The importance of contact network topology for the success of vaccination strategies

Abstract

The effects of a number of vaccination strategies on the spread of an SIR type disease are numerically investigated for several common network topologies including random, scale-free, small world, and meta-random networks. These strategies, namely, prioritized, random, follow links and contact tracing, are compared across networks using extensive simulations with disease parameters relevant for viruses such as pandemic influenza H1N1/09. Two scenarios for a network SIR model are considered. First, a model with a given transmission rate is studied. Second, a model with a given initial growth rate is considered, because the initial growth rate is commonly used to impute the transmission rate from incidence curves and to predict the course of an epidemic. Since a vaccine may not be readily available for a new virus, the case of a delay in the start of vaccination is also considered in addition to the case of no delay. It is found that network topology can have a larger impact on the spread of the disease than the choice of vaccination strategy. Simulations also show that the network structure has a large effect on both the course of an epidemic and the determination of the transmission rate from the initial growth rate. The effect of delay in the vaccination start time varies tremendously with network topology. Results show that, without the knowledge of network topology, predictions on the peak and the final size of an epidemic cannot be made solely based on the initial exponential growth rate or transmission rate. This demonstrates the importance of understanding the topology of realistic contact networks when evaluating vaccination strategies.

Fig. 1

Daily incidences (new infections) for the network topologies (see Section 2). On all networks, the average degree is 5, the population size is 200,000, the transmission rate is 0.06, the recovery rate is 0.2, and the initial number of infectious individuals is set to 100. Both graphs represent the same data but the left graph has a semi-log scale (highlighting the growth phase) while the right graph has a linear scale (highlighting the peak).

Fig. 2

Comparison of the incidence curves (as shown in (b)) on networks with identical disease parameters and degree distribution (as shown in (a)). The network topologies are the random, meta-random, and near neighbor networks. See Appendix B for details of the constructions of these networks.

Fig. 3

The effects of the vaccination strategies summarized in Table 3 for the network topologies in Table 1 given a fixed transmission rate $\beta$. There is no delay in the vaccination and parameters are equal to those used in Fig. 1.

Fig. 4

The results for Fig. 3 if the vaccination start is delayed by 40 days.

Fig. 5

Upper: the incidence curves with no vaccination on various networks, showing the epidemic peak time for comparison with the delay time. Lower: the final size reduction as a function of delay in random vaccination on various networks.

Fig. 6

Daily incidences for the network topologies in Table 1 without vaccination for the case where the initial growth rate is given. The transmission rates and initial number of infections for the various network topologies are given in Table 7, while the remaining parameters are the same as in Fig. 1. All the three sub-figures are based on the same data but show different representations for clarity, where the left-hand figures are in the log-linear scale (to emphasize the exponential growth) and the right-hand figure is in the linear scale (to emphasize the peak).

Fig. 7

The effects of the vaccination strategies for different topologies when the initial growth rate is given. The transmission rates $\beta$ are as indicated in Table 7, while the remaining parameters are identical to those in Fig. 6.

Fig. 8

As for Fig. 7 if the vaccination start is delayed by 40 days.