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

• * Corresponding author: Gabriel Turinici
• 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.

Mathematics Subject Classification: Primary: 92D30; Secondary: 92C42, 60J20, 91A13.

 Citation:

• 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 $1-\xi_\infty$ $0.00$ $5.04 \%$ $0.25$ $5.94 \%$ $0.50$ $7.02 \%$ $0.55$ $7.20\%$ $0.60$ $7.29\%$ $0.65$ $7.23 \%$ $0.75$ $5.74 \%$ $0.80$ $2.93 \%$ $0.85$ $0.00 \%$

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