Feasibility and optimisation results for elimination by mass trapping in a metapopulation model

Vector and Pest control is an important issue in terms of Food and Health security all around the World. In this paper, we consider the issue of mass trapping strategies for interconnected areas, where traps can only be deployed in some of them. Assuming linear dispersal between the areas, we consider and study a metapopulation model, and explore the global effect of a linear control, achieved by an (on purpose) increase of the mortality in certain areas. We show that the feasibility of population elimination is determined by an algebraic property on the Jacobian matrix at the origin of a so-called residual system. If elimination is not achievable, we then assess the smallest globally asymptotically stable equilibrium. Conversely when elimination is feasible, we study an optimisation problem consisting in achieving this task while minimising a certain cost function, chosen as a non-decreasing and convex function of the mortality rates added in the controlled areas. We show that such a minimisation problem admits a global minimiser, which is unique in the relevant cases. An interior point algorithm is proposed to compute the solution, using explicit formulas for the Jacobian matrix and the Hessian of the objective function of the unconstrained penalised problem. The results are illustrated and discussed with numerical simulations. • A metapopulation model is studied with patches linearly interconnected. • Mass trapping is considered as control method. • Global convergence to positive equilibrium or zero is characterized algebraically. • Optimal control allocation is studied and solved by interior point method. • The results hold for any given subset of controllable patches.