Managing infectious diseases over connected populations: a non-convex optimal control

Ceddia, M Graziano (2010): Managing infectious diseases over connected populations: a non-convex optimal control.

Abstract

The paper develops an optimal control model to analyse various management options for infectious diseases that occur in metapopulations, under both Nash and cooperative behaviour. As pathogens are renewable resources with negative value, the problem may be non-convex. Since the disease can be transmitted across various connected populations, externalities are involved. Both aspects deserve attention as two issues arise: a) is eradication of the disease in finite time preferable to indefinite treatment? b) are cooperative solutions well-behaved? The problem is solved numerically and the results indicate that while eradication is likely to be an optimal strategy when initial levels of infections are relatively low, the internalisation of between-population externalities (as indicated by the first order necessary conditions of the cooperative optimal control problem) might not always be possible. Also, ignoring these two aspects can lead to inadequate policy design.

• Item Type: MPRA Paper
• Language: English
• Keywords: infectious diseases; metapopulation; non-convexities; optimal control
• Subjects: C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C71 - Cooperative Games; Q - Agricultural and Natural Resource Economics ; Environmental and Ecological Economics > Q2 - Renewable Resources and Conservation > Q28 - Government Policy; C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C6 - Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C61 - Optimization Techniques ; Programming Models ; Dynamic Analysis; H - Public Economics > H0 - General > H00 - General; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

Anderson RM and May RM (1991) Infectious diseases of humans: dynamics and control. Oxford University Press.

Bailey NTJ (1975). The Mathematical Theory of Infectious Diseases and its Applications. London, Griffin.

Barrett S (2003) Global disease eradication. Journal of European Economic Association 1:591-600.

Barrett S and Hoel M (2007) Optimal disease eradication. Environment and Development Economics 12: 627-652.

Barucci E (2000) Differential games with nonconvexities and positive spillovers. European Journal of Operational Research 121:193-204.

Basar T, Losder GJ (1999) Dynamic noncooperative game theory. SIAM, Philadelphia.

Brock WA, Starrett D (2003) Managing systems with non-convex positive feedback. Environmental and Resource Economics 26:575-602.

Caputo MR (2005) Foundations of Dynamic Economic Analysis – Optimal Control Theory and Applications. Cambridge University Press.

D’Onofrio A, Manfredi P (2009) Information-related changes in contact patterns may trigger oscillations in the endemic prevalence of infectious diseases. Journal of Theoretical Biology 3:473-478.

Dockner EJ, Long NV (1993) International pollution control: cooperative versus noncooperative strategies. Journal of Environmental Economics and Management 24:13-29.

Engwerda J (2007) Multicriteria dynamic optimization problems and cooperative dynamic games. University of Tilburg, CentER working paper No. 2007-41. http://arno.uvt.nl/show.cgi?fid=60640. Cited 20 Apr 2010.

Fenichel EP, Horan RD and Hickling GJ et al. (2010) Bioeconomic management of invasive vector-borne diseases. Biological Invasions (in press). DOI 10.1007/s10530-010-9734-7.

Fenichel EP and Horan RD (2007a) Gender-based harvesting in wildlife disease management. American Journal of Agricultural Economics 89: 904-920.

Fenichel EP and Horan RD (2007b) Jointly-determined ecological thresholds and economic trade-offs in wildlife disease management. Natural Resource Modelling 20: 511-547.

Gersovitz M (2003) Births, recoveries, vaccinations and externalities. In: Arnott RJ et al. (ed) Essays in honor of Joseph Stiglitz, MIT, Cambridge, MA.

Gersovitz M, Hammer JS (2003) Infectious diseases, public policy and the marriage of economics and epidemiology. The World Bank Research Observer 8:129-157.

Goldman SM, Lightwood J (2002) Cost optimization in the SIS model of infectious disease with treatment. Topics in Economic Analysis and Policy 2(1), Article 4. http://www.bepress.com/bejeap/topics/vol2/iss1/art4. Cited 20 Apr 2010.

Hagenaars TJ, Donnelly CA and Ferguson NM (2004) Spatial heterogeneity and the persistence of infectious diseases. Journal of Theoretical Biology 229:349-359.

Judd KL (1998) Numerical methods in economics. The MIT Press.

Keeling MJ (1999) The effects of local spatial structure on epidemiological invasions. Proceedings of the Royal Society of London B 266: 859-867.

Keeling MJ, Eames KTD (2005) Networks and epidemic models. Journal of the Royal Society Interface 2:295-307.

Keeling MJ, Rohani P (2008) Modeling infectious diseases in humans and animals. Princeton University Press

Kilpatrick AM, Chmura AA, Gibbons DW et al (2006) Predicting the global spread of H5N1 avian influenza. Proceedings of the National Academy of Sciences of the United States of America 103:19368-19373.

Levin SA (1976) Population dynamic models in heterogeneous environments. Annual Review of Ecology and Systematics 7:287-310.

Lewis TR and Schmalensee R (1977) Nonconvexities and optimal exhaustion of renewable resources. International Economic Review 18:535-552.

Mbah MLN, Gilligan CA (2009) Optimization of control strategies for epidemics in heterogeneous populations with symmetric and asymmetric transmission. Journal of Theoretical Biology 262: 757-763.

McCallum H (2008) Landscape structure, disturbance and disease dynamics. In: Ostfeld S, Keesing F and Eviner VT (ed), Infectious disease ecology: effects of ecosystems on disease and disease on ecosystems. Princeton University Press.

Medical Research Council (2009) Environmental and Social Ecology of Human Infectious Diseases. http://www.mrc.ac.uk/Fundingopportunities/Calls/ESEI/index.htm. Cited 20 Apr 2010.

National Science Foundation (2009) Ecology of Infectious Diseases. http://www.nsf.gov/funding/pgm_summ.jsp?pims_id=5269. Cited 20 Apr 2010.

Park AW, Gubbings S and Gilligan CA (2001) Invasion and persistence of plant parasites in a spatially structured host population. Oikos 94:162-174.

Roberts MG and Heesterbeek JAP (2003) A new method for estimating the effort required to control an infectious disease. Proceedings of The Royal Society B 270: 1359-1364.

Rodriguez D, Torres-Sorando L (2001) Models of infectious diseases in spatially heterogeneous environments. Bulletin of Mathematical Biology 63:547-571.

Rondeau D (2001) Along the way back from the brink. Journal of Environmental Economics and Management 42:156-183.

Rowthorn RE, Laxminarayan R and Gilligan CA (2009) Optimal control of epidemics in metapopulations. Journal of the Royal Society Interface 6:1135-1144.

Sandler T, Arce DG (2002) A conceptual framework for understanding global and transnational public goods for health. Fiscal Studies 23:195-222.

Seierstad A, Sydsaeter K (1987) Optimal control theory with economic applications. North-Holland, Amsterdam, London,New York, Tokyo.

Shampine LF, Kierzenka J and Reichelt MW (2000). Solving boundary value problems for ordinary differential equations in Matlab with bvp4c. http://200.13.98.241/~martin/irq/tareas1/bvp_paper.pdf. Cited 20 Apr 2010.

Smith MD, Sanchirico JN and Wilen JE (2009) The economics of spatial-dynamic processes: applications to renewable resources. Journal of Environmental Economic and Management 57:104-121.

Stallknecht DE, Brown JD (2007) Wild birds and the epidemiology of avian influenza. Journal of Wildlife Diseases 43(3):S15-S20.

Su M., Hui C, Zhang Y et al (2009) How does the spatial structure of habitat loss affect the eco-epidemic dynamics. Ecological Modelling 220:51-59.

Tahvonen O, Salo S (1996) Non-convexities in optimal pollution accumulation. Journal of Environmental Economics and Management 31:160-177.

Waldenström J, Lundkvist A, Falk KI et al (2007) Migrating birds and tickborne encephalitis virus. Emerging Infectious Diseases 13(8):1215-1218.

White SH, Martin del Rey A and Rodriguez Sanchez G (2007) Modeling epidemics using cellular automata. Applied Mathematics and Computation 186:193-202.