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June 2018, 15(3): 629-652. doi: 10.3934/mbe.2018028

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

1. 

Université Paris-Dauphine, PSL Research University, CNRS UMR 7534, CEREMADE, 75016 Paris, France, & Università degli Studi di Pavia, Dipartimento di Matematica, 27100 Pavia, Italy,

2. 

Université Paris-Dauphine, PSL Research University, CNRS UMR 7534, CEREMADE, 75016 Paris, France, & Institut Universitaire de France, Paris, France

* Corresponding author: Gabriel Turinici

Received  March 2017 Accepted  July 29, 2017 Published  December 2017

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. Mathematical Biosciences & Engineering, 2018, 15 (3) : 629-652. doi: 10.3934/mbe.2018028
References:
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C. T. Bauch, Imitation dynamics predict vaccinating behaviour, Proc Biol Sci, 272 (2005), 1669-1675. doi: 10.1098/rspb.2005.3153.

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E. A. BelongiaM. E. SundaramD. L. McClureJ. K. MeeceJ. Ferdinands and 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. doi: 10.1016/j.vaccine.2014.06.052.

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show all references

References:
[1]

A. Abakuks, Optimal immunisation policies for epidemics, Advances in Appl. Probability, 6 (1974), 494-511. doi: 10.1017/S0001867800039963.

[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. doi: 10.1007/978-0-85729-115-8.

[5]

Y. BaiN. ShiQ. LuL. YangZ. WangL. LiH. HanD. ZhengF. LuoZ. Zhang and X. Ai, Immunological persistence of a seasonal influenza vaccine in people more than 3 years old, Human Vaccines & Immunotherapeutics, 11 (2015), 1648-1653. doi: 10.1080/21645515.2015.1037998.

[6]

C. T. Bauch and D. J. D. Earn, Vaccination and the theory of games, Proc. Natl. Acad. Sci. USA, 101 (2004), 13391-13394 (electronic). doi: 10.1073/pnas.0403823101.

[7]

C. T. BauchA. P. Galvani and D. J. D. Earn, Group interest versus self-interest in smallpox vaccination policy, Proceedings of the National Academy of Sciences, 100 (2003), 10564-10567. doi: 10.1073/pnas.1731324100.

[8]

C. T. Bauch, Imitation dynamics predict vaccinating behaviour, Proc Biol Sci, 272 (2005), 1669-1675. doi: 10.1098/rspb.2005.3153.

[9]

E. A. BelongiaM. E. SundaramD. L. McClureJ. K. MeeceJ. Ferdinands and 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. doi: 10.1016/j.vaccine.2014.06.052.

[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. doi: 10.1098/rsta.2013.0398.

[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. doi: 10.1103/PhysRevE.76.031127.

[12]

D. L. BritoE. Sheshinski and M. D. Intriligator, Externalities and compulsary vaccinations, Journal of Public Economics, 45 (1991), 69-90. doi: 10.1016/0047-2727(91)90048-7.

[13]

B. BuonomoA. d'Onofrio and D. Lacitignola, Global stability of an {SIR} epidemic model with information dependent vaccination, Mathematical Biosciences, 216 (2008), 9-16. doi: 10.1016/j.mbs.2008.07.011.

[14]

P. Cardaliaguet and S. Hadikhanloo, Learning in mean field games: The fictitious play, ESAIM Control Optim. Calc. Var., 23 (2017), 569-591. doi: 10.1051/cocv/2016004.

[15]

F. Carrat and A. Flahault, Influenza vaccine: The challenge of antigenic drift, Vaccine, 25 (2007), 6852-6862. doi: 10.1016/j.vaccine.2007.07.027.

[16]

F. H. Chen, A susceptible-infected epidemic model with voluntary vaccinations, Journal of Mathematical Biology, 53 (2006), 253-272. doi: 10.1007/s00285-006-0006-1.

[17]

M. L. Clements and 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. CodeçoP. M. LuzF. CoelhoA. P Galvani and 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. doi: 10.1371/journal.pcbi.1000425.

[20]

M.-G. Cojocaru, Dynamic equilibria of group vaccination strategies in a heterogeneous population, Journal of Global Optimization, 40 (2008), 51-63. doi: 10.1007/s10898-007-9204-7.

[21]

R. B. Couch and J. A. Kasel, Immunity to influenza in man, Annual Reviews in Microbiology, 37 (1983), 529-549. doi: 10.1146/annurev.mi.37.100183.002525.

[22]

N. Cox, Influenza seasonality: Timing and formulation of vaccines, Bulletin of the World Health Organization, 92 (2014), 311-311. doi: 10.2471/BLT.14.139428.

[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'OnofrioP. Manfredi and E. Salinelli, Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases, Theoretical Population Biology, 71 (2007), 301-317.

[26]

A. d'OnofrioP. Manfredi and E. Salinelli, Fatal SIR diseases and rational exemption to vaccination, Mathematical Medicine and Biology, 25 (2008), 337-357.

[27]

P. DoutorP. RodriguesM. do Céu Soares and F. A. C. C. Chalub, Optimal vaccination strategies and rational behaviour in seasonal epidemics, Journal of Mathematical Biology, 73 (2016), 1437-1465. doi: 10.1007/s00285-016-0997-1.

[28]

J. DushoffJ. B PlotkinC. ViboudD. J. D. Earn and 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. doi: 10.1093/aje/kwj024.

[29]

J. M. FerdinandsA. M. FryS. ReynoldsJ. G. PetrieB. FlanneryM. L. Jackson and 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 and J. A. Clarkson, Individual versus public priorities in the determination of optimal vaccination policies, American Journal of Epidemiology, 124 (1986), 1012-1020. doi: 10.1093/oxfordjournals.aje.a114471.

[31]

P. J. Francis, Optimal tax/subsidy combinations for the flu season, Journal of Economic Dynamics and Control, 28 (2004), 2037-2054. doi: 10.1016/j.jedc.2003.08.001.

[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. FunkM. Salathé and 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. doi: 10.1098/rsif.2010.0142.

[34]

A. P. GalvaniT. C. Reluga and 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. doi: 10.1073/pnas.0606774104.

[35]

P.-Y. Geoffard and T. Philipson, Disease eradication: Private versus public vaccination, The American Economic Review, 87 (1997), 222-230.

[36]

N. C. Grassly and C. Fraser, Seasonal infectious disease epidemiology, Proceedings of the Royal Society of London B: Biological Sciences, 273 (2006), 2541-2550. doi: 10.1098/rspb.2006.3604.

[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]

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Figure 1.  Two possible forms for the function $A$.
Figure 2.  Individual model.
Figure 5.  Results for Subsection 4.3.1. Top: the optimal converged strategy $\xi^{MFG}$ at times $\{t_0,...,t_{N-1} \}$. The weight of the non-vaccinating pure strategy (i.e., corresponding to time $t = \infty$) is $68\%$. Bottom: the corresponding cost $\mathcal{C}_{\xi^{MFG}}$. The red line corresponds to the cost of the non-vaccinating pure strategy $(\mathcal{C}_{\xi^{MFG}})_{N+1}$.
Figure 3.  The optimal converged strategy $\xi^{MFG}$ at times $\{t_0,...,t_{N-1} \}$ for subsection 4.2, case $\mathcal{M}_1$. The weight of the non-vaccinating pure strategy (i.e., corresponding to time $t = \infty$) is $88\%$; this means that $12\%$ of the population vaccinates.
Figure 4.  The optimal converged strategy $\xi^{MFG}$ at times $\{t_0,...,t_{N-1} \}$ for subsection 4.2, case $\mathcal{M}_2$. Here $15\%$ of the population vaccinates.
Figure 6.  Results of Subsection 4.3.1. Top: the evolution of the susceptible class $S_n$; bottom: the (total) infected class $I_n$.
Figure 7.  The decrease of the incentive to change strategy $E(\xi_k)$. Note that $E(\xi_k)$ does not decrease monotonically. In fact, there is no reason to expect such a behavior, since we are not minimizing $E(\cdot)$ in a monotonic fashion.
Figure 8.  Results of Subsection 4.3.2. Top: the optimal converged strategy $\xi^{MFG}$. The weight of the non-vaccinating pure strategy (i.e., corresponding to time $t = \infty$) is $91\%$. Bottom: the corresponding cost $\mathcal{C}_{\xi^{MFG}}$. The thin horizontal line corresponds to the cost of the non-vaccinating pure strategy $(\mathcal{C}_{\xi^{MFG}})_{N+1}$.
Table 1.  Results for the Subsection 4.4. Individual vaccination policy with respect to the failed vaccination rate of the vaccine.
Failed vaccination rate $f$ Vaccination rate $1-\xi_\infty$
$0.00$ $5.04 \%$
$0.25$ $5.94 \%$
$0.50$ $7.02 \%$
$0.55$ $7.20\%$
$0.60$ $7.29\%$
$0.65$ $7.23 \%$
$0.75$ $5.74 \%$
$0.80$ $2.93 \%$
$0.85$ $0.00 \%$
Failed vaccination rate $f$ Vaccination rate $1-\xi_\infty$
$0.00$ $5.04 \%$
$0.25$ $5.94 \%$
$0.50$ $7.02 \%$
$0.55$ $7.20\%$
$0.60$ $7.29\%$
$0.65$ $7.23 \%$
$0.75$ $5.74 \%$
$0.80$ $2.93 \%$
$0.85$ $0.00 \%$
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