Volatile opinions and optimal control of vaccine awareness campaigns: chaotic behaviour of the forward-backward sweep algorithm vs. heuristic direct optimization

•Opinions volatility is considered in modelling vaccination for infectious diseases.•We search for the optimal time-profile of vaccine awareness campaigns.•The model is non-linear in its control and the cost function is non-convex.•The optimization problem is numerically solved by both direct and indirect methods.•A non-convergence analysis is performed via methods for chaotic time-series. In modern societies the main sources of information are Internet-based social networks. Thus, the opinion of citizens on key topics, such as vaccines, is very volatile. Here, we explore the impact of volatility on the modelling of public response to vaccine awareness campaigns for favouring vaccine uptake. We apply a quasi-steady-state approximation to the model of spread and control of Susceptible-Infected-Removed diseases proposed in (d’Onofrio et al., PLoS One, 2012). This allows us to infer and analyze a new behavioural epidemiology model that is nonlinear in the control. Then, we investigate the efficient design of vaccine awareness campaigns by adopting optimal control theory. The resulting problem has important issues: (i) the integrand of its objective functional is non-convex; (ii) the application of forward-backward sweep (FBS) and gradient descent algorithms in some key cases does not work; (iii) analytical approaches provide continuous solutions that cannot rigorously be implemented since Public Health interventions cannot be fully flexible. Thus, on the one hand, we resort to direct optimization of the objective functional via heuristic stochastic optimization, in particular via particle swarm optimization and differential evolution algorithms. On the other hand, we investigate the non-convergence of the FBS algorithm with tools of the statistical theory of nonlinear chaotic time-series. Finally, since the direct optimization algorithms are stochastic, we provide a statistical assessment of the obtained solutions.