Optimal control of predator-prey mathematical model with infection and harvesting on prey

Abstract

This paper presents a predator-prey mathematical model with infection and harvesting on prey. The infection and harvesting only occur on the prey population and it assumed that the prey infection would not infect predator population. We analysed the mathematical model of predator-prey with infection and harvesting in prey. Optimal control, which is a prevention of the prey infection, also applied in the model and denoted as U. The purpose of the control is to increase the susceptible prey. The analytical result showed that the model has five equilibriums, namely the extinction equilibrium (E0), the infection free and predator extinction equilibrium (E1), the infection free equilibrium (E2), the predator extinction equilibrium (E3), and the coexistence equilibrium (E4). The extinction equilibrium (E0) is not stable. The infection free and predator extinction equilibrium (E1), the infection free equilibrium (E2), also the predator extinction equilibrium (E3), are locally asymptotically stable with some certain conditions. The coexistence equilibrium (E4) tends to be locally asymptotically stable. Afterwards, by using the Maximum Pontryagin Principle, we obtained the existence of optimal control U. From numerical simulation, we can conclude that the control could increase the population of susceptible prey and decrease the infected prey.