1. Invasive tree pests are increasingly threatening forest ecosystems. Understanding and controlling their spread presents a major challenge for managers, researchers, and policy makers. In 1999, the pine wood nematode Bursaphelenchus xylophilus, causal agent of pine wilt disease and native to North America, was first detected in Europe, more precisely in south western Portugal. Since then, it has spread to more than 30% of the country, producing large-scale damages in Portuguese forests. 2. We applied a new spatio-temporal network-based model to predict, at yearly intervals, the natural spread of the pine wood nematode on the Iberian Peninsula, as dispersed by the longhorn beetle Monochamus galloprovincialis, its only known vector in Europe. We compared the model predictions with independent observations of the natural spread of pine wilt disease in Portugal between 2005 and 2015. We simulated the effect of potential containment measures on the disease spread. 3. The model predicted the status (infected or non-infected) of forest areas with 93% accuracy, had 93% sensitivity and specificity, and AUC=0.96 (averages for 8 years with validation data). 4. Simulated clear-cut belts stopped the spread only if they were wider than 30 km, although thinner belts delayed invasion. Furthermore, clear-cuts were much more effective in slowing down the invasion when reinforced, in the adjacent areas, by the reduction of the vector beetle population through mass trapping and by the early detection and removal of infected trees. 5. Model projections identified areas through which the nematode is most likely to invade, by natural means, the phytosanitary buffer zone established along the Portuguese border, and Spain. 6. Synthesis and applications. In the absence of effective containment measures, the pine wood nematode may naturally spread into Spain in about 5 years. In less than 10 years, it may reach the major forest and climatic corridors that provide a gateway for subsequent expansion towards the rest of the Iberian Peninsula and, in the longer term, towards other European countries. Our model can considerably support management efforts by forecasting when areas are at highest risk of vector-mediated invasion as well as the effects of specific control measures on the disease spread.