Fig 1.
Convergence of FPSP policies and .
(Top) Time-behaviour of a 7-days FPSP policy with [X, Y] = [1, 6] with a single workday followed by six days lockdown. In the first 20 days (see the vertical orange dashed line) the virus invades a totally susceptible population generating a major epidemic with reproductive number . From the end of week 3 to the end of week 6 a major lockdown is enforced reducing the reproductive number to
. The FPSP is initiated in week 7 (see the vertical dashed light blue line). The epidemic has a trajectory that approximates that of a system with average
. (Bottom) Time-behaviours generated by five FPSPs of increasing frequency (corresponding to periods T ranging from 5 to 1 weeks) and the same duty-cycle DC = 1/7 = 2/14 = 3/21 = 4/28 = 5/35 ≃ 14.3%. In all cases, the epidemic behaviour seen in the first three weeks is suppressed and the outbreak dies out following closely the trajectory of an un-switched system with
(dashed blue line). As clearly show in the figure, smaller periods are associated with a higher vicinity to the average trajectory
(dashed blue line).
Fig 2.
The epidemic behaviour as a function of FPSP duty-cycle and period.
The total number of infected individuals plotted as a function of time for different FPSP policies. Colours distinguish duty-cycles. (Left) The success of the switching policy depends on whether or not the duty-cycle ensures , in which case the epidemic dies out. Smaller duty-cycles outperform larger ones in terms of virus growth, but they lead to longer lockdowns. (Right) Epidemic evolution for different periods, from T = 2weeks to T = 4weeks. Across the panels, the dynamics are qualitatively the same, but they show that shorter periods perform better. For comparison, the time-behaviour corresponding to a full lockdown is also reported (FPSP [0, 7]).
Fig 3.
Some instances of analysis of infection over time as a function of FPSP duty-cycle and associated modes of behaviours of the epidemic.
In all diagrams showing peak-values and peak-times, each point corresponds to a single policy and all policies have a period which is a multiple of seven days. Colours distinguish duty-cycles as indicated in the top-right panel. (Top-Left) Shows that infected peak-values increase with duty-cycle for fixed period lengths and, notably, with increased period length for fixed duty-cycles. (Top-right) shows that peak-times are small for policies attaining small and large peak values, while they are inversely related to the peak values for policies attaining middle peak values. (Bottom) highlights the time-behaviour of the epidemic for two choices of the period and four choices of the duty-cycle. These time-trajectories correspond to policies belonging to the two red rectangles shown at the bottom-left of the top-left panel.
Fig 4.
Full in depth analysis of infection over time as a function of FPSP duty-cycle and period and outer loop driven equilibria.
Analogously to the instances shown in Fig 3, in all diagrams showing peak-values and peak-times, each point corresponds to a single policy and all policies have a period which is a multiple of seven days. Markers/colours distinguish duty-cycles (e.g., blue diamonds in bottom diagrams denote policies with a duty-cycle between 20% and 25%) as seen in the top-right panel. (Top-Left) Shows that infected peak-values increase with duty-cycle for fixed period lengths and, notably, with increased period length for fixed duty-cycles. (Bottom-Left) shows that peak-times are small for policies attaining small and large peak values, while they are inversely related to the peak values for policies attaining middle peak values. Two distinct groups of policies, clustered on the basis of their peak-time behaviour, are highlighted in matching coloured regions in the (Top-Left) and (Bottom-Left) diagrams. (Bottom-right) highlights the duty-cycles to which the outer loop converges. Notably, these policies lie in the region of smallest peak-value (also, see Fig 5).
Fig 5.
Time behaviour of the outer supervisory control loop for different FPSP periods.
The periods range from 14 to 49 days. During non-lockdown, the epidemic evolves with parameters corresponding to a reproduction number . (Top) total amount of infected population. (Middle) Average reproductive number obtained with the duty-cycles shown in the lower panels. It can be noticed that, for each choice of the period T, the outer loop converges to a pair [X, Y] (whose exact values are highlighted with arrows of the corresponding colour in the two lower panels) that successfully suppresses the virus (see the convergence points in Fig 4).
Fig 6.
Illustrative scheme of the proposed mitigation strategy.
(Green) an example of an FPSP with period T = 7 days (one week) is shown on the left in which two consecutive periods with [X, Y] = [1, 6] corresponding to a duty-cycle DC = 1/7 ≃ 14.3% (green box on the left) and a subsequent period with a different FPSP policy [X, Y] = [2, 5] corresponding to a duty-cycle DC = 2/7 ≃ 28.5% are shown (dashed orange box). The application of this FPSP influences the actual dynamics of the epidemics as shown on the right (long left-right arrow on the top). (Orange) delayed and possibly uncertain empirical measurements are collected (green box on the right) and used by the adaptive outer supervisory controller to select the specific FPSP policy (bottom right to left orange arrow) to be used in the subsequent time-period (bottom to top vertical orange arrow pointing to the new FPSP policy in the dashed orange box).
Fig 7.
Percentage of peak infections parametrised by [X, Y] in a population of 107 individuals in the SIDARTHE model.
Fig 8.
Distribution of the peak times corresponding to Fig 7.
Fig 9.
Sensitivity analysis of the quantity I + D + A + R + T in the SIDARTHE model on quarantine effectiveness q.
(Top) The [2, 12] FPSP policy shows a good stabilising behaviour q ≤ 0.335, corresponding to approximately 66% reduction in infectious contacts during periodic quarantine days whereas a unsatisfactory behaviour is shown for higher values of q (see the trajectory for q = 0.375.) (Bottom). The outer loop action is added to the FPSP policy showing a stabilising behaviour also for q = 0.375.
Fig 10.
Sensitivity analysis of the quantity I + D + A + R + T in the SIDARTHE model on increased scaling factor (1+ d)β+ during periodic working days.
(Top) The [2, 12] open-loop FPSP policy remains stabilising for d ≤ 0.80, corresponding to a 80% increase in infectious contacts during periodic working days due to compensatory and anticipatory population behaviour but fails to stabilise for d = 1. (Bottom) The outer loop guarantees improved stabilising behaviour also in the case d = 1.
Fig 11.
Sensitivity analysis of the quantity I + D + A + R + T in the SIDARTHE model on uncertainty in epidemiological model parameters.
The y-axis has a linear scale for better visibility of differences across plots. Simulations in each row have fixed period length. Simulations in each column have fixed duty-cycle apart from the right-most column, which shows the effect of the outer loop. Parameters were sampled from zero-truncated Normal distributions with means σ1, …, σ16 and 10% standard deviation. β was estimated from samples of the basic reproduction number representing the consensus distribution in [39].
Fig 12.
Level sets as a function of the working days X obtained with an FPSP of period 7 days.
(Left) Changing in [1, 4.4]. (Centre) Changing quarantine effectiveness q in [0, 1]. (Right) Changing compensatory factor d in [0, 1].
Fig 13.
Level sets as a function of FPSP period T and with constant duty cycle DC = 2 / 7.
(Left) Changing in [1, 4.4]. (Centre) Changing quarantine effectiveness q in [0, 1]. (Right) Changing compensatory factor d in [0, 1].
Fig 14.
Threshold-based lockdown using delayed data.
(Left) Time-behaviour of the epidemic modelled by a SIR model under the action of a threshold-based lockdown executed based on data not affected by time-delays. (Right) Time-behaviour of the epidemic modelled by the same SIR model under the action of a threshold-based lockdown executed based on data now affected by a 2-weeks time-delay which cause the occurrence of a chain of out-of-control waves capable of infecting 3 − 5% of the population (of the order of 20 times the threshold).