Table 1.
Parameters used in this work to characterise the features of the infectious disease and individual preferences.
Fig 1.
Representative probability distributions of vaccination arrival time.
A) The distributions studied in this work, see Eq (32) with expected vaccination time 〈tv〉 = (n + 1)τ for n = 0, 1, 10, 40, for arbitrary τ. The higher the n, the more strongly peaked the distribution, with n = 0 yielding an exponentially decaying function with its peak at t = 0. B) The corresponding cumulative distribution functions, as defined by Eq (17).
Fig 2.
Self-organised social distancing is stronger the sooner the vaccination is expected.
A) Nash equilibrium behaviour k for a range of sharp vaccination times tv, as given in the legend of panel C). The behaviour is insensitive to tv for large values and indistinguishable from the behaviour if the vaccination is not expected to occur. When tv is comparable to or shorter than the duration of the epidemic without vaccination, equilibrium behaviour exhibits strong social distancing. The corresponding courses of the epidemic are shown in panel B for the susceptibles and C for the infected. D-F) Nash equilibrium behaviour k and course of the epidemic assuming that the vaccination arrival time is exponentially distributed, p = p0(t), see Eq (32), with an expected vaccination time 〈tv〉, see panel F for legend. The behaviour and corresponding course of epidemic arising from the assumption that vaccination will not occur are shown as gray solid lines. This is calculated from the case for precisely known vaccination time tv but with tv → ∞. G-I) Nash equilibrium behaviour k and course of the epidemic assuming that the vaccination arrival time is distributed according to p1, see Eq (32). See panel I for legend. Other parameters: infection cost α = 400 and no economic discounting, τecon → ∞.
Fig 3.
The number of expected vaccinations decreases with the expected vaccination time, whereas the peak of infections increases.
A) We show the vaccinated fraction of the population s(tv) for the equilibrium solution if the vaccination is known to occur at a precise tv in grey. The later the vaccination time, the smaller s(tv) becomes. For comparison, we are showing the expected number of vaccinations, given by the expectation value of the susceptible compartment, 〈s(tv)〉 as a function of the expected vaccination time 〈tv〉 for the a range of vaccination arrival distributions pn. These data follow the same qualitative trend as for certain vaccination timing, but the sharper the probability distribution (the higher the n), the more closely the result approaches the case of certain vaccination at t = tv. B) We show 〈s(tv)〉 as a function of 〈tv〉 again, but as a heat map for a greater range of probability distributions pn. Brighter colours indicate higher 〈s(tv)〉. Note the nonlinear increase of n on the y-axis. C) Relatedly, we show the peak of infections maxt(i(t)) as a heat map as function of the expected vaccination time 〈tv〉 for the same range of vaccine arrival distributions pn as in B). Brighter colours indicate higher maxt(i(t)). The sharper the probability distribution pn (the higher the n), the sharper is the increase in the infection peak maxt(i(t)) with expected vaccination time. Particularly high infection peaks are expected when the vaccination is certain to arrive outside of the expected duration of the epidemic, i.e. for large 〈tv〉 and large n.
Fig 4.
Social distancing is enhanced by higher certainty about vaccine arrival during the epidemic.
A) Nash equilibrium behaviour k for a range of vaccination time distributions pn, as given in the legend of panel C, all with expected vaccination time 〈tv〉 = 20. For reference, we show the equilibrium behaviour if vaccination is not expected to occur (dashed line labelled “no vax”). Even though the expected vaccination time is similar, and the expected no. of vaccinations is (almost) the same, see Fig 3, we observe that significant social distancing occurs for longer durations with increasing sharpness n of the vaccination distribution. The corresponding courses of the epidemic are shown in panel B for the susceptibles and C for the infectious (the “no vax” case peaks at max(i) ≈ 0.1).