Traveling waves in the Lethargic Crab Disease

Since 1997, the Lethargic Crab Disease (LCD) has decimated native populations of the mangrove land crab Ucides cordatus (Decapoda: Ocypodidae) along the Brazilian coast, spreading preferentially in the North–South direction and showing a periodic epidemic behavior. To study the spatial dissemination of LCD between estuaries, we propose a mathematical model using a system of partial differential reaction–diffusion equations. After a suitable change of variables, an analysis of the model shown that it presents four possible scenarios, namely, the trivial equilibrium, the disease-free equilibrium, endemic equilibrium, and limit cycles arising from a Hopf bifurcation. The threshold values depend on the basic reproductive number of crabs and fungi, and on the contact rate between these two population, modeled through mass action law. The existence of traveling wave solutions connecting disease free-equilibrium and endemic equilibrium is analyzed and the minimum wave speed for disease propagation obtained. A sensitivity analysis of the wave speed related to model parameters enables an understanding of how LCD can be controlled.