The mean and standard deviation of R across all 2,500 realizations were calculated as a function of cost of infection rinf, cost of vaccination rvac, vaccine efficacy ε, and incubation period ω for all three models. The standard deviation for these realizations is large since the neighbourhood size Q is approximately 10. However, the mean value of R varies within certain parameter regimes, as described below.
3.1 Simple Stochastic Model
In the simple stochastic model, all individuals make the same decision: all individuals either vaccinate or do not vaccinate at a given set of parameter values, because the payoff function and its constituent parameter values are the same for all contacts of the index case. Hence, the value of R can change suddenly as certain threshold parameter values are surpassed (). For instance, for low infection risk, none of the contacts of the index case vaccinate since PV < PN, and thus the average value of R is 2.3. However, for the cost of infection rinf > 0.25, PV > PN, all of the contacts vaccinate and the mean value of R becomes approximately 1 (). A similar effect appears in the plot of R versus rvac, cost of vaccination, () and ε, vaccine efficacy, (). The mean value of R is constant on either side of these thresholds for the plots of mean R versus rinf and rvac because these parameters influence vaccinating behaviour but not the probability that a susceptible or vaccinated person becomes infected. However, the mean value of R declines with increasing ε beyond the threshold in ε because beyond this threshold, all contacts vaccinate, and ring vaccination is more successful at higher vaccine efficacy.
Mean and +/− the standard deviation of values of R versus rinf, rvac, ε, ω with all other parameter values fixed at values in . “DSI” indicates the distributed stochastic model with imitation
Although a threshold is not observed in the plot of mean R versus the incubation period ω at the parameter values tested, the mean value of R increases as ω increases because contacts are exposed to infection for a longer period before symptoms appear in the index case, giving contacts the first opportunity to vaccinate ().
The deterministic predictions from (2)
in the case of no vaccination and (8)
in the case of vaccination agree with the mean values of the realizations of the simple stochastic case (results not shown).
3.2 Distributed Stochastic Model
In the distributed stochastic model, each individual is assigned a parameter value for: time to protective immunity λ, latent period σ, infectious period δ, incubation period ω, vaccine efficacy ε, cost of vaccination rvac, cost of infection rinf, and transmission probability p. The values are drawn from a lognormal distribution with the same mean value as in the simple stochastic model (see values in ). The distributed stochastic model is otherwise identical to the simple stochastic model. The resulting mean value of R is plotted against the mean parameter values for rinf, rvac, ε, and ω from the lognormal distribution (). The model predictions are qualitatively different from the simple stochastic model. Primarily, the thresholds in rvac, rinf, and ε appear to be “smeared out” relative to the simple stochastic model, because heterogeneity in the sampled parameter values means that the payoff functions for individuals are also variable. Therefore, in general there is no parameter value for which either PV > PN or PV < PN is true for all individuals. In general, for any given mean parameter value, PV > PN will hold for some individuals and PV < PN will hold for others. However, as the mean parameter values change, so does the mean behaviour: the mean value of R increases for increasing rvac and ω, because vaccination becomes less favourable as the perceived vaccine risk and the incubation period increase (). In contrast, the mean value of R decreases for increasing rinf and ε, because vaccination becomes more favourable as the disease risk and vaccine efficacy increase.
3.3 Distributed Stochastic Model with Imitation
In the imitation model, individuals consider both their own values of PV
as well as the inclination of other contacts (as measured by whether PV
) in making their decision about whether or not to vaccinate, as specified in (10)
. We use the stepwise functional form for g
) for our analyses, except where noted otherwise, because the impact of imitation is most clear with this functional form.
In the presence of imitation, the mean value of R (the average number of secondary infections) appears to be roughly the same as the mean value of R in the distributed stochastic model without imitation, for a broad range of parameters, including the lack of a threshold (). Moreover, the mean value of R does not change across a wide range of values of the imitation strength κ, under three different functional forms for the function g(V) (). We attribute this to the fact that imitation does not have a bias: individuals tend to imitate whichever strategy appears to be favoured.
Mean and +/− the standard deviation of values of R versus imitation strength κ for three different functional forms for g(V), with all of the other parameter values fixed at values in
However, if we examine how the values of R are distributed across the stochastic realizations, some interesting differences emerge. The distribution of R, and also of the number of individuals are who vaccinated, changes as the imitation strength κ increases (). For low values of κ, both distributions are unimodal and clustered around the same mean value as for the distributed stochastic model without imitation. However, as κ increases, the distribution of the number of vaccinated individuals becomes bimodal: for some parameter sets, vaccination is the favoured strategy in terms of what the payoff functions indicate for most individuals, and thus a strong majority of contacts opt for vaccination; for other parameter sets, non-vaccination is the favoured strategy and most contacts refuse vaccination (even those for whom the payoff to vaccinate exceeds the payoff not to vaccinate). This bimodal effect occurs at parameter values such that, on average, neither vaccination nor non-vaccination are favoured by a strong majority of contacts. (We note that the special case κ = 0 recovers the case of the distributed stochastic model without imitation, and simulations of the distributed stochastic model without imitation at the same parameter values fail to show bimodality (results not shown).)
The distribution of the number of secondary infections (left) and number of vaccinators (right) for the case of Q = 10 neighbours for different values of imitation strength κ and other parameter values as in
Interestingly, at the same parameter values where the number of contacts who vaccinate is bimodal, the distribution of secondary infections appears to remain relatively unimodal (). This is partly because imitation has larger impact on the first order effect of distribution of vaccinators than on the second order effect of the distribution of secondary infections. However, this is also because a relatively low number of contacts (Q = 10) does not provide sufficient resolution to distinguish two close peaks. Indeed, when the number of neighbours is increased to Q = 100 and parameter values are otherwise unchanged, clear bimodality in the distribution of secondary infections emerges as κ increases (, p = 0.05 results). Bimodality in the number of vaccinators remains dominant (in fact, with some appearance of trimodality) (, p = 0.05 results). The stronger unimodality in the distribution of the number of secondary infections compared to the distribution of the number of vaccinators also explains why the variance in the average number of secondary infections R is so similar for the distributed stochastic models with and without imitation (), despite the fact that the parameter ranges covered in include baseline parameter values where the number of vaccinators is known to be bimodal for the model with imitation.
The distribution of the number of secondary infections (left) and number of vaccinators (right) for the case of Q = 100 neighbours for different values of imitation strength κ and other parameter values as in
For all values of the imitation strength κ, the peaks in the distribution of secondary infections shift to higher values (i.e., more simulations with a large number of secondary infections) when the transmission p is increased from the baseline value p = 0.05 to a higher rate p = 0.2 (). This effect is not surprising because a higher transmission rate implies a greater number of secondary transmissions, even when vaccination is taken up and provides some reduction in secondary cases. However, what is more interesting is that the relative magnitude of the two peaks in the distribution of number vaccinated changes as p is increased: when p = 0.05 most individuals do not vaccinate, whereas when p = 0.2, most of them do (i.e., the relative size of the two peaks in the distribution of vaccinators is switched in the case for p = 0.05 compared to p = 0.2). An increase in p increases the probability of eventually becoming infected and thus experiencing disease penalties, hence vaccination becomes attractive for higher p, at least at these parameter values. This switch is again observed in the distribution of the number of secondary infections: when p = 0.05, the peak corresponding to more secondary infections (i.e., less vaccination) is larger, indicating that in most realizations, the majority of contacts do not vaccinate and the number of secondary infections increases. By comparison when p = 0.2, the peak corresponding to fewer secondary infections (i.e., more vaccination) is larger, indicating that in most realizations, the majority of contacts vaccinate.
As noted above, we used a step function to represent our imitation function g(V) in the case of distributed stochasticity with imitation (–). However, we also explored these results for a hyperbolic tangent function (results not shown) and found that instead of obtaining a bimodal distribution, we obtained a distribution that resembled a skewed normal distribution. This effect occurs because for most parameter values, the switch between favouring vaccination versus favouring non-vaccination is much sharper at the origin for the step function than for tanh under most parameter choices.
These results imply a fully-connected network where each contact of the index case is connected to—and exchanges information with—every other contact of the index case. To understand the impact of this assumption, we also explored the semi-connected case where individuals can only imitate the nearest plus or minus n neighbours in the ring. The introduction of semi-connectedness can change the results significantly for certain values of connectedness. For the special case of no connectedness (n = 0), the case of the distribution stochastic model without imitation is recovered and distributions are unimodal (results not shown). For n = 3, there was likewise little impact and the distributions remained unimodal (results not shown). For cases of intermediate connectedness (n = 12 and n = 25), the results change significantly (). The distribution of the number of vaccinators is no longer bimodal but becomes a highly skewed unimodal distribution with a high variance. The variance increases as the strength of imitation κ increases. This spreading effect occurs because in the semi-connected case, individuals are sampling a small proportion of the total number of contacts of the index case and, therefore, the average attractiveness of vaccinating versus not vaccinating is more highly variable than in the fully connected case, giving rise to greater variation in the level of vaccine adherence overall. Although the bimodality disappears, the greater variance still supports the conclusion that adding imitation can increase the variability in the predicted vaccine adherence, relative to the case of the distribution stochastic model without imitation. For the case n = 50, the fully-connected case is recovered, including bimodality (results not shown).
Fig. 6 The distribution of the number of secondary infections (left) and number of vaccinators (right) for the case of Q = 100 neighbours, who are imitating n neighbours to the right and n neighbours to the left, for different values of imitation strength κ (more ...)
The emergence of bimodality occurs for parameter sets such that the payoff to vaccinate is close to the payoff not to vaccinate. In such situations, individual variability in model parameters means that for some stochastic realizations, the payoff to vaccinate will be higher for the majority of contacts and hence vaccination tends to dominate. For other stochastic realizations, however, the payoff not to vaccinate will be higher and hence non-vaccination will dominate for the same mean parameter values. Moving away from this parameter regime sufficiently far means that either vaccination or non-vaccination will be favoured for all stochastic realizations, and imitation will only strengthen this tendency. This should cause a unimodal distribution of the number of secondary infections and the number of vaccinators.
This effect is seen in the distribution of secondary infections () and vaccinators () for a range of possible values for eight of the model parameters: λ, ε, ω, rvac
, δ, σ, p
. For most parameters, moving away from the baseline values collapses, the bimodal distribution function into a unimodal function that represents either dominant vaccination or dominant non-vaccination, depending on whether there has been an increase or a decrease relative to the baseline parameter value ( and ). For instance, increasing the cost of vaccination rvac
above the baseline value makes vaccination unattractive, collapsing the bimodal distribution to a unimodal distribution that represents dominant non-vaccination behaviour. In contrast, decreasing rvac
below the baseline value creates a unimodal distribution representing dominant vaccinating behaviour. However, for the transmission probability p
, the distribution remains bimodal across a broad range of parameter values before collapsing to a unimodal distribution. This is because p
appears in both the payoff to vaccinate and the payoff not to vaccinate ((5)
). Increasing p
decreases both payoffs because the individual is more likely to get infected for higher values of p
, thus the relative size of PV
does not change as much. Therefore, the distribution of secondary infections and vaccinators remains bimodal for a range of values of p
Fig. 7 The distribution of the infected individuals under full imitation for the case of Q = 100 for the parameters λ, ε, ω, rvac, rinf, p, σ, δ. The baseline parameter values are used to generate the gray distributions (more ...)
Fig. 8 The distribution of the vaccinated individuals under full imitation for the case of Q = 100 for the parameters λ, ε, ω, rvac, rinf, p, σ, δ. The baseline parameter values are used to generate the gray distributions (more ...)