Skip to main content

Main menu

  • Home
  • Articles
    • Current
    • Special Feature Articles - Most Recent
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • List of Issues
  • Front Matter
    • Front Matter Portal
    • Journal Club
  • News
    • For the Press
    • This Week In PNAS
    • PNAS in the News
  • Podcasts
  • Authors
    • Information for Authors
    • Editorial and Journal Policies
    • Submission Procedures
    • Fees and Licenses
  • Submit
  • Submit
  • About
    • Editorial Board
    • PNAS Staff
    • FAQ
    • Accessibility Statement
    • Rights and Permissions
    • Site Map
  • Contact
  • Journal Club
  • Subscribe
    • Subscription Rates
    • Subscriptions FAQ
    • Open Access
    • Recommend PNAS to Your Librarian

User menu

  • Log in
  • My Cart

Search

  • Advanced search
Home
Home
  • Log in
  • My Cart

Advanced Search

  • Home
  • Articles
    • Current
    • Special Feature Articles - Most Recent
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • List of Issues
  • Front Matter
    • Front Matter Portal
    • Journal Club
  • News
    • For the Press
    • This Week In PNAS
    • PNAS in the News
  • Podcasts
  • Authors
    • Information for Authors
    • Editorial and Journal Policies
    • Submission Procedures
    • Fees and Licenses
  • Submit
Research Article

Vaccination and the theory of games

Chris T. Bauch and David J. D. Earn
  1. †Department of Mathematics and Statistics, University of Guelph, Guelph, ON, Canada N1G 2W1; and §Department of Mathematics and Statistics, McMaster University, Hamilton, ON, Canada L8S 4K1

See allHide authors and affiliations

PNAS September 7, 2004 101 (36) 13391-13394; https://doi.org/10.1073/pnas.0403823101
Chris T. Bauch
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
David J. D. Earn
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
  1. Edited by Maurice R. Hilleman, Merck Institute for Vaccinology, West Point, PA, and approved July 21, 2004 (received for review May 28, 2004)

  • Article
  • Figures & SI
  • Info & Metrics
  • PDF
Loading

Abstract

Voluntary vaccination policies for childhood diseases present parents with a subtle challenge: if a sufficient proportion of the population is already immune, either naturally or by vaccination, then even the slightest risk associated with vaccination will outweigh the risk from infection. As a result, individual self-interest might preclude complete eradication of a vaccine-preventable disease. We show that a formal game theoretical analysis of this problem leads to new insights that help to explain human decision-making with respect to vaccination. Increases in perceived vaccine risk will tend to induce larger declines in vaccine uptake for pathogens that cause more secondary infections (such as measles and pertussis). After a vaccine scare, even if perceived vaccine risk is greatly reduced, it will be relatively difficult to restore prescare vaccine coverage levels.

The history of vaccination policy includes numerous bouts of public resistance, often in the form of vaccine scares (1-4). In the United Kingdom, for example, a pertussis vaccine scare in the 1970s caused a decline in the level of vaccine coverage, resulting in substantial increases in morbidity and mortality from whooping cough (4). Currently, measles-mumps-rubella vaccine uptake is declining in the United Kingdom, with mounting concern that widespread outbreaks of measles may recur (5).

In deciding whether to vaccinate their children, parents consider the risk of morbidity from vaccination, the probability that their child will become infected, and the risk of morbidity from such an infection. The decisions of individual parents are indirectly influenced by the decisions of all other parents, because the sum of these decisions yields the vaccine coverage levels in the population and hence the course of epidemics.

Game theory (6-9) attempts to predict individual behavior in such a setting, where the payoff to strategies chosen by individuals depends on the strategies adopted by others in the population. Here, we integrate epidemic modeling (10) into a game theoretical framework to analyze population behavior under voluntary vaccination policies for childhood diseases. This approach allows us to quantify how risk perception influences expected vaccine uptake and coverage levels and what role is played by the epidemiological characteristics of the pathogens.

The Vaccination Game

Description of Game. For simplicity, we imagine that all individuals are provided with the same information and use this information in the same way to assess risks. An individual's strategy is the probability P that s/he will choose to vaccinate. The vaccine uptake level in the population is the proportion of newborns who will be vaccinated and hence is the mean of all strategies adopted by individuals in the population. We ignore any delay between changes in vaccine uptake and corresponding changes in overall vaccine coverage in the population; consequently, if no disease-related or vaccine-related mortality occurs, then the proportion of the population vaccinated, p, will be equal to the vaccine uptake level.

The payoff to an individual will be greater when morbidity risk (probability of adverse consequences) is lower. We use r v and r i to denote the morbidity risks from vaccination and infection, respectively, and πp to denote the probability that an unvaccinated individual will eventually be infected if the vaccine coverage level in the population is p. With this notation, the payoff is -r v to a vaccinated individual and -r iπp to an unvaccinated individual. Thus, the strategy of vaccinating with probability P yields expected payoff Math

In the context of vaccination, parents act according to perceived morbidity risks, which may differ significantly from actual morbidity risks (3, 11). Consequently, we interpret r i and r v as the perceived morbidity risks from infection and vaccination and E(P, p) as the perceived payoff. The game is unchanged if we scale the payoff function by a constant. Therefore, we can eliminate one of the parameters, leaving only the relative risk, r = r v/r i. Thus, we can write Math

Characterization of Nash Equilibria. We now seek to identify which strategies are likely to be adopted. If most of the population adopts strategy P, and individuals that adopt any other strategy Q always obtain a lower payoff than those adopting P, then P is said to be a Nash equilibrium. In contrast, if most individuals adopt strategy Q, but individuals adopting a strategy that is closer than Q to P obtain a higher payoff than those adopting Q (and those adopting a strategy further from P obtain a lower payoff), and if this is true for any Q ≠ P, then P is said to be convergently stable. If P is a Nash equilibrium, and everyone is currently playing P, then no one should change strategy. If P is convergently stable, then regardless of what strategy is most common in the population, individuals should start to play strategies closer to P, and ultimately adopt P. It is generally expected that a strategy observed in a real population (12) must be a convergently stable Nash equilibrium (CSNE).

Suppose that a proportion ε of the population vaccinates with probability P and the remainder vaccinate with probability Q. Because we ignore any difference between vaccine uptake, εP + (1 - ε)Q, and overall vaccine coverage in the population, p, we can always write Math

The payoff to individuals playing P is, therefore, Math

whereas the payoff to individuals playing Q is Math

The payoff gain to an individual playing P in such a population is Math

The payoff gain ΔE is a measure of the incentive for an individual to change strategies from Q to P. For any given relative risk, r, there is a unique strategy P = P *, such that ΔE is strictly positive for all strategies Q ≠ P * and all proportions ε, where 0 ≤ ε < 1 (see Appendix for a proof). The special case of this fact for small proportions playing Q (ε near 1) implies that P * is a Nash equilibrium. We also show in Appendix that, if neither P nor Q is equal to the Nash equilibrium P *, but P is closer than Q to P *, then ΔE > 0, implying that P * is convergently stable and hence a CSNE.

The unique CSNE in this vaccination game is easily found (see Appendix). If the vaccine is perceived to be sufficiently risky (r ≥ π0) then the CSNE is “never vaccinate” (P * = 0). In contrast, if r < π0, then the CSNE is “vaccinate with nonzero probability P *” (0 < P * < 1). In the latter case, the CSNE is said to be mixed (as opposed to the pure strategies P = 0 and P = 1).

Incorporation of an Epidemic Model

To make more precise predictions, we must specify the infection probability πp. For this, we need an epidemiological model. We use a standard three-compartment model in which individuals are either susceptible to the disease (S), infectious (I), or recovered to a state of lifelong immunity (R). This SIR model, and variants thereof, are widely used in modeling childhood diseases (10, 13). The model is specified by the rates of change of the proportions of the population in each compartment. Math Math Math

Here, μ is the mean birth and death rate, β is the mean transmission rate, 1/γ is the mean infectious period, and p is the vaccine uptake level (assuming, for simplicity, that individuals are never infected before being vaccinated). Once a dynamical steady state is reached, the vaccine coverage level in the population will equal the uptake level. Because we shall focus on the steady-state solution of the model, our notation p for vaccine uptake is consistent with our notation in the game theoretical analysis (compare Eq. 3).

The third equation in the SIR model above is superfluous because S + I + R = 1. The remaining two equations can be written in a convenient, dimensionless form, Math Math

where τ = t/γ is time measured in units of the mean infectious period, f = μ/γ is the infectious period as a fraction of mean lifetime, and Math is the basic reproductive ratio (the average number of secondary cases produced by a typical primary case in a fully susceptible population). For childhood diseases, f < 0.001 and Math (e.g., ref. 10).

The predictions of the SIR model depend on the critical coverage level that eliminates the disease from the population (10), p crit, which itself is a function of Math. Math

If p ≥ p crit, then the system converges to the disease-free state (Ŝ, Î) = (1 - p, 0), whereas if p < p crit, it converges to a stable endemic state given by Math

and Math

Because S and I are constant in this situation, the probability that an unvaccinated individual eventually becomes infected can be expressed, using Eqs. 10-14, as the proportion of susceptible individuals becoming infected versus dying in any unit time, Math

(Note that the parameter f does not appear in this expression for πp, so the CSNE will not depend on the birth rate or the infectious period of the disease.) The condition r < π0, which yields a mixed CSNE, can therefore be written Math

The value of the mixed CSNE P *, obtained by solving the equation r = πP* for P *, is Math

Results and Discussion

For any perceived relative risk r > 0, the expected vaccine uptake is less than the eradication threshold, i.e., P * < p crit (Fig. 1). This finding formalizes an argument that has previously been made qualitatively (8, 14); namely, it is impossible to eradicate a disease through voluntary vaccination when individuals act according to their own interests. In situations where vaccination is perceived to be more risky than contracting the disease (r > 1), one would expect, even without the aid of a model, that no parents would vaccinate their children. Our game theoretical analysis shows that, in fact, the threshold in perceived relative risk beyond which all parents should cease vaccinating depends on Math. In particular, parents can be expected to play a pure nonvaccinator strategy if r > π0, i.e., if Math

Fig. 1.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 1.

Vaccine coverage p * at the CSNE versus relative risk r, from Eq. 17, for various values of Embedded Image. Dashed horizontal lines demarcate the critical coverage level p crit that eliminates the disease from the population (Eq. 12). In the limit of very large Embedded Image, the plot of p * versus r approaches a step function with a step at r = 1 (Eq. 17).

For childhood diseases, this relative-risk threshold is close to 1, but for diseases with relatively small Math, the threshold could be substantially smaller.

With knowledge of the perceived relative risk, r, we can thus predict vaccine coverage levels under voluntary policies. However, risk perception (and hence the value of r) can change over time in response to a variety of factors, such as media coverage and the activities of antivaccination groups (3, 11, 15, 16, ¶). Under normal circumstances, the relative risk is perceived to be very low (typically much lower than the relative-risk threshold, r ≪ π0 < 1). During a vaccine scare, the perceived risk of vaccination will rise (by definition) and hence relative risk will increase to some new level r′> r. Note that a reduction in the perceived risk of morbidity from natural infection has the same effect. In either case, the qualitative nature of our predictions depends on whether the new risk ratio exceeds the relative-risk threshold; if r < r′ ≪ π0, then behavioral changes will be relatively minor during a scare, whereas if r ≪ π0 < r′, then dramatic changes in vaccine uptake can occur (see Table 1 and Fig. 2).

Fig. 2.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 2.

Analysis of vaccine scares: payoff gain, ΔE, and change in vaccine uptake, ΔP, after a shift in risk perception from r < π0 to r′ (see Table 1). For this figure, r = 0.1 and the proportion of individuals currently adopting the new CSNE is ε = 0 (corresponding to the start of a vaccine scare); the shapes of the curves are qualitatively similar for other values of r and ε.

View this table:
  • View inline
  • View popup
Table 1. Payoff gain ΔE (Eq. 6) to an individual adopting the new CSNE P′ (associated with perceived relative risk r′) when a proportion ε of the population does the same, and the remainder play the strategy P (which is the CSNE associated with relative risk r)

Several lines of evidence suggest that it is likely that r′ > π0 during a vaccine scare. Many parents currently have concerns about the safety of the measles-mumps-rubella vaccine (17, 18) and other vaccines (19), and many parents (in developed countries) believe that diseases such as measles and whooping cough are essentially harmless (20). (Together, these observations indicate that r′ > 1 for measles, mumps, and rubella at present in the United Kingdom.) Targeted surveys show that among subscribers to a parenting magazine (21) and among inhabitants of specific areas in the United Kingdom (22), a significant proportion of parents believe vaccines entail more risk than the diseases against which they protect (r′ > 1) and this perception is correlated with not vaccinating (21).

When r ≪ π0 < r′, the degree to which a vaccine scare is likely to have an impact on vaccination behavior depends sensitively on the value of Math. The payoff gain ΔE that measures the incentive to switch from the previous CSNE P (associated with r ≪ π0) to the new CSNE P′ (associated with r′ > π0) is always larger for diseases with larger Math. Consequently, we would expect individuals to be convinced more rapidly to change their vaccination behavior in the face of a vaccine scare for measles or whooping cough (for which Math) than for less transmissible infections. In general, for a given increase in risk perception, we expect precipitous reductions in vaccine uptake to be more common for diseases with higher Math.

If Math is large, individuals are also likely to be more responsive to any reductions in the perceived relative risk of vaccination that occur after a vaccine scare (Fig. 3 and Table 1). Such reductions in r might result from media coverage of a few severe cases of disease (which are more likely as vaccine uptake drops and disease incidence rises). More importantly, education programs that aim to increase public confidence in vaccines after a scare are likely to be most effective for precisely the vaccines for which scares have the greatest impact.

Fig. 3.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 3.

Analysis of public education programs to counteract vaccine scares: payoff gain, ΔE, and change in vaccine uptake, ΔP, after a shift in risk perception from r > π0 to r′. As in Fig. 2, ε = 0 here. The results are independent of r (because the CSNE is always P = 0 when r > π0). The shapes of the curves are qualitatively similar for other values of ε, but the maximum of ΔE goes to zero as ε increases to 1.

Unfortunately, the effectiveness of education programs is constrained in a way that vaccine scares are not. During a vaccine scare, the payoff gain ΔE is given by the expression in the second row of Table 1; this expression is bounded below by a positive number for all ε (even for ε = 1), so the incentive not to vaccinate remains substantial even as the vaccine coverage approaches zero. In contrast, during successful education programs to combat a vaccine scare, risk perception will shift from r > π0 to r′ < π0, and the proportion of the population vaccinated will climb to the new CSNE level as more and more individuals are vaccinated. In this case, the payoff gain for adopting the new CSNE is given by the third row of Table 1, which implies (regardless of Math) that ΔE → 0 as ε → 1; this means that the incentive to vaccinate diminishes as the vaccine coverage approaches the new CSNE level. We conclude that, in general, it will be relatively easy to induce a drop in vaccine uptake during a scare, but relatively difficult to restore uptake levels afterward. This prediction is consistent with the history of the pertussis vaccination scare during the 1970s in Britain (23), for which vaccine uptake dropped much more quickly than it later recovered after the scare. All else being equal, we anticipate that when the current measles-mumps-rubella scare in Britain is over, vaccine uptake will rise more slowly than it declined.

We have demonstrated previously that game theory can be a useful tool for evaluating schemes to prepare for the potential reintroduction of a pathogen that has been eradicated globally through mass vaccination (9). Here, we have investigated the feedback between individual vaccination decisions and population-level processes that determine vaccine uptake and herd immunity for an endemic disease, bearing in mind that vaccination decisions are strongly influenced by incorrect risk perception (11, ¶). Because our goal has been to elucidate the most fundamental issues, we have focused on the simplest possible epidemiological model appropriate for childhood diseases and have assumed implicitly that transient dynamics (13), seasonal forcing (13, 24), and stochasticity (13, 25) all have negligible effects. We have also ignored variance in risk perception and any effects of risk perception spreading nonhomogeneously through social networks. All these features of real systems merit further investigation.

Acknowledgments

We thank Sigal Balshine, Junling Ma, and two anonymous referees for helpful comments. C.T.B. is supported by the Natural Sciences and Engineering Research Council of Canada. D.J.D.E. is supported by the Natural Sciences and Engineering Research Council of Canada, the Canadian Institutes of Health Research, the Canada Foundation for Innovation, the Ontario Innovation Trust, and an Ontario Premier's Research Excellence Award.

Footnotes

  • ↵ ‡ To whom correspondence should be addressed. E-mail: cbauch{at}uoguelph.ca.

  • This paper was submitted directly (Track II) to the PNAS office.

  • Abbreviation: CSNE, convergently stable Nash equilibrium.

  • ↵ ¶ Fox, F., MMR Learning Lessons, Meeting hosted by the Science Media Centre at the Royal Institution, London, U.K., May 2, 2002. Available at: www.sciencemediacentre.org/mmr_report.htm.

  • Copyright © 2004, The National Academy of Sciences

References

  1. ↵
    Durbach, N. (2000) Soc. Hist. Med. 13 (1), 45-62. pmid:11624425
    OpenUrlAbstract
  2. Albert, M. R., Ostheimer, K. G. & Breman, J. G. (2001) N. Engl. J. Med. 344 , 375-379. pmid:11172172
    OpenUrlCrossRefPubMed
  3. ↵
    Streefland, P. H. (2001) Health Policy 55 , 159-172. pmid:11164965
    OpenUrlCrossRefPubMed
  4. ↵
    Baker, J. P. (2003) Vaccine 21 , 4003-4010. pmid:12922137
    OpenUrlCrossRefPubMed
  5. ↵
    Jansen, V. A. A., Stollenwerk, N., Jensen, H. J., Ramsay, M. E., Edmunds, W. J. & Rhodes, C. J. (2003) Science 301 , 804. pmid:12907792
    OpenUrlFREE Full Text
  6. ↵
    von Neumann, J. & Morgenstern, O. (1944) Theory of Games and Economic Behavior (Princeton Univ. Press, Princeton).
  7. Maynard-Smith, J. (1982) Evolution and the Theory of Games (Cambridge Univ. Press, Cambridge, U.K.).
  8. ↵
    May, R. M. (2000) Science 287 , 601-602. pmid:10691541
    OpenUrlFREE Full Text
  9. ↵
    Bauch, C. T., Galvani, A. P. & Earn, D. J. D. (2003) Proc. Natl. Acad. Sci. USA 100 , 10564-10567. pmid:12920181
    OpenUrlAbstract/FREE Full Text
  10. ↵
    Anderson, R. M. & May, R. M. (1991) Infectious Diseases of Humans (Oxford Univ. Press, Oxford).
  11. ↵
    Bellaby, P. (2003) Br. Med. J. 327 , 725-728. pmid:14512482
    OpenUrlFREE Full Text
  12. ↵
    Eshel, I. (1996) J. Math. Biol. 34 , 485-510. pmid:8691082
    OpenUrlCrossRefPubMed
  13. ↵
    Bauch, C. T. & Earn, D. J. D. (2003) Proc. R. Soc. London Ser. B 270 , 1573-1578.
    OpenUrlCrossRefPubMed
  14. ↵
    Geoffard, P. & Philipson, T. (1997) Am. Econ. Rev. 87 , 222-230.
    OpenUrl
  15. ↵
    Taylor, B., Miller, E., Lingam, R., Andrews, N., Simmons, A. & Stowe, J. (2002) Br. Med. J. 324 , 393-396. pmid:11850369
    OpenUrlAbstract/FREE Full Text
  16. ↵
    Gangarosa, E. J., Galazka, A. M., Wolfe, C. R., Phillips, L. M., Gangarosa, R. E., Miller, E. & Chen, R. T. (1998) Lancet 351 , 356-361. pmid:9652634
    OpenUrlCrossRefPubMed
  17. ↵
    Roberts, R. J., Sandifer, Q. D., Evans, M. R., Nolan-Farrell, M. Z. & Davis, P. M. (1995) Br. Med. J. 310 , 1629-1639. pmid:7795447
    OpenUrlAbstract/FREE Full Text
  18. ↵
    Evans, M., Stoddart, H., Condon, L., Freeman, E., Grizzell, M. & Mullen, R. (2001) Br. J. Gen. Prac. 51 , 904-910.
    OpenUrlAbstract/FREE Full Text
  19. ↵
    Lashuay, N., Tjoa, T., de Nuncio, M. L. Z., Franklin, M., Elder, J. & Jones, M. (2000) Prev. Med. 31 , 522-528. pmid:11071832
    OpenUrlCrossRefPubMed
  20. ↵
    Schmitt, H. J. (2002) Vaccine 20 , S2-S4.
    OpenUrlCrossRef
  21. ↵
    Asch, D. A., Baron, J., Hershey, J. C., Kunreuther, H., Meszaros, J., Ritov, I. & Spranca, M. (1994) Med. Decis. Making 14 , 118-123. pmid:8028464
    OpenUrlAbstract/FREE Full Text
  22. ↵
    Smailbegovic, M. S., Laing, G. J. & Bedford, H. (2003) Child: Care Health Dev. 29 , 303-311. pmid:12823336
    OpenUrlCrossRefPubMed
  23. ↵
    Miller, E. & Gay, N. J. (1997) Dev. Biol. Stand. 89 , 15-23. pmid:9272331
    OpenUrlPubMed
  24. ↵
    Earn, D. J. D., Rohani, P., Bolker, B. M. & Grenfell, B. T. (2000) Science 287 , 667-670. pmid:10650003
    OpenUrlAbstract/FREE Full Text
  25. ↵
    Bartlett, M. S. (1960) Stochastic Population Models in Ecology and Epidemiology (Methuen, London).
PreviousNext
Back to top
Article Alerts
Email Article

Thank you for your interest in spreading the word on PNAS.

NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. We do not capture any email address.

Enter multiple addresses on separate lines or separate them with commas.
Vaccination and the theory of games
(Your Name) has sent you a message from PNAS
(Your Name) thought you would like to see the PNAS web site.
CAPTCHA
This question is for testing whether or not you are a human visitor and to prevent automated spam submissions.
Citation Tools
Vaccination and the theory of games
Chris T. Bauch, David J. D. Earn
Proceedings of the National Academy of Sciences Sep 2004, 101 (36) 13391-13394; DOI: 10.1073/pnas.0403823101

Citation Manager Formats

  • BibTeX
  • Bookends
  • EasyBib
  • EndNote (tagged)
  • EndNote 8 (xml)
  • Medlars
  • Mendeley
  • Papers
  • RefWorks Tagged
  • Ref Manager
  • RIS
  • Zotero
Request Permissions
Share
Vaccination and the theory of games
Chris T. Bauch, David J. D. Earn
Proceedings of the National Academy of Sciences Sep 2004, 101 (36) 13391-13394; DOI: 10.1073/pnas.0403823101
del.icio.us logo Digg logo Reddit logo Twitter logo CiteULike logo Facebook logo Google logo Mendeley logo
  • Tweet Widget
  • Facebook Like
  • Mendeley logo Mendeley
Proceedings of the National Academy of Sciences of the United States of America: 101 (36)
Table of Contents

Submit

Sign up for Article Alerts

Jump to section

  • Article
    • Abstract
    • The Vaccination Game
    • Incorporation of an Epidemic Model
    • Results and Discussion
    • Acknowledgments
    • Footnotes
    • References
  • Figures & SI
  • Info & Metrics
  • PDF

You May Also be Interested in

Setting sun over a sun-baked dirt landscape
Core Concept: Popular integrated assessment climate policy models have key caveats
Better explicating the strengths and shortcomings of these models will help refine projections and improve transparency in the years ahead.
Image credit: Witsawat.S.
Model of the Amazon forest
News Feature: A sea in the Amazon
Did the Caribbean sweep into the western Amazon millions of years ago, shaping the region’s rich biodiversity?
Image credit: Tacio Cordeiro Bicudo (University of São Paulo, São Paulo, Brazil), Victor Sacek (University of São Paulo, São Paulo, Brazil), and Lucy Reading-Ikkanda (artist).
Syrian archaeological site
Journal Club: In Mesopotamia, early cities may have faltered before climate-driven collapse
Settlements 4,200 years ago may have suffered from overpopulation before drought and lower temperatures ultimately made them unsustainable.
Image credit: Andrea Ricci.
Steamboat Geyser eruption.
Eruption of Steamboat Geyser
Mara Reed and Michael Manga explore why Yellowstone's Steamboat Geyser resumed erupting in 2018.
Listen
Past PodcastsSubscribe
Birds nestling on tree branches
Parent–offspring conflict in songbird fledging
Some songbird parents might improve their own fitness by manipulating their offspring into leaving the nest early, at the cost of fledgling survival, a study finds.
Image credit: Gil Eckrich (photographer).

Similar Articles

Site Logo
Powered by HighWire
  • Submit Manuscript
  • Twitter
  • Facebook
  • RSS Feeds
  • Email Alerts

Articles

  • Current Issue
  • Special Feature Articles – Most Recent
  • List of Issues

PNAS Portals

  • Anthropology
  • Chemistry
  • Classics
  • Front Matter
  • Physics
  • Sustainability Science
  • Teaching Resources

Information

  • Authors
  • Editorial Board
  • Reviewers
  • Subscribers
  • Librarians
  • Press
  • Site Map
  • PNAS Updates
  • FAQs
  • Accessibility Statement
  • Rights & Permissions
  • About
  • Contact

Feedback    Privacy/Legal

Copyright © 2021 National Academy of Sciences. Online ISSN 1091-6490