Abstract
The theory of the basic reproduction ratio \(R_{0}\) and its computation formulae for almost periodic compartmental epidemic models are established. It is shown that the disease-free almost periodic solution is stable if \(R_{0}<1\), and unstable if \(R_{0}>1\). We also apply the developed theory to a patchy model with almost periodic population dispersal and disease transmission coefficients to obtain a threshold type result for uniform persistence and global extinction of the disease.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Altizer, S., Dobson, A., Hosseini, P., Hudson, P., Pascual, M., Rohani, P.: Seasonality and the dynamics of infectious diseases. Ecol. Lett. 9, 467–484 (2006)
Bacaër, N., Guernaoui, S.: The epidemic threshold of vector-borne diseases with seasonality. J. Math. Biol. 53, 421–436 (2006)
Bacaër, N.: Approximation of the basic reproduction number \(R_{0}\) for vector-borne diseases with a periodic vector population. Bull. Math. Biol. 69, 1067–1091 (2007)
Bacaër, N., Khaladi, M.: On the basic reproduction number in a random environment. J. Math. Biol. doi:10.1007/s00285-012-0611-0
Cordova-Lepe, F., Robledo, G., Pinto, M., Gonzalez-Olivares, E.: Modeling pulse infectious events irrupting into a controlled context: a SIS disease with almost periodic parameters. Appl. Math. Model. 36, 1323–1337 (2012)
Corduneanu, C.: Almost Periodic Functions. Chelsea Publishing Company, New York (1989)
Diekmann, O., Heesterbeek, J.A.P., Metz, J.A.J.: On the definition and the computation of the basic reproduction ratio \(R_{0}\) in the models for infectious disease in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990)
Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Fink, A.M.: Compact families of almost periodic functions and an application of the Schauder fixed-point theorem. SIAM J. Appl. Math. 17, 1258–1262 (1969)
Fink, A.M.: Almost Periodic Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (1974)
Hale, J.K.: Ordinary Differential Equations. Robert E. Krieger Publishing Company, INC, Malabar (1980)
Hale, J.K.: Asymptotic Behavior of Dissipative Systems. Math. Surveys and Monographs 25. American Mathematical Society. Providence (1988)
Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics, Series 247. Longman Scientific and Technical (1991)
Hethcote, H.W.: The mathematics of infectious diseases. SIAM Rev 42, 599–653 (2000)
Hutson, V., Shen, W., Vichers, G.T.: Estimates for the principal spectrum point for certain time-dependent parabolic operators. Proc. Am. Math. Soc. 129, 1669–1679 (2000)
Inaba, H.: On a new perspective of the basic reproduction number in heterogeneous environments. J. Math. Biol. 65, 309–348 (2012)
Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)
Sacker, R., Sell, G.: A spectral theory for linear systems. J. Differ. Equ. 27, 320–358 (1978)
Sell, G.: Topological Dynamics and Ordinary Differential Equations. Van Nostrand Reinhold, London (1971)
Shen, W., Yi, Y.: Almost automorphic and almost periodic dynamics in skew-product semiflows. Memoirs Am. Math. Soc. 136(647), 1–93 (1998)
Smith, H.L., Waltman, P.: The Theory of the Chemostat. Cambridge University Press, Cambridge (1995)
Thieme, H.R.: Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. Biosci. 166, 173–201 (2000)
Thieme, H.R.: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188–211 (2009)
van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)
Wang, W., Zhao, X.-Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20, 699–717 (2008)
Zhao, X.-Q.: Persistence in almost periodic predator-prey reaction-diffusion systems. Fields Inst Commun 36, 259–268 (2003)
Zhao, X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003)
Acknowledgments
Bin-Guo Wang would like to thank the China Scholarship Council and Lanzhou University for financial support during the period of his overseas study, and to express his gratitude to the Department of Mathematics and Statistics, Memorial University of Newfoundland for its kind hospitality. The research is supported in part by the NSF of China under the Grants 10926091 and 11271172. The research is supported in part by the NSERC of Canada and the URP fund of Memorial University.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wang, BG., Zhao, XQ. Basic Reproduction Ratios for Almost Periodic Compartmental Epidemic Models. J Dyn Diff Equat 25, 535–562 (2013). https://doi.org/10.1007/s10884-013-9304-7
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10884-013-9304-7