Journal of Mathematical Biology

, Volume 66, Issue 4–5, pp 1045–1064 | Cite as

Characterizing the next-generation matrix and basic reproduction number in ecological epidemiology

  • M. G. RobertsEmail author
  • J. A. P. Heesterbeek


We address the interaction of ecological processes, such as consumer-resource relationships and competition, and the epidemiology of infectious diseases spreading in ecosystems. Modelling such interactions seems essential to understand the dynamics of infectious agents in communities consisting of interacting host and non-host species. We show how the usual epidemiological next-generation matrix approach to characterize invasion into multi-host communities can be extended to calculate \(\mathcal{R _{0}}\), and how this relates to the ecological community matrix. We then present two simple examples to illustrate this approach. The first of these is a model of the rinderpest, wildebeest, grass interaction, where our inferred dynamics qualitatively matches the observed phenomena that occurred after the eradication of rinderpest from the Serengeti ecosystem in the 1980s. The second example is a prey-predator system, where both species are hosts of the same pathogen. It is shown that regions for the parameter values exist where the two host species are only able to coexist when the pathogen is present to mediate the ecological interaction.


Epidemiological stability Ecological stability Infectious diseases 

Mathematics Subject Classification (2000)

92D30 92D40 



The authors thank two anonymous referees whose suggestions led to improvements in the manuscript. The first author received financial support from the Marsden Fund under contract MAU1106.


  1. Beltrami E, Carroll TO (1994) Modelling the role of viral disease in recurrent phytoplankton blooms. J Math Biol 32:857–863Google Scholar
  2. Boldin B (2006) Introducing a population into a steady community: the critical case, the centre manifold and the direction of bifurcation. SIAM J Appl Math 66:1424–1453MathSciNetzbMATHCrossRefGoogle Scholar
  3. Chattopadhyay J, Arino O (1999) A predator prey model with disease in the prey. Nonlinear Anal 36:747–766MathSciNetzbMATHCrossRefGoogle Scholar
  4. Diekmann O, Heesterbeek JAP, Roberts MG (2010) The construction of next-generation matrices for compartmental epidemic systems. J R Soc Interface 7:873–885CrossRefGoogle Scholar
  5. Dizney LJ, Ruedas LA (2009) Increased host species diversity and decreased prevalence of Sin Nombre virus. Emerg Infect Dis 15:1012–1018CrossRefGoogle Scholar
  6. Fenton A, Rands SA (2006) The impact of parasite manipulation and predator foraging behavior on predator-prey communities. Ecology 87:2832–2841CrossRefGoogle Scholar
  7. Getz WM (2011) Biomass transformation webs provide a unified approach to consumer-reource modelling. Ecol Lett 14:113–124CrossRefGoogle Scholar
  8. Hadeler KP, Freedman HI (1989) Predator-prey populations with parasitic infection. J Math Biol 27: 609–631MathSciNetzbMATHCrossRefGoogle Scholar
  9. Han L, Ma Z, Hethcote HW (2001) Four predator prey models with infectious diseases. Math Comput Model 34:849–858MathSciNetzbMATHCrossRefGoogle Scholar
  10. Han L, Pugliese A (2009) Epidemics in two competing species. Nonlinear Anal RWA 10:723–744MathSciNetzbMATHCrossRefGoogle Scholar
  11. Haque M, Venturino E (2006) The role of transmissible diseases in Holling-Tanner predator-prey model. Theor Pop Biol 70:273–288zbMATHCrossRefGoogle Scholar
  12. Hatcher MJ, Dick JTA, Dunn AM (2006) How parasites affect interactions between competitors and predators. Ecol Lett 9:1253–1271CrossRefGoogle Scholar
  13. Hilker F, Langlais M, Malchow H (2009) The Allee effect and infectious diseases: extinction, multistability, and the disappearance of oscillations. Am Nat 173:72–88CrossRefGoogle Scholar
  14. Hilker FM, Langlais M, Petrovskii SV, Malchow H (2007) A diffusive SI model with Allee effect and application to FIV. Math Biosci 206:61–80MathSciNetzbMATHCrossRefGoogle Scholar
  15. Hilker FM, Malchow H (2006) Strange periodic attractors in a preypredator system with infected prey. Math Popul Stud 13:119–134MathSciNetzbMATHCrossRefGoogle Scholar
  16. Hilker FM, Schmitz K (2008) Disease-induced stabilization of predator-prey oscillations. J Theor Biol 255:299–306CrossRefGoogle Scholar
  17. Holdo RM et al (2009) A disease-mediated trophic cascade in the Serengeti and its implications for ecosystem C. PLoS Biol 7(9):e1000210CrossRefGoogle Scholar
  18. Hsieh YH, Hsiao CK (2008) Predator-prey model with disease infection in both populations. Math Med Biol 25:247–266zbMATHCrossRefGoogle Scholar
  19. Jones KE, Patel NG, Levy MA, Storeygard A, Balk D, Gittleman JL, Daszak P (2008) Global trends in emerging infectious diseases. Nature 451:990–993CrossRefGoogle Scholar
  20. Kacha A, Hbid MH (2009) Mathematical study of a bacteria model with level of infection structure. Nonlinear Anal Real World Appl 10:1662–1678MathSciNetzbMATHCrossRefGoogle Scholar
  21. Keesing F et al (2010) Impacts of biodiversity on the emergence and transmission of infectious diseases. Nature 468:647–652CrossRefGoogle Scholar
  22. Lafferty KD et al (2006) Parasites dominate food web links. PNAS 103:11211–11216CrossRefGoogle Scholar
  23. Lafferty KD et al (2008) Parasites in food webs: the ultimate missing links. Ecol Lett 11:533–546CrossRefGoogle Scholar
  24. Malchow H, Petrovskii S, Venturino E (2008) Spatiotemporal patterns in Ecology and Epidemiology. Chapman and Hall/CRC, LondonzbMATHGoogle Scholar
  25. Malchow H, Hilker FM, Sarkar RR, Brauer K (2005) Spatiotemporal patterns in an excitable plankton system with lysogenic viral infection. Math Comput Modell 42:1035–1048MathSciNetzbMATHCrossRefGoogle Scholar
  26. Matchett MR et al (2010) Enzootic Plague reduces black-footed ferret (Mustela nigripes) survival in Montana. Vector-Borne and Zoonotic Dis 10(1):27–35CrossRefGoogle Scholar
  27. McCann KS (2012) Food webs. Princeton University Press, PrincetonGoogle Scholar
  28. Morozov AYu (2011) Revealing the role of predator-dependent disease transmission in the epidemiology of a wildlife infection: a model study. Theor Ecol (published online 3 Nov 2011)Google Scholar
  29. Munson L et al (2008) Climate extremes promote fatal co-infections during Canine distemper epidemics in African lions. PLoS One 3(6):e2545CrossRefGoogle Scholar
  30. Oliveira NF, Hilker (2010) Modelling disease introduction as biological control of invasive predators to preserve endangered prey. Bull Math Biol 72:444–468MathSciNetzbMATHCrossRefGoogle Scholar
  31. Ostfeld RS, Keesing F, Eviner VT (eds) (2008) Infectious disease ecology: the effects of ecosystems on disease and of disease on ecosystems. Princeton University Press, PrincetonGoogle Scholar
  32. Randolph SE, Dobson ADM (2012) Pangloss revisited: a critique of the dilution effect and the biodiversity-buffers-disease paradigm. Parasitology (published online 16 Feb 2012)Google Scholar
  33. Redpath SM et al (2006) Testing the role of parasites in driving the cyclic population dynamics of a gamebird. Ecol Lett 9:410–418CrossRefGoogle Scholar
  34. Sieber M, Malchow H, Schimansky-Geier L (2007) Constructive effects of environmental noise in an excitable preypredator plankton system with infected prey. Ecol Complex 4:223–233CrossRefGoogle Scholar
  35. Siekmann I, Malchow H, Venturino E (2010) On competition of predators and prey infection. Ecol Complex 7:446–457CrossRefGoogle Scholar
  36. Súzan G et al (2009) Experimental evidence for reduced rodent diversity causing increased hantavirus prevalence. PLoS One 4:e5461CrossRefGoogle Scholar
  37. Taylor LH, Latham SM, Woolhouse ME (2001) Risk factors for human disease emergence. Phil Trans R Soc B 356:983–989CrossRefGoogle Scholar
  38. Telfer S et al (2005) Disruption of a host-parasite system following the introduction of an exotic species. Parasitology 130:661–668CrossRefGoogle Scholar
  39. Telfer S et al (2010) Species interactions in a parasite community drive infection risk in a wildlife population. Science 330:243–246CrossRefGoogle Scholar
  40. Thorne ET, Williams ES (1988) Disease and endangered species: the black-footed ferret as a recent example. Conserv Biol 2:66–74CrossRefGoogle Scholar
  41. Venturino E (1994) The influence of diseases on Lotka-Volterra systems. Rocky Mt J Math 24:381–402MathSciNetzbMATHCrossRefGoogle Scholar
  42. Venturino E (1995) Epidemics in predator-prey models: disease among the prey. In: Arino O, Axelrod D, Kimmel M, Langlais M (eds) Mathematical population dynamics: analysis of heterogeneity, vol 1. Wuertz Publishing Ltd, Winnipeg, pp 381–393Google Scholar
  43. Venturino E (2002) Epidemics in predator-prey models: disease in the predators. IMA J Math Appl Med Biol 19:185–205zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Institute of Information and Mathematical Sciences, New Zealand Institute for Advanced Study, and Infectious Disease Research CentreMassey UniversityAucklandNew Zealand
  2. 2.Department of Farm Animal Health, Faculty of Veterinary MedicineUniversity of Utrecht UtrechtThe Netherlands

Personalised recommendations