Fig 1.
Layout of simple system used for testing.
a) The thickness of the connecting line indicates the mean force of infection between the two subpopulations. b) Progress of an epidemic without control (infected hosts against time for each subpopulation) simulated 100 times.
Table 1.
Epidemic, simulation and control parameterisations for the 2x2 and 4x4 systems. Co = Controllable, Ch = Challenging, Int = Intermediate. Units are all abstract but included to clarify dimensionality (t = time). *The number of hosts in each subpopulation was randomly generated so expressing the parameterisation of costs as shown in the optimisation description is not meaningful. Translating between the notations:
Fig 2.
OC controller performance for the controllable 2x2 system.
Violin plots of a) rewards and b) eradication times. Counterintuitively, the more constrained proportional control outperforms the less constrained absolute control. The height of the plots is scaled separately for each of the controllers. The central black box in each violin shows the data quartiles and the whiskers show the farthest datapoint within 1 . 5 × IQR of the upper and lower quartiles. 1000 simulations were run for all experiments on the eradication time plot but the number of samples included in the plot (N) is shown separately for each of the controls because the disease was not eradicated in all simulations.
Fig 3.
Comparing the state predictions of the absolute optimal control variants for the controllable system with the results in the equivalent stochastic system.
a) State predictions for the controllable system. b) Results from the stochastic system. (c) and (d) show the same comparison but focusing on the first two control steps. For the absolute case, the optimal control predicts that the disease will be eradicated and no further effort is required but the stochastic plots demonstrate that this is not reliable in the more realistic model and the disease persists to the end of the simulation in a large number of trajectories. Looking at the state predictions of the proportional optimal control variant (e), it is unable to expend enough resource to eradicate the disease in the first control step and so adopts a more conservative strategy with a portion of the initial resource expended to thin the population (shown in purple) to reduce the longer term impact of the epidemic. Note that the y-axis scales are different for the 5 subplots.
Fig 4.
Controller performance for the controllable 2x2 system.
Violin plot of rewards (a) and eradication times (b). The comparison shows priority list and MPC controllers alongside equal per host control and no control references. The equivalent OC controller is also shown in grey for comparison. The priority list controllers outperform MPC – even when it is using the shortest control horizon – but it does not appear any of the priority list controllers are different to the others. The height of the plots is scaled separately for each of the controllers. Inner box and whisker plot definitions are as per Fig 2. 100 simulations were run for all experiments but the number of samples included in the plot (N) is shown separately for each of the controls on the eradication time plot because the disease was not eradicated in all simulations. (c) shows the number of infected hosts and the culling control summed across all subpopulations for an example simulation for MPC absolute horizon 1. The “removed” values shown are not cumulative – they are the number of hosts that died of the disease or were culled within that timestep. The “expected removed” line is an estimation of how many hosts would have been expected to be removed by the control correcting for hosts dying naturally (adding the average of γI across the control step to the specified culling rate) and times during the control period when there were no hosts to cull (for each subpopulation, subtracting the control rate multiplied by the duration when I was zero in that control step). The optimal control removes all infected hosts in the system for each timestep but no more. (d) is an equivalent plot for the prioritise I controller showing the controller overspending to remove the infection more reliably.
Fig 5.
Results summary for the 2x2 challenging system. Equivalent to Fig 4.
When eradication is not feasible, MPC and optimal control both outperform the priority based controllers (although by a smaller margin). Again, the way in which the subpopulations are prioritised for the priority based controllers appears to be unimportant. The example disease progress curves for the overall system show that both optimal control and priority based controls are using all the available control resource until close to the end of the outbreak. The violin plot of rewards (a) is shown excluding the small number of high scoring cases (reward = 8,000 to 10,000). Each row is a summary of 100 simulations and N is the number of datapoints displayed within the range of the plot.
Fig 6.
Performance results for a 4x4 grid.
Sample size of 100 simulations per row, N is the number of results within the range of the plot. For the controllable system (a and b), the results are equivalent to the small system with eradications from the priority based methods allowing them to significantly outperform the MPC control. For the intermediate system (c and d), the priority based methods are able to eradicate the disease within the first few control steps in a significant fraction of the simulations but the optimal control is always removing the final infected hosts after the epidemic peak. For the challenging system (e and f), the MPC control has done better than any of the priority based methods. However, in this case, the overall benefit of any of the optimised methods is relatively small compared to the equal per host baseline.