5.5 Numerical investigations
5.5.2 Effect of global contacts on the critical vaccination cover-
In this section we investigate the effect of global contacts on the uniform, ac- quaintance and optimal vaccination strategies. To do this we fix the degree distribution and match the network and global models without any vaccination (i.e. c = 0) by either R0 (as in Chapter 4) or by the final size of a major outbreak.
We can then compare the critical vaccination coverages of the three vaccination strategies in the different the models.
We consider the extreme cases of acquaintance vaccination with either pS = 1
or pC = 1 since, as we discuss in Section 5.1.4, for a fixed vaccination coverage
RA0 is strictly increasing in pC. However, the difference in critical vaccination
coverage between these two acquaintance vaccination strategies is small, as illustrated in Figures 5.8, 5.9 and 5.10.
The addition of global contacts (while fixing either R0 or the final size of
a major outbreak) decreases the difference in critical vaccination coverage of the optimal, acquaintance and uniform vaccination strategies, as illustrated in Figures 5.8, 5.9 and 5.10. This is as we would intuitively expect; the effect of vaccination strategies targeting individuals with large degrees will be diluted when the epidemic depends less upon the network structure. Furthermore, a larger decrease in the difference between critical vaccination coverages is ob- served when the network and global model has a heavy-tailed degree distribution. We also note that with the addition of global contacts the critical vaccination coverage of the acquaintance vaccination strategy often becomes closer to the critical vaccination coverage of the optimal vaccination strategy rather than the critical vaccination coverage of the uniform vaccination strategy, especially when considering heavy-tailed degree distributions, as illustrated in Figure 5.11.
Consider a network and global model with a given fixed degree distribution, infectious period and R0 and assume that λG = 0. With the knowledge that
the critical vaccination coverage in the network and global model depends only on R0 we expect that, as λN is decreased and λG is increased to fix R0, the
critical vaccination coverage under the optimal and acquaintance vaccination will converge from below to 1 − 1/R0. This effect can be observed in Figures 5.8
(a) λG= 0 and pN = 0.41.
(b) λG= 2 and pN = 0.33.
(c) λG= 4 and pN = 0.23.
Figure 5.8: The effect of global contacts on vaccination strategies on network and global models matched with R0 = 6.5. The other parameters are I ∼ Const(1)
(a) λG= 0 and pN = 0.9.
(b) λG= 3.2 and pN = 0.5.
(c) λG= 6.4 and pN = 0.1.
Figure 5.9: The effect of global contacts on vaccination strategies on network and global models matched with R0 = 7.2. The other parameters are I ∼ Const(1)
(a) λG= 0 and pN = 0.9, so R0 = 14.
(b) λG= 0.57 and pN = 0.5, so R0 = 8.3.
(c) λG= 1.75 and pN = 0.1, so R0 = 2.9.
Figure 5.10: The effect of global contacts on vaccination strategies on network and global models matched by final size, with parameters I ∼ Const(1) and
(a) λG= 0 and pN = 0.9, so R0 = 78.
(b) λG= 0.6525 and pN = 0.5, so R0 = 44.
(c) λG= 1.75 and pN = 0.1, so R0 = 9.
Figure 5.11: The effect of global contacts on vaccination strategies on network and global models matched by final size, with parameters I ∼ Const(1) and
cination. Thus (assuming there is a sizeable variance in the degree distribution so that the acquaintance vaccination strategy has a smaller critical vaccination coverage than the uniform vaccination strategy) implementing an acquaintance vaccination strategy will lead to an underestimation of the required critical vacci- nation coverage if we have accurately estimated R0 but incorrectly assumed the
epidemic spreads primarily via the network, and instead the epidemic spreads via global contacts.
In contrast, if models are matched by the final size of a major outbreak the effect of global contacts on the critical vaccination coverage is less clear, and we discuss some possible interactions in the following paragraphs. Similarly to Chapter 4, in which we consider the difference between the final size of a major outbreak in two network and global models with the same degree distribution matched by R0, when we match two network and global models with the same
degree distribution by the final size of a major outbreak they are likely to have different basic reproduction numbers, unless σ2D = µD.
Since the critical vaccination coverage of the uniform vaccination strategy depends only on R0, the effect of global contacts on the critical vaccination
coverage under the uniform vaccination strategy when the models are matched by the final size of a major outbreak is clear: if R0 increases then the critical
vaccination coverage of the uniform vaccination strategy will increase, and if R0
decreases then the critical vaccination coverage will decrease, as illustrated in Figure 5.10.
Consideration of the optimal and acquaintance vaccination strategies is more complex. We expect both these strategies to be most effective when there is sizeable variance in the degree distribution. However, if there is a sizeable variance in the degree distribution then R0 is generally increasing as the λG is
decreased while pN is increased to maintain the final size of a major outbreak.
This creates a tradeoff, in which the model may require a larger critical vaccina- tion coverage under the optimal and acquaintance vaccination strategies due to the increased R0, or a smaller critical vaccination coverage owing to the optimal
and acquaintance vaccination strategies being more effective in the network due to the large variability in the degree distribution.
by the final size of a major outbreak, the critical vaccination coverage under the optimal or acquaintance vaccination strategy in the model with small λG
can be an overestimate or an underestimate of the critical vaccination coverage of the vaccination strategy when λG is increased, as illustrated in Figures 5.10
and 5.9. Furthermore, it is possible that the critical vaccination coverage of the optimal or acquaintance vaccination strategy on the standard network model is an overestimate of the critical vaccination coverage of the strategy when a small number of global contacts are introduced (compare Figures 5.11a and 5.11b for the optimal vaccination strategy) but also an underestimate of the critical vaccination coverage of the vaccination strategy when a large number of global contacts are introduced (compare Figures 5.11b and 5.11c for the optimal vaccination strategy).
Finally, recall that the final size of a major outbreak in the network and global model with a Poisson degree distribution depends on R0 but not the
specific value of λG and pN (see Proposition 4.1). However the specific value
of pN and λG does have an impact on the critical vaccination coverages of the
acquaintance and optimal vaccination strategies, as illustrated in Figure 5.9.
5.6
Concluding remarks
In this chapter we consider three vaccination strategies on the network and global model, specifically the acquaintance, uniform and optimal vaccination strategies. For each vaccination strategy, we show how to find a threshold parameter determining whether a major outbreak can occur and the final size of a major outbreak.
Under the acquaintance vaccination strategy we prove that, for a fixed vacci- nation coverage, maximising pC will maximise R0A and the final size of a major
outbreak. Under the uniform vaccination strategy, we show that the critical vaccination coverage of the network and global model is equal to 1 − 1/R0, where
R0 is the basic reproduction number of the network and global model under no
vaccination. We prove that the critical vaccination coverage of the acquaintance vaccination strategy is not always smaller than the critical vaccination coverage of the uniform vaccination strategy, and we give conditions under which the acquaintance vaccination strategy has a larger critical vaccination coverage than the uniform vaccination strategy.
We compare our asymptotic calculations to simulations of the epidemic in finite populations. Our results are as expected under the optimal and uniform vaccination strategies, with the asymptotic calculation of the final size of a major outbreak being an overestimation of the simulated final size of a major outbreak in finite populations. However, this relationship is reversed under the acquaintance vaccination strategy, with the asymptotic calculation of the final size of a major outbreak being an underestimation of the simulated final size of a major outbreak in finite populations. Further analysis suggests that the underestimation is caused by the correlation between the vaccination coverage under the acquaintance vaccination strategy and both the indicator function for whether a major outbreak occurs and the final size of a major outbreak.
Finally we investigate the effect of global contacts on the critical vaccination coverage of the optimal, acquaintance and uniform vaccination strategies by comparing models with either R0 or the final size of a major outbreak kept fixed.
We show that the addition of global contacts will lead to a decreased difference in critical vaccination coverage between the vaccination strategies, thus diluting the benefit of the acquaintance vaccination strategy. Furthermore, we show that the critical vaccination coverage in the standard network model under the optimal and acquaintance vaccination strategies matched by R0 or the final
size of a major outbreak can be either an underestimate or overestimate of the critical vaccination coverage when global contacts are added.
Throughout this chapter we assume that we vaccinate individuals with a perfect vaccine. However, this is an unrealistic assumption in many practical applications. Some vaccines never result in full immunity, only reducing the probability of infection, and sometimes vaccinated individuals will not become immune at all. However, the perfect vaccine assumption allows us to analyse the network and global model under the three vaccination strategies with two-type branching processes whereas under the generalised vaccine reaction model we would require the use of 12-type branching processes, which are more difficult to analyse. An extension of this model to the generalised vaccine action model would be an interesting topic for further research.
Although we do not consider the single-neighbour acquaintance vaccination strategy, we conjecture that global contacts would cause a similar dilution to the
effects found for the acquaintance vaccination strategy, and thus the acquain- tance vaccination strategy and the single-neighbour acquaintance vaccination strategy would provide similar results, as in Ball and Sirl (2013).
Finally, we acknowledge that the acquaintance vaccination strategy pre- sented here would be very difficult, and indeed potentially morally questionable, to implement in practice in human populations. However, we note that the acquaintance vaccination strategy could be very effective in static computer networks, in which moral questions do not arise. Nevertheless, the study of acquaintance vaccination is still very useful to understand the impact of vaccina- tion strategies that target high degree individuals. Furthermore, understanding exactly why the acquaintance vaccination strategy can underperform compared to the uniform vaccination strategy and why the asymptotic final size of a major outbreak in the network and global model under the acquaintance vaccination strategy is an underestimate for finite populations may result in the construction of better vaccination strategies that can be applied to human populations.