# American Institute of Mathematical Sciences

June  2020, 25(6): 2143-2183. doi: 10.3934/dcdsb.2019222

## Spreading speeds of rabies with territorial and diffusing rabid foxes

 1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287-1804, USA 2 Northern Border University, Saudi Arabia 3 Department of Applied Mathematics, AGH University of Science and Technology, Kraków, Poland

* Corresponding author

Dedicated to the memory of Hans F. Weinberger

Received  December 2018 Revised  April 2019 Published  June 2020 Early access  September 2019

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

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.

Citation: Khalaf M. Alanazi, Zdzislaw Jackiewicz, Horst R. Thieme. Spreading speeds of rabies with territorial and diffusing rabid foxes. Discrete and Continuous Dynamical Systems - B, 2020, 25 (6) : 2143-2183. doi: 10.3934/dcdsb.2019222
##### References:

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##### References:
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 $\omega = 5.3$
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 $200 \,[ \rm{km}^2/ \rm{year}]$
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 $100 \,[ \rm{km}^2/ \rm{year}]$
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 $\rm{(b)} \, D = 40 \,[ \rm{km}^2/ \rm{year}]$
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]
 $S^\circ$ [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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