The analysis of laboratory fire experiments led to the development of a reaction-diffusion model for the spread of fire across a fuel bed in windless and slopeless conditions. A method for the determination of coefficients in this model based on the dynamic features of a spreading fire is given. The numerical study of the mathematical problem proposed allows us to predict the rate of spread, the fire front perimeter and the temperature distribution for line-ignition and point-ignition fires. These results are compared with success to experimental data. Furthermore, the model allows us to estimate the acceleration of spread for a point-ignition fire in its initial stage and in the steady-state phase.