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2014, 11(5): 1045-1063. doi: 10.3934/mbe.2014.11.1045

## Optimal control of vaccination dynamics during an influenza epidemic

 1 Department of Physiology, McGill University, Montreal, Quebec, H3G 1Y6, Canada 2 Agent-Based Modelling Laboratory, York University, Toronto, Ontario, M3J 1P3, Canada

Received  July 2013 Revised  April 2014 Published  June 2014

For emerging diseases like pandemic influenza, several factors could impact the outcome of vaccination programs, including a delay in vaccine availability, imperfect vaccine-induced protection, and inadequate number of vaccines to sufficiently lower the susceptibility of the population by raising the level of herd immunity. We sought to investigate the effect of these factors in determining optimal vaccination strategies during an emerging influenza infection for which the population is entirely susceptible. We developed a population dynamical model of disease transmission and vaccination, and analyzed the control problem associated with an adaptive time-dependent vaccination strategy, in which the rate of vaccine distribution is optimally determined with time for minimizing the total number of infections (i.e., the epidemic final size). We simulated the model and compared the outcomes with a constant vaccination strategy in which the rate of vaccine distribution is time-independent. When vaccines are available at the onset of epidemic, our findings show that for a sufficiently high vaccine efficacy, the adaptive and constant vaccination strategies lead to comparable outcomes in terms of the epidemic final size. However, the adaptive vaccination requires a vaccine coverage higher than (or equivalent to) the constant vaccination regardless of the rate of vaccine distribution, suggesting that the latter is a more cost-effective strategy. When the vaccine efficacy is below a certain threshold, the adaptive vaccination could substantially outperform the constant vaccination, and the impact of adaptive strategy becomes more pronounced as the rate of vaccine distribution increases. We observed similar results when vaccines become available with a delay during the epidemic; however, the adaptive strategy may require a significantly higher vaccine coverage to outperform the constant vaccination strategy. The findings indicate that the vaccine efficacy is a key parameter that affects optimal control of vaccination dynamics during an epidemic, raising an important question on the trade-off between effectiveness and cost-effectiveness of vaccination policies in the context of limited vaccine quantities.
Citation: 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
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