Susceptibility Propagation in Network Epidemics

• Susceptibility propagation is a modeling approach that quantifies how infection probabilities spread through complex networks while incorporating realistic behavioral responses.
• The modified degree‐based mean-field model integrates mitigation factors—similar to logistic growth constraints—to capture nonmonotonic interaction probabilities and dynamic threshold behavior.
• Analytical results reveal that although scale-free networks exhibit a zero epidemic threshold, behavioral mitigation can slow epidemic saturation and guide effective intervention strategies.

Susceptibility propagation in the context of network epidemiology refers to how the probability of being infected transmits through complex networks, particularly under interventions that reduce the effective participation of infected individuals. The recent degree-based mean-field analysis of the susceptible-infected-susceptible (SIS) model with a mitigation factor formalizes the effect of realistic behavioral responses—such as isolation or hospitalization—on disease propagation in heterogeneous (Barabási–Albert) networks. By analytically modifying the key interaction probability, the paper quantifies both threshold and saturation regimes in epidemic prevalence and elucidates how mitigation fundamentally alters the feedback between network heterogeneity and epidemic persistence (Kim et al., 8 Jan 2025).

1. Degree-Based Mean-Field Formulation of SIS Dynamics

The classic heterogeneous mean-field (HMF) model for SIS dynamics defines the probability ρₖ(t) that a node of degree k is infected as evolving according to: $$\frac{\partial \rho_k(t)}{\partial t} = -\rho_k(t) + \lambda k [1 - \rho_k(t)] \Theta(t)$$ where λ is the infection rate and Θ(t) denotes the probability that an edge from a node leads to an infected node. In the steady state,

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and, for uncorrelated networks,

$$\Theta = \sum_k q(k) \rho_k\quad \text{with} \quad q(k) = \frac{k P(k)}{\langle k \rangle}$$

For a Barabási–Albert (BA) network, characterized by $P(k) = \frac{2m^2}{k^3}$, the self-consistency equation simplifies to: $$\Theta = \lambda m \Theta \ln\left(1+\frac{1}{\lambda m \Theta}\right)$$ The solution structure includes both the disease-free state (Θ = 0) and a nontrivial endemic state for any λ > 0, reflecting the classic result that scale-free networks with diverging second moment possess λ_c = 0 (no finite epidemic threshold). The network prevalence follows: $$\rho = 2 (\lambda m \Theta)^2 \left [ \frac{1}{\lambda m \Theta} - \ln\left(1 + \frac{1}{\lambda m \Theta}\right) \right ]$$

2. Incorporating Mitigation: Model Extension and Mathematical Structure

The modified HMF (mHMF) model introduces a mitigation factor reflecting the notion that, as the fraction of infected nodes rises, their participation in spreading the infection is reduced—mimicking behavioral responses such as self-isolation or hospitalization. The key alteration in the model is: $$\Theta = \sum_k q(k) \rho_k (1 - \rho_k) = \frac{1}{\langle k \rangle}\sum_k k P(k) \rho_k(1 - \rho_k)$$ On the BA network, the modified self-consistency equation becomes: $$\Theta = \lambda m \Theta \left[ \ln\left(1 + \frac{1}{\lambda m \Theta}\right) - \frac{1}{1 + \lambda m \Theta} \right]$$ This additional (1 - ρₖ) factor is structurally analogous to logistic growth limitations in population dynamics and leads to new qualitative behavior in Θ.

3. Nonmonotonic Interaction Probability and Analytical Characterization

Unlike the original model, where Θ increases monotonically with λ, the mHMF model exhibits nonmonotonicity: Θ(λ) increases, reaches a maximum at some intermediate λ_p m, and then decreases for larger infection rates. The peak value Θ_p and its position λ_p m are determined by: $$\Theta_p = \frac{\sqrt{\lambda_p m} - 1}{\lambda_p m}$$

$$1 + \sqrt{\lambda_p m} = \lambda_p m \ln \left( \frac{\sqrt{\lambda_p m}}{\sqrt{\lambda_p m} - 1} \right)$$

This result formally captures how mitigation mechanisms can reverse the trend of increasing infectious contact probability, even as λ grows. The nonmonotonicity in Θ—effectively, the expected probability of transmitting infection along a random edge—encodes the feedback between escalating infection and behavioral or structural suppression.

4. Prevalence Behavior and Control Implications

Despite the nonmonotonicity in Θ, the overall prevalence ρ remains a monotonically increasing function of λ in both HMF and mHMF models. The derivative is given by: $$\frac{d\rho}{d\lambda} = 2m(\Theta + \lambda \frac{d\Theta}{d\lambda}) \Psi_2(\lambda m \Theta)$$ where: $\Psi_2(x) = 2 - 2x \ln(1 + 1/x) - \frac{1}{1+x}$ Both terms are strictly positive, ensuring monotonic prevalence growth. However, due to the mitigation effect, for large λ, the prevalence may increase more slowly or saturate. The probability that an edge leads to an infected node (Θ) is damped at high prevalence levels, offering a mechanistic route by which epidemics can be self-limiting due to the reduced effective connectivity of the infected population.

5. Broader Modeling and Public Health Relevance

The inclusion of a mitigation factor in the mHMF model offers a framework for analytically quantifying the impact of realistic behavioral modifications—such as isolation or hospitalization—on epidemic outcomes in heterogeneous networks. Even for networks with λ_c = 0 in the classic SIS scenario, mitigation-induced nonmonotonicity in Θ introduces new dynamical phases: initial spread, maximal transmission regime, and subsequent suppression as infected fraction rises. This result provides a mathematical explanation for observed epidemic slowdowns under isolation or capacity saturation and informs design of interventions that aim to reduce the effective network degree of infected individuals (e.g., through aggressive self-isolation or prioritization of hospital admission).

6. Analytical Summary and Outlook

Model	Θ(λ) Behavior	Epidemic Threshold (λ_c)	Prevalence ρ(λ)
HMF	Monotonically increasing	λ_c = 0 (scale-free)	Monotonically increases
mHMF	Nonmonotonic: rises then falls	λ_c = 0 (scale-free)	Monotonically increases

This mathematical extension refines the classical epidemic modeling paradigm by precisely quantifying how containment, isolation, or behavioral mitigation alter susceptibility propagation, even in highly heterogeneous contact structures. The analytical results underscore that, in scale-free systems, targeted reduction of infected contacts can introduce effective saturation and potential dynamical control, altering the progression and persistence of epidemic processes (Kim et al., 8 Jan 2025).