Hesitancy and refusal of vaccines preventing childhood diseases are spreading due to 'pseudo-rational' behaviours: parents overweigh real and imaginary side effects of vaccines. Nonetheless, the 'Public Health System' (PHS) may enact public campaigns to favour vaccine uptake. To determine the optimal time profiles for such campaigns, we apply the optimal control theory to an extension of the susceptible-infectious-removed (SIR)-based behavioural vaccination model by d'Onofrio et al. (2012, PLoS ONE, 7, e45653). The new model is of susceptible-exposed-infectious-removed (SEIR) type under seasonal fluctuations of the transmission rate. Our objective is to minimize the total costs of the disease: the disease burden, the vaccination costs and a less usual cost: the economic burden to enact the PHS campaigns. We apply the Pontryagin minimum principle and numerically explore the impact of seasonality, human behaviour and latency rate on the control and spread of the target disease. We focus on two noteworthy case studies: the low (resp. intermediate) relative perceived risk of vaccine side effects and relatively low (resp. very low) speed of imitation. One general result is that seasonality may produce a remarkable impact on PHS campaigns aimed at controlling, via an increase of the vaccination uptake, the spread of a target infectious disease. In particular, a higher amplitude of the seasonal variation produces a higher effort and this, in turn, beneficially impacts the induced vaccine uptake since the larger is the strength of seasonality, the longer the vaccine propensity remains large. However, such increased effort is not able to fully compensate the action of seasonality on the prevalence.

中文翻译：

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行为疫苗接种模型中的最佳公共卫生干预措施：季节性，行为和潜伏期之间的相互作用。

由于“伪理性”行为，对预防儿童疾病的疫苗的犹豫和拒绝正在蔓延：父母超过了疫苗的真实和想象中的副作用。但是，“公共卫生系统”（PHS）可能会开展公众运动以支持疫苗的使用。为了确定此类运动的最佳时间分布，我们将最佳控制理论应用于d'Onofrio等人对基于易感性感染去除（SIR）的行为疫苗接种模型的扩展。（2012，PLoS ONE，7，e45653）。在传播速率的季节性波动下，该新模型具有易感暴露-传染去除（SEIR）类型。我们的目标是最大程度地降低疾病的总成本：疾病负担，疫苗接种成本和较少的日常花费：开展PHS运动的经济负担。我们应用庞特里亚金极小值原理，并在数值上探索季节性，人类行为和潜伏率对目标疾病的控制和传播的影响。我们着重研究两个值得注意的案例：疫苗副作用的相对风险相对较低（中间），相对较低的仿制速度（相对极低）。一个普遍的结果是，季节性可能会对旨在通过增加疫苗接种摄入量控制目标传染病传播的PHS活动产生重大影响。尤其是，季节性变化的幅度越大，产生的作用就越大，这反过来会有利地影响诱导的疫苗摄入，因为季节性强度越大，疫苗的倾向性就越长。然而，