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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

Received  December 2018 Revised  April 2019 Published  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 & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019222
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show all references

References:
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K. M. Alanazi, A Rabies Model with Distributed Latent Period and Territorial and Diffusing Rabid Foxes, Ph.D. thesis, Arizona State University, Tempe, 2018.  Google Scholar

[2]

K. M. AlanaziZ. Jackiewicz and H. R. Thieme, Numerical simulations of the spread of rabies in a spatially distributed fox population, Mathematics and Computers in Simulation, 159 (2019), 161-182.  doi: 10.1016/j.matcom.2018.11.010.  Google Scholar

[3]

K. M. AlanaziZ. Jackiewicz and H. R. Thieme, Numerical simulations of the spread of rabies in two-dimensional space, Applied Numerical Mathematics, 135 (2019), 87-98.  doi: 10.1016/j.apnum.2018.08.009.  Google Scholar

[4]

R. M. AndersonH. C. JacksonR. M. May and A. M. Smith, Population dynamics of fox rabies in Europe, Nature, 289 (1981), 765-771.  doi: 10.1038/289765a0.  Google Scholar

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L. AndralM. ArtoisM. F. A. Aubert and J. Blancou, Radio-pistage de renards enrages, Comp. Immun. Microbiol. Infect. Dis., 5 (1982), 285-291.  doi: 10.1016/0147-9571(82)90050-9.  Google Scholar

[6]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, In: Nonlinear Diffusion, Pitman, London, 1977, 1–23.  Google Scholar

[7]

D. G. Aronson and H. R. Weinberger, Nonlinear Diffusion in population genetics, combustion, and nerve pulse propagation, Partial Differential Equations and Related Topics (J. Goldstein, ed.), 5–49, Lecture Notes in Mathematics 446, Springer, Berlin 1975.  Google Scholar

[8]

C. BeaumontJ.-B. BurieA. Ducrot and P. Zongo, Propagation of Salmonella within an industrial hens house, SIAM J. Appl. Math., 72 (2012), 1113-1148.  doi: 10.1137/110822967.  Google Scholar

[9]

K. BögelH. MoegleF. KnorppA. ArataK. Dietz and P. Diethelm, Characteristics of the spread of a wildlife rabies epidemic in Europe, Bull. World Health. Organ., 54 (1976), 433-447.   Google Scholar

[10]

E. BouinJ. GarnierC. Henderson and F. Patout, Thin front limit of an integro-differential Fisher-KPP equation with fat-tailed kernels, SIAM J. Math. Anal., 50 (2018), 3365-3394.  doi: 10.1137/17M1132501.  Google Scholar

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O. Diekmann, Limiting behaviour in an epidemic model, Nonlinear Analysis, TMA, 1 (1977), 459-470.  doi: 10.1016/0362-546X(77)90011-6.  Google Scholar

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O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.  doi: 10.1007/BF02450783.  Google Scholar

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[14]

A. DucrotM. Langlais and P. Magal, Multiple travelling waves for an SI-epidemic model, Netw. Heterog. Media, 8 (2013), 171-190.  doi: 10.3934/nhm.2013.8.171.  Google Scholar

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A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. Roy. Soc. Edinburgh Sect A, 139 (2009), 459-482.  doi: 10.1017/S0308210507000455.  Google Scholar

[16]

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[17]

J. FangX. Yu and X.-Q. Zhao, Traveling waves and spreading speeds for time-space periodic monotone systems, J. Funct. Anal., 272 (2017), 4222-4262.  doi: 10.1016/j.jfa.2017.02.028.  Google Scholar

[18]

A. P. FarrellJ. P. CollinsA. L. Greer and H. R. Thieme, Times from infection to disease-induced death and their influence on final population sizes after epidemic outbreaks, Bull. Math. Biol., 80 (2018), 1937-1961.  doi: 10.1007/s11538-018-0446-y.  Google Scholar

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Z. Feng, Applications of Epidemiological Models to Public Heath Policymaking. The Role of Heterogeneity in Model Predictions, World Scientific, Singapore, 2014. Google Scholar

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J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.  Google Scholar

[21]

J. Garnier and M. A. Lewis, Expansion under climate change: The genetic consequences, Bull Math. Biol., 78 (2016), 2165-2185.  doi: 10.1007/s11538-016-0213-x.  Google Scholar

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M. G. Garroni and J. L. Menaldi, Green Functions for Second Order Parabolic Integro-Differential Problems, Longman Scientific & Technical, Essex 1992.  Google Scholar

[23]

S. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM J. Appl. Math., 65 (2005), 550-566.  doi: 10.1137/S0036139903436613.  Google Scholar

[24]

D. A. JonesG. RöstH. L. Smith and H. R. Thieme, On spread of phage infection of bacteria in a petri dish, SIAM J. Applied Math., 72 (2012), 670-688.  doi: 10.1137/110848360.  Google Scholar

[25]

D. A. JonesH. L. Smith and H. R. Thieme, Spread of viral infection of immobilized bacteria, Networks Heterog Media, 8 (2013), 327-342.  doi: 10.3934/nhm.2013.8.327.  Google Scholar

[26]

D. A. JonesH. L. Smith and H. R. Thieme, Spread of phage infection of bacteria in a petri dish, Disc. Cont. Dyn. Syst. Ser. B, 21 (2016), 471-496.  doi: 10.3934/dcdsb.2016.21.471.  Google Scholar

[27]

D. LambinetJ. F. BoisvieuxM. Artois and L. Andral, Modele mathématique de la propagation d'une épizootie de rage vulpine, Rev. Epidém. et Santé Publ., 26 (1978), 9-28.   Google Scholar

[28]

H. Liu, Spatial Spread of Rabies in Wildlife, Dissertation, Arizona State University, December 2013.  Google Scholar

[29]

H. G. Lloyd, The Red Fox, Batsford LTD, London, 1980. Google Scholar

[30] D. W. Macdonald, Rabies and Wildlife. A biologist's perspective, Oxford University Press, Oxford, 1980.   Google Scholar
[31]

J. A. J. Metz and F. van den Bosch, Velocities of epidemic spread, In: Epidemic Models: Their Structure and Relation to Data (Mollison D., ed.), 150–186, Cambridge University Press, Cambridge 1995. Google Scholar

[32]

J. A. J. Metz, D. Mollison and F. van den Bosch, The dynamics of invasion waves, In: The Geometry of Ecological Interactions: Simplifying Spatial Complexity (Dieckmann U, Law R, Metz JAJ, eds.), 482-512, Cambridge Univ. Press, Cambridge, 2000. Google Scholar

[33]

H. MoegleF. KnorppK. BögelA. ArataK. Dietz and P. Diethelm, Zur Epidemiologie der Wildtiertollwut, Zbl. Vet. Med. B, 21 (1974), 647-659.   Google Scholar

[34]

D. Mollison, Dependence of epidemic and population velocities on basic parameters, Math. Biosci., 107 (1991), 255-287.  doi: 10.1016/0025-5564(91)90009-8.  Google Scholar

[35]

J. D. Murray, Mathematical Biology, Springer, Berlin Heidelberg, 1989.  Google Scholar

[36]

J. D. MurrayE. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes, Proc. R. Soc. Lond. B, 229 (1986), 111-150.   Google Scholar

[37]

J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity, J. theor. Biol., 156 (1992), 327-348.  doi: 10.1016/S0022-5193(05)80679-4.  Google Scholar

[38]

C. Ou amd J. Wu, Spatial spread of rabies revisited: Influence of age-dependent diffusion on nonlinear dynamics, SIAM J. Appl. Math., 67 (2006), 138-163.  doi: 10.1137/060651318.  Google Scholar

[39]

L. Rass and J. Radcliffe, Spatial Deterministic Epidemics, AMS, Providence 2003. doi: 10.1090/surv/102.  Google Scholar

[40]

M. G. Roberts and R. R. Kao, The dynamics of an infectious disease in a population with birth pulses, Math. Biosci., 149 (1998), 23-36.  doi: 10.1016/S0025-5564(97)10016-5.  Google Scholar

[41]

S. Ruan, Spatial-temporal dynamics in nonlocal epidemic models, Mathematics for Life Science and Medicine, 97–122, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007.  Google Scholar

[42]

S. Ruan, Modeling the transmission dynamics and control of rabies in China, Math. Biosci., 286 (2017), 65-93.  doi: 10.1016/j.mbs.2017.02.005.  Google Scholar

[43]

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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]
$ 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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