The impact of social media advertisements and treatments on the dynamics of infectious diseases with optimal control strategies

The dissemination of public health information through television and social media posts is essential for informing the public about the transmission of contagious diseases, which is crucial in preventing the spread of various infectious diseases. In this paper, we propose a non-linear mathematical model to assess the effect of advertisements through social media in creating awareness and limiting treatment on spreading infectious diseases. These initiatives may alter population behaviour and divide the susceptible population into subgroups. In addition, to comprehend these dynamics better, we use half-saturation constant rates for media coverage and treatment. The model’s well-posedness and feasibility are evaluated. The possible biological equilibrium points are calculated. Local and global stability are carried out. The objective of our study is to produce the model’s bifurcation. Transcritical, Saddle–node, Hopf bifurcation of codimension 1 and Cusp, Generalized-Hopf (Bautin), and Bogdanov–Takens (BT) bifurcation of codimension 2 are studied for this purpose. Due to the limited medical resources and supply efficiency, the model exhibits backward bifurcation, resulting in bistability. Moreover, the occurrence condition for stability and direction of Hopf bifurcation is discussed. This model study demonstrates that the system is significantly influenced by the pace with which awareness programmes are implemented and that raising this value above a threshold may result in continuous oscillation. Sensitivity analysis employs the normalized forward sensitivity index of the basic reproduction number to provide a comprehensive understanding of the effect of various parameters on accelerating and limiting disease spread. Further, the minimum possible cost is determined by formulating an optimal control system based on sensitivity analysis and applying Pontryagin’s maximum principle. Methods of cost-effectiveness, such as ACER and ICER, are used to determine the most cost-effective control intervention strategy among all the strategies. Numerical simulations have been done to support all theoretical findings.