Advanced Search
Article Contents
Article Contents

Spreading speeds of rabies with territorial and diffusing rabid foxes

  • * Corresponding author

    * Corresponding author

Dedicated to the memory of Hans F. Weinberger

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

Abstract Full Text(HTML) Figure(4) / Table(1) Related Papers Cited by
  • 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.


    \begin{equation} \\ \end{equation}
  • 加载中
  • 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]
     | Show Table
    DownLoad: CSV
  • [1] K. M. Alanazi, A Rabies Model with Distributed Latent Period and Territorial and Diffusing Rabid Foxes, Ph.D. thesis, Arizona State University, Tempe, 2018.
    [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.
    [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.
    [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.
    [5] 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.
    [6] D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, In: Nonlinear Diffusion, Pitman, London, 1977, 1–23.
    [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.
    [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.
    [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. 
    [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.
    [11] O. Diekmann, Limiting behaviour in an epidemic model, Nonlinear Analysis, TMA, 1 (1977), 459-470.  doi: 10.1016/0362-546X(77)90011-6.
    [12] O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 6 (1978), 109-130.  doi: 10.1007/BF02450783.
    [13] O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Diff. Eqns., 33 (1979), 58-73.  doi: 10.1016/0022-0396(79)90080-9.
    [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.
    [15] 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.
    [16] A. DucrotP. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Ration. Mech. Anal., 195 (2010), 311-331.  doi: 10.1007/s00205-008-0203-8.
    [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.
    [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.
    [19] Z. Feng, Applications of Epidemiological Models to Public Heath Policymaking. The Role of Heterogeneity in Model Predictions, World Scientific, Singapore, 2014.
    [20] J. Garnier, Accelerating solutions in integro-differential equations, SIAM J. Math. Anal., 43 (2011), 1955-1974.  doi: 10.1137/10080693X.
    [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.
    [22] M. G. Garroni and J. L. Menaldi, Green Functions for Second Order Parabolic Integro-Differential Problems, Longman Scientific & Technical, Essex 1992.
    [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.
    [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.
    [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.
    [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.
    [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. 
    [28] H. Liu, Spatial Spread of Rabies in Wildlife, Dissertation, Arizona State University, December 2013.
    [29] H. G. Lloyd, The Red Fox, Batsford LTD, London, 1980.
    [30] D. W. MacdonaldRabies and Wildlife. A biologist's perspective, Oxford University Press, Oxford, 1980. 
    [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.
    [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.
    [33] H. MoegleF. KnorppK. BögelA. ArataK. Dietz and P. Diethelm, Zur Epidemiologie der Wildtiertollwut, Zbl. Vet. Med. B, 21 (1974), 647-659. 
    [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.
    [35] J. D. Murray, Mathematical Biology, Springer, Berlin Heidelberg, 1989.
    [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. 
    [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.
    [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.
    [39] L. Rass and J. Radcliffe, Spatial Deterministic Epidemics, AMS, Providence 2003. doi: 10.1090/surv/102.
    [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.
    [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.
    [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.
    [43] W. Rudin, Real and Complex Analysis, McGraw-Hill 1966, New York, 1974.
    [44] A. B. Sargeant, Red fox spatial characteristics in relation to waterfowl predation, J. Wildlife Mgmt., 36 (1972), 225-236.  doi: 10.2307/3799055.
    [45] P. E. Sartwell, The distribution of incubation periods of infectious diseases, Am. J. Hyg., 51 (1950), 310-318. 
    [46] P. E. Sartwell, The incubation period and the dynamics of infectious disease, Am. J. Epid., 83 (1966), 204-216.  doi: 10.1093/oxfordjournals.aje.a120576.
    [47] N. ShigesadaK. Kawasaki and H. F. Weinberger, Spreading speeds of invasive species in a periodic patchy environment: Effects of dispersal based on local information and gradient-based taxis, Jpn. J. Ind. Appl. Math., 32 (2015), 675-705.  doi: 10.1007/s13160-015-0191-7.
    [48] H. ShuX. PanX.-S. Wang and J. Wu, Traveling waves in epidemic models: Non-monotone diffusive systems with non-monotone incidence rates, J. Dyn. Diff. Eqns., 31 (2019), 883-901.  doi: 10.1007/s10884-018-9683-x.
    [49] H. L. Smith and X.-Q. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 31 (2000), 514-534.  doi: 10.1137/S0036141098346785.
    [50] J. W.-H. SoJ. Wu and X. Zou, A reaction-diffusion model for a single species with age structure, Ⅰ. Travelling wavefronts on unbounded domains, Proc. Roy. Soc. London Ser. A, 457 (2001), 1841-1853.  doi: 10.1098/rspa.2001.0789.
    [51] F. Steck and A. Wandeler, The epidemiology of fox rabies in Europe, Epidemiol. Rev., 2 (1980), 71-96.  doi: 10.1093/oxfordjournals.epirev.a036227.
    [52] H. R. Thieme, A model for the spatial spread of an epidemic, J. Math. Biology, 4 (1977), 337-351.  doi: 10.1007/BF00275082.
    [53] H. R. Thieme, Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations, J. Reine Angew. Math., 306 (1979), 94-121. 
    [54] H. R. Thieme, Some mathematical considerations of how to stop the spatial spread of a rabies epidemic, Biological Growth and Spread (W. Jäger, H. Rost, P. Tautu, eds.), 310–319. Lecture Notes in Biomathematics, 38. Springer, 1980.
    [55] H. R. ThiemeMathematics in Population Biology,, Princeton University Press, Princeton, 2003. 
    [56] H. R. Thieme, Book report on [39], Math. Biosci., 202 (2006), 218-225. 
    [57] H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Diff. Eqn., 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X.
    [58] B. C. Tian and R. Yuan, Traveling waves for a diffusive SEIR epidemic mode with standard incidences, Science China Math., 60 (2017), 813-832.  doi: 10.1007/s11425-016-0487-3.
    [59] B. Toma and L. Andral, Epidemiology of fox rabies, Adv. Virus Res., 21 (1977), 1-36.  doi: 10.1016/S0065-3527(08)60760-5.
    [60] F. van den BoschJ. A. J. Metz and O. Diekmann, The velocity of spatial population expansion, J. Math. Biol., 28 (1990), 529-565.  doi: 10.1007/BF00164162.
    [61] H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion systems, J. Nonl. Sci., 21 (2011), 747-783.  doi: 10.1007/s00332-011-9099-9.
    [62] X. WangY. ShiZ. Feng and J. Cui, Evaluations of interventions using mathematical models with exponential and non-exponential distributions for disease stages: the case of ebola, Bull. Math. Biol., 79 (2017), 2149-2173.  doi: 10.1007/s11538-017-0324-z.
    [63] Z.-C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc. A, 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.
    [64] H. F. Weinberger, On spreading speeds and traveling waves for growth and migration models in a periodic habitat, J. Math. Biol., 45 (2002), 511-548.  doi: 10.1007/s00285-002-0169-3.
    [65] H. F. WeinbergerM. A. Lewis and B. Li, Anomalous spreading speeds of cooperative recursion systems, J. Math. Biol., 55 (2007), 207-222.  doi: 10.1007/s00285-007-0078-6.
    [66] R. Wu and X.-Q. Zhao, Propagation dynamics for a spatially periodic integrodifference competition model, J. Diff. Eqns., 264 (2018), 6507-6534.  doi: 10.1016/j.jde.2018.01.039.
    [67] Z. Xu, Traveling waves in an SEIR epidemic model with the variable total population, Disc. Cont. Dyn. Syst. Ser. B, 21 (2016), 3723-3742.  doi: 10.3934/dcdsb.2016118.
    [68] G.-B. ZhangY. Li and Z. Feng, Exponential stability of traveling waves in a nonlocal dispersal epidemic model with delay, J. Comput. Appl. Math., 344 (2018), 47-72.  doi: 10.1016/j.cam.2018.05.018.
    [69] J. ZhangZ. JinG.-Q. SunT. Zhou and S. Ruan, Analysis of rabies in China: Transmission dynamics and control, Plos One, 6 (2011), e20891.  doi: 10.1371/journal.pone.0020891.
    [70] J. ZhangZ. JinG. SunX. Sun and S. Ruan, Spatial spread of rabies in China, J. Appl. Anal. Comput., 2 (2012), 111-126. 
    [71] J. ZhangZ. JinG.-Q. SunX.-D. Sun and S. Ruan, Modeling seasonal rabies epidemics in China, Bull. Math. Biol., 75 (2013), 206-211.  doi: 10.1007/s11538-012-9720-6.
    [72] X.-Q. Zhao and D. Xiao, The asymptotic speed of spread and traveling waves for a vector disease model, J. Dyn. Diff. Eqns., 18 (2006), 1001-1019.  doi: 10.1007/s10884-006-9044-z.
    [73] J. ZinsstagS. DürrM. A. PennyR. MindekemF. RothS. Menendez GonzalezS. Naissengar and J. Hattendorf, Transmission dynamics and economics of rabies control in dogs and humans in an African city, PNAS, 106 (2009), 14996-15001.  doi: 10.1073/pnas.0904740106.
  • 加载中




Article Metrics

HTML views(712) PDF downloads(398) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint