Mathematical modeling for relation between parents' health education and vaccine uptake

We formulate susceptible-infectious-susceptible (SIS) mathematical models to assess health education roles in enhancing human papillomavirus (HPV) vaccines uptake in both childhood and adult ages. The model is considered in logistically growing populations. We derive the basic reproduction number for the model and use it to describe the local-asymptotic stability of the disease-free equilibrium (DFE). The contribution of educated and uneducated in basic reproduction number is estimated too. We estimate the time series solution of the model for uneducated, low educated and high educated scenarios. The higher the education level the more the vaccine uptake, the less the risky behavior, and the shorter the time needed to control HPV. Late education for adults is more effective than early education for children as late education is applied with vaccine and that could increase vaccine uptake for parents and their kids. Moreover, we introduce an optimal problem to search for the best introduction of early and late educations that maximize the vaccines uptake and control HPV with the least possible cost. In the time period 20 years, the time-dependent optimal educations are as follows: Full coverage of early education for the full period regardless of vaccine efficacy used. This needs to be supported with maximum education rate (one unit) for the first 13 years for vaccine with 90% efficacy, which needs to be extended for three more years when vaccine efficacy is reduced to 65%. The optimal educations have almost the same cost and the same effect in reducing HPV cases regardless of vaccine efficacy. This means that increasing health education is a viable low-cost strategy to support vaccines that have low efficacy.