Rational social distancing policy during epidemics with limited healthcare capacity
Fig 3
The rational behaviour in the presence of a healthcare threshold depends on the maximum cost of infection.
(A) The infection cost α(i) for a range of healthcare thresholds ihc, see Eq 6 with steepness σ = 300. The colours encode the position of the healthcare threshold ihc for the whole figure. For comparison, two scenarios where α(i) = α0 (grey line) and α(i) = α1 (black) are considered as well. The base infection cost is kept constant throughout, α0 = 100, whereas α1 is varied in the following panels. (B) Typical example for the equilibrium behaviour of population k(t) (B1) and the corresponding infectious cases i (B2) over time for low maximum infection cost α1. We compare the behaviour for a healthcare threshold, α0 = 100, α1 = 1.75α2 with ihc = 0.1 (yellow) to the behaviour for constant infection costs α(i) = 100 (grey) and α(i) = 1.75α0 (black) These cases are also marked in panel (C) by correspondingly coloured circles beneath the letters B. (C) The peak of the epidemic max(i) as a function of the maximum cost of being infectious α1 corresponding to the infection cost scenarios shown in (A). We also mark the data points corresponding to the examples shown in panels (B1–2) and (D1–2) with circles and corresponding labels above. (D) Typical example for the equilibrium behaviour of population k(t) (D1) and the corresponding infectious cases i (D2) over time for high maximum infection cost α1. We compare the behaviour for a healthcare threshold, α0 = 100, α1 = 4α2 with ihc = 0.1 (yellow) to the behaviour for constant infection costs α(i) = 100 (grey) and α(i) = 4α0 (black) These cases are also marked in panel (C) by correspondingly coloured circles beneath the letters D. For the same data as in panel (C): (E) Total number of cases after the epidemic has run its course. (F) Duration of the epidemic as defined by the time interval for which i > 10−4. (G) Total cost of the epidemic −U for equilibrium behaviour in units of the minimal infection cost α0. In the inset, the epidemic cost is shown in units of the maximum infection cost. Lines in (D-G) serve as guides to the eye.