Regional optimization problem for a general reaction–diffusion malaria model with temperature and rainfall

Abstract

This work primarily addresses the regional optimization problem of a malaria model in heterogeneous environments, which incorporates general incidence rate, vaccination, temperature and rainfall. To reflect the differences in resource allocation across different regions and times, the upper bound of control is extended to a bounded function related to spatial location and time. On this basis, a control problem is investigated, with the goal of minimizing the discrepancy between the actual and desired control outcomes for mosquitoes and infectious humans, and the associated costs. Firstly, the application of truncation technique and semigroup theory enables us to obtain the existence, uniqueness and some estimates of the positive strong solution for the controlled system. Despite the intricate nature of the controlled system, which makes proving global solvability challenging, this hurdle is successfully overcome. Subsequently, applying the technique of minimizing sequence, the existence of optimal pair is demonstrated. By utilizing the differentiability of control-to-state mapping, we establish the first-order necessary optimality condition. Specifically, deeper insights are offered in the left neighborhood of the terminal moment. Furthermore, provide the second-order necessary and sufficient optimality conditions. Finally, explore the spread and control of malaria in The Gambia. The analysis shows that the upper bound of control possesses a remarkable influence on its optimal form. Moreover, temperature and rainfall can significantly affect costs and benefits of control measures, thereby affecting the evaluation process. These outcomes enrich and generalize the study on regional optimal control problems of disease models, and provide new perspectives for related research.