer, Dordrecht, 1992.

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