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On constant-sign periodic solutions in modelling the spread of interdependent epidemics

Published online by Cambridge University Press:  17 February 2009

Ravi P. Agarwal ,
Donal O'Regan  and
Patricia J. Y. Wong
Ravi P. Agarwal
Affiliation:
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, Florida 32901–6975, USA; e-mail: [email protected].
Donal O'Regan
Affiliation:
Department of Mathematics, National University of Ireland, Galway, Ireland.
Patricia J. Y. Wong
Affiliation:
School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798, Singapore; e-mail: [email protected].
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Abstract

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We consider the following model that describes the spread of n types of epidemics which are interdependent on each other:

Our aim is to establish criteria such that the above system has one or multiple constant-sign periodic solutions (u1, u2, …, un), that is, for each 1 ≤ i ≤ n, ui, is periodic and θiui ≥ 0 where θi, ε (1, −1) is fixed. Examples are also included to illustrate the results obtained.

Keywords

periodic solutionsintegral equationsepidemicsfixed point theorems
Type
Research Article
Information
The ANZIAM Journal , Volume 47 , Issue 3 , January 2006 , pp. 309 - 332
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Agarwal, R. P., O'Regan, D. and Wong, P. J. Y., Positive solutions of differential, difference and integral equations (Kluwer, Dordrecht, 1999).Google Scholar
[2]Agarwal, R. P., O'Regan, D. and Wong, P. J. Y., “Constant-sign Lp solutions for a system of integral equations”, Results Math. 46 (2004) 195–219.Google Scholar
[3]Agarwal, R. P., O'Regan, D. and Wong, P. J. Y., “Constant-sign solutions of a system of Fredholm integral equations”, Acta Appl. Math. 80 (2004) 57–94.CrossRefGoogle Scholar
[4]Agarwal, R. P., O'Regan, D. and Wong, P. J. Y., “Eigenvalues of a system of Fredholm integral equations”, Math. Comput. Modelling 39 (2004) 1113–1150.Google Scholar
[5]Agarwal, R. P., O'Regan, D. and Wong, P. J. Y., “Triple solutions of constant sign for a system of Fredholm integral equations”, Cubo 6 (2004) 1–45.Google Scholar
[6]Agarwal, R. P., O'Regan, D. and Wong, P. J. Y., “Constant-sign solutions of a system of integral equations: The semipositone and singular case”, Asymptot. Anal. 43 (2005) 47–74.Google Scholar
[7]Agarwal, R. P., O'Regan, D. and Wong, P. J. Y., “Constant-sign periodic and almost periodic solutions for a system of integral equations”, Acta Appl. Math. (to appear).Google Scholar
[8]Agarwal, R. P. and Wong, P. J. Y., Advanced topics in difference equations (Kluwer, Dordrecht, 1997).Google Scholar
[9]Cooke, K. L. and Kaplan, J. L., “A periodicity threshold theorem for epidemics and population growth”, Mat. Biosc. 31 (1976) 87–104.Google Scholar
[10]Erbe, L. H., Hu, S. and Wang, H., “Multiple positive solutions of some boundary value problems”, J. Math. Anal. Appl. 184 (1994) 640–648.CrossRefGoogle Scholar
[11]Erbe, L. H. and Wang, H., “On the existence of positive solutions of ordinary differential equations”, Proc. Amer. Math. Soc. 120 (1994) 743–748.Google Scholar
[12]Guo, D. and Lakshmikantham, V., Nonlinear problems in abstract cones (Academic Press, San Diego, 1988).Google Scholar
[13]Krasnosel'skii, M. A., Positive solutions of operator equations (Noordhoff, Groningen, 1964).Google Scholar
[14]Leggett, R. W. and Williams, L. R., “Multiple positive fixed points of nonlinear operators on ordered Banach spaces”, Indiana Univ. Math. J. 28 (1979) 673–688.Google Scholar
[15]Leggett, R. W. and Williams, L. R., “A fixed point theorem with application to an infectious disease model”, J. Math. Anal. Appl. 76 (1980) 91–97.CrossRefGoogle Scholar
[16]Lian, W., Wong, F. and Yeh, C., “On the existence of positive solutions of nonlinear second order differential equations”, Proc. Amer Math. Soc. 124 (1996) 1117–1126.Google Scholar
[17]Nussbaum, R., “A periodicity threshold theorem for some nonlinear integral equations”, SIAM J. Anal. 9 (1978) 356–376.CrossRefGoogle Scholar
[18]O'Regan, D. and Meehan, M., Existence theory for nonlinear integral and integrodifferential equations (Kluwer, Dordrecht, 1998).CrossRefGoogle Scholar
[19]Smith, H., “On periodic solutions of delay integral equations modelling epidemics and population growth”, Ph. D. Thesis, University of Iowa City, 1976.Google Scholar
[20]Williams, L. R. and Leggett, R. W., “Nonzero solutions of nonlinear integral equations modeling infectious disease”, SIAM J. Anal. 13 (1982) 112–121.Google Scholar