# 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:

show all references

##### 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]
 [1] Hui Cao, Yicang Zhou. The basic reproduction number of discrete SIR and SEIS models with periodic parameters. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 37-56. doi: 10.3934/dcdsb.2013.18.37 [2] Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595 [3] Nitu Kumari, Sumit Kumar, Sandeep Sharma, Fateh Singh, Rana Parshad. Basic reproduction number estimation and forecasting of COVID-19: A case study of India, Brazil and Peru. Communications on Pure and Applied Analysis, , () : -. doi: 10.3934/cpaa.2021170 [4] Gerardo Chowell, R. Fuentes, A. Olea, X. Aguilera, H. Nesse, J. M. Hyman. The basic reproduction number $R_0$ and effectiveness of reactive interventions during dengue epidemics: The 2002 dengue outbreak in Easter Island, Chile. Mathematical Biosciences & Engineering, 2013, 10 (5&6) : 1455-1474. doi: 10.3934/mbe.2013.10.1455 [5] E. Almaraz, A. Gómez-Corral. On SIR-models with Markov-modulated events: Length of an outbreak, total size of the epidemic and number of secondary infections. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2153-2176. doi: 10.3934/dcdsb.2018229 [6] Yan Hong, Xiuxiang Liu, Xiao Yu. Global dynamics of a Huanglongbing model with a periodic latent period. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021302 [7] Tianhui Yang, Lei Zhang. Remarks on basic reproduction ratios for periodic abstract functional differential equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (12) : 6771-6782. doi: 10.3934/dcdsb.2019166 [8] Yu Yang, Dongmei Xiao. Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models. Discrete and Continuous Dynamical Systems - B, 2010, 13 (1) : 195-211. doi: 10.3934/dcdsb.2010.13.195 [9] Attila Dénes, Gergely Röst. Single species population dynamics in seasonal environment with short reproduction period. Communications on Pure and Applied Analysis, 2021, 20 (2) : 755-762. doi: 10.3934/cpaa.2020288 [10] Tom Burr, Gerardo Chowell. The reproduction number $R_t$ in structured and nonstructured populations. Mathematical Biosciences & Engineering, 2009, 6 (2) : 239-259. doi: 10.3934/mbe.2009.6.239 [11] Tianhui Yang, Ammar Qarariyah, Qigui Yang. The effect of spatial variables on the basic reproduction ratio for a reaction-diffusion epidemic model. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3005-3017. doi: 10.3934/dcdsb.2021170 [12] Edoardo Beretta, Dimitri Breda. An SEIR epidemic model with constant latency time and infectious period. Mathematical Biosciences & Engineering, 2011, 8 (4) : 931-952. doi: 10.3934/mbe.2011.8.931 [13] Horst R. Thieme. Distributed susceptibility: A challenge to persistence theory in infectious disease models. Discrete and Continuous Dynamical Systems - B, 2009, 12 (4) : 865-882. doi: 10.3934/dcdsb.2009.12.865 [14] Gaofei Wu, Yuqing Zhang, Xuefeng Liu. New complementary sets of length $2^m$ and size 4. Advances in Mathematics of Communications, 2016, 10 (4) : 825-845. doi: 10.3934/amc.2016043 [15] Xuefeng San, Yuan He. Traveling waves for a two-group epidemic model with latent period and bilinear incidence in a patchy environment. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3299-3318. doi: 10.3934/cpaa.2021106 [16] Gerardo Chowell, Catherine E. Ammon, Nicolas W. Hengartner, James M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Mathematical Biosciences & Engineering, 2007, 4 (3) : 457-470. doi: 10.3934/mbe.2007.4.457 [17] Ling Xue, Caterina Scoglio. Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences & Engineering, 2015, 12 (3) : 565-584. doi: 10.3934/mbe.2015.12.565 [18] Gabriela Marinoschi. Identification of transmission rates and reproduction number in a SARS-CoV-2 epidemic model. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022128 [19] Yoshiaki Muroya, Yoichi Enatsu, Huaixing Li. A note on the global stability of an SEIR epidemic model with constant latency time and infectious period. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 173-183. doi: 10.3934/dcdsb.2013.18.173 [20] Tom Reichert. Pandemic mitigation: Bringing it home. Mathematical Biosciences & Engineering, 2011, 8 (1) : 65-76. doi: 10.3934/mbe.2011.8.65

2021 Impact Factor: 1.497