Existence of optimal control and time optimality of a stochastic age-space structured tuberculosis model

In this paper, considering the influences of stochastic noise and age structure on the spread of tuberculosis, a stochastic age-space structured tuberculosis model that integrates both prevention and treatment control strategies is formulated. To be specific, the positivity and uniqueness of the mild solution of model are acquired. After that, from a cost-benefit perspective, an optimal control problem of tuberculosis is formulated to minimize the number of infected population and control cost by virtue of Gronwall's inequality and Mazur's lemma. The time optimal control is further discussed. Based on this analysis, some concrete examples are given for verify theoretical results.