Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection

  • Received: 01 March 2017 Accepted: 29 July 2017 Published: 01 June 2018
  • MSC : Primary: 92D30; Secondary: 92C42, 60J20, 91A13

  • We analyze a model of agent based vaccination campaign against influenza with imperfect vaccine efficacy and durability of protection. We prove the existence of a Nash equilibrium by Kakutani's fixed point theorem in the context of non-persistent immunity. Subsequently, we propose and test a novel numerical method to find the equilibrium. Various issues of the model are then discussed, such as the dependence of the optimal policy with respect to the imperfections of the vaccine, as well as the best vaccination timing. The numerical results show that, under specific circumstances, some counter-intuitive behaviors are optimal, such as, for example, an increase of the fraction of vaccinated individuals when the efficacy of the vaccine is decreasing up to a threshold. The possibility of finding optimal strategies at the individual level can help public health decision makers in designing efficient vaccination campaigns and policies.

    Citation: Francesco Salvarani, Gabriel Turinici. Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection[J]. Mathematical Biosciences and Engineering, 2018, 15(3): 629-652. doi: 10.3934/mbe.2018028

    Related Papers:

  • We analyze a model of agent based vaccination campaign against influenza with imperfect vaccine efficacy and durability of protection. We prove the existence of a Nash equilibrium by Kakutani's fixed point theorem in the context of non-persistent immunity. Subsequently, we propose and test a novel numerical method to find the equilibrium. Various issues of the model are then discussed, such as the dependence of the optimal policy with respect to the imperfections of the vaccine, as well as the best vaccination timing. The numerical results show that, under specific circumstances, some counter-intuitive behaviors are optimal, such as, for example, an increase of the fraction of vaccinated individuals when the efficacy of the vaccine is decreasing up to a threshold. The possibility of finding optimal strategies at the individual level can help public health decision makers in designing efficient vaccination campaigns and policies.


    加载中
    [1] [ A. Abakuks, Optimal immunisation policies for epidemics, Advances in Appl. Probability, 6 (1974): 494-511.
    [2] [ R. M. Anderson and R. M. May, Infectious Diseases of Humans Dynamics and Control, Oxford University Press, 1992.
    [3] [ J. Appleby, Getting a flu shot? it may be better to wait, CNN, September 15, http://edition.cnn.com/2016/09/26/health/wait-for-flu-shot/index.html, 2016.
    [4] [ N. Bacaër, A Short History of Mathematical Population Dynamics, Springer-Verlag London, Ltd., London, 2011.
    [5] [ Y. Bai,N. Shi,Q. Lu,L. Yang,Z. Wang,L. Li,H. Han,D. Zheng,F. Luo,Z. Zhang,X. Ai, Immunological persistence of a seasonal influenza vaccine in people more than 3 years old, Human Vaccines & Immunotherapeutics, 11 (2015): 1648-1653.
    [6] [ C. T. Bauch,D. J. D. Earn, Vaccination and the theory of games, Proc. Natl. Acad. Sci. USA, 101 (2004): 13391-13394 (electronic).
    [7] [ C. T. Bauch,A. P. Galvani,D. J. D. Earn, Group interest versus self-interest in smallpox vaccination policy, Proceedings of the National Academy of Sciences, 100 (2003): 10564-10567.
    [8] [ C. T. Bauch, Imitation dynamics predict vaccinating behaviour, Proc Biol Sci, 272 (2005): 1669-1675.
    [9] [ E. A. Belongia,M. E. Sundaram,D. L. McClure,J. K. Meece,J. Ferdinands,J. J. VanWormer, Waning vaccine protection against influenza a (h3n2) illness in children and older adults during a single season, Vaccine, 33 (2015): 246-251.
    [10] [ Adrien Blanchet and Guillaume Carlier, From Nash to Cournot-Nash equilibria via the Monge-Kantorovich problem Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 372 (2014), 20130398, 11pp.
    [11] [ R. Breban, R. Vardavas and S. Blower, Mean-field analysis of an inductive reasoning game: Application to influenza vaccination Phys. Rev. E, 76 (2007), 031127.
    [12] [ D. L. Brito,E. Sheshinski,M. D. Intriligator, Externalities and compulsary vaccinations, Journal of Public Economics, 45 (1991): 69-90.
    [13] [ B. Buonomo,A. d'Onofrio,D. Lacitignola, Global stability of an {SIR} epidemic model with information dependent vaccination, Mathematical Biosciences, 216 (2008): 9-16.
    [14] [ P. Cardaliaguet,S. Hadikhanloo, Learning in mean field games: The fictitious play, ESAIM Control Optim. Calc. Var., 23 (2017): 569-591.
    [15] [ F. Carrat,A. Flahault, Influenza vaccine: The challenge of antigenic drift, Vaccine, 25 (2007): 6852-6862.
    [16] [ F. H. Chen, A susceptible-infected epidemic model with voluntary vaccinations, Journal of Mathematical Biology, 53 (2006): 253-272.
    [17] [ M. L. Clements,B. R. Murphy, Development and persistence of local and systemic antibody responses in adults given live attenuated or inactivated influenza a virus vaccine, Journal of Clinical Microbiology, 23 (1986): 66-72.
    [18] [ C. T. Codeço,P. M. Luz,F. Coelho,A. P Galvani,C. Struchiner, Vaccinating in disease-free regions: a vaccine model with application to yellow fever, Journal of The Royal Society Interface, 4 (2007): 1119-1125.
    [19] [ F. Coelho and C. T. Codeço, Dynamic modeling of vaccinating behavior as a function of individual beliefs PLoS Comput Biol, 5 (2009), e1000425, 10pp.
    [20] [ M.-G. Cojocaru, Dynamic equilibria of group vaccination strategies in a heterogeneous population, Journal of Global Optimization, 40 (2008): 51-63.
    [21] [ R. B. Couch,J. A. Kasel, Immunity to influenza in man, Annual Reviews in Microbiology, 37 (1983): 529-549.
    [22] [ N. Cox, Influenza seasonality: Timing and formulation of vaccines, Bulletin of the World Health Organization, 92 (2014): 311-311.
    [23] [ O. Diekmann and J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2000.
    [24] [ Josu Doncel, Nicolas Gast, and Bruno Gaujal, Mean-Field Games with Explicit Interactions, working paper or preprint, 2016.
    [25] [ A. d'Onofrio,P. Manfredi,E. Salinelli, Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theoretical Population Biology, 71 (2007): 301-317.
    [26] [ A. d'Onofrio,P. Manfredi,E. Salinelli, Fatal SIR diseases and rational exemption to vaccination, Mathematical Medicine and Biology, 25 (2008): 337-357.
    [27] [ P. Doutor,P. Rodrigues,M. do Céu Soares,F. A. C. C. Chalub, Optimal vaccination strategies and rational behaviour in seasonal epidemics, Journal of Mathematical Biology, 73 (2016): 1437-1465.
    [28] [ J. Dushoff,J. B Plotkin,C. Viboud,D. J. D. Earn,L. Simonsen, Mortality due to influenza in the United States-an annualized regression approach using multiple-cause mortality data, American journal of epidemiology, 163 (2006): 181-187.
    [29] [ J. M. Ferdinands,A. M. Fry,S. Reynolds,J. G. Petrie,B. Flannery,M. L. Jackson,E. A. Belongia, Intraseason waning of influenza vaccine protection: Evidence from the us influenza vaccine effectiveness network, 2011-2012 through 2014-2015, Clinical Infectious Diseases, 64 (2017): p544.
    [30] [ P. E. M. Fine,J. A. Clarkson, Individual versus public priorities in the determination of optimal vaccination policies, American Journal of Epidemiology, 124 (1986): 1012-1020.
    [31] [ P. J. Francis, Optimal tax/subsidy combinations for the flu season, Journal of Economic Dynamics and Control, 28 (2004): 2037-2054.
    [32] [ D. Fudenberg and D. K. Levine, The Theory of Learning in Games volume 2 of MIT Press Series on Economic Learning and Social Evolution, MIT Press, Cambridge, MA, 1998.
    [33] [ S. Funk,M. Salathé,V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, Journal of The Royal Society Interface, 7 (2010): 1247-1256.
    [34] [ A. P. Galvani,T. C. Reluga,G. B. Chapman, Long-standing influenza vaccination policy is in accord with individual self-interest but not with the utilitarian optimum, Proceedings of the National Academy of Sciences, 104 (2007): 5692-5697.
    [35] [ P.-Y. Geoffard,T. Philipson, Disease eradication: Private versus public vaccination, The American Economic Review, 87 (1997): 222-230.
    [36] [ N. C. Grassly,C. Fraser, Seasonal infectious disease epidemiology, Proceedings of the Royal Society of London B: Biological Sciences, 273 (2006): 2541-2550.
    [37] [ S. Greenland and R. R. Frerichs, On measures and models for the effectiveness of vaccines and vaccination programmes, International Journal of Epidemiology, 17 (1988), p456.
    [38] [ M. E. Halloran, I. M. Longini and C. J. Struchiner, Design and Analysis of Vaccine Studies, Statistics for Biology and Health. Springer New York, 2009.
    [39] [ H. W. Hethcote,P. Waltman, Optimal vaccination schedules in a deterministic epidemic model, Mathematical Biosciences, 18 (1973): 365-381.
    [40] [ M. Huang,R. P. Malhamé,P. E. Caines, Nash equilibria for large-population linear stochastic systems of weakly coupled agents, In Elkébir Boukas and Roland P. Malhamé, editors,, Analysis, Control and Optimization of Complex Dynamic Systems, Springer US,, 4 (2005): 215-252.
    [41] [ M. Huang,R. P. Malhamé,P. E. Caines, Large population stochastic dynamic games: Closed-loop mckean-vlasov systems and the Nash certainty equivalence principle, Commun. Inf. Syst., 6 (2006): 221-252.
    [42] [ R. Jordan,D. Kinderlehrer,F. Otto, The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29 (1998): 1-17.
    [43] [ S. Kakutani, A generalization of Brouwer's fixed point theorem, Duke Math. J., 8 (1941): 457-459.
    [44] [ E. Kissling, B. Nunes, C. Robertson, M. Valenciano, A. Reuss, A. Larrauri, J. M. Cohen, B. Oroszi, C. Rizzo, A. Machado, D. Pitigoi, L. Domegan, I. Paradowska-Stankiewicz, U. Buchholz, A. Gherasim, I. Daviaud, J. K. Horvath, A. Bella, E. Lupulescu, J. O'Donnell, M. Korczynska, A. Moren and I.-MOVE case-control study team, I-move multicentre casecontrol study 2010/11 to 2014/15: Is there within-season waning of influenza type/subtype vaccine effectiveness with increasing time since vaccination?, Euro Surveill., 21 (2016), 30201.
    [45] [ A. Lachapelle,J. Salomon,G. Turinici, Computation of mean field equilibria in economics, Math. Models Methods Appl. Sci., 20 (2010): 567-588.
    [46] [ L. Laguzet,G. Turinici, Global optimal vaccination in the SIR model: Properties of the value function and application to cost-effectiveness analysis, Mathematical Biosciences, 263 (2015): 180-197.
    [47] [ L. Laguzet,G. Turinici, Individual vaccination as Nash equilibrium in a SIR model with application to the 2009-2010 influenza A (H1N1) epidemic in France, Bulletin of Mathematical Biology, 77 (2015): 1955-1984.
    [48] [ J.-M. Lasry,P.-L. Lions, Lions, Jeux à champ moyen. I: Le cas stationnaire,, C. R., Math., Acad. Sci. Paris, 343 (2006): 619-625.
    [49] [ J.-M. Lasry,P.-L. Lions, Lions, Jeux à champ moyen. II: Horizon fini et contrôle optimal,, C. R., Math., Acad. Sci. Paris, 343 (2006): 679-684.
    [50] [ J.-M. Lasry,P.-L. Lions, Mean field games, Japanese Journal of Mathematics, 2 (2007): 229-260.
    [51] [ A. S. Monto,S. E. Ohmit,J. G. Petrie,E. Johnson,R. Truscon,E. Teich,J. Rotthoff,M. Boulton,J. C. Victor, Comparative efficacy of inactivated and live attenuated influenza vaccines, New England Journal of Medicine, 361 (2009): 1260-1267.
    [52] [ R. Morton,K. H. Wickwire, On the optimal control of a deterministic epidemic, Advances in Appl. Probability, 6 (1974): 622-635.
    [53] [ J. Müller, Optimal vaccination strategies-for whom?, Mathematical Biosciences, 139 (1997): 133-154.
    [54] [ S. Ng,V. J. Fang,D. K. M. Ip,K.-H. Chan,G. M. Leung,J. S. Malik Peiris,B. J. Cowling, Estimation of the association between antibody titers and protection against confirmed influenza virus infection in children, Journal of Infectious Diseases, 208 (2013): 1320-1324.
    [55] [ K. L. Nichol,A. Lind,K. L. Margolis,M. Murdoch,R. McFadden,M. Hauge,S. Magnan,M. Drake, The effectiveness of vaccination against influenza in healthy, working adults, New England Journal of Medicine, 333 (1995): 889-893.
    [56] [ M. T Osterholm,N. S. Kelley,A. Sommer,E. A. Belongia, Efficacy and effectiveness of influenza vaccines: A systematic review and meta-analysis, The Lancet Infectious Diseases, 12 (2012): 36-44.
    [57] [ T. C. Reluga,C. T. Bauch,A. P. Galvani, Evolving public perceptions and stability in vaccine uptake, Math. Biosci., 204 (2006): 185-198.
    [58] [ T. C. Reluga,A. P. Galvani, A general approach for population games with application to vaccination, Mathematical Biosciences, 230 (2011): 67-78.
    [59] [ S. P. Sethi,P. W. Staats, Optimal control of some simple deterministic epidemic models, J. Oper. Res. Soc., 29 (1978): 129-136.
    [60] [ E. Shim,G. B. Chapman,J. P. Townsend,A. P. Galvani, The influence of altruism on influenza vaccination decisions, Journal of The Royal Society Interface, 9 (2012): 2234-2243.
    [61] [ D. M. Skowronski,S. Aleina Tweed,S. Aleina Tweed,G. De Serres, Rapid decline of influenza vaccine-induced antibody in the elderly: Is it real, or is it relevant?, The Journal of Infectious Diseases, 197 (2008): 490-502.
    [62] [ N. M. Smith, J. S. Bresee, D. K. Shay, T. M. Uyeki, N. J. Cox and R. A. Strikas, Prevention and control of influenza: Recommendations of the advisory committee on immunization practices (acip), MMWRRecomm Rep, 55 (2006), 1-42. https://www.cdc.gov/mmwr/preview/mmwrhtml/rr5510a1.htm.
    [63] [ P. G. Smith, L. C. Rodrigues and P. E. M. Fine, Assessment of the protective efficacy of vaccines against common diseases using case-control and cohort studies, International Journal of Epidemiology, 13 (1984), 87-93.
    [64] [ C. J. Struchiner, M. E. Halloran, J. M. Robins and A. Spielman, The behaviour of common measures of association used to assess a vaccination programme under complex disease transmission patterns-a computer simulation study of malaria vaccines, International Journal of Epidemiology, 19 (1990), 187-196.
    [65] [ I. Swiecicki, T. Gobron and D. Ullmo, Schrödinger approach to mean field games, Phys. Rev. Lett., 116(2016), 128701.
    [66] [ J. D Tamerius, J. Shaman, W. J. Alonso, K. Bloom-Feshbach, C. K. Uejio, An. Comrie and C. Viboud, Environmental predictors of seasonal influenza epidemics across temperate and tropical climates, PLoS Pathog, 9 (2013), e1003194.
    [67] [ J. J. Treanor,H. K. Talbot,S. E. Ohmit,L. A. Coleman,M. G. Thompson,P.-Y. Cheng,J. G. Petrie,G. Lofthus,J. K. Meece,J. V. Williams,L. Berman,C. Breese Hall,A. S. Monto,M. R. Griffin,E. Belongia,D. K. Shay, Effectiveness of seasonal influenza vaccines in the United States during a season with circulation of all three vaccine strains, Clinical Infectious Diseases, 55 (2012): 951-959.
    [68] [ G. Turinici, Metric gradient flows with state dependent functionals: the Nash-MFG equilibrium flows and their numerical schemes, Nonlinear Analysis 165 (2017) 163-181.
    [69] [ R. Vardavas, R. Breban and S. Blower, Can influenza epidemics be prevented by voluntary vaccination?, PLoS Comput Biol, 3 (2007), e85.
    [70] [ G. A. Weinberg and P. G. Szilagyi, Vaccine epidemiology: Efficacy, effectiveness, and the translational research roadmap, Journal of Infectious Diseases, 201 (2010), 1607-1610
    [71] [ X. Zhao,V. J. Fang,S. E. Ohmit,A. S. Monto,A. R. Cook,B. J. Cowling, Quantifying protection against influenza virus infection measured by hemagglutination-inhibition assays in vaccine trials,, Epidemiology, 27 (2016): 143-151.
  • Reader Comments
  • © 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(3810) PDF downloads(649) Cited by(6)

Article outline

Figures and Tables

Figures(8)  /  Tables(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog