SIS-Vo Framework: Epidemic & Opinion Dynamics

• SIS-Vo framework is an integrative model combining epidemic and opinion dynamics, emphasizing how demographic factors shape vaccine uptake.
• It employs coupled differential equations over contact and signed opinion networks to simulate disease spread and targeted messaging effects.
• The model provides fixed-point, stability, and control analyses to guide robust, microtargeted vaccination strategies against misinformation.

The SIS-Vo framework is an epistemically informed mathematical model that couples classical epidemic (SIS) dynamics with the propagation of vaccine-related opinions on a signed social network. Designed to capture the interplay between demographic structure, opinion formation, and vaccination capacity, SIS-Vo provides fixed-point, stability, and control-theoretic characterizations for robust vaccination policy in the face of demographic-dependent misinformation (Casu et al., 6 Dec 2025).

1. Model Structure: State Variables and Compartmental Representation

SIS-Vo extends standard compartmental models by augmenting each node $i$ (subpopulation) with an explicit opinion state. For node $i$ $(i = 1, \ldots, n)$:

• $s_i(t)$: Fraction of susceptibles.
• $x_i(t)$: Fraction of infecteds.
• $v_i(t)$: Fraction of vaccinated (permanently immune).
• $o_i(t) \in (-\frac{1}{2}, \frac{1}{2})$ (generally $\mathbb{R}$): Average opinion toward the vaccine ($o_i<0$ indicates hesitancy, $o_i>0$ indicates support). These satisfy $s_i(t) + x_i(t) + v_i(t) = 1$ at all times.

The population is modeled on two networks:

• Contact network $B = [\beta_{ij}]$ defining epidemic transmission rates.
• Signed opinion network $A = [\alpha_{ij}]$ encoding demographic-affinity-based influence, where $\alpha_{ij} = d_i^T d_j$ and $d_i \in \mathbb{R}^k$ is the demographic feature vector for group $i$.

2. Dynamical System: Coupled SIS and Opinion ODEs

The SIS-Vo system is governed at the node level by the following ODEs:

$$\begin{align*} \dot{s}_i &= \gamma_i x_i - s_i \left( \sum_{j=1}^n \beta_{ij} x_j + \rho \left[1 - \frac{v_i}{\kappa_i}\right] \right) \ \dot{x}_i &= s_i \sum_{j=1}^n \beta_{ij} x_j - \gamma_i x_i \ \dot{v}_i &= \rho \left(1 - \frac{v_i}{\kappa_i}\right) s_i \ \end{align*}$$

with $s_i = 1 - x_i - v_i$. Here, $\gamma_i$ is the recovery rate and $\rho$ sets the timescale of vaccination. The carrying capacity $\kappa_i$ of vaccination is shaped dynamically by $o_i$.

Opinion evolution follows:

$$\dot{o}_i = (x_i - o_i - \tfrac{1}{2}) + \sum_{j=1}^n |\alpha_{ij}| \left[ \mathrm{sign}(\alpha_{ij}) o_j - o_i \right]$$

or compactly with the signed Laplacian $\widetilde{A}$,

$$\dot{o} = x - o - \tfrac{1}{2} \mathbf{1} - \widetilde{A} o$$

where $\widetilde{A}_{ij} = \sum_j |\alpha_{ij}|$ for $i = j$, $- \alpha_{ij}$ otherwise.

3. Opinion–Vaccination Coupling and Demographic Heterogeneity

The critical innovation is the coupling between opinion and vaccination. The vaccination carrying capacity map is defined:

$$\kappa_i(o_i) = \max \left( v_i, \frac{1 + 2 o_i}{1 - 2 o_i} \right)$$

Hence, positive shifts in $o_i$ (more supportive opinions) increase $\kappa_i$ toward or above unity, enhancing maximum possible vaccination. Negative $o_i$ depress $\kappa_i$, potentially bottlenecking vaccine coverage below herd immunity.

Demographically structured external messaging enters as $u \in \mathbb{R}^k$, $\,\|u\| = 1$, with the node-specific effect $u^T d_i$ influencing $\dot{o}_i$. By selecting $u$ in the span of selected $d_i$, “microtargeting” is achieved: messaging can be biased toward particular demographic groups within the population.

4. Equilibrium Analysis: Fixed Points and Stability Criteria

Disease-Free (“Healthy”) Equilibrium

At the disease-free fixed point ($x_i^* = 0$), the states solve: $v_i^* = \min(1,\,\kappa_i^*)$

$$(\widetilde{A} + I) o^* = D^T u - \tfrac{1}{2} \mathbf{1}$$

with $D^T u$ the demographic–message projection. The equilibrium configuration is: $$(s_i^*,x_i^*,v_i^*,o_i^*) = \begin{cases} (0,0,1,\frac{\kappa_i^*-1}{2(\kappa_i^*+1)}), & \kappa_i^* \geq 1 \ (1-\kappa_i^*,0,\kappa_i^*,\frac{\kappa_i^*-1}{2(\kappa_i^*+1)}), & \kappa_i^* < 1 \end{cases}$$

Endemic Equilibrium

With $x_i^* > 0$ for some $i$, the equilibrium is determined by the coupled fixed-point equations:

• $$x_i^* = (1-v_i^*) \frac{\sum_j \beta_{ij} x_j^*}{\gamma_i + \sum_j \beta_{ij} x_j^*}$$
• $v_i^* = \min(1, \kappa_i^*)$
• $$(\widetilde{A} + I) o^* = D^T u - \tfrac{1}{2} \mathbf{1} + x^*$$

Stability of the Healthy State

By linearization about the disease-free fixed point with $v = v^*$, $o = o^*$ fixed, the subsystem for $x$ reduces to:

$$\dot{x} \approx M x, \quad M = \mathrm{diag}(1-v^*) B - \mathrm{diag}(\gamma)$$

A sufficient condition for local asymptotic stability is:

$$\max_i \left\{ \lambda_{\max}(B) (1 - \kappa_i^*) \right\} < \gamma_{\min}, \quad \kappa_i^* < 1 \ \forall i$$

where $\lambda_{\max}(B)$ is the spectral radius of $B$ and $\gamma_{\min} = \min_i \gamma_i$. If satisfied, all eigenvalues of $M$ have negative real part, and the disease-free equilibrium attracts all nearby states.

5. Numerical Illustration: Microtargeted Messaging and Regime Transition

Numerical experiments demonstrate the population-level impact of targeted intervention. With random demographic assignment and specified networks $B$, $A$, three message strategies $u$ are compared:

• Aligned: $u$ parallel to $\sum_i d_i$ (“majority-demographic targeting”).
• Anti-aligned: $u$ anti-parallel.
• Random: $u$ random in $\mathbb{R}^k$.

Alignment delivers $o_i^* > 0$ for many subpopulations, driving $\kappa_i^* \geq 1$ and saturating vaccination ($v_i^* = 1$), thus $x_i^* = 0$ (disease-free regime). Anti-alignment depresses $o_i^*$, $\kappa_i^*$, and results in endemic persistence for most groups. Random $u$ yields intermediate outcomes. This demonstrates the efficacy of microtargeted messaging campaigns for shifting equilibrium from endemicity to a healthy regime.

6. Control-Theoretic Perspective and Robust Policy Design

SIS-Vo is naturally suited to control-theoretic approaches. The design task can be formalized as minimizing the endemic burden

$J(u) = \sum_{i=1}^n N_i x_i^*(u)$

subject to equilibrium constraints for $(x^*, v^*, o^*)$, which depend nonlinearly on $u$. Alternatively, one can optimize for the largest region in demographic space where the spectral stability conditions hold, maximizing robust health outcomes.

Misinformation attacks are represented by bounded adversarial perturbations $w_i$ added to the opinion ODEs, leading to a robust min–max optimization: $$\min_{u}\;\max_{\|w\|\leq\delta} \sum_i N_i\,x_i^*(u,w)$$ By leveraging the convexity and network structure of the stability constraints, tractable sufficient conditions (e.g., linear matrix inequalities) can be derived to guarantee that the healthy equilibrium remains robust against the worst-case demographic-specific misinformation up to strength $\delta$.

7. Significance and Applications

SIS-Vo synthesizes epidemiological compartmental dynamics, signed-network opinion evolution, demographic stratification, and targeted control into a unified ODE framework. Its fixed-point and spectral analyses provide explicit criteria and interventions for designing vaccination campaigns that are robust to targeted misinformation in high-risk demographic subpopulations. The framework yields actionable insights for microtargeted public health messaging and exposes the vulnerabilities associated with opinion–epidemiology coupling, particularly in the presence of homophily-driven antagonistic influence and heterogeneous vaccine acceptance (Casu et al., 6 Dec 2025).