Analysis of a chronological age-structured epidemic model with a pair of optimal treatment controls

Abstract

Emerging infectious diseases are one of the core concerns in epidemiology, and they often lead to a global public health emergency and consequently affect socio-economic sectors. The risk of contracting an infection varies with different age groups in a population. In this investigation, we present a continuous age-structured epidemic model that incorporates a chronological age-dependent bilinear disease incidence rate. We establish the model's feasibility from the population perspective and derive the basic reproduction number, . The analysis reveals that the model consistently exhibits a disease-free equilibrium, and a unique endemic equilibrium emerges whenever . Furthermore, the value of determines the stability of the age-structured model. We further formulate an optimal control problem by introducing a pair of age-dependent control variables, namely, (i) pre-cautions or medical care to the latently infected individuals and (ii) treatments or hospitalization of the infected individuals. We aim to minimize the cost of implementing these two controls so that the severity of an epidemic can be mitigated. We derived the adjoint equations to the age-dependent optimal control problem using Gateaux derivatives, then proved the existence and uniqueness of optimal control solution pair using Ekeland's principle. Finally, numerical simulations are conducted to verify the analysis and visualize the solution profiles of the model. Our observations suggest that while both control measures are effective in reducing the impact of the disease, taking precautions proves to be significantly more effective in mitigating the spread of the epidemic.