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

Abstract

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.

References

  1. Alexander HK, Day T (2010) Risk factors for the evolutionary emergence of pathogens. J R Soc Interface 7: 1455–1474

    Article  Google 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: 245–259

    Article  Google Scholar 

  3. Allen LJS (2008) An introduction to stochastic epidemic models. In: Mathematical epidemiology, Lecture Notes in Math, Springer, Berlin, vol 1945, pp 81–130

  4. Anderson RM, May RM (1982) Coevolution of hosts and parasites. Parasitology 85: 411–426

    Article  Google Scholar 

  5. Anderson RM, May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University Press, Oxford

    Google Scholar 

  6. Andersson H, Britton T (2000) Stochastic epidemic models and their statistical analysis. Lecture Notes in Statistics, 151. Springer, New York

    Google Scholar 

  7. André J-B, Hochberg ME (2005) Virulence evolution in emerging infectious diseases. Evolution 59: 1406–1412

    Google Scholar 

  8. Athreya KB, Ney PE (1972) Branching processes. Springer, New York

    Google Scholar 

  9. Ball FG, Donelly P (1995) Strong approximations for epidemic models. Stoch Process Appl 55: 1–21

    MATH  Article  Google Scholar 

  10. Bartlett MS (1949) Some evolutionary stochastic processes. J R Stat Soc B 11: 211–229

    MathSciNet  MATH  Google Scholar 

  11. Bartlett MS (1955) An introduction to stochastic processes. Cambridge Univeristy Press, Cambridge

    Google Scholar 

  12. Becker NG (1989) Analysis of infectious disease data. Chapman and Hall, London

    Google 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 90–103

  14. Bergh O (2007) The dual myths of the healthy wild fish and the unhealthy farmed fish. Dis Aquat Org 75: 159–164

    Article  Google 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 1–45

  16. Bharucha-Reid AT (1958) Comparison of populations whose growth can be described by a branching stochastic process. Sankhyā 19: 1–14

    MATH  Google Scholar 

  17. Biggs PM (1985) Infectious animal disease and its control. Phil Trans R Soc Lond B 310: 259–274

    Article  Google Scholar 

  18. Bolker BM, Nanda A, Shah D (2010) Transient virulence of emerging pathogens. J R Soc Interface 7: 811–822

    Article  Google Scholar 

  19. Bull JJ, Ebert D (2008) Invasion thresholds and the evolution of non-equilibrium virulence. Evol Appl 1: 172–182

    Article  Google Scholar 

  20. Crawford D (2007) Deadly companions: how microbes shaped our history. Oxford University Press, Oxford

    Google Scholar 

  21. Daley DJ, Gani J (1999) Epidemic modelling: an introduction. Cambridge University Press, Cambridge

    Google Scholar 

  22. Day T (2002) On the evolution of virulence and the relationship between various measures of mortality. Proc R Soc Lond B 269: 1317–1323

    Article  Google Scholar 

  23. Day T (2003) Virulence evolution and the timing of disease life-history events. Trends Ecol Evol 18: 113–118

    Article  Google 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 23–43

  25. Day T, Proulx SR (2004) A general theory for the evolutionary dynamics of virlence. Am Nat 163: E41–E63

    Article  Google 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: 7489–7494

    Article  Google 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 167–204

    Google 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 161–172

    Google Scholar 

  29. Ewald PW (1983) Host–parasite relations, vectors, and the evolution of disease severity. Annu Rev Ecol Syst 14: 465–485

    Article  Google Scholar 

  30. Ewald PW (1994) Evolution of infectious disease. Oxford University Press, Oxford

    Google Scholar 

  31. Frank SA (1996) Models of parasite virulence. Q Rev Biol 71: 37–78

    Article  Google Scholar 

  32. Fraser D (2005) Animal welfare and the intensification of animal production: an alternative interpretation. UN Food and Agriculture Organization, Rome

    Google 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: 17441–17446

    Article  Google Scholar 

  34. Gandon S, Mackinnon MJ, Nee S, Read AF (2001) Imperfect vaccines and the evolution of pathogen virulence. Nature 414: 751–756

    Article  Google Scholar 

  35. Gantmakher FR (1989) The theory of matrices, vol 1. Chelsea, New York

    Google 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-Alfort

  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 87–109

  38. Girvan M, Callaway DS, Newman MEJ, Strogatz SH (2002) A simple model of epidemics with pathogen mutation. Phys Rev E 65: 031915

    Article  Google Scholar 

  39. Greenwood M (1931) On the statistical measure of infectiousness. J Hyg Camb 31: 336–351

    Article  Google 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: 276–278

    Article  Google Scholar 

  41. Haccou P, Jagers P, Vatutin VA (2005) Branching processes: variation, growth, and extinction of populations. Cambridge University Press, Cambridge

    Google Scholar 

  42. Hanski, IA, Gaggiotti, OE (eds) (2004) Ecology, genetics and evolution of metapopulations. Elsevier Academic Press, Burlington, MA

    Google Scholar 

  43. Harvell CD, Kim K, Burkholder JM et al (1999) Emerging marine diseases—climate links and anthropogenic factors. Science 285: 1505–1510

    Article  Google 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: 375–382

    Article  Google Scholar 

  45. Heinzmann D (2009) Extinction times in multitype Markov branching processes. J Appl Probab 46: 296–307

    MathSciNet  MATH  Article  Google Scholar 

  46. Hethcote HW (1994) A thousand and one epidemic models. In: Lecture Notes in Biomathematics 100, pp 504–515

  47. Jagers P (1975) Branching processes with biological applications. Wiley, London

    Google 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: 495–503

    Article  Google Scholar 

  49. Karlin S, Taylor HM (1981) A second course in stochastic processes. Academic Press, New York

    Google 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 149–165

  51. Kermack WO, McKendrick AG (1927) A contribution to the mathematcial theory of epidemics. Proc R Soc Lond A 115: 700–721

    MATH  Article  Google Scholar 

  52. Kijima M (1997) Markov processes for stochastic modeling. Chapman and Hall, London

    Google Scholar 

  53. Knolle H (1989) Host density and the evolution of parasite virulence. J Theor Biol 136: 199–207

    MathSciNet  Article  Google Scholar 

  54. Lenski RE, May RM (1994) The evolution of virulence in parasites and pathogens: reconciliation between two competing hypotheses. J Theor Biol 169: 253–265

    Article  Google Scholar 

  55. Lion S, Boots M (2010) Are parasites ‘prudent’ in space. Ecol Lett 13: 1245–1255

    Article  Google Scholar 

  56. Lipsitch M, Moxon ER (1997) Virulence and transmissibility of pathogens: what is the relationship. Trends Microbiol 5: 31–37

    Article  Google Scholar 

  57. Ludwig D (1975) Qualitative behavior of stochastic epidemics. Math Biosci 23: 47–73

    MathSciNet  MATH  Article  Google Scholar 

  58. Massad E (1987) Transmission rates and the evolution of pathogenicity. Evolution 41: 1127–1130

    Article  Google Scholar 

  59. Mathews JD, Chesson JM, McCaw JM, McVernon J (2009) Understanding influenza transmission, immunity and pandemic threats. Influenza Other Respir Viruses 3(4): 143–149

    Article  Google Scholar 

  60. Matthews P (1988) Covering problems for Brownian motion on spheres. Ann Probab 18: 189–199

    MathSciNet  Article  Google Scholar 

  61. McKendrick AG (1926) Applications of mathematics to medical problems. Proc Edinb Math Soc 14: 98–130

    Google 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: 178–184

    MathSciNet  MATH  Article  Google Scholar 

  63. Mode CT, Sleeman CK (2000) Stochastic processes in epidemiology. World Scientific, Singapore

    Google 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: 593–600

    Article  Google 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 47

  66. Sniezko S (1974) The effects of environmental stress on outbreaks of infectious diseases of fishes. J Fish Biol 6: 197–208

    Article  Google Scholar 

  67. Thieme HR (2003) Mathematics in population biology. Princeton University Press, Princeton, NJ

    Google 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 85–103

    Google Scholar 

  69. Witter RL (1997) Increased virulence of Marek’s disease virus field isolates. Avian Dis 41: 149–163

    Article  Google Scholar 

  70. Whittle P (1955) The outcome of a stochastic epidemic—a note on Bailey’s paper. Biometrica 42: 116–122

    MathSciNet  MATH  Google 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: 3075–3083

    Article  Google Scholar 

  72. Yersin A (1894) La peste bubonique á Hong Kong. Ann Inst Pasteur 8: 662–667

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to K. Borovkov.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Borovkov, K., Day, R. & Rice, T. High host density favors greater virulence: a model of parasite–host dynamics based on multi-type branching processes. J. Math. Biol. 66, 1123–1153 (2013). https://doi.org/10.1007/s00285-012-0526-9

Download citation

Keywords

  • Epidemics
  • Virulence
  • Multitype branching process

Mathematics Subject Classification

  • Primary 60J85
  • Secondary 90D30
  • 60J80
  • 92B99