Modeling Epidemic Dynamics of Mutant Strains with Evolutionary Game-based Vaccination Behavior

TL;DR

The paper addresses the need to jointly model epidemic spread with mutant strains and adaptive vaccination behavior. It introduces an extended SIRS framework with a vaccinated state $V$ and a mutant-infected state $I_2$, coupled to an evolutionary game on networks that governs vaccination decisions, all embedded in a microscopic Markov chain-based coupling (MMCA). Key contributions include a mechanistic vaccination-update mechanism driven by neighbor interactions and global epidemic indicators, explicit MMCA state-transition equations, and comprehensive simulations showing how risk perception, herd behavior, and vaccination parameters shape outbreak control. The work provides quantitative guidance for public health policies aimed at lowering vaccine costs, boosting efficacy, and minimizing side effects to enhance vaccination coverage and epidemic containment. Practical implications span policymaking and individual protective strategies, with the framework applicable to scenarios featuring moderate-to-low infection and mutation rates or rapid containment under strong herd effects.

Abstract

The outbreak of mutant strains and vaccination behaviors have been the focus of recent epidemiological research, but most existing epidemic models failed to simultaneously capture viral mutation and consider the complexity and behavioral dynamics of vaccination. To address this gap, we develop an extended SIRS model that distinguishes infections with the original strain and a mutant strain, and explicitly introduces a vaccinated compartment state. At the behavioral level, we employ evolutionary game theory to model individual vaccination decisions, where strategies are determined by both neighbors' choices and the current epidemiological situation. This process corresponds to the time-varying vaccination rate of susceptible individuals transitioning to vaccinated individuals at the epidemic spreading level. We then couple the epidemic and vaccination behavioral processes through the microscopic Markov chain approach (MMCA) and finally investigate the evolutionary dynamics via numerical simulations. The results show that our framework can effectively mitigate outbreaks across different disease scenarios. Sensitivity analysis further reveals that vaccination uptake is most strongly influenced by vaccine cost, efficacy, and perceived risk of side effects. Overall, this behavior-aware modeling framework captures the co-evolution of viral mutation and vaccination behavior, providing quantitative and theoretical support for designing effective public health vaccination policies.

Figures (7)

• Figure 1: Coupled dynamics of strategic vaccination and expanded SIRS epidemic spreading. This figure illustrates the core mechanisms of strategic vaccination (above layer) and epidemic spread (below layer) and their relationship. The above subfigure illustrates a strategy update for a current individual (gray node $i$). Its decision is influenced by the two choices of its surrounding neighbors (connected to it by solid lines)---to get vaccinated (red nodes) or not (blue nodes). Below (Disease Layer): Shows the state transitions during epidemic spreading. Solid arrows represent state transitions governed by rates ($\beta_1$, $\beta_2$, $\mu_1$, $\mu_2$) and vaccination effect ($\lambda$). The key coupling of these two layers is that the vaccination decision from the game layer decides the process of an individual in state S transitioning to the V state, then influences the whole spreading process (demonstrated via the dashed arrows). This figure is used to display how vaccination behavior co-evolves with the epidemic, as individuals' strategies directly alter their disease state probabilities.
• Figure 2: Mechanism of the vaccination game: This schematic elucidates the process of individual vaccination choices in our model and the details of the vaccination game among neighbors. The top row (Decision process) illustrates the population-level evolution from left to right: the initial distribution of vaccinated (red nodes) and unvaccinated (blue nodes) individuals; their intermediate states after being influenced by the epidemic dynamics before the vaccination game; and the configuration after the vaccination game. Each individual's decision is determined by both the overall epidemic influence (indicated by dashed arrows) and the outcomes of its neighbor games (detailed in the bottom row). Bottom row (Game among neighbors) tells the mechanism of the game among neighbors based on the three distinct neighbor environments (yellow, blue, and purple sets) from above. It shows that for a focal decision-maker (gray node), the vaccination strategy is analyzed for the vaccination choices of neighbors that yield distinct fitness values for it, which then inform its decision based on the corresponding payoff scenarios.
• Figure 3: Impact of dual-strain virus infection rates on different perceived risk groups and mutation rates, and their mutual combined effects. This figure systematically investigates how the infection rates of dual strains (original strain $\beta_1$, mutant strain $\beta_2$), perceived risk levels, and virus mutation rates jointly affect epidemic transmission and vaccination outcomes. For the details, subfigures (a) and (b) show the relationship between the infection rate of the original strain $\beta_1$ and its infection density (represented by the vertical axis), and the relationship between the infection rate of the mutant strain $\beta_2$ and its infection density under different perceived risk scenarios. Subfigures (c) and (d) show the same relationship described above at different mutation rate settings. Subfigures (e) to (h) analyze the combined impact of the infection rate of the two strains on disease transmission. Subfigures (e) and (f) include the impact of the global game on vaccination decisions, and subfigures (g) and (h) represent the scenario without the impact of the global game $W=0$. And subfigures (e) and (g) show the density of individuals S, and subfigures (f) and (h) show the density of individuals V.
• Figure 4: Frequency of infected population. This figure presents simulation results regarding the occurrence frequency of different infected population numbers, modulated by three key parameters: infection rate, recovery rate, and perceived risk (each parameter is categorized into low, medium, and high levels). Subfigures (a), (b), and (c) respectively correspond to the three parameter groups ($\beta_1,\beta_2$, $\gamma_1,\gamma_2$, and perceived risk). In each subfigure, the red, green, and blue lines correspond to the low, medium, and high levels within each group, respectively. The horizontal axis represents the number of infected individuals observed throughout all iterations, and the vertical axis indicates the frequency of occurrence for each infected count during the simulation.
• Figure 5: Temporal evolution of epidemic spread and strategy selection under different herding coefficients. This figure characterizes epidemic dynamics by three core metrics, the number of infected individuals, the densities of S and V individuals, and further reveals how herding behavior modulates the temporal evolution of epidemic spread and individual vaccination strategy selection. The epidemic dynamics are characterized by the number of infected individuals and the densities of S and V individuals. In subfigures (a)–(c), the green dotted, red solid, blue solid, and cyan dashed lines represent the time-varying densities of I, S, V, and R individuals, respectively. Among them, subfigures (a)–(c) correspond to three scenarios of low, medium, and high conformity tendencies, with the conformity coefficient $\omega_1$ being 0.1, 0.6, and 0.9, respectively.