We study the dynamics of a disease under administration of a vaccine and antiviral drug, where the disease transmits directly from the parents to the offspring (vertical transmission) and also through contact with infective individuals (horizontal transmission). While vaccination to those susceptible reduces the horizontal transmission, administration of the antiviral drug to infected individuals lessens the chance of vertical transmission. Thus the vaccine and antiviral drug play different roles in controlling the disease, which has both vertical and horizontal transmission. We develop a 3D model with Susceptible–Infected–Recovered under vaccination to the susceptible and antiviral treatment to the infected and consider a control theoretic approach using the Pontryagin maximum principle to analyse the costeffectiveness of the control process. Our results demonstrate that a mixed intervention strategy of vaccination and antiviral drug in a proper ratio is the most effective way to control the disease. We show that cost-effectiveness of both intervention strategies intimately depends on disease-related parameters, such as force of infection, probability of being infected to offspring from infected mothers, loss of immunity or reinfection and also on cost of treatment.

中文翻译：

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垂直传播疾病的最优控制：一种综合方法

我们研究了在疫苗和抗病毒药物的管理下疾病的动力学，其中疾病直接从父母传播给后代（垂直传播），也通过与感染者的接触（水平传播）传播。虽然对易感人群进行疫苗接种可以减少水平传播，但将抗病毒药物施用于受感染的人却减少了垂直传播的机会。因此，疫苗和抗病毒药物在控制疾病方面起着不同的作用，具有垂直和水平两种传播方式。我们开发了一种3D模型，在接种疫苗后对感染者进行了易感性和抗病毒治疗，并在感染后恢复了易感性，并考虑使用Pontryagin最大原理的控制理论方法来分析控制过程的成本效益。我们的结果表明，以适当比例混合疫苗和抗病毒药物进行干预是控制该病最有效的方法。我们表明，这两种干预策略的成本效益都直接取决于与疾病相关的参数，例如感染力，被感染母亲感染后代的可能性，免疫力或再感染的丧失以及治疗费用。