Skip to main content
SpringerLink
Log in

Spatiotemporal Patterns of Disease Spread: Interaction of Physiological Structure, Spatial Movements, Disease Progression and Human Intervention

  • Chapter
Structured Population Models in Biology and Epidemiology

Part of the book series: Lecture Notes in Mathematics ((LNMBIOS,volume 1936))

In this article we review some recent literature on the mathematical modelling of vector-borne diseases with special reference to West Nile virus and with particular focus on the role of the developmental stages of hosts in determining the transmission dynamics, the effectiveness of different approaches to controlling the vector and the spatial spread of an epidemic. A possible model incorporating the developmental stages of avian hosts is discussed which consists of equations for infective and susceptible juvenile and adult hosts and infected adult vectors. Conditions for the system to evolve to the disease free state are presented. These elucidate the role of, for example, the various death rates involved. We also review a mathematical model which incorporates culling the vector at either the larval or the adult stage and the effectiveness of the two approaches is compared. Conditions are given that are sufficient for eradication and this leads insights into the required minimum frequency of culling. Very infrequent culling is no better than no culling at all and can actually increase the time average of the number of infected vectors. We also review a reaction-diffusion extension of the model which can be used to estimate the speed at which an epidemic moves through space. Finally, we review some recent work on the use of patch models of a West Nile virus epidemic. These models are arguably easier to relate to surveillance data which is organised according to administrative regions or landscape. The patch model is used to study the situation when the dispersal of birds is not symmetric.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. J. S. Blackmore and R. P. Dow, Differential feeding of culex tarsalis on nestling and adult birds, Mosq News 18 (1958), 15–17

    Google Scholar 

  2. C. Bowman, A. B. Gumel, P. van den Driessche, J. Wu, and H. Zhu, A mathematical model for assessing control strategies against west nile virus, Bull. Math. Biol. 67 (2005), 1107–1133

    Article  MathSciNet  Google Scholar 

  3. N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math. 50 (1990), 1663–1688

    Article  MATH  MathSciNet  Google Scholar 

  4. S. A. Gourley and N. F. Britton, A predator prey reaction diffusion system with nonlocal effects, J. Math. Biol. 34 (1996), 297–333

    MATH  MathSciNet  Google Scholar 

  5. S. A. Gourley, J. W. H. So, and J. Wu, Non-locality of reaction–diffusion equations induced by delay: biological modeling and nonlinear dynamics, J. Math. Sci. 124 (2004), 5119–5153

    Article  MathSciNet  Google Scholar 

  6. S. A. Gourley and J. Wu, Delayed non-local diffusive systems in biological invasion and disease spread, Fields Inst. Commun. 48 137–200, Amer. Math. Soc., Providence, RI, 2006

    Google Scholar 

  7. S. A. Gourley, R. Liu, and J. Wu, Some vector borne diseases with structured host populations: extinction and spatial spread, SIAM J. Appl. Math. 67 (2006/07), 408–433

    Google Scholar 

  8. S. Gourley, R. Liu, and J. Wu, Eradicating vector-borne diseases via age-structured culling, J. Math. Biol. 54 (2007), 309–335

    Article  MATH  MathSciNet  Google Scholar 

  9. W. S. C. Gurney, S. P. Blythe, and R. M. Nisbet, Nicholson’s blowflies revisited, Nature 287 (1980), 17–21

    Article  Google Scholar 

  10. L. C. Harrington, T. W. Scott, K. Lerdthusnee, R. C. Coleman, A. Costero, G. G. Clark, J. J. Jones, S. Kitthawee, P. Kittayapong, R. Sithiprasasna, and J. D. Edman, Dispersal of the dengue vector aedes aegypti within and between rural communities, Am. J. Trop. Med. Hyg. 72(2) (2005), 209–220

    Google Scholar 

  11. Y. Kuang, Delay differential equations with applications in population dynamics, vol. Mathematics in Science and Engineering 191, Academic Press, Boston, 1993

    MATH  Google Scholar 

  12. M. Lewis, J. Renclawowicz, and P. van den Driessche, Traveling waves and spread rates for a west nile virus model, Bull. Math. Biol. 68 (2006), 3–23

    Article  MathSciNet  Google Scholar 

  13. R. Liu, J. Shuai, H. Zhu, and J. Wu, Modeling spatial spread of west nile virus and impact of directional dispersal of birds., Math. Biosci. Eng. 3 (2006), 145–160

    Google Scholar 

  14. C. C. Lord and J. F. Day, Simulation studies of st. louis encephalitis and west nile virues: the impact of bird mortality, Vector Borne Zoonotic Dis. 1 (4) (2001), 317–329

    Article  Google Scholar 

  15. C. C. Lord and J. F. Day, Simulation studies of st. louis encephalitis virus in south florida, Vector Borne Zoonotic Dis. 1 (4) (2001), 299–315

    Article  Google Scholar 

  16. P. J. McCall and D. W. Kelly, Learning and memory in disease vectors, Trends Parasitol. 18 (2002), 429–433

    Article  Google Scholar 

  17. A. Okubo, Diffusion-type models for avian range expansion, University of Ottawa Press, Ottawa 1998

    Google Scholar 

  18. C. Ou and J. Wu, Spatial spread of rabies revisited: influence of age-dependent diffusion on nonlinear dynamics, SIAM J. Appl. Math. 67 (2006), 138–163

    Article  MATH  MathSciNet  Google Scholar 

  19. M. A. Pozio, Behaviour of solutions of some abstract functional differential equations and application to predator-prey dynamics, Nonlinear Anal. 4 (1980), 917–938

    Article  MATH  MathSciNet  Google Scholar 

  20. M. A. Pozio, Some conditions for global asymptotic stability of equilibria of integro-differential equations, J. Math. Anal. Appl. 95 (1983), 501–527

    Article  MATH  MathSciNet  Google Scholar 

  21. R. Redlinger, Existence theorems for semilinear parabolic systems with functionals, Nonlinear Anal. 8 (1984), 667–682

    Article  MATH  MathSciNet  Google Scholar 

  22. R. Redlinger, On volterra’s population equation with diffusion, SIAM J. Math. Anal. 16 (1985), 135–142

    Article  MATH  MathSciNet  Google Scholar 

  23. M. R. Sardelis and M. J. Turell, Ochlerotatus j. japonicus in frederick county, maryland: Discovery, distribution, and vector competence for west nile cirus, J. Am. Mosq. Control Assoc. 17 (2001), 137–141

    Google Scholar 

  24. T. W. Scott, L. H. Lorenz, and J. D. Edman, Effects of house sparrow age and arbovirus infection on attraction of mosquitoes, J Med Entomol 27 (1990), 856–863

    Google Scholar 

  25. M. W. Service, Effects of wind on the behaviour and distribution of mosquitoes and blackflies, Int. J. Biometeorol. 24 (1980), 347–353

    Article  Google Scholar 

  26. R. R. L. Simons and S. A. Gourley, Extinction criteria in stage-structured population models with impulsive culling, SIAM. J. Appl. Maths 66 (2006), 1853–1870

    Article  MATH  MathSciNet  Google Scholar 

  27. H. L. Smith and H. R. Thieme, Strongly order preserving semiflows generated by functional-differential equations, J. Diff. Eqns. 93 (1991), 332–363

    Article  MATH  MathSciNet  Google Scholar 

  28. J. W. H. So, J. Wu, and X. Zou, A reaction diffusion model for a single species with age structure, i. travelling wave fronts on unbounded domains, Proc. Roy. Soc. Lond. Ser. A. 457 (2001), 1841–1853

    Google Scholar 

  29. A. J. Tatem, S. I. Hay, and D. J. Rogers, Global traffic and disease vector dispersal, Proc. Nat. Acad. Sci. 103 (2006), 6242–6247

    Article  Google Scholar 

  30. H. R. Thieme and X. -Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differ. Equ. 195 (2) (2003), 430–470

    Article  MATH  MathSciNet  Google Scholar 

  31. M. J. Turell, M. O’Guinn, and J. Oliver, Potential for new york mosquitoes to transmit west nile virus, Am. J. Trop. Med. Hyg. 62 (2002), 413–414

    Google Scholar 

  32. M. J. Turell, M. L. O’Guinn, D. J. Dohm, and J. W. Jones, Vector competence north american mosquitoes (diptera: Cullocidae) for west nile virus, J. Med. Entomol. 38 (2001), 130–134

    Article  Google Scholar 

  33. P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48

    Article  MATH  MathSciNet  Google Scholar 

  34. M. P. Ward, A. Raim, S. Yaremych-Hamer, R. Lampman, and R. J. Novak, Does the roosting behavior of birds affect transmission dynamics of west nile virus?, Am. J. Trop. Med. Hyg. 75 (2006), 350–355

    Google Scholar 

  35. M. J. Wonham, T. de Camino-Beck, and M. Lewis, An epidemiological model for west nile virus: Invasion analysis and control applications, Proc. R. Soc. Lond., Ser. B 271 (1538) (2004), 501–507

    Article  Google Scholar 

  36. Y. Yamada, Asymptotic stability for some systems of semilinear volterra diffusion equations, J. Diff. Eqns. 52 (1984), 295–326

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Surrey, Guildford, Surrey, GU2 7XH, UK

    S. A. Gourley

  2. Department of Mathematics, Purdue University, 150 N, University Street, West Lafayette, IN, 47907-2067, USA

    R. Liu

  3. Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, ON, Canada, M3J 1P3

    J. Wu

Authors
  1. S. A. Gourley
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. University of Le Havre, 25 rue philippe Lebon, 76058, Le Havre, France

    Pierre Magal

  2. Department of Mathematics, University of Miami, Coral Gables, FL, 33124-4250, USA

    Shigui Ruan

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Gourley, S.A., Liu, R., Wu, J. (2008). Spatiotemporal Patterns of Disease Spread: Interaction of Physiological Structure, Spatial Movements, Disease Progression and Human Intervention. In: Magal, P., Ruan, S. (eds) Structured Population Models in Biology and Epidemiology. Lecture Notes in Mathematics, vol 1936. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78273-5_4

Download citation

Publish with us

Policies and ethics

Search

Navigation