Bulletin of Mathematical Biology

, Volume 75, Issue 10, pp 1961–1984 | Cite as

Equilibria of an Epidemic Game with Piecewise Linear Social Distancing Cost

  • Timothy C. Reluga
Original Article


Around the world, infectious disease epidemics continue to threaten people’s health. When epidemics strike, we often respond by changing our behaviors to reduce our risk of infection. This response is sometimes called “social distancing.” Since behavior changes can be costly, we would like to know the optimal social distancing behavior. But the benefits of changes in behavior depend on the course of the epidemic, which itself depends on our behaviors. Differential population game theory provides a method for resolving this circular dependence. Here, I present the analysis of a special case of the differential SIR epidemic population game with social distancing when the relative infection rate is linear, but bounded below by zero. Equilibrium solutions are constructed in closed-form for an open-ended epidemic. Constructions are also provided for epidemics that are stopped by the deployment of a vaccination that becomes available a fixed-time after the start of the epidemic. This can be used to anticipate a window of opportunity during which mass vaccination can significantly reduce the cost of an epidemic.


Epidemiological games Social distancing SIR Differential population game 



I am grateful to the reviewers for their diligence and helpful comments. This research was partially supported by NSF grant DMS-0920822, NIH grant PAR-08-224, and Bill and Melinda Gates Foundation Grant 49276.


  1. Arrow, K. J., & Kurz, M. (1970). Public investment, the rate of return, and optimal fiscal policy. Baltimore: Johns Hopkins Press. Google Scholar
  2. Auld, M. (2003). Choices, beliefs, and infectious disease dynamics. J. Health Econ., 22, 361–377. CrossRefGoogle Scholar
  3. Bressan, A., & Piccoli, B. (2007). Introduction to the mathematical theory of control. Palo Alto: American Institute of Mathematical Sciences. zbMATHGoogle Scholar
  4. Chen, F. (2012). A mathematical analysis of public avoidance behavior during epidemics using game theory. J. Theor. Biol., 302, 18–28. doi: 10.1016/j.jtbi.2012.03.002. CrossRefMathSciNetzbMATHGoogle Scholar
  5. Chen, F., Jiang, M., Rabidoux, S., & Robinson, S. (2011). Public avoidance and epidemics: insights from an economic model. J. Theor. Biol., 278, 107–119. doi: 10.1016/j.jtbi.2011.03.007. MathSciNetCrossRefzbMATHGoogle Scholar
  6. Clark, C. W. (1976). Mathematical bioeconomics. New York: Wiley. zbMATHGoogle Scholar
  7. Corless, R. M., Gonnet, G. H., Hare, D. E. G., Jeffrey, D. J., & Knuth, D. E. (1996). On the Lambert W function. Adv. Comput. Math., 5, 329–359. doi: 10.1007/BF02124750. MathSciNetCrossRefzbMATHGoogle Scholar
  8. Fudenberg, D., & Tirole, J. (1991). Game theory. Cambridge: MIT Press. zbMATHGoogle Scholar
  9. Isaacs, R. (1965). Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization. New York: Wiley. zbMATHGoogle Scholar
  10. Kermack, W. O., & McKendrick, A. G. (1927). Contributions to the mathematical-theory of epidemics. Proc. R. Soc. Lond., 115, 700–721. CrossRefzbMATHGoogle Scholar
  11. Kevorkian, J. (1990). Partial differential equations: analytic solution techniques. London: Chapman & Hall. CrossRefzbMATHGoogle Scholar
  12. Lenhart, S., & Workman, J. T. (2007). Optimal control applied to biological models. London: Chapman & Hall/CRC. zbMATHGoogle Scholar
  13. Reluga, T. C. (2010). Game theory of social distancing in response to an epidemic. PLoS Comput. Biol., 6, e1000793. doi: 10.1371/journal.pcbi.1000793. MathSciNetCrossRefGoogle Scholar
  14. Reluga, T. C., & Galvani, A. P. (2011). A general approach for population games with application to vaccination. Math. Biosci., 230, 67–78. doi: 10.1016/j.mbs.2011.01.003. MathSciNetCrossRefzbMATHGoogle Scholar
  15. Sandholm, W. (2011). Population games and evolutionary dynamics. Cambridge: MIT Press. zbMATHGoogle Scholar

Copyright information

© Society for Mathematical Biology 2013

Authors and Affiliations

  1. 1.Department of Mathematics, Department of BiologyPennsylvania State UniversityUniversity ParkUSA

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