Fig. 4: The effect of initial conditions and of containment measures on subthreshold epidemics in Italy.

The SEPIAR model has been numerically integrated for a timespan τ = 90 days starting from different initial conditions, while assuming that spatially homogeneous containment measures are in place from the beginning of the epidemic. a Total number of infected individuals in the community, evaluated as \(Z(t)=\mathop{\sum }\nolimits_{i = 1}^{n}{E}_{i}(t)+{P}_{i}(t)+{I}_{i}(t)+{A}_{i}(t)\), for outbreaks started by seeding one by one each of the 107 Italian provinces (solid lines: across-province median; shadings: min-max envelope) with an initial number of exposed individuals E_i(0) = 100, assuming an otherwise fully susceptible population (S_i(0) = N_i − E_i(0)). b Provinces in the map are color-coded according to the cumulated number of cases over the whole national territory up to the end of the simulation timespan for a subthreshold epidemic seeded in the considered province and for the intermediate-control scenario (yellow) of (a). c Same as (a) for a simulated outbreak obtained by seeding the provinces where the ten busiest Italian airports are located (“Methods”). A total number of 100 exposed individuals has been allocated proportionally to the total passenger flux at the beginning of the simulation. d Map of the projected infections in each province up to the end of the simulation period for the intermediate-control scenario (yellow) of (c). Parameters as in Table 2, with ϵ_i = ϵ (numerical values are given in c), ξ_ij = 0.5, and \({\chi }_{i}^{X}=0.1\) days⁻¹ for all i’s, j’s, and X ∈ {E, P, I, A}. For the three combinations of the control parameters shown in a and c, we find \({{\mathcal{R}}}_{{{c}}}\approx 0.99\), 0.90, and 0.67 for ϵ = 0.26, 0.33, and 0.50, respectively, with e_c (evaluated for w^X = 1, X ∈ {E, P, I, A}) ≈ 0.13, 0.099, and 0.017 day⁻¹.