Basic Reproduction Ratios for Almost Periodic Compartmental Epidemic Models

Journal of Dynamics and Differential Equations

Volumn 25, Issue 2, 2013, Pages 535-562

  Wang, Bin Guo ^a   Zhao, Xiao Qiang ^b  

^a LANZHOU UNIVERSITY   (China)

^b MEMORIAL UNIVERSITY OF NEWFOUNDLAND   (Canada)

Author keywords

Almost periodicity; Compartmental models; Reproduction ratio; Skew product semiflow; Threshold dynamics

Indexed keywords

EID: 84877801520     PISSN: 10407294     EISSN: 15729222     Source Type: Journal    
DOI: 10.1007/s10884-013-9304-7     Document Type: Article

References (27)

1. 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).
2. Bacaër, N., Guernaoui, S.: The epidemic threshold of vector-borne diseases with seasonality. J. Math. Biol. 53, 421-436 (2006).
3. 0 for vector-borne diseases with a periodic vector population. Bull. Math. Biol. 69, 1067-1091 (2007).
4. Bacaër, N., Khaladi, M.: On the basic reproduction number in a random environment. J. Math. Biol. doi: 10. 1007/s00285-012-0611-0.
5. 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).
6. Corduneanu, C.: Almost Periodic Functions. Chelsea Publishing Company, New York (1989).
7. 0 in the models for infectious disease in heterogeneous populations. J. Math. Biol. 28, 365-382 (1990).
8. Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000).
9. 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).
10. Fink, A. M.: Almost Periodic Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (1974).
11. Hale, J. K.: Ordinary Differential Equations. Robert E. Krieger Publishing Company, INC, Malabar (1980).
12. Hale, J. K.: Asymptotic Behavior of Dissipative Systems. Math. Surveys and Monographs 25. American Mathematical Society. Providence (1988).
13. Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics, Series 247. Longman Scientific and Technical (1991).
14. Hethcote, H. W.: The mathematics of infectious diseases. SIAM Rev 42, 599-653 (2000).
15. 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).
16. Inaba, H.: On a new perspective of the basic reproduction number in heterogeneous environments. J. Math. Biol. 65, 309-348 (2012).
17. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976).
18. Sacker, R., Sell, G.: A spectral theory for linear systems. J. Differ. Equ. 27, 320-358 (1978).
19. Sell, G.: Topological Dynamics and Ordinary Differential Equations. van Nostrand Reinhold, London (1971).
20. Shen, W., Yi, Y.: Almost automorphic and almost periodic dynamics in skew-product semiflows. Memoirs Am. Math. Soc. 136(647), 1-93 (1998).
21. Smith, H. L., Waltman, P.: The Theory of the Chemostat. Cambridge University Press, Cambridge (1995).
22. Thieme, H. R.: Uniform persistence and permanence for non-autonomous semiflows in population biology. Math. Biosci. 166, 173-201 (2000).
23. Thieme, H. R.: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188-211 (2009).
24. 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).
25. Wang, W., Zhao, X.-Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20, 699-717 (2008).
26. Zhao, X.-Q.: Persistence in almost periodic predator-prey reaction-diffusion systems. Fields Inst Commun 36, 259-268 (2003).
27. Zhao, X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003).