Journal of Mathematical Biology

, Volume 66, Issue 7, pp 1527–1553

Games of age-dependent prevention of chronic infections by social distancing

Article

Abstract

Epidemiological games combine epidemic modelling with game theory to assess strategic choices in response to risks from infectious diseases. In most epidemiological games studied thus-far, the strategies of an individual are represented with a single choice parameter. There are many natural situations where strategies can not be represented by a single dimension, including situations where individuals can change their behavior as they age. To better understand how age-dependent variations in behavior can help individuals deal with infection risks, we study an epidemiological game in an SI model with two life-history stages where social distancing behaviors that reduce exposure rates are age-dependent. When considering a special case of the general model, we show that there is a unique Nash equilibrium when the infection pressure is a monotone function of aggregate exposure rates, but non-monotone effects can appear even in our special case. The non-monotone effects sometimes result in three Nash equilibria, two of which have local invasion potential simultaneously. Returning to a general case, we also describe a game with continuous age-structure using partial-differential equations, numerically identify some Nash equilibria, and conjecture about uniqueness.

Keywords

Epidemiological games Social distancing Age structure 

Mathematics Subject Classification

91A13 91A10 92D30 92D25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramowitz, M, Stegun, IA (eds) (1972) Handbook of mathematical functions with formulas, graphs, and mathematical tables. 10th edn. National Bureau of Standards, Washington, DCMATHGoogle Scholar
  2. Arias E (2006) United states life tables, 2003. National Vital Statistics Reports 54(14)Google Scholar
  3. Aubin JP (1979) Mathematical methods of game and economic theory, DoverGoogle Scholar
  4. Brito DL, Sheshinski E, Intriligator MD (1991) Externalities and compulsory vaccinations. J Public Econ 45: 69–90CrossRefGoogle Scholar
  5. Bunimovich-Mendrazitsky S, Stone L (2005) Modeling polio as a disease of development. J Theor Biol 237(3): 302–315MathSciNetCrossRefGoogle Scholar
  6. Charlesworth B (1994) Evolution in age-structured populations. Cambridge University Press, New YorkMATHCrossRefGoogle Scholar
  7. Chen FH (2004) Rational behavioral response and the transmission of stds. Theor Pop Biol 66(1): 307–316MATHCrossRefGoogle Scholar
  8. Chen FH (2006) A susceptible-infected epidemic model with voluntary vaccinations. J Math Biol 53(1): 253–272MathSciNetMATHCrossRefGoogle Scholar
  9. Clark CW, Mangel M (2000) Dynamics State Variable Models in Ecology: Methods and Applications. Oxford University Press, New York, NYGoogle Scholar
  10. Cornforth DM, Reluga TC, Shim E, Bauch CT, Galvani AP, Meyers LA (2011) Erratic flu vaccination emerges from short-sighted behavior in contact networks. PLOS Comput BiolGoogle Scholar
  11. Diekmann O, Heesterbeek JAP, Metz JAJ (1990) On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J Math Biol 28(4): 365–382MathSciNetMATHCrossRefGoogle Scholar
  12. Donoghue JF, Golowich E, Holstein BR (1996) Dyamics of the standard model. Cambridge University Press, CambridgeGoogle Scholar
  13. Houston A, McNamara J (1999) Models of adaptive behaviour. Cambridge University Press, CambridgeGoogle Scholar
  14. Kermack WO, McKendrick AG (1933) Contributions to the mathematical-theory of epidemics. III. further studies of the problem of endemicity. Proc Royal Soc Lond 141(843): 94–122MATHCrossRefGoogle Scholar
  15. McNamara JM, Houston AI, Collins E.J. (2001) Optimality models in behavioral biology. SIAM Rev 43(3): 413–466MathSciNetMATHCrossRefGoogle Scholar
  16. Reluga T (2010) Game theory of social distancing in response to an epidemic. PLOS Comput Biol 6(5): e1000793. doi:10.1371/journal.pcbi.1000793 MathSciNetCrossRefGoogle Scholar
  17. Reluga TC (2009) An SIS game with two subpopulations. J Biol Dyn 3(5): 515–531MathSciNetCrossRefGoogle Scholar
  18. Reluga TC, Galvani AP (2011) A general approach for population games with application to vaccination. Math Biosci 230(2): 67–78MathSciNetMATHCrossRefGoogle Scholar
  19. Reluga TC, Medlock J, Poolman E, Galvani AP (2007) Optimal timing of disease transmission in an age-structured population. Bull Math Biol 69(8): 2711–2722MathSciNetMATHCrossRefGoogle Scholar
  20. Rzewuski J (1969) Field theory. Hafner Publishing Company, New YorkGoogle Scholar
  21. Schenzle D (1984) An age-structured model of pre-and post-vaccination measles transmission. Math Med Biol 1(2): 169–191MathSciNetMATHCrossRefGoogle Scholar
  22. van den Driessche P, Watmough J (2000) A simple SIS epidemic model with a backward bifurcation. J Math Biol 40(1): 525–540MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

Personalised recommendations