Erratic Flu Vaccination Emerges from Short-Sighted Behavior in Contact Networks

The effectiveness of seasonal influenza vaccination programs depends on individual-level compliance. Perceptions about risks associated with infection and vaccination can strongly influence vaccination decisions and thus the ultimate course of an epidemic. Here we investigate the interplay between contact patterns, influenza-related behavior, and disease dynamics by incorporating game theory into network models. When individuals make decisions based on past epidemics, we find that individuals with many contacts vaccinate, whereas individuals with few contacts do not. However, the threshold number of contacts above which to vaccinate is highly dependent on the overall network structure of the population and has the potential to oscillate more wildly than has been observed empirically. When we increase the number of prior seasons that individuals recall when making vaccination decisions, behavior and thus disease dynamics become less variable. For some networks, we also find that higher flu transmission rates may, counterintuitively, lead to lower (vaccine-mediated) disease prevalence. Our work demonstrates that rich and complex dynamics can result from the interaction between infectious diseases, human contact patterns, and behavior.


contact of degree k does not become infected is
Combining all these terms, we find that ζ must satisfy the infinite-order polynomial equation Alternatively, using generating function methods, one can derive ζ as the probability that a node at the end of a random edge in a Bethe lattice with given degree distribution and connectivity is a member of a finite cluster.
2 Proof that there are no p-cycles for p > 2 when s = 0 Theorem 1. If decisions are only made based on previous season (s = 0), it is impossible to have an oscillation that is a p-cycle for any p > 2, regardless of the transmissibility T .
Proof. Let A be the Nash Equilibrium. Now suppose we have x 1 , x 2 be the first two ζ's in a cycle. Suppose x 1 < A. We note that this means x 2 > A since the response of x 1 will have more vaccinators than A since the lower the initial ζ is, the greater the response ζ must be.
As a result, subsequent x's must be going ever-closer to A or every-farther, and cannot be finite in number. So limit cycles must have one element (if it is A) or two elements.
We note that this statement relies on the fact that higher ζ's lead to lower ζ's in response which is true when vaccines lead to a fractional decrease in infection probability, but not when the vaccine protection is a per-contact reduction as described below in Section 4. This does not mean the limit cycles are unique. There may be three or more period-2 cycles for given parameter values. The topology implies that the number of cycles is always odd, with every pair of neighbouring stable cycles separated by an unstable cycle. Related results are found in [1].

Model
Given the main paper Eqs. (3), (4), and (6) for the payoff function, α k , δ k , and ζ, respectively, then a degree k individual's best-response strategy (that which maximizes their payoff for a where H() is the Heaviside function The general existence of a solution to this equation can be established using the Ky Fan inequality or Kakutani fixed point theorem [2]. Note that rather than having an arbitrary given by Eqs. (4) and (5) allows us to treat the best response function v B as a function of the scalar probability η(v) for random graphs; v * ∈ v B (η(v * )). If we apply formulas for the probability of escaping infection η(v) to both sides, we get the scalar inclusion relation . Substituting e = η(v * ), e ∈ η(v B (e)) and equivalently 0 ∈ η(v B (e))−e.
We can show that there is a unique solution e * to this inclusion relation. First, we observe that the correspondence η(v B (e)) is continuous. η is non-negative so if e = 0, η is a probability less than 1, so if e = 1, η(v B (1)) − 1 ≤ 0. Suppose I 1 and I 2 are two subsets of the real numbers. We say I 1 ≤ I 2 if and only if, for every x i ∈ I 1 and y j ∈ I 2 , x i ≤ y j . Now, the best-response vaccination probabilities are weakly increasing as the risk of infection increases in the sense that for each component k, . We also know that the less vaccination there is, the greater the risk of infection, so e 1 < e 2 implies η(v B (e 1 )) ≥ η(v B (e 2 )) with equality holding only when v B (e 1 ) = v B (e 2 ). Subtracting, we conclude that η(v B (e)) − e is strictly decreasing in e.
These conditions imply that there is exactly one solution e * ∈ [0, 1] such that Thus far, we have derived a necessary condition for a Nash equilibrium: if v * is a Nash equilibrium, then η(v * ) = e * . Obviously, there are many combinations of vaccination strategies that can result in an overall risk e * . However, a Nash equilibrium must also be a best response, and the set of best responses (a subset of [0, 1] N ) is a continuous well-ordered set under the component-wise ordering relation. Specifically, if e 1 and e 2 are forces of infection with e 1 < e 2 , then for w 1 ∈ v B k (e 1 ) and w 2 ∈ v B k (e 2 ), w 1,k ≤ w 2,k for all k but w 1 = w 2 . This is ordering relies on α k being strictly decreasing.
We also know that η(v B ) is continuous, e * is in the range of η, and η is strictly decreasing Theorem 3. Suppose v * (T ) is the Nash equilibrium strategy set as a function of the transmissibility T . If best responses are given by Eq. (S1), then for every degree class k, Proof. We know η(v * (T 1 ), T 1 ) ≤ η(v * (T 1 ), T 2 ) where η(v * (T 1 ), T 2 ) is the resulting risk of the strategy set v * (T 1 ) in a network with edge transmissibility of T 2 for instance. The for all k because the best response is increasing in risk for fixed transmissibility. Further, for all k since for a fixed risk, the best response is increasing in T . So and using the vaccine efficacy term r from above, the infection probability for a vaccinated individual is Given r and T and assuming that a proportion zeta of contacts are not infected, the critical cost of vaccination above which individuals with degree k will not vaccinate is given by For small k, and the limit is approached from above with the vaccine efficacy r controlling the rate of convergence to the asymptotic critical vaccine cost. Thus, c * V has a maximum value as a function of k so long as the vaccine provides partial protection (r > 0) on a per-contact basis.  This window-effect breaks the conditions used to prove uniqueness of the Nash equilibrium above; and we conjecture that there may be multiple Nash equilibria in some scenarios.
This also means that best responses are not generally monotone functions of the risk ζ, which also violates Theorem 3 (Nash equilibrium vaccination increases with transmissibility). The efficacies used in the main text were not reported this way, but rather as a proportional reduction in likelihood of infection, independent of degree.