Stochastic games of parental vaccination decision making and bounded rationality

TL;DR

This work develops a stochastic game-theoretic framework for parental vaccination decisions by coupling a two-strategy stochastic replicator dynamic with a stochastic SIR model, capturing bounded rationality through noise in perceived utilities and in transmission. The authors derive a coupled system with a stochastic basic reproduction number $R_0^s=\dfrac{\beta}{\mu+\gamma+\frac{1}{2}\sigma_1^2}$ and analyze equilibria, revealing how noise can generate bistability and lower stochastic herd immunity thresholds $HIT^s$ compared to the deterministic case. They establish almost-sure stability results for disease-free states under specified conditions and present a stochastic optimal control framework that yields an explicit, bounded-control law to reduce vaccination costs and achieve full uptake under uncertainty. Overall, the paper demonstrates how bounded rationality and social norms interact with epidemic dynamics and provides a control mechanism to counteract non-rational vaccination behavior in noisy environments.

Abstract

Vaccination is an effective strategy to prevent the spread of diseases. However, hesitancy and rejection of vaccines, particularly in childhood immunizations, pose challenges to vaccination efforts. In that case, according to rational decision-making and classical utility theory, parents weigh the costs of vaccination against the costs of not vaccinating their children. Social norms influence these parental decision-making outcomes, deviating their decisions from rationality. Additionally, variability in values of utilities stemming from stochasticity in parents' perceptions over time can lead to further deviations from rationality. In this paper, we employ independent white noises to represent stochastic fluctuations in parental perceptions of utility functions of the decisions over time, as well as in the disease transmission rates. This approach leads to a system of stochastic differential equations of a susceptible-infected-recovered (SIR) model coupled with a stochastic replicator equation. We explore the dynamics of these equations and identify new behaviors emerging from stochastic influences. Interestingly, incorporating stochasticity into the utility functions for vaccination and nonvaccination leads to a decision-making model that reflects the bounded rationality of humans. Noise, like social norms, is a two-sided sword that depends on the degree of bounded rationality of each group. We also perform a stochastic optimal control as a discount to the cost of vaccination to counteract bounded rationality.

Figures (7)

• Figure 3.1: Simulation of susceptible $S$, infected $I$ and vaccinated $x$, when (a) $R_0^s<1$ and $\sigma_1^2\leq \beta$ at $\beta=31$ and $\sigma_1^2=30$ giving $R_0=1.87$ and $R_0^s=.98$, and (b) $\frac{\sigma_1^2}{\beta}>\max\left(1,\dfrac{R_0}{2}\right)$ at $\beta=33.3$ and $\sigma_1^2=50$ giving $R_0=2$ and $R_0^s=.8$ (even when it is more than one). The rest of the parameters are $\mu=1/50$ year$^{-1}$, $\gamma=365/22$ year$^{-1}$, $\kappa=1.69$ year$^{-1}$, $\omega=.0015$, $\delta=.0005$, $\sigma_2^2=.0008$, and $\sigma_3^2=.0006$.
• Figure 3.2: (a) Parameter plane of $\sigma_2^2$ versus $\sigma_3^2$ based on Theorem \ref{['thm:vax']}. Other parameter values are $\beta = 100$, $\mu = 1/50$, $\gamma = 365/22$, $\kappa = 1.69$, $\omega = 0.1$, $\delta = 0.5$, and $\sigma_1^2=.16$. In that case, $R_0^s=6$. (b) Simulation of the susceptible, infected, and vaccinated when $\sigma_2^2=1.5$ and $\sigma_3^2=.2$. The endemic equilibrium of $\mathcal{E}_4$ type is stable. (c) Simulation of the susceptible, infected, and vaccinated when $\sigma_2^2=.2$ and $\sigma_3^2=1.5$. The disease-free equilibrium of $\mathcal{E}_1$ type is stable.
• Figure 3.3: Simulation of the susceptible, infected, and vaccinated without control in (a) and with control in (b). (c) The control $u^*(t)$ is shown for $u_{max}=.8$, $T_f=150$, $\alpha_1=0$, $\alpha_2=1000$, and $\alpha_3=100$. The rest of the parameter values are $\beta = 100$, $\mu = 1/50$, $\gamma = 365/22$, $\kappa = 1.69$, $\omega = 2$, $\delta = 0.1$, $\sigma_1^2=.01$, $\sigma_2^2=.5$ and $\sigma_3^2=1.4$. The initial values are $S(0)=.9$, $I(0)=.1$ and $x(0)=.1$. In that case, $R_0^s=6$.
• Figure 3.4: Simulation of the susceptible, infected, and vaccinated when $\sigma_2^2=.15$ and $\sigma_3^2=.2$ in (a) and (b) with initial $x(0)=.8$. The disease-free equilibrium of $\mathcal{E}_1$ type and the disease endemic equilibrium of $\mathcal{E}_4$ type are stable -- a Bernoulli stationary distribution. The rest of the parameter values are $\beta = 100$, $\mu = 1/50$, $\gamma = 365/22$, $\kappa = 1.69$, $\omega = 0.1$, $\delta = 0.5$, and $\sigma_1^2=.16$. In that case, $R_0^s=6$.
• Figure 3.5: Estimate of the probability $P\left(\lim_{t\to\infty}x(t)=0\vert x(0)\right)$ using $200$ simulation runs at $T=100$, $dt = 0.001$ and different initial values of $x(0)$, with parameter values $\beta = 100$, $\mu = 1/50$, $\gamma = 365/22$, $\kappa = 1.69$, $\omega = 0.1$, $\delta = 0.5$, $\sigma_1^2 = .16$, and initial values $S(0)=0.4$ and $I(0)=0.4$.

Theorems & Definitions (13)

• Definition 1: Almost Sure Exponential Stability
• Definition 2: Almost Sure Logistic Stability
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 1
• Lemma 2: Itô's formula
• Lemma 3