Fig. 6.

Target state estimation in epidemics. (A) US air transportation network, where vertices represent cities and edges represent the direct flights. The target (red) and sensor (blue) cities are highlighted. (B) Network flow between three cities (Upper) and the corresponding inference graph (Lower) of the dynamical system (Eq. 9). The dynamics are taken into account by expanding each vertex i as a set of SIRD state nodes, where links represent the linear (solid lines) and nonlinear (dashed lines) interactions between the state variables in the differential equations. (C) Box plot of the error between the time $t_{\text{p}}$ of the epidemic peak in each target city and the predicted peak time ${\widehat{t}}_{\text{p}}$. The red bars show predictions from free-run simulations, while the blue bars show predictions given by the estimates of the designed functional observer, both for 100 independent realizations with arbitrary initial conditions. The bottom, middle, and top of each box represent the 25th, 50th (median), and 75th percentiles of the sample, respectively; the whiskers mark the 5th and 95th percentiles. For illustration purposes, this example assumed an outbreak initiated in Miami, Florida, and its spreading dominated by domestic air transportation. Materials and Methods has simulation and modeling details.