Restricted access
Research articles
Research articles
Imitation dynamics of vaccination behaviour on social networks
Published:28 July 2010https://doi.org/10.1098/rspb.2010.1107
Abstract
The problem of achieving widespread immunity to infectious diseases by voluntary vaccination is often presented as a public-goods dilemma, as an individual's vaccination contributes to herd immunity, protecting those who forgo vaccination. The temptation to free-ride brings the equilibrium vaccination level below the social optimum. Here, we present an evolutionary game-theoretic approach to this problem, exploring the roles of individual imitation behaviour and population structure in vaccination. To this end, we integrate an epidemiological process into a simple agent-based model of adaptive learning, where individuals use anecdotal evidence to estimate costs and benefits of vaccination. In our simulations, we focus on parameter values that are realistic for a flu-like infection. Paradoxically, as agents become more adept at imitating successful strategies, the equilibrium level of vaccination falls below the rational individual optimum. In structured populations, the picture is only somewhat more optimistic: vaccination is widespread over a range of low vaccination costs, but coverage plummets after cost exceeds a critical threshold. This result suggests parallels to historical scenarios in which vaccination coverage provided herd immunity for some time, but then rapidly dropped. Our work sheds light on how imitation of peers shapes individual vaccination choices in social networks.
References
- 1
Colgrave J. . 2006 State of immunity: the politics of vaccination in twentieth-century America. Berkeley, CA: University of California Press. Google Scholar - 2
Fine P.& Clarkson J. . 1986 Individual versus public priorities in the determination of optimal vaccination polices. Am. J. Epidemiol. 124, 1012–1020. Crossref, PubMed, ISI, Google Scholar - 3
Galvani A. P., Reluga T. C.& Chapman G. B. . 2007 Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum. Proc. Natl Acad. Sci. USA 104, 5692–5697.doi:10.1073/pnas.0606774104 (doi:10.1073/pnas.0606774104). Crossref, PubMed, ISI, Google Scholar - 4
Basu S., Chapman G. B.& Galvani A. P. . 2008 Integrating epidemiology, psychology, and economics to achieve HPV vaccination targets. Proc. Natl Acad. Sci. USA 105, 19 018–19 023. Crossref, ISI, Google Scholar - 5
Bauch C. T., Galvani A. P.& Earn D. J. D. . 2003 Group interest versus self interest in smallpox vaccination policy. Proc. Natl Acad. Sci. USA 100, 10 564–10 567.doi:10.1073/pnas.1731324100 (doi:10.1073/pnas.1731324100). Crossref, ISI, Google Scholar - 6
Bauch C. T.& Earn D. J. D. . 2004 Vaccination and the theory of games. Proc. Natl Acad. Sci. USA 101, 13 391–13 394.doi:10.1073/pnas.0403823101 (doi:10.1073/pnas.0403823101). Crossref, ISI, Google Scholar - 7
Vardavas R., Breban R.& Blower S. . 2007 Can influenza epidemics be prevented by voluntary vaccination? PLoS Comp. Biol. 3, e85.doi:10.1371/journal.pcbi.0030085 (doi:10.1371/journal.pcbi.0030085). Crossref, PubMed, ISI, Google Scholar - 8
Breban R., Vardavas R.& Blower S , 2007 Mean-field analysis of an inductive reasoning game: application to influenza vaccination. Phys. Rev. E 76, 031127.doi:10.1103/PhysRevE.76.031127 (doi:10.1103/PhysRevE.76.031127). Crossref, ISI, Google Scholar - 9
van Boven M., Klinkenberg D., Pen I., Weissing F. J.& Heesterbeek H. . 2008 Self-interest versus group-interest in antiviral control. PLoS ONE 3, e1558.doi:10.1371/journal.pone.0001558 (doi:10.1371/journal.pone.0001558). Crossref, PubMed, ISI, Google Scholar - 10
Bauch C. T. . 2005 Imitation dynamics predict vaccinating behaviour. Proc. R. Soc. B 272, 1669–1675.doi:10.1098/rspb.2005.3153 (doi:10.1098/rspb.2005.3153). Link, ISI, Google Scholar - 11
Cojocaru M. G. . 2008 Dynamic equilibria of group vaccination strategies in a heterogeneous population. J. Glob. Optim. 40, 51–63.doi:10.1007/s10898-007-9204-7 (doi:10.1007/s10898-007-9204-7). Crossref, ISI, Google Scholar - 12
Reluga T. C., Bauch C. T.& Galvani A. P. . 2006 Evolving public perceptions and stability in vaccine uptake. Math. Biosci. 204, 185–198.doi:10.1016/j.mbs.2006.08.015 (doi:10.1016/j.mbs.2006.08.015). Crossref, PubMed, ISI, Google Scholar - 13
Jansen V. A., Stollenwerk N., Jensen H. J., Ramsay M. E., Edmunds W.J.& Rhodes C. J. . 2003 Measles outbreaks in a population with declining vaccine uptake. Science 301, 804.doi:10.1126/science.1086726 (doi:10.1126/science.1086726). Crossref, PubMed, ISI, Google Scholar - 14
Hardin G. . 1968 The tragedy of the commons. Science 162, 1243–1248.doi:10.1126/science.162.3859.1243 (doi:10.1126/science.162.3859.1243). Crossref, PubMed, ISI, Google Scholar - 15
Perisic A.& Bauch C. T. . 2008 Social contact networks and disease eradicability under voluntary vaccination. PLoS Comp. Biol. 5, e1000280. Crossref, ISI, Google Scholar - 16
Tversky A.& Kahneman D. . 1973 Availability: a heuristic for judging frequency and probability. Cogn. Psychol. 5, 207–232.doi:10.1016/0010-0285(73)90033-9 (doi:10.1016/0010-0285(73)90033-9). Crossref, ISI, Google Scholar - 17
Johnson E. J.& Tversky A. . 1983 Affect, generalization, and the perception of risk. J. Pers. Soc. Psychol. 45, 20–31.doi:10.1037/0022-3514.45.1.20 (doi:10.1037/0022-3514.45.1.20). Crossref, ISI, Google Scholar - 18
Palekar R., Pettifor A., Behets F.& MacPhail C. . 2008 Association between knowing someone who died of AIDS and behavior change among South African youth. AIDS Behav. 12, 903–912.doi:10.1007/s10461-007-9325-5 (doi:10.1007/s10461-007-9325-5). Crossref, PubMed, ISI, Google Scholar - 19
Nicoll A., Elliman D.& Ross E. . 1998 MMR vaccination and autism 1998: déjà vu–pertussis and brain damage 1974? Br. Med. J. 316, 715–716. Crossref, PubMed, Google Scholar - 20
Eames K. T. D. . 2009 Networks of influence and infection: parental choices and childhood disease. J. R. Soc. Interface 6, 811–814.doi:10.1098/rsif.2009.0085 (doi:10.1098/rsif.2009.0085). Link, ISI, Google Scholar - 21
Maynard-Smith J. . 1982 Evolution and the theory of games. Cambridge, UK: Cambridge University Press. Crossref, Google Scholar - 22
Nowak M. A.& Sigmund K. . 2004 Evolutionary dynamics of biological games. Science 303, 793–799.doi:10.1126/science.1093411 (doi:10.1126/science.1093411). Crossref, PubMed, ISI, Google Scholar - 23
Nowak M. A. . 2006 Evolutionary dynamics: exploring the equations of life. Cambridge, MA: Harvard University Belknap Press. Crossref, Google Scholar - 24
Levin B. R., Lipsitch M.& Bonhoeffer S. . 1999 Population biology, evolution, and infectious disease: convergence and synthesis. Science 283, 806–809.doi:10.1126/science.283.5403.806 (doi:10.1126/science.283.5403.806). Crossref, PubMed, ISI, Google Scholar - 25
Diekmann O.& Heesterbeek J. A. P. . 2000 Mathematical epidemiology of infectious diseases: model building, analysis, and interpretation. Chichester, UK: Wiley. Google Scholar - 26
Blume L. E. . 1993 The statistical mechanics of strategic interaction. Games Econ. Behav. 5, 387–424.doi:10.1006/game.1993.1023 (doi:10.1006/game.1993.1023). Crossref, ISI, Google Scholar - 27
Szabó G.& Tőke C. . 1998 Evolutionary prisoner's dilemma game on a square lattice. Phys. Rev. E 58, 69.doi:10.1103/PhysRevE.58.69 (doi:10.1103/PhysRevE.58.69). Crossref, ISI, Google Scholar - 28
Traulsen A., Pacheco J. M.& Nowak M. A. . 2007 Pairwise comparison and selection temperature in evolutionary game dynamics. J. Theor. Biol. 246, 522–529.doi:10.1016/j.jtbi.2007.01.002 (doi:10.1016/j.jtbi.2007.01.002). Crossref, PubMed, ISI, Google Scholar - 29
Traulsen A., Semmann D., Sommerfeld R. D., Krambeck H.-J.& Milinski M. . 2010 Human strategy updating in evolutionary games. Proc. Natl Acad. Sci. USA 107, 2962–2966.doi:10.1073/pnas.0912515107 (doi:10.1073/pnas.0912515107). Crossref, PubMed, ISI, Google Scholar - 30
Erdős P.& Rényi A. . 1959 On random graphs. Publ. Math. Debrecen 6, 290–297. Crossref, Google Scholar - 31
Barabási A. L.& Albert R. . 1999 Emergence of scaling in random networks. Science 286, 509–512.doi:10.1126/science.286.5439.509 (doi:10.1126/science.286.5439.509). Crossref, PubMed, ISI, Google Scholar - 32
Hofbauer J.& Sigmund K. . 1998 Evolutionary games and population dynamics. Cambridge, UK: Cambridge University Press. Crossref, Google Scholar - 33
Schuster P.& Sigmund K. . 1983 Replicator dynamics. J. Theor. Biol. 100, 533–538.doi:10.1016/0022-5193(83)90445-9 (doi:10.1016/0022-5193(83)90445-9). Crossref, ISI, Google Scholar - 34
Taylor P. D.& Jonker L. B. . 1978 Evolutionarily stable strategies and game dynamics. Math. Biosci. 40, 145–156.doi:10.1016/0025-5564(78)90077-9 (doi:10.1016/0025-5564(78)90077-9). Crossref, ISI, Google Scholar - 35
Pastor-Satorras R.& Vespignani A. . 2002 Immunization of complex networks. Phys. Rev. E 65, 036104.doi:10.1103/PhysRevE.65.036104 (doi:10.1103/PhysRevE.65.036104). Crossref, ISI, Google Scholar - 36
Schlag K. H. . 1998 Why imitate, and if so, how? A boundedly rational approach to multi-armed bandits. J. Econ. Theory 78, 130–156.doi:10.1006/jeth.1997.2347 (doi:10.1006/jeth.1997.2347). Crossref, ISI, Google Scholar - 37
- 38
Bikhchandani S., Hirshleifer D.& Welch I. . 1992 A theory of fads, fashion, custom, and cultural change as informational cascades. J. Pol. Econ. 107, 797–817. Google Scholar - 39
Barton A. M. . 2009 Application of cascade theory to online systems: a study of email and Google cascades. Minn. J.L. Sci. Tech. 10, 473–502. Google Scholar - 40
Szabó G.& Hauert C. . 2002 Phase transitions and volunteering in spatial public goods games. Phys. Rev. Lett. 89, 11810.doi:10.1103/PhysRevLett.89.118101 (doi:10.1103/PhysRevLett.89.118101). Crossref, ISI, Google Scholar - 41
Hauert C.& Doebeli M. . 2004 Spatial structure often inhibits the evolution of cooperation in the Snowdrift game. Nature 428, 643–646.doi:10.1038/nature02360 (doi:10.1038/nature02360). Crossref, PubMed, ISI, Google Scholar - 42
Nowak M. A.& May R. M. . 1992 Evolutionary games and spatial chaos. Nature 359, 826–829.doi:10.1038/359826a0 (doi:10.1038/359826a0). Crossref, ISI, Google Scholar - 43
Nowak M. A. . 2006 Five rules for the evolution of cooperation. Science 314, 1560–1563.doi:10.1126/science.1133755 (doi:10.1126/science.1133755). Crossref, PubMed, ISI, Google Scholar - 44
Nowak M. A., Tarnita C. E.& Antal T. . 2010 Evolutionary dynamics in structured populations. Phil. Trans. R. Soc. B 365, 19–30.doi:10.1098/rstb.2009.0215 (doi:10.1098/rstb.2009.0215). Link, ISI, Google Scholar - 45
Ohtsuki H., Hauert C., Lieberman E.& Nowak M. A. . 2006 A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505.doi:10.1038/nature04605 (doi:10.1038/nature04605). Crossref, PubMed, ISI, Google Scholar - 46
Tarnita C. E., Antal T., Ohtsuki H.& Nowak M. A. . 2009 Evolutionary dynamics in set structured populations. Proc. Natl Acad. Sci. USA 106, 8601–8604.doi:10.1073/pnas.0903019106 (doi:10.1073/pnas.0903019106). Crossref, PubMed, ISI, Google Scholar - 47
Tarnita C. E., Ohtsuki H., Antal T., Fu F.& Nowak M. A. . 2009 Strategy selection in structured populations. J. Theor. Biol. 259, 570–581.doi:10.1016/j.jtbi.2009.03.035 (doi:10.1016/j.jtbi.2009.03.035). Crossref, PubMed, ISI, Google Scholar - 48
Bansal S., Pourbohloul B.& Meyers L. A. . 2006 A comparative analysis of influenza vaccination programs. PLoS Med. 3, e387.doi:10.1371/journal.pmed.0030387 (doi:10.1371/journal.pmed.0030387). Crossref, PubMed, ISI, Google Scholar - 49
Capolongo M. J., DiBonaventura M. D.& Chapman G. B. . 2006 Physician vaccinate thyself: why influenza vaccination rates are higher among clinicians than among nonclinicians. Ann. Behav. Med. 31, 288 296.doi:10.1207/s15324796abm3103_11 (doi:10.1207/s15324796abm3103_11). Crossref, ISI, Google Scholar - 50
Palfrey T. R.& Rosenthal H. . 1984 Participation and the provision of discrete public goods: a strategic analysis. J. Pub. Econ. 24, 171–193.doi:10.1016/0047-2727(84)90023-9 (doi:10.1016/0047-2727(84)90023-9). Crossref, ISI, Google Scholar - 51
Donald A.& Muthu V. . 2002 MMR links with autism and inflammatory bowel disease. Clin. Evid. 7, 331–340. Google Scholar
