Vaccination Strategies for Small Worlds

Abstract

Infectious diseases are a serious threat to our health. Vaccination often can prevent their spread, but typically it is not feasible to vaccinate absolutely everyone. Sometimes it is necessary to carefully target the group of individuals to whom a limited supply of vaccine should be administered in order to achieve the largest amount of overall protection for the whole population. A method for choosing the group to be targeted for maximal effect is called a vaccination strategy. The development of optimal vaccination strategies leads to interesting mathematical problems and requires some knowledge of the contact network of the given population. Here we focus on contact networks that are so-called small-world networks. Through a sequence of background information and preliminary exercises, we take the readers to the point where they can explore some open research problems that are formulated in the last section.

Appendix: Hints for Selected Exercises

Hint for Exercises 1 and 2

In Exercise 1, the large components will always be of the same size; in Exercise 2, their sizes will most likely be similar, but not exactly the same.

Hint for Exercise 3

If you followed instructions on batch processing from [13], choosing the output option Table output, the data you will need (total number of removed turtles) should be in the rightmost column of your output file. You may wish to sort your data in this column from lowest to highest. All hosts who are removed by the end of the simulation must be in the same connected component as the index case j ^∗. In the majority of simulations, these components should be large, as in Exercise 2.

Hint for Exercise 4

You can use the mean size of the components that you classified as “large” for your estimate of ϱ(1. 5).

Hint for Exercise 12(a)

Choose an N large enough so that you can test several values of d. Find a pattern, dependent on d, for the maximum degree of any node.

Hint for Exercise 15

The results of Milgram’s experiment do not tell us anything about the maximum or mean distance. They give a statistically significant estimate of some percentile of the distances, but not necessarily of the median.

Hint for Exercise 21

The mean degrees will a.a.s. approach 〈k〉 ≈ 2d + λ in G _SW ¹(N, d, λ) and 〈k〉 ≈ 4 + λ in G _SW ²(N ², 1, λ).

Hint for Exercise 23(a)

Partition the vertex set V of the small-world model into pairwise disjoint sets of consecutively numbered nodes V _i of roughly equal size. Then form a new graph G _I by making each of the sets V _i a vertex and drawing an edge {V _i , V _j } if, and only if, there is at least one v ^∗ ∈ V _i and at least one v ^∗∗ ∈ V _j such that {v ^∗, v ^∗∗} forms an edge in the original small-world network. For an illustration of this construction see the hint at the very end of Module [7].

It can be shown that (a slight modification of) this construction will give Erdős-Rényi random graphs G _I with sufficiently large mean degrees.

Hint for Exercise 24

• (a) You may still see a few outbreaks that look fairly major, which is due to random effects in a relatively small population.

• (b) Results will vary, but you may see results similar to the following:

  Table 14

Hint for Exercise 25

Note: Do NOT duplicate your experiment from Exercise 24, as this will change the parameter settings. Instead, create a New experiment. It is highly unlikely to see any major outbreaks. We got: mean prop. of secondary infections = 0.04; max prop. of secondary infections = 0.15; prop. of runs with no secondary infections = 0.31. Your results should be in the same ballpark.

Hint for Exercise 26

If you predicted you will still see major outbreaks, you would be correct. For the complete network, we got: mean prop. of secondary infections = 0.90; max prop. of secondary infections = 0.97; and prop. of runs with no secondary infections = 0.05. For the G _NN ¹(120, 2) network, we got: mean prop. of secondary infections = 0.65; max prop. of secondary infections = 0.99; and prop. of runs with no secondary infections = 0. Your results may be similar. After sorting the results from the lowest to the highest value of the output column, you will most likely observe a dramatic separation between major and minor outbreaks for the complete network, and a gradual increase without a distinctive gap between minor and major outbreaks for the G _NN ¹(120, 2) network.

Hint for Exercise 27

The optimal strategy would create barriers that the pathogens cannot cross by vaccinating evenly spaced groups of two adjacent hosts. One vector that implements this type of strategy is

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