Optimizing pharmaceutical and non-pharmaceutical interventions in an age-structured epidemic model

Besides being transmitted through direct contact with infected individuals, infectious diseases are often transmitted through contact with contaminated inanimate objects, known as fomites. Therefore, in addition to conventional pharmaceutical interventions, environmental sanitation plays a crucial role in minimizing disease transmission. This study examines two different ways of disease transmission by incorporating an additional compartment into an age-dependent SIRS model framework, which represents the concentration of pathogens in fomites. The basic reproduction number, R0 is obtained for the model. It is found that R0 is a monotonic decreasing function with respect to the half-saturation level of environmental contamination. It possesses two steady states, disease-free and endemic. Their stability and uniform persistence criterion are obtained with respect to the threshold value R0, as it crosses the unit value. The model is further extended to an optimal control problem (OCP), which includes two control interventions (i) the age-dependent treatment of infected individuals and (ii) environmental sanitation. The existence and uniqueness of the OCP are examined utilizing the concept of Gâteaux derivative. The model is numerically simulated considering feasible age-dependent parameters to visualize the analytical results. Moreover, several solutions to the OCP are plotted considering both the presence or absence of the single or both control variables. It is found that each of the control interventions is capable of containing the disease spread. However, a combined control strategy with low-cost controls is most effective in minimizing disease transmission while the two routes of infection are considered.