Save the pine forests of wilt disease using a fractional optimal control strategy

There is no doubt that the pine trees contribute greatly to the strengthening of the economy and the domestic wealth in many countries, where they produce dense grain, maintain the soil, especially in the sand slopes, uses wood in fuel and is one of the most important sources of recreation. However, over the past few years, pine forests have been plagued by many diseases, notably the Pine Wilt Disease (PWD), which poses a major threat to these forests. So, in this article, we investigate the best strategy to reduce and eliminate this disease using fractional optimal control strategy. Here, we introduce a mathematical system of equations contains the fractional order derivative with respect to time which describes the transmission dynamics of PWD. We calculate the general basic reproduction number R0 and discuss the stability of a disease-free and endemic equilibrium in the proposed model. Furthermore, a Fractional Optimal Control Problem (FOCP) with three proposed controls is formulated, and the fractional order necessary conditions of optimality by using the Ponntryagin maximum principle are derived. We apply both analytical and numerical techniques on the FOCP with suggested controls to demonstrate the effective control strategies to prevent the transmission of the PWD. The numerical simulation of the suggested FOCP is presented and these results are expected to be helpful in the early treatment of PWD-infected trees.