Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Here X.S is used to indicate the S component of a position vector X. This notation is the same as that used commonly in object oriented computer languages and will make our analysis more readable than the use of arbitrary vector indices.
Within the literature on correspondences, the terms upper semicontinuity and upper hemicontinuity are used as synonyms for outer continuity, while lower semicontinuity and lower hemicontinuity are used as synonyms for inner continuity. These terms have various draw-backs that the inner–outer dichotomy avoids.
References
Arrow, K. J., & Kurz, M. (1970). Public investment, the rate of return, and optimal fiscal policy. Baltimore: Johns Hopkins Press.
Auld, M. (2003). Choices, beliefs, and infectious disease dynamics. J. Health Econ., 22, 361–377.
Bressan, A., & Piccoli, B. (2007). Introduction to the mathematical theory of control. Palo Alto: American Institute of Mathematical Sciences.
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.
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.
Clark, C. W. (1976). Mathematical bioeconomics. New York: Wiley.
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.
Fudenberg, D., & Tirole, J. (1991). Game theory. Cambridge: MIT Press.
Isaacs, R. (1965). Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization. New York: Wiley.
Kermack, W. O., & McKendrick, A. G. (1927). Contributions to the mathematical-theory of epidemics. Proc. R. Soc. Lond., 115, 700–721.
Kevorkian, J. (1990). Partial differential equations: analytic solution techniques. London: Chapman & Hall.
Lenhart, S., & Workman, J. T. (2007). Optimal control applied to biological models. London: Chapman & Hall/CRC.
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.
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.
Sandholm, W. (2011). Population games and evolutionary dynamics. Cambridge: MIT Press.
Acknowledgements
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Reluga, T.C. Equilibria of an Epidemic Game with Piecewise Linear Social Distancing Cost. Bull Math Biol 75, 1961–1984 (2013). https://doi.org/10.1007/s11538-013-9879-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-013-9879-5