Journal
SIAM JOURNAL ON APPLIED MATHEMATICS
Volume 76, Issue 4, Pages 1633-1657Publisher
SIAM PUBLICATIONS
DOI: 10.1137/15M1029564
Keywords
disease spread; wavelike environment; Fisher-KPP wave; pulse wave; generalized eigenvalues
Categories
Funding
- Natural Sciences and Research Council of Canada (NSERC)
- Canadian Institute for Health Research (CIHR)
- Canada Research Chair program (CRC)
- International Development and Research Centre (IDRC) through its IDRC-CRC International Research Chair program
- NSF of China [11371111]
- Fundamental Research Funds for Central Universities of China [AUGA5710050313]
- NSFC [11301442]
- RGC [PolyU 253004/14P]
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We consider the Fisher-KPP equation in a wavelike shifting environment for which the wave profile of the environment is given by a monotonically decreasing function changing signs (shifting from favorable to unfavorable environment). This type of equation arises naturally from the consideration of pathogen spread in a classical susceptible-infected-susceptible epidemiological model of a host population where the disease impact on host mobility and mortality is negligible. We conclude that there are three different ranges of the disease transmission rate where the disease spread has distinguished spatiotemporal patterns: extinction; spread in pace with the host invasion; spread not in a wave format and slower than the host invasion. We calculate the disease propagation speed when disease does spread. Our analysis for a related elliptic operator provides closed form expressions for two generalized eigenvalues in an unbounded domain. The obtained closed forms yield unsolvability of the related elliptic equation in the critical case, which relates to the open problem 4.6 in [H. Berestycki and L. Rossi, J. Eur. Math. Soc. (JEMS), 8 (2006), pp. 195-215].
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