Publications and preprints (most recent first)

• Arik, A., Cairns, A.J.G., Dodd, E., Macdonald, A.S., and Streftaris, G. (2024)
  The effect of the COVID-19 health disruptions on breast cancer mortality for older women: A semi-Markov modelling approach.
  To appear in Scandinavian Actuarial Journal
  • PDF postprint
• Andrew J.G. Cairns, Jie Wen and Torsten Kleinow (2023)
  Drivers of Mortality: Risk Factors and Inequality
  Journal of the Royal Statistical Society, Series A.
  • high resolution PDF
  • PDF (journal website; open access)
    
• Mayhew, L., Chen, M.S., and Cairns, A.J.G. (2023)
  Levelling up - the great health challenge.
  To appear in Geneva Papers on Risk and Insurance - Issues and Practice.
  • Postprint (PDF)
• Cairns, A.J.G. (2023)
  The Common Cohort Effect Model for Cause of Death Data.
  
  • Working paper (PDF)
• Cairns, A.J.G. and Redondo Loures, C. (2023)
  Higher-Age US Mortality By Education and Cause of Death: Trends, Inequality and Controllable Risk Factors.
  
  • Living to 100 Symposium paper (PDF)
  • Slides (PDF), Living to 100 Symposium, January 2023.
  • Video (MP4).
• Cairns, A.J.G., Wen, J, and Kleinow, T. (2023)
  Estimating neighbourhood death rates using the random forest algorithm. Longevity Bulletin, 15: 22-29.
  • Full issue (PDF)
  • LIFE App, Version 3
• Wen, J., Cairns, A.J.G., and Kleinow, T. (2023)
  Modelling Socio-Economic Mortality at Neighbourhood Level
  Working Paper (REVISED) ASTIN Bulletin, 53: 285-310. (Open access journal version)
  (Postprint)
• Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G., Jones, O., and Rowney, J. (2022)
  A General Framework for Analysing the Mortality Experience of a Large Portfolio of Lives: With an Application to the UK Universities Superannuation Scheme.
  European Actuarial Journal, (Journal open access PDF)
• Cairns, A.J.G., Blake, D., Kessler, A.R., and Kessler, M. (2020)
  The Impact of Covid-19 on Future Higher-Age Mortality.
  Working paper, Heriot-Watt University.
  Full PDF available on ResearchGate.
• Redondo Loures, C., and Cairns, A.J.G. (2021)
  Cause of Death Specific Cohort Effects in U.S. Mortality .
  (Postprint) Insurance: Mathematics and Economics, 99: 190-199.
• Dowd, K., Cairns, A.J.G., and Blake, D. (2020)
  CBDX: A Workhorse Mortality Model from the Cairns-Blake-Dowd Family
  Annals of Actuarial Science 14: 445-460. (PDF postprint)
• Redondo Loures, C., and Cairns, A.J.G. (2019)
  Mortality In The US By Education Level.
  Annals of Actuarial Science, 14: 384-419. (PDF postprint)
• Wen, J., Kleinow, T., and Cairns, A.J.G. (2020)
  Trends in Canadian Mortality By Pension Level: Evidence From the CPP and QPP. (PDF file)
  North American Actuarial Journal, 24: 533-561.
  Journal version.
  
• Wen, J., Kleinow, T., and Cairns, A.J.G. (2019)
  Trends in Canadian Mortality By Pension Level: Evidence From the CPP and QPP. (PDF file)
  (This is the original, longer and more detailed version of Wen, Kleinow and Cairns (2020).)
  
• Wen, J., Cairns, A.J.G., and Kleinow, T., (2021)
  Fitting Multi-Population Mortality Models to Socio-Economic Groups
  Annals of Actuarial Science, 15: 144-172. (PDF postprint)
  
• Cairns, A.J.G., Kallestrup-Lamb, M., Rosenskjold, C.P.T., Blake, D., and Dowd, K., (2019)
  Modelling Socio-Economic Differences in the Mortality of Danish Males Using a New Affluence Index.
  ASTIN Bulletin 49: 555-590. (Postprint PDF file)
  
• Blake, D., Cairns, A.J.G., Dowd, K., and Kessler, A.R. (2019)
  Still living with mortality: the longevity risk transfer market after one decade
  Presented at a Sessional Research Meeting of the Institute and Faculty of Actuaries, Edinburgh, 29 January, 2018.
  British Actuarial Journal, 24: 1-80. (PDF file)
  
• Cairns, A.J.G., and El Boukfaoui, G. (2021)
  Basis Risk in Index Based Longevity Hedges: A Guide For Longevity Hedgers
  (PDF file: postprint) North American Actuarial Journal, 25:sup1, S97-S118.
  DOI: 10.1080/10920277.2019.1651658
• Chen, L., Cairns, A.J.G., and Kleinow, T. (2017) (Revised)
  Small Population Bias and Sampling Effects in Stochastic Mortality Modelling
  European Actuarial Journal 7: 193-230. (Postprint - PDF file)
• Mavros, G., Cairns, A.J.G., Streftaris, G., and Kleinow, T. (2017)
  Stochastic Mortality Modelling: Key Drivers and Dependent Residuals
  North American Actuarial Journal 21: 343-368.
  (Postprint: PDF file)
• Cairns, A.J.G., Blake, D., Dowd, K., and Kessler, A.R. (2016)
  Phantoms Never Die: Living with Unreliable Population Data
  Journal of the Royal Statistical Society, Series A, 179: 975-1005. (Journal PDF file)
  Wiley/JRSS-A Journal Website.
  • Earlier working paper, Heriot-Watt University (2014) (PDF file).
    
• Enchev, V., Kleinow, T., and Cairns, A.J.G. (2017)
  Multi-population mortality models: Fitting, Forecasting and Comparisons.
  Scandinavian Actuarial Journal,
  2017: 319-342. (Postprint: PDF file) DOI: 10.1080/03461238.2015.1133450
• Dowd, D., Blake, D., and Cairns A.J.G. (2016)
  The Myth of Methuselah and the Uncertainty of Death: The Mortality Fan Charts.
  Risks, 4: 21.
• van Loon, P.R.F., Cairns, A.J.G., McNeil, A.J., and Veys, A. (2015)
  Modelling the Liquidity Premium on Corporate Bonds.
  Annals of Actuarial Science, 9: 264-289.
  (Postprint: PDF file)
• Karabey, U., Kleinow, T., and Cairns, A.J.G. (2014)
  Factor risk quantification in annuity models.
  Insurance: Mathematics and Economics, 58: 34-45.
  Please e-mail me for an official reprint.
• Cairns, A.J.G., Blake, D., Dowd, K., and Kessler, A. (2014)
  Phantoms Never Die: Living with Unreliable Mortality Data
  (PDF file) Working paper, Heriot-Watt University.
• Sahin, S., Cairns A.J.G., Kleinow, T., and Wilkie, A.D., (2014)
  A yield-only model for the term structure of interest rates.
  Annals of Actuarial Science, 8: 99-130. Journal web page.
  DOI: 10.1017/S1748499513000146
• Blake, D., Boardman, T., and Cairns, A., (2014)
  Sharing Longevity Risk: Why Governments Should Issue Longevity Bonds.
  North American Actuarial Journal, 18: 258-277. Journal version.
  DOI: 10.1080/10920277.2014.883229
• Cairns, A.J.G., Blake, D., Dowd, K., and Coughlan, G.D., (2014)
  Longevity Hedge Effectiveness: A Decomposition.
  Quantitative Finance, 14: 217-235. (PDF file - Final version) (Journal online version)
  DOI: 10.1080/14697688.2012.748986
• Blake, D., Cairns, A.J.G., Coughlan, G., Dowd, K., and MacMinn, R. (2013)
  The New Life Market,
  Journal of Risk and Insurance, 80: 501-558.
  DOI: 10.1111/j.1539-6975.2012.01514.x
  Republished in French in Revue d'economie financiere, 122: 129-164.
• Kleinow, T., and Cairns A.J.G. (2013)
  Mortality and smoking prevalence: An empirical investigation in ten developed countries.
  British Actuarial Journal, 18: 452-466. Journal online version.
  Official PDF, reproduced with permission from Cambridge University Press and the Institute and Faculty of Actuaries. Copyright Institute and Faculty of Actuaries.
• Cairns, A.J.G. (2014)
  Modeling and Management of Longevity Risk,
  in P. B. Hammond, R. Maurer, and O. S. Mitchell, eds., Recreating Sustainable Retirement: Resilience, Solvency, and Tail Risk. Oxford, U.K.: Oxford University Press, pp. 71-88.
  (PDF file - First version - March 2013)
• Cairns, A.J.G. (2013)
  Robust hedging of longevity risk.
  Journal of Risk and Insurance, 80: 621-648.
  (PDF file - Final version - 30/1/2013)
• Leuwattanachotinan, C., Cairns, A.J.G., and Streftaris, G. (2012)
  Model Fitting of a Two-Factor Arbitrage-Free Model For the Term Structure of Interest Rates using Markov Chain Monte Carlo: Part 1: Theory and Methodology
  Working paper
• Leuwattanachotinan, C., Cairns, A.J.G., and Streftaris, G. (2012)
  Model Fitting of a Two-Factor Arbitrage-Free Model for the Term Structure of Interest Rates using Markov Chain Monte Carlo: Part 2: Applications with UK Data
  Working paper
• Cairns, A.J.G. (2011)
  Modelling and management of longevity risk: approximations to survival functions and dynamic hedging.
  Insurance: Mathematics and Economics, 49: 438-453. (PDF file - REVISED)
  Please e-mail me for an official reprint.
• Dowd, K., Blake, D., Cairns, A.J.G., Coughlan, G.D., and Khalaf-Allah, M. (2011a)
  A gravity model of mortality rates for two related populations.
  North American Actuarial Journal, 15: 334-356. (Please e-mail me for an official reprint.)
• Dowd, K., Blake, D., and Cairns, A.J.G. (2011)
  A computationally efficient algorithm for estimating the distribution of future annuity values under interest-rate and longevity risks.
  North American Actuarial Journal, 15: 237-247. (Please e-mail me for an official reprint.)
• Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., and Khalaf-Allah, M. (2011)
  Bayesian Stochastic Mortality Modelling for Two Populations.
  ASTIN Bulletin 41: 29-59. (PDF file)
  (Original working paper, Heriot-Watt University. PDF file.)
• Guy D. Coughlan, Marwa Khalaf-Allah, Yijing Ye, Sumit Kumar, Andrew J.G. Cairns, David Blake and Kevin Dowd, (2011)
  Longevity hedging: A framework for longevity basis risk analysis and hedge effectiveness.
  North American Actuarial Journal, 15: 150-176. (Please e-mail me for an official reprint.)
  Working Paper, Heriot-Watt University. PDF file.
• Wilkie, A.D., Sahin, S., Cairns, A.J.G., and Kleinow, T. (2010)
  Yet more on a stochastic economic model: Part 1: Updating and refitting, 1995 to 2009.
  Annals of Actuarial Science, 5: 53-99. Copyright Institute and Faculty of Actuaries. (PDF)
• Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., and Khalaf-Allah, M. (2011)
  Mortality density forecasts: an analysis of six stochastic mortality models.
  Insurance: Mathematics and Economics, 48: 355-367. (PDF file.)
  Please e-mail me for an official reprint.
• MacDonald, B.-J. and Cairns, A.J.G., (2011)
  Three Retirement Models for Defined Contribution Pension Plan Members: A Simulation Study.
  Insurance: Mathematics and Economics, 48: 1-18. (PDF)
  (Please e-mail me for a reprint of the IME version.)
• Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D., and Khalaf-Allah, M. (2010)
  Evaluating the Goodness of Fit of Stochastic Mortality Models.
  Insurance: Mathematics and Economics, 47: 255-265.
• Dowd, K., Cairns, A.J.G., Blake, D., Coughlan, G.D., Epstein, D., and Khalaf-Allah, M. (2010)
  Backtesting Stochastic Mortality Models: An Ex-Post Evaluation of Multi-Period-Ahead Density Forecasts.
  North American Actuarial Journal, 14: 281-298.
• Dowd, K., Blake, D., and Cairns, A.J.G. (2010)
  Facing up to Uncertain Life Expectancy: The Longevity Fan Charts.
  Demography 47: 67-78.
• Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., and Khalaf-Allah, M. (2010)
  Bayesian Stochastic Mortality Modelling for Two Populations.
  Working Paper, Heriot-Watt University. PDF file.
• Dawson, P., Dowd, K., Cairns, A.J.G., and Blake, D. (2010)
  Survivor Derivatives: A Consistent Pricing Framework.
  Journal of Risk and Insurance, 77: 579-596.
• MacDonald, B.-J. and Cairns, A.J.G., (2009)
  Getting feedback on defined contribution pension plans.
  Journal of Risk and Insurance. 76(2): 385-417.
  (PDF file.) [Original version: 10/2006]
• Blake, D., Cairns, A.J.G. and Dowd, K. (2009)
  Designing a defined-contribution pension plan: what to learn from aircraft designers.
  Financial Analysts Journal, 65(1): 37-42.
• Cairns, A.J.G., Blake, D., Dowd, K., Coughlan, G.D., Epstein, D., Ong, A., and Balevich, I. (2009)
  A quantitative comparison of stochastic mortality models using data from England and Wales and the United States.
  
  • North American Actuarial Journal 13(1): 1-35. (Reprint.)
    Copyright 2009 by the Society of Actuaries, Schaumburg, Illinois. Reprinted with permission.
  • Longer version (preprint, 2007) (PDF file.)
• Dawson, P., Dowd, K., Cairns, A.J.G., and Blake, D. (2009)
  Options on normal underlyings with an application to the pricing of survivor swaptions. Journal of Futures Markets, 29(8): 757-774.
• Blake, D., Cairns, A.J.G., and Dowd, K. (2008)
  The birth of the life market.
  Alternative Investment Quarterly. (PDF)
• Blake, D., Cairns, A.J.G., and Dowd, K. (2008)
  Longevity risk and the grim reaper's toxic tail: the survivor fan charts.
  Insurance: Mathematics and Economics, 42: 1062-1066.
• Cairns, A.J.G., Blake, D., Dowd, K. (2008)
  Modelling and management of mortality risk: a review.
  Scandinavian Actuarial Journal 108(2-3): 79-113. (PDF file.)
  DOI: 10.1080/03461230802173608
• Cairns, A.J.G., (2007)
  A Multifactor Generalisation of the Olivier-Smith Model for Stochastic Mortality.
  In Proceedings of the 1st IAA Life Colloquium, Stockholm, 2007. (PDF file.)
• Byrne, A., Blake, D., Cairns, A., and Dowd, K. (2007) Default funds in UK defined-contribution plans. Financial Analysts Journal 64(4): 40-51.
• Blake, D., Cairns, A.J.G., and Dowd, K. (2007) The Impact of Occupation and Gender on the Pensions from Defined Contribution Plans,
  Geneva Papers on Risk and Insurance, 32: 458-82 (.)
• MacDonald, B.-J. and Cairns, A.J.G., (2007)
  The impact of DC pension systems on population dynamics.
  North Americal Actuarial Journal 11(1): 17-48. (PDF file.)
  Copyright 2007 by the Society of Actuaries, Schaumburg, Illinois. Reprinted with permission.
  (A previous version of this paper under the title "DC pension plans for all: What if" was presented at the AFIR Colloquium in Zurich, September 2005.)
• Byrne, A., Blake, D., Cairns, A.J.G., and Dowd, K. (2006) There's no time like the present: the cost of delaying retirement saving. Financial Services Review 15: 213-231.
• Blake, D., Cairns, A.J.G.,, Dowd, K., and MacMinn, R. (2006) Longevity Bonds: Financial Engineering, Valuation and Hedging. Journal of Risk and Insurance, 73: 647-672.
• Cairns, A.J.G., Blake, D., and Dowd, K., (2006) Stochastic Lifestyling: Optimal Dynamic Asset Allocation for Defined Contribution Pension Plans Journal of Economic Dynamics and Control 30(5): 843-877. (Please e-mail me for a reprint.)
• Blake, D., Cairns, A.J.G., and Dowd, K. (2006) Living with mortality: longevity bonds and other mortality-linked securities British Actuarial Journal 12: 153-197.
  Discussed at the Faculty of Actuaries on 16 January, 2006 and at the Institute of Actuaries on 27 February, 2006. (PDF file.)
• Cairns, A.J.G., Blake, D., and Dowd, K., (2006) Pricing Death: Frameworks for the Valuation and Securitization of Mortality Risk ASTIN Bulletin, 36: 79-120. (PDF file.)
  Earlier version in Proceedings of the International AFIR Colloquium, Boston, November 2004, pages 509-540:
  
  • Cairns, A.J.G., Blake, D., and Dowd, K., (2004) Pricing Frameworks for Securitization of Mortality Risk Preprint, 32 pages: (PDF file.)
• Dowd, K., Blake, D., Cairns, A.J.G., and Dawson, P. (2006) Survivor swaps. Journal of Risk and Insurance, 73: 1-17.
• Dowd, K., Blake, D., and Cairns, A. J. G., (2006) The Grave Problem of Longevity Risk. Financial Engineering News, No. 49, May-June 2006, pp. 19, 30.
• Cairns, A.J.G., Blake, D., and Dowd, K., (2006) A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration. Journal of Risk and Insurance 73: 687-718. (PDF file.)
• Dowd, K. Cairns, A.J.G., and Blake, D. (2006) Mortality-dependent financial risk measures. Insurance: Mathematics and Economics 38: 427-440. (Please e-mail me for a reprint.)
• Huang, H.-C., and Cairns, A.J.G. (2005) On the control of defined-benefit pension plans. Insurance: Mathematics and Economics. 38: 113-131. (Please e-mail me for a reprint.)
• Dowd, K., Blake, D., and Cairns, A.J.G.PensionMetrics: designing defined-contribution pension plans. Risk May 2005: 81-82.
• Cairns, A.J.G., Blake, D., Dawson, P., and Dowd, K., (2005) Pricing the Risk on Longevity Bonds. Life and Pensions October: 41-44. (PDF file.)
• Cairns, A.J.G., and Garcia Rosas, S.A. (2003) A Family Of Term-structure Models with Stochastic Volatility Preprint, 24 pages. (PDF file.)
• Cairns, A.J.G., (2004) A family of term-structure models for long-term risk management and derivative pricing Mathematical Finance, 14: 415-444. (PDF file.)
• Huang, H-C., and Cairns, A.J.G., (2004) Valuation and Hedging of Limited Price Indexed Liabilities. British Actuarial Journal, 10: 627-663. (Please e-mail me for a reprint.)
  
• Cairns, A.J.G. (2004) Pension-Fund Mathematics. To appear in the Encyclopaedia of Actuarial Science. (PDF file)
• Cairns, A.J.G. (2004) Interest-Rate Models To appear in the Encyclopaedia of Actuarial Science. (PDF file)
• Blake, D., Cairns, A.J.G., and Dowd, K., (2003) Pensionmetrics II: Stochastic pension plan design during the distribution phase. Insurance: Mathematics and Economics 33: 29-47. (Please e-mail me for a reprint.)
  (PDF file.)
• Cairns, A.J.G., (2001) From financial economics to fair valuation. In Proceedings of the 11th International AFIR Colloquium, Toronto, September 2001, Volume 1, pp135-166. (PDF file )
  
• Cairns, A.J.G. (2001) An alternative derivation of the Vasicek model. Unpublished note. (PDF )
• Blake, D., Cairns, A.J.G., and Dowd, K., (2001) Pensionmetrics: Stochastic pension plan design and value-at-risk during the accumulation phase. Insurance: Mathematics and Economics, 29: 187-215. (Please e-mail me for a reprint.)
  (PDF file.)
• Cairns, A.J.G., and Pritchard, D.J., (2001) Stability of descriptive models for the term structure of interest rates with application to German market data. British Actuarial Journal 7: 467-507. (PDF file - 342 Kb)
• Cairns, A.J.G., Blake, D., and Dowd, K., (2000) Optimal dynamic asset allocation for defined-contribution pension plans (Preprint - updated 6/11/2000) 25 pages. (PDF file )
  An earlier version appears in Proceedings of the 10th AFIR Colloquium, Tromsoe, June, 2000, pp131-154.
  A substantially updated version of this paper appears above as "Stochastic lifestyling ....".
  
• Cairns, A.J.G., (2000) A multifactor model for the term structure and inflation for long-term risk management with an extension to the equities market. Preprint 31 pages. (PDF file - 367 Kb)
  (An earlier version of this paper appears in Proceedings of the 9th AFIR Colloquium, Tokyo, August, 1999, Volume 3: 93-113.)
• Cairns, A.J.G., (2000) A Discussion of Parameter and Model Uncertainty in Insurance Insurance: Mathematics and Economics, 27: 313-330. (Reprints) (PDF file.)
• Cairns, A.J.G., (2000) Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time ASTIN Bulletin 30: 19-55. (Please e-mail me for a reprint or they can be downloaded from the ASTIN Bulletin website.)
  ERRATUM: PDF file.
• Head, S.J., Adkins, D., Cairns, A.J.G., Corvesor, A., Cule, D., Exley, J., Johnson, I., Spain, J., and Wise, A. (2000) Pension fund valuations and market values. (Discussed at sessional meetings of the Institute of Actuaries, October 1999 and the Faculty of Actuaries, November, 2000.) British Actuarial Journal, 6: 55-118 (with discussion on pages 119-141). (PDF file WARNING 32 MB - may take ages to download!)
• Cairns, A.J.G., (1999) An analysis of the level of security provided by the Minimum Funding Requirement British Actuarial Journal, 5: 585-610. (PDF file)
• Cairns, A.J.G., Dickson, D.C.M., Macdonald, A.S., Waters, H.R., and Willder, M., (1998) Stochastic processes: learning the language. (Presented to the Faculty of Actuaries Students' Society, November, 1998.) (PDF file - 1.1 Mb!!!!) (postscript file)
• Cairns, A.J.G., (1998) Descriptive bond-yield and forward-rate models for the British government securities' market. (with discussion) British Actuarial Journal, 4(2): 265-321 and 350-383. (PDF - 1.88Mb !)
• Feldman, K.S., Bergman, B., Cairns, A.J.G., Chaplin, G.B., Gwilt, G.D., Lockyer, P.R., and Turley, F.B., (1998) Report of the Fixed Interest Working Group. (with discussion) British Actuarial Journal 4(2): 213-263 and 350-383.
• Macdonald, A.S., Cairns, A.J.G., Gwilt, P.L., and Miller, K.A., (1998) An international comparison of recent trends in population mortality. British Actuarial Journal 4: 3-141. (Please e-mail me for a reprint.)
• Cairns, A.J.G., and Parker, G., (1997) Stochastic pension fund modelling. Insurance: Mathematics and Economics, 21: 43-79. (Please e-mail me for a reprint or from the IME website.)
• Cairns, A.J.G., Sensitivity analysis of epidemic models, (1998) in The Encyclopaedia of Biostatistics, Volume 5, pages 4056-4069, Wiley.
• Cairns, A.J.G., (1996) Continuous-Time Pension-Fund Modelling in Proceedings of the 6th AFIR Colloquium, Nuremberg, October 1996, 1: 609-624. (PDF copy)
  (See also the paper Some Notes on Stochastic Pension Fund Models in Continuous Time above.)
• Cairns, A.J.G., (1994) An introduction to stochastic pension fund modelling. (Paper for the Vancouver Interest Rate Risk Workshop) (PDF copy).
• Cairns, A.J.G., (1995) Pension funding in a stochastic environment: the role of objectives in selecting an asset allocation strategy. Proceedings of the 5th AFIR International Symposium, Brussels, 1: 429-453. (PDF copy).
• Cairns, A.J.G., The present value of a series of cash flows: convergence in a random environment. ASTIN Bulletin, 25: 81-94, 1995. (Please e-mail me for a reprint.)
• Cairns, A.J.G., (1995) Primary components of epidemic models, in Epidemic Models editor D.Mollison, Cambridge University Press, pages 350-371.
• Cairns, A.J.G., (1991) Model fitting and projection of the AIDS epidemic, Mathematical Biosciences, 107: 451-489.
• Cairns, A.J.G., (1990) Epidemics in heterogeneous populations II: non-exponential incubation periods and variable infectiousness, IMA Journal of Mathematics Applied in Medicine and Biology, 7: 219-230.
• Cairns, A.J.G., (1989) Epidemics in heterogeneous populations: aspects of optimal vaccination policies, IMA Journal of Mathematics Applied in Medicine and Biology, 6: 137-159.