Immunization Success Rate (ISR)

• Immunization Success Rate (ISR) is a quantitative metric defining the fraction or probability that susceptible nodes become immune via vaccination, infection, or engineered strategies.
• It is estimated using domain-specific methodologies such as epidemiological compartment models, network dynamics, and seroprevalence surveillance to inform outbreak thresholds.
• ISR serves as a critical parameter in setting epidemic thresholds, enabling targeted immunization strategies and guiding practical interventions to achieve herd immunity.

Immunization Success Rate (ISR) is a quantitative metric and modeling parameter that measures the effectiveness of immunization interventions—by vaccination, infection, or engineered immunity—across diverse domains including epidemiology, stochastic population dynamics, and network-based spreading processes. ISR encodes the fraction or probability that a susceptible population, individual, or targeted node set is rendered immune against a given hazard (e.g., disease, information spread, or system perturbation), thereby directly modulating outbreak thresholds, extinction probabilities, and the macroscopic evolution of system dynamics.

1. Formal Definitions and Domain-Specific Instantiations

The definition of Immunization Success Rate (ISR) is domain-specific, adjusting to the mechanistic details and sampling frameworks in different modeling regimes:

• Epidemiological Compartment Models: In classical and extended SIR-type models (e.g., SIRVVD, SAIVR), ISR typically denotes the per-individual vaccine efficacy (probability of immunity post-vaccination) or, in bulk interventions, the overall proportion of susceptibles immunized by a campaign (Omae et al., 2022, Angeli et al., 2021, Dong et al., 2020).
• Network-based Infection Models: ISR corresponds to the probability that targeted nodes in a network are successfully immunized, either by actual coverage (model A) or by reduction in susceptibility (model B) (Tanimoto, 2011, Tanimoto, 2011).
• Stochastic Tumor Models: ISR is defined as the stationary probability mass in the extinction (zero population) state, a function of immunization strength and noise parameters (Bose et al., 2008).
• Seroprevalence Surveillance: ISR is operationalized as the estimated current fraction of a regional population possessing functional immunity, assessed via serological markers for antibodies, regardless of source (natural infection or vaccine) (Si et al., 2022).

These definitions are operationalized via explicit system equations, often as threshold-controlling parameters or as estimable quantities in statistical inference.

2. Analytical Frameworks Incorporating ISR

ISR fundamentally enters dynamical equations as a coefficient for transitions from susceptible to immune states or as a modifier of effective transmission rates. Representative analytic frameworks include:

• SIRVVD Model: Two-dose COVID-19 models introduce success rates ε₁, ε₂, where ε_j is the fractional reduction in susceptibility after j doses:

$$\begin{align*} \dot{V}_1 &= \alpha_1 S - \theta_1 V_1 - \alpha_2 V_1 - (1 - \epsilon_1)\beta V_1 I \ \dot{V}_2 &= \alpha_2 V_1 - \theta_2 V_2 - (1 - \epsilon_2)\beta V_2 I \end{align*}$$

with thresholds set by $$R_\text{eff} = \beta(N' - \epsilon_1 V_1 - \epsilon_2 V_2)/\gamma$$; the epidemic is suppressed iff $R_\text{eff} < 1$ (Omae et al., 2022).

• SAIVR Model: The instantaneous successful immunization flux is $ISR(t) = \lambda \epsilon V(t)$, converging to $ISR \approx \lambda \delta$ in the regime of steady daily vaccine rollout $\delta$ with per-dose efficacy $\lambda$ (Angeli et al., 2021).
• Network Models (SIR/SIS): For targeted immunization subsets $T$ in a degree-heterogeneous network, ISR $r$ modifies the epidemic threshold:

$$\beta_c = \frac{\langle l \rangle}{\langle k l \rangle - r \langle k l \rangle_T}$$

Only $r=1$ for hub nodes is sufficient to restore a finite threshold when the degree distribution is scale-free with $2 < \gamma < 3$ (Tanimoto, 2011). For bipartite contacts, the threshold depends bilinearly on $(\alpha_1, \alpha_2)$ ISR pair (Tanimoto, 2011).

• Stochastic Population Dynamics: ISR is computed as the integral of the stationary probability density at extinction:

$\text{ISR} = \int_0^{x_{\text{th}}} P_s(x) dx$

where $P_s(x)$ depends on the immunization strength parameter $\beta$ among others (Bose et al., 2008).

• Statistical Serosurveillance: ISR is inferred from MRP-adjusted seroprevalence time series, $$ISR(t) = \sum_s (N_s / N_{\mathrm{total}}) \hat{p}_s(t)$$, with natural- and vaccine-derived immunity components partitioned via specific IgG markers (Si et al., 2022).

3. Estimation Methodologies for ISR

Reliable ISR quantification requires domain-adapted estimation procedures:

• Epidemiological Time-Series and HMMs: For SIAs (e.g., measles), ISR (denoted $p$ or $\varphi$) is inferred via a two-stage approach: first, estimate reporting probability ($\rho$) using OLS on no-campaign stretches; second, fit a TSIR HMM with MCMC to recover a posterior over ISR, summarized by the posterior median and credible intervals (Dong et al., 2020).
• ODE Parameter Fitting: ISR is treated as a fixed parameter (e.g., efficacy $\epsilon_j$), empirically determined from trial data or through adjustment to epidemic curves (Omae et al., 2022, Angeli et al., 2021).
• Network Immunization Strategies: ISR (fraction $r$ or vector $(\alpha_1, \alpha_2)$) is fixed ex ante as the chosen coverage or success rate of immunization in targeted degree or bipartite classes (Tanimoto, 2011, Tanimoto, 2011).
• Seroprevalence MRP Modeling: Bayesian hierarchical logistic regression poststratifies antibody status to produce a real-time ISR time series, with corrections for sensitivity/specificity and partitioning by antibody type for natural vs. vaccine-derived immunity (Si et al., 2022).
• Stochastic Model Analysis: ISR is a function of controllable parameters (e.g., $\beta$), evaluated numerically via integration of analytic $P_s(x)$ or approximate expressions for extinction times (Bose et al., 2008).

4. ISR in Epidemic Thresholds, Extinction, and Outbreak Control

ISR serves as the critical control parameter for system-level transitions:

• Epidemic Suppression: In deterministic and networked epidemic models, increasing ISR directly raises the epidemic threshold $\beta_c$ (minimum transmission needed for persistent outbreaks). In scale-free networks, a partial ISR below unity in hub nodes is insufficient—strictly $r = 1$ is required for epidemic control (Tanimoto, 2011, Tanimoto, 2011).
• Herd Immunity Calculations: SIRVVD and SAIVR frameworks demonstrate that higher ISR (i.e., improved vaccine efficacy or broader coverage) substantially reduces the critical coverage fraction for herd immunity, particularly when facing high-transmissibility variants (Omae et al., 2022, Angeli et al., 2021).
• Stochastic Extinction: In tumor-growth models, ISR quantifies the probability mass at extinction; increasing ISR via immunization strength $\beta$ or by minimizing noise correlations accelerates extinction and raises the likelihood of total tumor eradication (Bose et al., 2008).
• Real-World Surveillance: In community seroprevalence, ISR tracks changing population-level immunity in near real time, revealing gaps between vaccine coverage and total functional immunity, as well as quantifying the impact of natural infection waves and waning immunity (Si et al., 2022).

5. Representative Parameter Ranges and Empirical Results

The empirical magnitude and interpretation of ISR are dataset- and context-dependent:

Domain / Model	Typical ISR Formulation	Empirical Range (where reported)
SIRVVD (two-dose vaccine)	$\epsilon_1$, $\epsilon_2$	0.356 (dose 1), 0.880 (dose 2) (Omae et al., 2022)
TSIR (measles campaign)	$\varphi = p$ (campaign efficacy)	0.499 (median, 0.145–0.859 CI) (Dong et al., 2020)
SAIVR (COVID-19 ODE model)	$\lambda\delta$	0.009–0.0285 per day (0.9%–2.85%) (Angeli et al., 2021)
Complex Net/Targeted Immunization	$r$ (target coverage)	Effective only for $r=1$ in hubs (Tanimoto, 2011)
Tumor extinction probability	$\int_{0}^{x_{th}} P_s(x) dx$	ISR($\beta$) increases from 0.5 to 0.9 as $\beta$ raised from 0.0 to $\sim$1.2 (Bose et al., 2008)
SARS-CoV-2 community serology	Bayesian MRP-adjusted prevalence	74% immune (summer 2021), 45% vaccine coverage (Si et al., 2022)

ISR is often subject to considerable uncertainty due to underreporting, reporting lag, model assumptions, and real-world heterogeneity.

6. Interplay of ISR with Heterogeneity, Targeting, and Network Structure

Heterogeneity in contact patterns, susceptibility, and immunization coverage is fundamental in ISR-mediated control:

• Heterogeneous Network Topologies: In highly skewed degree distributions, immunizing solely random or low-degree nodes is ineffective. ISR must reach unity in the highest-degree class to enforce a finite outbreak threshold (Tanimoto, 2011). In bipartite networks, perfect ISR in one critical class allows moderate ISR in the other (Tanimoto, 2011).
• Targeted Campaigns and Partial Efficacy: Partial ISR achieved through campaign inefficiency, vaccine refusal, or biological limits cannot guarantee extinction in heavy-tailed systems unless aligned with structural leverage points (e.g., removing all "superspreaders") (Tanimoto, 2011, Tanimoto, 2011).
• Time-Dependent and Waning ISR: In systems where immunity wanes or is acquired over time (e.g., via ongoing vaccination or infection), ISR becomes a dynamic, temporally resolved quantity best captured via continuous statistical surveillance (Si et al., 2022, Angeli et al., 2021).

ISR thus encapsulates both the biological efficacy and operational reach of immunization interventions, serving as the critical lever in disease elimination, outbreak prevention, or stochastic extinction.

7. Practical Impact and Implementation Guidelines

• Optimal Immunization Strategies: Analytical thresholds yield closed-form guidance on minimal ISR necessary for epidemic control in distinct systems: increase both the efficacy and coverage, target high-degree nodes or influential populations, and monitor for variant-induced shifts in required ISR (Omae et al., 2022, Tanimoto, 2011, Tanimoto, 2011).
• Data-Driven ISR Estimation: Effective estimation pipelines (e.g., MRP serology, TSIR HMM) must account for sampling bias, test errors, and underreporting, providing robust ISR trajectories for policy input (Dong et al., 2020, Si et al., 2022).
• Validation and Surveillance: ISR as an operational metric should be validated against independent outbreak data (PCR positivity, ED visits) and cross-referenced to official immunization records, especially where vaccine-induced and natural immunity must be disambiguated (Si et al., 2022).
• Adaptation to System Evolution: Ongoing parameter recalibration is essential as system properties (variant transmission, immunity duration) fluctuate (Angeli et al., 2021, Omae et al., 2022).

ISR, when rigorously defined and empirically estimated, directly determines the threshold for successful immunization-driven control or elimination across domain boundaries, guiding both modeling and operational intervention.