Threshold phenomena with respect to the initiation of depopulation in a simple model of foot-and-mouth disease

Depopulation is one of the important interventions for the outbreak of animal diseases. Simulation models using actual case scenarios conclude that early depopulation is the most efficient in preventing the spread of foot-and-mouth disease (FMD). However, the long delay in its initiation was often seen in the actual cases and the theoretical analyses of FMD epidemiology with depopulation needs further elaboration. Here, we investigated the qualitative features of epidemic models when depopulation at a fixed capacity was delayed. We built a simple deterministic model for FMD based on state-transition, the SEIIR model whose unit is a single farm. The model settings and parameters were determined using the data from the 2010 epidemic in Miyazaki, Japan. By numerical calculation, we showed the existence of the threshold phenomenon with respect to delays in the initiation of depopulation and if the initiation of full-fledged depopulation surpasses the certain critical timing, the final size of the epidemic rapidly increases leading to a "catastrophic situation". We also revealed the mechanism of the threshold phenomenon from the relationship between the depopulation capacity and the increasing rate of infection. Although it can be delayed with lower transmission coefficients, the threshold phenomenon still exists. Thus, the existence of the critical timing for depopulation appears to be a universal feature of FMD epidemiology when depopulation is used as the main treatment for disease control.