  Human mobility contributes to the fast spatio-temporal propagation of
infectious diseases. During an outbreak, monitoring the infection situation on
either side of an international border is very crucial as there is always a
higher risk of disease importation associated with cross-border migration.
Mechanistic models are effective tools to investigate the consequences of
cross-border mobility on disease dynamics and help in designing effective
control strategies. However, in practice, due to the unavailability of
cross-border mobility data, it becomes difficult to propose reliable,
model-based strategies. In this study, we propose a method for estimating
cross-border mobility flux between any pair of regions that share an
international border from the observed difference in the timing of the
infection peak in each region. Assuming the underlying disease dynamics is
governed by a Susceptible-Infected-Recovered (SIR) model, we employ stochastic
simulations to obtain the maximum likelihood cross-border mobility estimate for
any pair of regions where the difference in peak time can be measured. We then
investigate how the estimate of cross-border mobility flux varies depending on
the disease transmission rate, which is a key epidemiological parameter. We
further show that the uncertainty in mobility flux estimates decreases for
higher disease transmission rates and larger observed differences in peak
timing. Finally, as a case study, we apply the method to some selected regions
along the Poland-Germany border which are directly connected through multiple
modes of transportation and quantify the cross-border fluxes from the COVID-19
cases data during the period $20^{\rm th}$ February $2021$ to $20^{\rm th}$
June $2021$.

Populations and Evolution;Physics and Society