Dynamics of a metapopulation epidemic model with localized culling

A two-patches metapopulation mathematical model, describing the dynamics of Susceptibles and Infected in wildlife diseases, is presented. The two patches are identical in absence of control, and culling activities are performed in only one of them. Firstly, the dynamics of the system in absence of control is investigated. Then, two types of localized culling strategies (proactive end reactive) are considered. The proactive control is modeled by a constant culling effort, and for the ensuing model the disease free equilibrium is characterized and existence of the endemic equilibrium is discussed in terms of a suitable control reproduction number. The localized reactive control is modeled by a piecewise constant culling effort function, that introduces an extra-mortality when the number of infected individuals in the patch overcomes a given threshold. The reactive control is then analytically and numerically investigated in the frame of Filippov systems. We find that localized culling may be ineffective in controlling diseases in wild populations when the infection affects host fecundity in addition to host mortality, even leading to unexpected increases in the number of infected individuals in the nearby areas.