Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

On some epidemic models


Author: Gustaf Gripenberg
Journal: Quart. Appl. Math. 39 (1981), 317-327
MSC: Primary 92A15; Secondary 45M05
DOI: https://doi.org/10.1090/qam/636238
MathSciNet review: 636238
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The qualitative behavior of the solution $ x$ of the equation

$\displaystyle x\left( t \right) = k\left( {p\left( t \right) - \smallint _0^tA\... ...right)\left( {f\left( t \right) + \smallint _0^ta(t - s)x(s)ds} \right),t \ge 0$

is studied. This equation arises in the study of the spread of an infectious disease that does not induce permanent immunity.

References [Enhancements On Off] (What's this?)

  • [1] N. T. J. Bailey, The mathematical theory of infectious diseases and its applications, 2d ed., Hafner Press, New York, 1975 MR 0452809
  • [2] K. L. Cooke and J. A. Yorke, Some equations modelling growth processes and gonorrhea epidemics, Math. Biosci. 16, 75-101 (1973) MR 0312923
  • [3] O. Diekmann, Limiting behavior in an epidemic model, Nonlinear Anal. Theory, Methods, Appl. 1, 459-470 (1977) MR 0624451
  • [4] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Diff. Eqs. 33, 58-73 (1979) MR 540817
  • [5] G. Gripenberg, A Volterra equation with nonintegrable resolvent, Proc. Amer. Math. Soc. 73, 57-60 (1979) MR 512058
  • [6] G. Gripenberg, On the resolvents of Volterra equations with nonincreasing kernels, J. Math. Anal. Appl., 76, 134-145 (1980) MR 586652
  • [7] G. Gripenberg, Periodic solutions of an epidemic model, J. Math. Biol., 10, 271-280 (1980) MR 599811
  • [8] G. S. Jordan and R. L. Wheeler, A generalization of the Wiener-Levy theorem applicable to some Volterra equations, Proc. Amer. Math. Soc. 57, 109-114 (1976) MR 0405023
  • [9] R. K. Miller, Nonlinear Volterra integral equations, W. A. Benjamin, Menlo Park, 1971 MR 0511193
  • [10] D. F. Shea and S. Wainger, Variants of the Wiener-Levy theorem with applications to stability problems for some Volterra integral equations, Amer. J. Math. 97, 312-343 (1975) MR 0372521
  • [11] H. Smith, Periodic solutions of an epidemic model with a threshold, Rocky Mountain J. Math. 9, 131-142 (1979) MR 517979
  • [12] P. Waltman, Deterministic threshold models in the theory of epidemics, Lecture Notes in Biomathematics, Vol. 1, Springer-Verlag, New York, 1974 MR 0359874
  • [13] F. J. S. Wang, Asymptotic behavior of some deterministic epidemic models, SIAM J. Math. Anal. 9, 529-534 (1978) MR 0475993

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 92A15, 45M05

Retrieve articles in all journals with MSC: 92A15, 45M05


Additional Information

DOI: https://doi.org/10.1090/qam/636238
Article copyright: © Copyright 1981 American Mathematical Society

American Mathematical Society