  The spatial distributions of diverse facilities are often understood in terms
of the optimization of the commute distance or the economic profit.
Incorporating more general objective functions into such optimization framework
may be useful, helping the policy decisions to meet various social and economic
demands. As an example, we consider how hospitals should be distributed to
minimize the total fatalities of tuberculosis (TB). The empirical data of Korea
shows that the fatality rate of TB in a district decreases with the areal
density of hospitals, implying their correlation and the possibility of
reducing the nationwide fatalities by adjusting the hospital distribution
across districts. Approximating the fatality rate by the probability of a
patient not to visit a hospital in her/his residential district for the
duration period of TB and evaluating the latter probability in the random-walk
framework, we obtain the fatality rate as an exponential function of the
hospital density with a characteristic constant related to each district's
effective lattice constant estimable empirically. This leads us to the optimal
hospital distribution which finds the hospital density in a district to be a
logarithmic function of the rescaled patient density. The total fatalities is
reduced by 13\% with this optimum. The current hospital density deviates from
the optimized one in different manners from district to district, which is
analyzed in the proposed model framework. The assumptions and limitations of
our study are also discussed.
