Optimal control of vertically transmitted disease: an integrated approach.

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 cost-effectiveness 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.