Mean-field analysis of an inductive reasoning game: Application to influenza vaccination

Recently we have introduced an inductive reasoning game of voluntary yearly vaccination to establish whether or not a population of individuals acting in their own self-interest would be able to prevent influenza epidemics. Here, we analyze our model to describe the dynamics of the collective yearly vaccination uptake. We discuss the mean-field equations of our model and first order effects of fluctuations. We explain why our model predicts that severe epidemics are periodically expected even without the introduction of pandemic strains. We find that fluctuations in the collective yearly vaccination uptake induce severe epidemics with an expected periodicity that depends on the number of independent decision makers in the population. The mean-field dynamics also reveal that there are conditions for which the dynamics become robust to the fluctuations. However, the transition between fluctuation-sensitive and fluctuation-robust dynamics occurs for biologically implausible parameters. We also analyze our model when incentive-based vaccination programs are offered. When a family-based incentive is offered, the expected periodicity of severe epidemics is increased. This results from the fact that the number of independent decision makers is reduced, increasing the effect of the fluctuations. However, incentives based on the number of years of prepayment of vaccination may yield fluctuation-robust dynamics where severe epidemics are prevented. In this case, depending on prepayment, the transition between fluctuation-sensitive and fluctuation-robust dynamics may occur for biologically plausible parameters. Our analysis provides a practical method for identifying how many years of free vaccination should be provided in order to successfully ameliorate influenza epidemics.

Figure 1

Diagram illustrating the evaluation tree for each selfish individual. An individual that decides to vaccinate [branch (a)] will judge their choice depending on whether there was an epidemic that season. If the coverage is equal or greater than the critical coverage $p_n⩾{\pi }_c$ [branch (a1)], they will conclude that their choice to get vaccinated that season was not necessary to prevent infection. Otherwise, if the coverage is lower than the critical coverage $p_n<{\pi }_c$ [branch (a2)], they will conclude that their choice was beneficial for avoiding infection that season. An individual that decides not to vaccinate that season [branch (b)] will judge their choice based on whether they were infected. If they do get infected [branch (b1)] they will conclude that their choice of not vaccinating was detrimental and that vaccination was necessary for avoiding infection. Instead, if by chance they avoid infection [branch (b2)], they will conclude that vaccination was unnecessary.

Figure 2

Coverage (i.e., fraction of vaccinated individuals) and incidence (i.e., fraction of infected individuals) dynamics for a population of $N={10}^5$ individuals using a memory parameter $s=0.7$ and a probability $q\left(0\right)=0.8$ of getting infected when $p=0$. (a1) For ${\pi }_c=0.6$, the dynamics of $p$ is approximately cyclic: as $p$ approaches ${\pi }_c$ from below, it eventually fluctuates above ${\pi }_c$ and then abruptly drops below ${\pi }_c$. (b1) For ${\pi }_c=0.6$, the dynamics of $p$ is very well approximated by a period two orbit. (a2) and (b2) The dynamics of the incidence $I∕N$ mirrors that of the coverage. A low coverage yields a severe incidence, while a high coverage yields a mild incidence or if $p>{\pi }_c$ epidemics are prevented and therefore the incidence is zero.

Figure 3

Bifurcation diagram of the map given by Eq. (7) versus ${\pi }_c$. We have chosen $s=0.7$ and $q\left(0\right)=0.8$. The dotted line indicates the approximated position of ${\Pi }_0$, the ${\pi }_c$ value where, due to the discontinuity of the map, a period two orbit is created. We note that numerical noise greatly perturbs the dynamics of the piecewise smooth map since ${\pi }^{*}∊\partial {\mathcal{I}}_2$. To improve the stability of the numerics, we slightly modified the map in order for ${\pi }^{*}∊{\mathcal{I}}_2$; in turn, this slightly changes the threshold value of ${\pi }_c$ and reduces the ${\pi }_c$ interval for which the period two orbit and the ${\pi }^{*}$ fixed point coexist as attractors.

Figure 4

(a) Schematics of the map when the critical orbit is created. The map is represented by the thick line. The graphical iteration of the critical orbit is represented by a loop. The basin of ${\pi }^{*}$ is the whole domain [0, 1], excluding the points of the period two orbit. (b) Schematics of the map right after the critical orbit is created. The basin of ${\pi }^{*}$ is the open interval in the shaded area, while the basin of the period two orbit consists of all the other points in [0, 1].

Figure 5

Schematics illustrating the evaluation tree for each family head. A family head that decides to vaccinate themselves and their family [branch (a)] will judge their choice depending on whether there was an epidemic that season. If the coverage was equal or greater than the critical coverage $p_n⩾{\pi }_c$ [branch (a1)], they will conclude that their choice to get vaccinated (and have their families vaccinated) that season was unnecessary for preventing infection. Otherwise, if the coverage was lower than the critical coverage $p_n<{\pi }_c$ [branch (a2)], they will conclude that their choice was beneficial for avoiding infection that season. A family head that decides not to vaccinate themselves and their family [branch (b)] will judge their choice based on how many of their family members were infected. If $k$ members get infected [branch (bk)], they will conclude that vaccination was necessary only for $k$ members of their family.

Figure 6

Coverage dynamics for two public health incentives in a population of $N={10}^5$ noncommunicating selfish individuals using a memory parameter $s=0.7$, a critical coverage ${\pi }_c=0.6$, and a probability $q\left(0\right)=0.8$ of getting infected when the coverage $p=0$. (a) The head of the family makes the decision as to whether or not their family vaccinates. The coverage dynamics when the family size is eight $(C=8)$ is shown in black; the coverage dynamics when individuals make vaccination decisions independently is shown in gray for comparison. Similar results were obtained for family sizes of two and four. (b) Individuals that pay for one vaccination are given three extra years of vaccination $(y=3)$; the coverage dynamics is shown in black and it’s time average is ameliorated. The coverage dynamics when individuals pay for every year of vaccination is shown in gray for comparison.

Figure 7

Bifurcation diagram of the dynamical system given by Eqs. (39, 40, 41) versus ${\pi }_c$ [$y=1$, $s=0.7$ and $q\left(0\right)=0.8$]. The dotted lines, from left to right, indicate the approximated position of ${\Pi }_0\approx 0.205$ $(k=2)$, ${\Pi }_1\approx 0.435$ $(k=5)$, ${\Pi }_2\approx 0.551$ $(k=5)$, and ${\Pi }_3\approx 0.647$ $(k=6)$. The dashed line marks our chosen value of ${\pi }_c=0.6$. In this case, incentive 2 with $y=1$ or $y=2$ does not induce a qualitative change in the dynamics of the coverage, while for $y=3$ such a change occurs through a border-collision bifurcation.