# American Institute of Mathematical Sciences

• Previous Article
Delay induced spatiotemporal patterns in a diffusive intraguild predation model with Beddington-DeAngelis functional response
• MBE Home
• This Issue
• Next Article
Dynamics of an ultra-discrete SIR epidemic model with time delay
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:

show all references

##### References:
Two possible forms for the function $A$.
Individual model.
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 $(\mathcal{C}_{\xi^{MFG}})_{N+1}$.
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.
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.
Results of Subsection 4.3.1. Top: the evolution of the susceptible class $S_n$; bottom: the (total) infected class $I_n$.
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.
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 $(\mathcal{C}_{\xi^{MFG}})_{N+1}$.
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 $1-\xi_\infty$ $0.00$ $5.04 \%$ $0.25$ $5.94 \%$ $0.50$ $7.02 \%$ $0.55$ $7.20\%$ $0.60$ $7.29\%$ $0.65$ $7.23 \%$ $0.75$ $5.74 \%$ $0.80$ $2.93 \%$ $0.85$ $0.00 \%$
 [1] Fabio Camilli, Elisabetta Carlini, Claudio Marchi. A model problem for Mean Field Games on networks. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4173-4192. doi: 10.3934/dcds.2015.35.4173 [2] Sunmi Lee, Romarie Morales, Carlos Castillo-Chavez. A note on the use of influenza vaccination strategies when supply is limited. Mathematical Biosciences & Engineering, 2011, 8 (1) : 171-182. doi: 10.3934/mbe.2011.8.171 [3] Dennis L. Chao, Dobromir T. Dimitrov. Seasonality and the effectiveness of mass vaccination. Mathematical Biosciences & Engineering, 2016, 13 (2) : 249-259. doi: 10.3934/mbe.2015001 [4] Alan J. Terry. Pulse vaccination strategies in a metapopulation SIR model. Mathematical Biosciences & Engineering, 2010, 7 (2) : 455-477. doi: 10.3934/mbe.2010.7.455 [5] Qianqian Cui, Zhipeng Qiu, Ling Ding. An SIR epidemic model with vaccination in a patchy environment. Mathematical Biosciences & Engineering, 2017, 14 (5-6) : 1141-1157. doi: 10.3934/mbe.2017059 [6] Pierre Cardaliaguet, Jean-Michel Lasry, Pierre-Louis Lions, Alessio Porretta. Long time average of mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 279-301. doi: 10.3934/nhm.2012.7.279 [7] Martin Burger, Marco Di Francesco, Peter A. Markowich, Marie-Therese Wolfram. Mean field games with nonlinear mobilities in pedestrian dynamics. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1311-1333. doi: 10.3934/dcdsb.2014.19.1311 [8] Folashade B. Agusto, Abba B. Gumel. Theoretical assessment of avian influenza vaccine. Discrete & Continuous Dynamical Systems - B, 2010, 13 (1) : 1-25. doi: 10.3934/dcdsb.2010.13.1 [9] Urszula Ledzewicz, Heinz Schättler. On optimal singular controls for a general SIR-model with vaccination and treatment. Conference Publications, 2011, 2011 (Special) : 981-990. doi: 10.3934/proc.2011.2011.981 [10] Shujing Gao, Dehui Xie, Lansun Chen. Pulse vaccination strategy in a delayed sir epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2007, 7 (1) : 77-86. doi: 10.3934/dcdsb.2007.7.77 [11] Olivier Guéant. New numerical methods for mean field games with quadratic costs. Networks & Heterogeneous Media, 2012, 7 (2) : 315-336. doi: 10.3934/nhm.2012.7.315 [12] Juan Pablo Maldonado López. Discrete time mean field games: The short-stage limit. Journal of Dynamics & Games, 2015, 2 (1) : 89-101. doi: 10.3934/jdg.2015.2.89 [13] Martino Bardi. Explicit solutions of some linear-quadratic mean field games. Networks & Heterogeneous Media, 2012, 7 (2) : 243-261. doi: 10.3934/nhm.2012.7.243 [14] Yves Achdou, Victor Perez. Iterative strategies for solving linearized discrete mean field games systems. Networks & Heterogeneous Media, 2012, 7 (2) : 197-217. doi: 10.3934/nhm.2012.7.197 [15] Diogo A. Gomes, Gabriel E. Pires, Héctor Sánchez-Morgado. A-priori estimates for stationary mean-field games. Networks & Heterogeneous Media, 2012, 7 (2) : 303-314. doi: 10.3934/nhm.2012.7.303 [16] Eunha Shim. Prioritization of delayed vaccination for pandemic influenza. Mathematical Biosciences & Engineering, 2011, 8 (1) : 95-112. doi: 10.3934/mbe.2011.8.95 [17] Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299 [18] Sherry Towers, Katia Vogt Geisse, Chia-Chun Tsai, Qing Han, Zhilan Feng. The impact of school closures on pandemic influenza: Assessing potential repercussions using a seasonal SIR model. Mathematical Biosciences & Engineering, 2012, 9 (2) : 413-430. doi: 10.3934/mbe.2012.9.413 [19] Min Zhu, Zhigui Lin. Modeling the transmission of dengue fever with limited medical resources and self-protection. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 957-974. doi: 10.3934/dcdsb.2018050 [20] Majid Jaberi-Douraki, Seyed M. Moghadas. Optimal control of vaccination dynamics during an influenza epidemic. Mathematical Biosciences & Engineering, 2014, 11 (5) : 1045-1063. doi: 10.3934/mbe.2014.11.1045

2016 Impact Factor: 1.035