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Spreading speeds of rabies with territorial and diffusing rabid foxes

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    * Corresponding author

Dedicated to the memory of Hans F. Weinberger

The first author was supported by a scholarship from Northern Border University (Saudi Arabia)

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  • A mathematical model is formulated for the fox rabies epidemic that swept through large areas of Europe during parts of the last century. Differently from other models, both territorial and diffusing rabid foxes are included, which leads to a system of partial differential, functional differential and differential-integral equations. The system is reduced to a scalar Volterra-Hammerstein integral equation to which the theory of spreading speeds pioneered by Aronson and Weinberger is applied. The spreading speed is given by an implicit formula which involves the space-time Laplace transform of the integral kernel. This formula can be exploited to find the dependence of the spreading speed on the model ingredients, in particular on those describing the interplay between diffusing and territorial rabid foxes and on the distribution of the latent period.

    Mathematics Subject Classification: Primary: 92D30, 45K05; Secondary: 45M99.


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  • Figure 11.1.  Spreading speed $ c^* $ dependence on $ S^\circ $ if it affects a normally distributed home-range size and all rabid foxes are territorial and the length of the latent period is exponentially distributed. We use (11.8) to solve (9.16), $ 1/\gamma = 33.44 \, [ \rm{day}], $ $ 1/\nu = 5 \, [ \rm{day}], $ $ \beta = 0.5 \,[ \rm{km}^2/ \rm{day}], $ and $ \omega = 5.3 $

    Figure 11.2.  Spreading speed $ c^* $ dependence on $ S^\circ $ if it affects home-range size. We use (11.16) with $ \hat \Upsilon(s) = e^{- \tau s} $ to solve (9.16). The average durations of the latent and infectious periods are chosen to be $ 33.44 \, [ \rm{day}] $ and $ 5 \, [ \rm{day}] $, respectively, while the diffusion rate is chosen to be $ 200 \,[ \rm{km}^2/ \rm{year}] $

    Figure 11.3.  Spreading speed $ c^* $ dependence on $ S^\circ $ if it influences home-range size. We use (11.16) with $ \hat \Upsilon(s) = e^{- \tau s} $ to solve (9.16). The average durations of the latent and infectious periods are chosen to be $ 33.44 \, [ \rm{day}] $ and $ 5 \, [ \rm{day}] $, respectively, while the diffusion rate is chosen to be $ 100 \,[ \rm{km}^2/ \rm{year}] $

    Figure 13.1.  Spreading speed $ c^* $ dependence on $ p_1. $ We use (9.15) with $ \hat \Upsilon(s) = e^{- \tau s} $ to solve (9.16). Here, $ L = 5 \,[ \rm{day}], $ $ \tau = 28 \,[ \rm{day}], $ $ b = 5/ \pi^2 \,[ \rm{km}^2], $ $ {\mathcal R}_0 = 4.6, $ $ \rm{(a)} \, D = 34 \,[ \rm{km}^2/ \rm{year}] $ and $ \rm{(b)} \, D = 40 \,[ \rm{km}^2/ \rm{year}] $

    Table 1.  The spreading speed $ c^* $ as a function of fox density compared to wave speeds for a model with population turn-over. The initial fox density $ S^\circ $ is equal to the fox carrying capacity $ K $ in [35,36,37]. The other parameters are chosen as therein though the symbols may be different

    $ S^\circ $ [foxes/km$ ^2 $] $ c^* $ [km/year] comparative speed [km/year]
    1.5 36 35   [36,Table 3]
    2.0 52 50   [36,Table 3]
    2.5 65 70   [36,Table 3]
    3.0 76 80   [36,Table 3]
    4.6 103 103   [37,Table 2]
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