Jump-diffusion Model for the Global Spread of an Amphibian Disease

A system of jump-diffusion stochastic differential equations is considered for modelling the dynamics of the spread of an amphibian disease. In this investigation, it is assumed that the amphibians are located in M regions which are widely and uniformly spaced on the surface of the earth and that the disease is present initially in only one region. Within each region, the amphibians live in N separate patches. A jump-diffusion stochastic system is derived for the number of infected patches in each of the M regions. Computational simulations are performed and compared with results predicted by a deterministic SIS model, a continuous-trajectory stochastic differential equation model, and Monte Carlo calculations. It is seen that the rate of spread predicted by the jump-diffusion model agrees well with that predicted by Monte Carlo calculations. Indeed, if there is a step increase in the transmission rates or a step decrease in the recovery rates, then the disease can spread globally from region to region at an exponential rate.