A two-group epidemic model with heterogeneity in cognitive effects.

During the outbreak of new infectious diseases, media information and medical resources play crucial roles in shaping the dynamics of disease transmission. To investigate the combined impact of media information and limited medical resources on disease spread, we proposed a two-group compartmental model. This model divided the population into two groups based on their ability to receive information. We derived the basic reproduction number, analyzed the local stability of the disease-free equilibrium, and examined the conditions under which disease extinction or persistence occured. For control strategies, we explored both constant and optimal control approaches under the constraint of limited media resources. Numerical simulations indicated that enhancing the population's responsiveness to media and medical resources helped reduce the infection rate. The model also exhibited complex dynamical behaviors, such as backward bifurcation, forward-backward bifurcation, and homoclinic bifurcation, which presented significant challenges for disease control. Furthermore, we conducted numerical simulations of the optimal control problem to validate and support our theoretical findings. In the case of constant control, as the disparity between the two populations increases, media resources should be increasingly allocated to the information-insensitive group. For optimal control, we employed the forward-backward sweep method, where media resources were increasingly allocated to information-insensitive groups as population heterogeneity rises. This study established an empirical framework for optimizing media-driven public health communication strategies, offering critical insights into the strategic allocation of limited media resources across heterogeneous populations.