Inter-city infections and the role of size heterogeneity in containment strategies

This study examines the effectiveness of regional lockdown strategies in mitigating pathogen spread across regional units, termed cities hereinafter. We develop simplified models to analyze infection spread across cities within a country during an epidemic wave. Isolation of a city is initiated when infection numbers within the city surpass defined thresholds. We compare two strategies: strategy $\left(P\right)$ consists in prescribing thresholds proportionally to city sizes, while the same threshold is used for all cities under strategy $\left(U\right)$. Given the heavy-tailed distribution of city sizes, strategy $\left(P\right)$ may result in more secondary infections from larger cities than strategy $\left(U\right)$.

Random graph models are constructed to represent infection spread as a percolation process. In particular, we consider a model in which mobility between cities only depends on city sizes. We assess the relative efficiency of the two strategies by comparing the ratios of the number of individuals under isolation to the total number of infections by the end of the epidemic wave under strategy $\left(P\right)$ and $\left(U\right)$. Additionally, we derive analytical formulas for disease prevalence and basic reproduction numbers.

Our models are calibrated using mobility data from France, Poland and Japan, validated through simulation. The findings indicate that mobility between cities in France and Poland is mainly determined by city sizes. However, a poor fit was observed with Japanese data, highlighting the importance to include other factors like e.g. geography for some countries in modeling. Our analysis suggest similar effectiveness for both strategies in France and Japan, while strategy $\left(U\right)$ demonstrates distinct merits in Poland.