Journal of Mathematical Biology

, Volume 66, Issue 6, pp 1123–1153 | Cite as

High host density favors greater virulence: a model of parasite–host dynamics based on multi-type branching processes

  • K. BorovkovEmail author
  • R. Day
  • T. Rice


We use a multitype continuous time Markov branching process model to describe the dynamics of the spread of parasites of two types that can mutate into each other in a common host population. While most mathematical models for the virulence of infectious diseases focus on the interplay between the dynamics of host populations and the optimal characteristics for the success of the pathogen, our model focuses on how pathogen characteristics may change at the start of an epidemic, before the density of susceptible hosts decline. We envisage animal husbandry situations where hosts are at very high density and epidemics are curtailed before host densities are much reduced. The use of three pathogen characteristics: lethality, transmissibility and mutability allows us to investigate the interplay of these in relation to host density. We provide some numerical illustrations and discuss the effects of the size of the enclosure containing the host population on the encounter rate in our model that plays the key role in determining what pathogen type will eventually prevail. We also present a multistage extension of the model to situations where there are several populations and parasites can be transmitted from one of them to another. We conclude that animal husbandry situations with high stock densities will lead to very rapid increases in virulence, where virulent strains are either more transmissible or favoured by mutation. Further the process is affected by the nature of the farm enclosures.


Epidemics Virulence Multitype branching process 

Mathematics Subject Classification

Primary 60J85 Secondary 90D30 60J80 92B99 


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  1. Alexander HK, Day T (2010) Risk factors for the evolutionary emergence of pathogens. J R Soc Interface 7: 1455–1474CrossRefGoogle Scholar
  2. Alizon S, Hurford A, Mideo N, van Baalen M (2009) Virulence evolution and the trade-off hypothesis: history, current state of affairs and the future. J Evol Biol 22: 245–259CrossRefGoogle Scholar
  3. Allen LJS (2008) An introduction to stochastic epidemic models. In: Mathematical epidemiology, Lecture Notes in Math, Springer, Berlin, vol 1945, pp 81–130Google Scholar
  4. Anderson RM, May RM (1982) Coevolution of hosts and parasites. Parasitology 85: 411–426CrossRefGoogle Scholar
  5. Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University Press, OxfordGoogle Scholar
  6. Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis. Lecture Notes in Statistics, 151. Springer, New YorkCrossRefGoogle Scholar
  7. André J-B, Hochberg ME (2005) Virulence evolution in emerging infectious diseases. Evolution 59: 1406–1412Google Scholar
  8. Athreya KB, Ney PE (1972) Branching processes. Springer, New YorkzbMATHCrossRefGoogle Scholar
  9. Ball FG, Donelly P (1995) Strong approximations for epidemic models. Stoch Process Appl 55: 1–21zbMATHCrossRefGoogle Scholar
  10. Bartlett MS (1949) Some evolutionary stochastic processes. J R Stat Soc B 11: 211–229MathSciNetzbMATHGoogle Scholar
  11. Bartlett MS (1955) An introduction to stochastic processes. Cambridge Univeristy Press, CambridgezbMATHGoogle Scholar
  12. Becker NG (1989) Analysis of infectious disease data. Chapman and Hall, LondonGoogle Scholar
  13. Becker NG, Marschner I (1990) The effect of heterogeneity on the spread of disease. In: Gabriel J-P et al (eds) Stochastic processes in epidemic theory 86. Lecture Notes in Biomathematics, pp 90–103Google Scholar
  14. Bergh O (2007) The dual myths of the healthy wild fish and the unhealthy farmed fish. Dis Aquat Org 75: 159–164CrossRefGoogle Scholar
  15. Bernoulli D (1760) Essai d’une nouvelle analyse de la mortalité causée par la petit vérole et des advanteges de l’inoculation pour la prévenir. Mém. Math Phys Acad R Sci, Paris, pp 1–45Google Scholar
  16. Bharucha-Reid AT (1958) Comparison of populations whose growth can be described by a branching stochastic process. Sankhyā 19: 1–14zbMATHGoogle Scholar
  17. Biggs PM (1985) Infectious animal disease and its control. Phil Trans R Soc Lond B 310: 259–274CrossRefGoogle Scholar
  18. Bolker BM, Nanda A, Shah D (2010) Transient virulence of emerging pathogens. J R Soc Interface 7: 811–822CrossRefGoogle Scholar
  19. Bull JJ, Ebert D (2008) Invasion thresholds and the evolution of non-equilibrium virulence. Evol Appl 1: 172–182CrossRefGoogle Scholar
  20. Crawford D (2007) Deadly companions: how microbes shaped our history. Oxford University Press, OxfordGoogle Scholar
  21. Daley DJ, Gani J (1999) Epidemic modelling: an introduction. Cambridge University Press, CambridgezbMATHGoogle Scholar
  22. Day T (2002) On the evolution of virulence and the relationship between various measures of mortality. Proc R Soc Lond B 269: 1317–1323CrossRefGoogle Scholar
  23. Day T (2003) Virulence evolution and the timing of disease life-history events. Trends Ecol Evol 18: 113–118CrossRefGoogle Scholar
  24. Day T, Gandon S (2006) Insights from Price’s equation into evolutionary epidemiology. In: Feng Z, Dieckmann U, Levin SA (eds) Disease evolution: models, concepts, and data analyses 71. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, , pp 23–43Google Scholar
  25. Day T, Proulx SR (2004) A general theory for the evolutionary dynamics of virlence. Am Nat 163: E41–E63CrossRefGoogle Scholar
  26. de Roode JC, Yates AJ, Altizer S (2008) Virulence—transmission trade-offs and population divergence in virulence in a naturally occurring butterfly parasite. Proc Natl Acad Sci USA 105: 7489–7494CrossRefGoogle Scholar
  27. Dietz K, Schenke D (1985) Mathematical models for infectious disease statistics. In: Atkinson AC, Fienberg SE (eds) A celebration of statistics. Springer, New York, pp 167–204CrossRefGoogle Scholar
  28. Ebert D (1999) The evolution and expression of parasite virulence. In: Stearns SC (eds) Evolution in health and disease. Oxford University Press, Oxford, pp 161–172Google Scholar
  29. Ewald PW (1983) Host–parasite relations, vectors, and the evolution of disease severity. Annu Rev Ecol Syst 14: 465–485CrossRefGoogle Scholar
  30. Ewald PW (1994) Evolution of infectious disease. Oxford University Press, OxfordGoogle Scholar
  31. Frank SA (1996) Models of parasite virulence. Q Rev Biol 71: 37–78CrossRefGoogle Scholar
  32. Fraser D (2005) Animal welfare and the intensification of animal production: an alternative interpretation. UN Food and Agriculture Organization, RomeGoogle Scholar
  33. Fraser C, Hollingsworth TD, Chapman R, de Wolf F, Hanage WP (2007) Variation in HIV-1 set-point viral load: epidemiological analysis and an evolutionary hypothesis. Proc Natl Acad Sci USA 104: 17441–17446CrossRefGoogle Scholar
  34. Gandon S, Mackinnon MJ, Nee S, Read AF (2001) Imperfect vaccines and the evolution of pathogen virulence. Nature 414: 751–756CrossRefGoogle Scholar
  35. Gantmakher FR (1989) The theory of matrices, vol 1. Chelsea, New YorkGoogle Scholar
  36. Gerbier G (1999) Effect of animal density on FMD spread. European Commission for the Control of Foot-and-Mouth Disease Report. Research Group of the Standing Technical Committee, Maisons-AlfortGoogle Scholar
  37. Getz WM, Lloyd-Smith JO (2006) Basic methods for modeling the invasion and spread of contagious diseases. In: Disease evolution, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 71. Am. Math. Soc., Providence, RI, pp 87–109Google Scholar
  38. Girvan M, Callaway DS, Newman MEJ, Strogatz SH (2002) A simple model of epidemics with pathogen mutation. Phys Rev E 65: 031915CrossRefGoogle Scholar
  39. Greenwood M (1931) On the statistical measure of infectiousness. J Hyg Camb 31: 336–351CrossRefGoogle Scholar
  40. Guan Y, Zheng BJ, He YQ et al (2003) Isolation and characterization of viruses related to the SARS coronavirus from animals in southern China. Science 302: 276–278CrossRefGoogle Scholar
  41. Haccou P, Jagers P, Vatutin VA (2005) Branching processes: variation, growth, and extinction of populations. Cambridge University Press, CambridgezbMATHCrossRefGoogle Scholar
  42. Hanski, IA, Gaggiotti, OE (eds) (2004) Ecology, genetics and evolution of metapopulations. Elsevier Academic Press, Burlington, MAGoogle Scholar
  43. Harvell CD, Kim K, Burkholder JM et al (1999) Emerging marine diseases—climate links and anthropogenic factors. Science 285: 1505–1510CrossRefGoogle Scholar
  44. Harvell D, Aronson R, Baron N et al (2004) The rising tide of ocean diseases: unsolved problems and research priorities. Front Ecol Environ 2: 375–382CrossRefGoogle Scholar
  45. Heinzmann D (2009) Extinction times in multitype Markov branching processes. J Appl Probab 46: 296–307MathSciNetzbMATHCrossRefGoogle Scholar
  46. Hethcote HW (1994) A thousand and one epidemic models. In: Lecture Notes in Biomathematics 100, pp 504–515Google Scholar
  47. Jagers P (1975) Branching processes with biological applications. Wiley, LondonzbMATHGoogle Scholar
  48. Jeltsch F, Müller MS, Grimm V, Wissel C, Brandl R (1997) Pattern formation triggered by rare events: lessons from the spread of rabies. Proc R Soc Lond B 264: 495–503CrossRefGoogle Scholar
  49. Karlin S, Taylor HM (1981) A second course in stochastic processes. Academic Press, New YorkzbMATHGoogle Scholar
  50. Kendall D (1956) Deterministic and stochastic epidemics in closed popultaions. In: Proceedings of third Berkeley symposium in mathematics and statistics and probability 4, pp 149–165Google Scholar
  51. Kermack WO, McKendrick AG (1927) A contribution to the mathematcial theory of epidemics. Proc R Soc Lond A 115: 700–721zbMATHCrossRefGoogle Scholar
  52. Kijima M (1997) Markov processes for stochastic modeling. Chapman and Hall, LondonzbMATHGoogle Scholar
  53. Knolle H (1989) Host density and the evolution of parasite virulence. J Theor Biol 136: 199–207MathSciNetCrossRefGoogle Scholar
  54. Lenski RE, May RM (1994) The evolution of virulence in parasites and pathogens: reconciliation between two competing hypotheses. J Theor Biol 169: 253–265CrossRefGoogle Scholar
  55. Lion S, Boots M (2010) Are parasites ‘prudent’ in space. Ecol Lett 13: 1245–1255CrossRefGoogle Scholar
  56. Lipsitch M, Moxon ER (1997) Virulence and transmissibility of pathogens: what is the relationship. Trends Microbiol 5: 31–37CrossRefGoogle Scholar
  57. Ludwig D (1975) Qualitative behavior of stochastic epidemics. Math Biosci 23: 47–73MathSciNetzbMATHCrossRefGoogle Scholar
  58. Massad E (1987) Transmission rates and the evolution of pathogenicity. Evolution 41: 1127–1130CrossRefGoogle Scholar
  59. Mathews JD, Chesson JM, McCaw JM, McVernon J (2009) Understanding influenza transmission, immunity and pandemic threats. Influenza Other Respir Viruses 3(4): 143–149CrossRefGoogle Scholar
  60. Matthews P (1988) Covering problems for Brownian motion on spheres. Ann Probab 18: 189–199MathSciNetCrossRefGoogle Scholar
  61. McKendrick AG (1926) Applications of mathematics to medical problems. Proc Edinb Math Soc 14: 98–130Google Scholar
  62. Meester R, de Koning J, de Jong MCC, Diekmann O (2002) Modeling and real-time prediction of classical swine fever epidemics. Biometrics 58: 178–184MathSciNetzbMATHCrossRefGoogle Scholar
  63. Mode CT, Sleeman CK (2000) Stochastic processes in epidemiology. World Scientific, SingaporezbMATHGoogle Scholar
  64. Pulkinnen K, Suomalainen L-R, Read AF, Ebert D, Rintamäki P, Valtonen ET (2010) Intensive fish farming and the evolution of pathogen virulence: the case of coumnaris disease in Finland. Proc R Soc B 277: 593–600CrossRefGoogle Scholar
  65. Slingenbergh J, Glibert M (2010) Do old and new forms of poultry go together. In: FAO abstracts: poultry production in the 21st century. FAO, Rome, p 47Google Scholar
  66. Sniezko S (1974) The effects of environmental stress on outbreaks of infectious diseases of fishes. J Fish Biol 6: 197–208CrossRefGoogle Scholar
  67. Thieme HR (2003) Mathematics in population biology. Princeton University Press, Princeton, NJzbMATHGoogle Scholar
  68. van Baalen M (2002) Contact networks and the evolution of virulence. In: Dieckmann U, Metz JAJ, Sabelis MW, Sigmund K (eds) Adaptive dynamics of infectious diseases: in pursuit of virulence management. Cambridge University Press, Cambridge, pp 85–103CrossRefGoogle Scholar
  69. Witter RL (1997) Increased virulence of Marek’s disease virus field isolates. Avian Dis 41: 149–163CrossRefGoogle Scholar
  70. Whittle P (1955) The outcome of a stochastic epidemic—a note on Bailey’s paper. Biometrica 42: 116–122MathSciNetzbMATHGoogle Scholar
  71. Yates A, Antia R, Regoes RR (2006) How do pathogen evolution and host heterogeneity interact in disease emergence?. Proc R Soc Lond B 273: 3075–3083CrossRefGoogle Scholar
  72. Yersin A (1894) La peste bubonique á Hong Kong. Ann Inst Pasteur 8: 662–667Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of MelbourneParkvilleAustralia
  2. 2.Department of ZoologyThe University of MelbourneParkvilleAustralia

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