Tricritical behavior in epidemic dynamics with vaccination

TL;DR

The paper investigates a minimal vaccination-epidemic (MVE) model to understand how vaccination influences nonequilibrium phase transitions. By combining mean-field analysis, Monte Carlo simulations, and finite-size scaling across complete graphs and structured networks, it shows a dichotomy: bistable, first-order transitions with a tricritical directed-percolation (TDP) universality on infinite-dimensional structures, and a continuous, directed-percolation (DP)–like transition with DP exponents on 2D square lattices. The tricritical point and crossovers conform to mean-field TDP scaling, with explicit exponents and crossover behavior quantified for both complete graphs and 2D lattices. These results reveal universal features of epidemic dynamics under vaccination, highlighting how network structure governs the nature of phase transitions and providing a framework for analyzing vaccination strategies in complex populations.

Abstract

We scrutinize the phenomenology arising from a minimal vaccination-epidemic (MVE) dynamics using three methods: mean-field approach, Monte Carlo simulations, and finite-size scaling analysis. The mean-field formulation reveals that the MVE model exhibits either a continuous or a discontinuous active-to-absorbing phase transition, accompanied by bistability and a tricritical point. However, on square lattices, we detect no signs of bistability, and we disclose that the active-to-absorbing state transition has a scaling invariance and critical exponents compatible with the continuous transition of the directed percolation universality class. Additionally, our findings indicate that the tricritical and crossover behaviors of the MVE dynamics belong to the universality class of mean-field tricritical directed percolation.

Figures (6)

• Figure 1: Sketch of the vaccination dynamics with an illustration of the probabilities $\{\lambda,\eta,\gamma,\sigma\}$ described in the text. For brevity, we call this model as minimal vaccination-epidemic (MVE) dynamics. The MVE model is composed by four compartmental transitions: (i) two activated by a pairwise interaction ($S+I \stackrel{}{\rightarrow} 2I$ and $V+I \stackrel{}{\rightarrow} 2I$); and (ii) two non-mediated by interactions ($S \stackrel{}{\rightarrow} V$ and $I \stackrel{}{\rightarrow} S$).
• Figure 2: Mean-field phase diagram of the minimal vaccination-epidemic (MVE) model, as depicted in Fig. \ref{['fig:min_vac_sudden_model_sketch']}, for $\gamma=0.07$, and: $\eta=0.1$ (a) and $\eta=0.5$ (b). The shaded area corresponds to the bistable (B) phase where the dynamics has three fixed points: two stable ($\rho_{\infty}^{+}$ and $\rho_{\infty}^o$) and one unstable ($\rho_{\infty}^{-}$). In the Disease-Free phase $\rho_{\infty}^o$ is the only the stable fixed point. In turn, $\rho_{\infty}^{+}$ is the unique stable fixed point in the Endemic phase. The coordinates of the tricritical point (TCP) are obtained from Eq. \ref{['eq:tcp']}. From the thresholds of $\rho_{\infty}^{\pm}$ in Eq. \ref{['Eq:sol3_Iplus']}, we obtain the solid and dashed curves that represent continuous and discontinuous transitions, respectively. The bistability can be suppressed if the vaccination probability $\eta$ is above the threshold given by the Eq. \ref{['eq:eta_threshold']}.
• Figure 3: Stationary infected density $\rho$ versus the transmissibility $\lambda$ for the MVE model on: (a) complete graphs, (b) random $k$-regular networks with $k=20$, (c) square lattice. Simulations performed with $t_{max}=10^5$ on structures with $N=10^4$ individuals each. The control parameter $\lambda$ is increased (red triangles) and decreased (blue circles) in the range $0.49\leq \lambda\leq 0.7$ at constant intervals of $\Delta \lambda = 0.01$. In the upper branch the simulations started with $\lambda=0.7$ from $\rho_o=1$ (fully active initial condition), whereas in the lower branch the simulations started with $\lambda=0.49$ from $\rho_o=1/N$ (localized initial condition). The stable and unstable theoretical solutions, obtained from the Eq. \ref{['Eq:sol3_Iplus']}, are represented in the solid and dashed lines, respectively.
• Figure 4: Time evolution of the order parameter $\rho$ for several initial conditions $\rho_o$. (a) $\lambda=0.50$, complete graphs, (b) $\lambda=0.51$, random $k$-regular networks with $k=20$, (c) $\lambda=0.595$, square lattice. All structures with $N=10^4$ individuals. Each panel has $100$ samples, and for better detecting eventual bistability we plot the time series for each parameter setting without averages. The time series in panels (a-b) exhibit bistable solutions, whereas the panel (c) exhibits a single stable steady state. In panels (a-b) note that in some dynamic scenarios, the presence of pronounced epidemic peaks in the short-term does not necessarily promote persistence of the transmission chain in the long-term.
• Figure 5: Critical behavior of the MVE on square lattices. All panels are in log-log scale. Panel (a): $\rho(t)$ versus $t$ for simulations performed with $L=100$ and several $\lambda$ around the critical point; the black dashed line corresponds to the fit with the least curvature. Panel (b): $\rho^{*} = \rho (t) t^{\beta/\nu_{||} }$ versus $\Delta^{*} = t (\lambda-\lambda_c)^{\nu_{||}}$ for simulations with $L=100$; the color legend of panel (a) also applies to panel (b). Panel (c): $\Tilde{\rho} = \rho(t) L^{\beta/\nu_{\perp}}$ versus $\Tilde{\Delta} = t L^{-z}$ for simulations performed with $L=\{75,100,150,200\}$. In panels (b-c) we use the $\lambda_c$ estimated with the procedure shown in panel (a). In all cases we use $\rho_o=1$ and we let the simulations run until the dynamics reaches the stationary state. The estimated critical quantities are shown inside each panel and the best estimates are compiled in the Table \ref{['tab:exponents_MVE_2d']}.