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We study the effectiveness of regional lockdown strategies to mitigate the spread of a pathogen across regional units, in the following called cities, within a country or region for a single infection wave. The heterogeneity in the epidemically relevant connectivity is defined via a random network model with cities as nodes, where the city's sizes determine their connectivity via a gravity type kernel function. Isolation of a whole city is initiated when infection numbers surpass defined thresholds. We consider two basic strategies for the lockdowns. Strategy~$(P)$ isolates cities based on a proportional threshold of infections, while stra\-tegy~\((U)\) uses a uniform infection threshold for all cities. Given the heavy-tailed distribution of city sizes, strategy \((P)\) can potentially result in more secondary infections from larger cities than strategy \((U)\). As an efficiency measure we use the ratio of individuals under lockdown and the number of infected individuals. Additionally, we analytically derive formulas for the basic reproduction numbers and prevalences. Our model is fitted to mobility data from France, Japan, and Poland, and validated through simulations. The findings indicate that while the model aligns well with data from France and Poland, it deviates in Japan, highlighting the importance of geographical nuances in pathogen spread modeling. Furthermore, it suggests that for France (and Japan) both strategies perform equally well, while for Poland strategy \((U)\) outperforms strategy \((P)\)., Comment: main text: 48 pages, supplementary material: 15p, 109 figures (including the supplementary material)