Abstract
Parasites reproduce and are subject to natural selection at several different, but intertwined, levels. In the recent paper, Gilchrist and Coombs (Theor. Popul. Biol. 69:145–153, 2006) relate the between-host transmission in the context of an SI model to the dynamics within a host. They demonstrate that within-host selection may lead to an outcome that differs from the outcome of selection at the host population level. In this paper we combine the two levels of reproduction by considering the possibility of superinfection and study the evolution of the pathogen’s within-host reproduction rate p. We introduce a superinfection function φ = φ(p,q), giving the probability with which pathogens with trait q, upon transmission to a host that is already infected by pathogens with trait p, “take over” the host. We consider three cases according to whether the function q → φ(p,q) (i) has a discontinuity, (ii) is continuous, but not differentiable, or (iii) is differentiable in q = p. We find that in case (i) the within-host selection dominates in the sense that the outcome of evolution at the host population level coincides with the outcome of evolution in a single infected host. In case (iii), it is the transmission to susceptible hosts that dominates the evolution to the extent that the singular strategies are the same as when the possibility of superinfections is ignored. In the biologically most relevant case (ii), both forms of reproduction contribute to the value of a singular trait. We show that when φ is derived from a branching process variant of the submodel for the within-host interaction of pathogens and target cells, the superinfection functions fall under case (ii). We furthermore demonstrate that the superinfection model allows for steady coexistence of pathogen traits at the host population level, both on the ecological, as well as on the evolutionary time scale.
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References
Alizon, S.: Parasite virulence evolution: insights from embedded models. PhD thesis, University of Paris, France (2006)
Alizon S. and Baalen M. (2005). Emergence of a convex trade-off between transmission and virulence. Am. Nat. 165: 155–167
Anderson R.M. and May R.M. (1982). Coevolution of hosts and parasites. Parasitology 85: 411–426
Anderson R.M. and May R.M. (1991). Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford
Antia R., Levin B.R. and May R.M. (1994). Within-host population dynamics and the evolution and maintenance of macroparasite virulence. Am. Nat. 144: 457–472
Day T. and Proulx S.R. (2004). A general theory for the evolutionary dynamics of virulence. Am. Nat. 163: 40–63
De Leenheer P. and Smith H.L. (2003). Virus dynamics: a global analysis. SIAM J. Appl. Math. 63(4): 1313–1327
Dieckmann U. and Metz J.A.J. (2006). Surprising evolutionary predictions from enhanced ecological realism. Theor. Popul. Biol. 69(3): 263–281
Dieckmann, U., Metz, J.A.J., Sabelis, M.W., Sigmund, K.: Adaptive Dynamics of Infectious Diseases: In Pursuit of Virulence Management. Cambridge Studies in Adaptive Dynamics, Cambridge University Press, Cambridge (2002)
Diekmann, O.: A beginner’s guide to adaptive dynamics. In: Mathematical Modelling of Population Dynamics of Banach Center Publ., vol. 63, pp. 47–86. Polish Acad. Sci., Warsaw (2004)
Diekmann O., Gyllenberg M. and Metz J.A.J. (2003). Steady-state analysis of structured population models. Theor. Popul. Biol. 63(4): 309–338
Diekmann, O., Heesterbeek, J.A.P.: Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation. Wiley Series in Mathematical and Computational Biology. Wiley, Chichester (2000)
Ewald P.W. (1983). Host–parasite relations, vectors and the evolution of disease severity. Ann. Rev. Ecol. Syst. 14: 465–485
Ewald P.W. (1994). Evolution of Infectious Disease. Oxford University Press, Oxford
Ganusov V.V. and Antia R. (2003). Trade-offs and the evolution of virulence of microparasites: do details matter? Theor. Popul. Biol. 64(2): 211–220
Geritz S.A.H. (2005). Resident-invader dynamics and the coexistence of similar strategies. J. Math. Biol. 50(1): 67–82
Geritz S.A.H., Gyllenberg M., Jacobs F.J.A. and Parvinen K. (2002). Invasion dynamics and attractor inheritance. J. Math. Biol. 44: 548–560
Geritz S.A.H., Kisdi E., Meszena G. and Metz J.A.J. (1998). Evolutionary singular strategies and the adaptive growth and branching of the evolutionary tree. Evol. Ecol. 12: 35–57
Gilchrist M.A. and Coombs D. (2006). Evolution of virulence: interdependence, constraints and selection using nested models. Theor. Popul. Biol. 69: 145–153
Gomes, G.M., Medley, G.F.: Dynamics of multiple strains of infectious agents coupled by cross-immunity: a comparison of models. In: Castillo-Chavez, C., et al. (eds.) Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory. Proceedings of a Workshop, Integral part of the IMA Program on Mathematics in Biology. IMA Vol. Math. Appl. 126, pp. 171–191. Springer, New York (2002)
Grenfell B.T., Pybus O.G., Gog J.R., Wood J.L.N., Daly J.M., Mumford J.A. and Holmes E.C. (2004). Unifying the epidemiological and evolutionary dynamics of pathogens. Science 303: 327–332
Haccou, P., Jagers, P., Vatutin, V.A.: Branching Processes: Variation, Growth, and Extinction of Populations. Cambridge Studies in Adaptive Dynamics. Cambridge University Press, Cambridge (2005)
Hochberg M.E. and Holt R.D. (1990). The coexistence of competing parasites. I. The role of cross-species infection. Am. Nat. 136: 517–541
Klinkenberg D. and Heesterbeek J.A.P. (2005). A simple model for the within-host dynamics of a protozoan parasite. Proc. Roy. Soc. B 272: 593–600
Lenski R.E. and May R.M. (1994). The evolution of virulence in parasites and pathogens: reconciliation between the competing hypotheses. J. Theor. Biol. 169: 253–265
Levin, S.A.: Coevolution. In: Freedman H., Strobeck C. (eds.) Population Biology. Lecture notes in Biomathematics 52, pp. 328–334 (1983)
Levin, S.A.: Some approaches to the modelling of coevolutionary interactions. In: Nitecki M. (ed.) Coevolution, pp. 21–65 (1983)
Levin S.A. and Pimentel D. (1981). Selection of intermediate rates of increase in parasite-host systems. Am. Nat. 117: 308–315
Matessi C. and Di Pasquale C. (1996). Long-term evolution of multilocus traits. J. Math. Biol. 34: 613–653
May R.M. and Anderson R.M. (1983). Epidemiology and genetics in the coevolution of parasites and hosts. Proc. Roy. Soc. Lond. B 219: 281–313
Metz, J.A.J., Geritz, S.A.H., Meszéna, G., Jacobs, F.J.A., van Heerwaarden, J.S.: Adaptive dynamics, a geometrical study of the consequences of nearly faithful reproduction. In: van Strien, S.J., et al. (eds.) Stochastic and spatial structures of dynamical systems. Proceedings of the Meeting, Amsterdam, Netherlands, January 1995. Verh. Afd. Natuurkd., Amsterdam, 1. Reeks, K. Ned. Akad. Wet. 45, pp. 183–231 (1996)
Metz, J.A.J., Mylius, S.D., Diekmann, O.: When does evolution optimise? On the relation between types of density dependence and evolutionarily stable life histories. IIASA working paper WP-96-04, (1996). http://www.iiasa.ac.at/cgi-bin/pubsrch?WP96004
Meyers L.A., Levin B.R., Richardson A.R. and Stojiljkovic I. (2003). Epidemiology, hypermutation, within-host evolution and the virulence of neisseria meningitidis. Proc. Roy. Soc. Lond. B 270: 1667–1677
Mosquera J. and Adler F.R. (1998). Evolution of virulence: a unified framework for coinfection and superinfection. J. Theor. Biol. 195: 293–313
Murase A., Sasaki T. and Kajiwara T. (2005). Stability analysis of pathogen-immune interaction dynamics. J. Math. Biol. 51(3): 247–267
Mylius S.D. and Diekmann O. (1995). On evolutionarily stable life histories, optimization and the need to be specific about density dependence. Oikos 74: 218–224
Nowak M.A. and May R.M. (1994). Superinfection and the evolution of parasite virulence. Proc. Roy. Soc. Lond. B 255: 81–89
Nowak M.A. and May R.M. (2000). Virus Dynamics: Mathematical Principles of Immunology and Virology. Oxford University Press, Oxford
Perelson A.S., Kirschner D.E. and Boer R. (1993). Dynamics of HIV infection of CD4+ T cells. Math. Biosci. 114(1): 81–125
Pugliese, A.: Evolutionary dynamics of virulence. Available online at: http://www.science.unitn.it/pugliese/
Pugliese A. (2002). On the evolutionary coexistence of parasite strains. Math. Biosci. 177/178: 355–375
Saldaña J., Elena S.F. and Solé R.V. (2003). Coinfection and superinfection in RNA virus population: a selection-mutation model. Math. Biosci. 183: 135–160
Smith V.H. and Holt R.D. (1996). Resource competition and within-host disease dynamics. Tree 11: 386–389
Thieme, H.R.: Pathogen competition and coexistence and the evolution of virulence. In: Mathematics for Life Sciences and Medicine. Springer, Heidelberg (2007, in press)
van Baalen M. and Sabelis M.W. (1995). The milker-killer dilemma and spatially structured predator-prey interactions. Oikos 74: 391–400
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Boldin, B., Diekmann, O. Superinfections can induce evolutionarily stable coexistence of pathogens. J. Math. Biol. 56, 635–672 (2008). https://doi.org/10.1007/s00285-007-0135-1
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DOI: https://doi.org/10.1007/s00285-007-0135-1
Keywords
- Evolution
- Virulence
- Evolutionary branching
- ESS
- CSS
- Basic reproduction ratio
- Superinfection
- Invasibility
- Optimization
- Within-host dynamics
- Pathogen
- Nested model
- Pairwise invasibility plot
- Coexistence
- Evolutionary suicide
- Singular strategy
- Dimorphism
- Virus