Statistical physics of vaccination

1.1. Background

• Human behavior, including vaccine scares and inadequate public support, can cause disease resurgence or undermine local elimination and global eradication despite available control technologies.Examples include measles-mumps-rubella and oral polio vaccine scares, as well as failures during the 2014 West African Ebola outbreaks.Source: §1.1 · p. 2

1.2. Motivation and objective

• Coupled behavior-disease models aim to capture the two-way interplay between human behavior and infectious-disease dynamics, potentially improving insight and predictions over models that neglect behavior.The report also considers one-directional coupling from behavior to disease or disease to behavior.Source: §1.2 · p. 2
• The report reviews physics-informed mathematical models of behavior-disease interactions and identifies open questions and promising directions for future research.It emphasizes methods and concepts from physics, especially statistical physics, and selects references intended to represent the field’s breadth.Source: §1.2 · p. 2

1.3. Outline of the report

• The report progresses from epidemiological foundations and mean-field models to network and spatial models, coupled behavior-disease dynamics, behavioral data, digital data, and concluding perspectives.Sections 3–5 establish disease-transmission models without explicit behavior before later sections address behavioral coupling and validation.Source: §1.3 · pp. 2–3

Pathogen biology

• Microparasites are the main modeling focus because their short generation times, direct within-host reproduction, and extensive long-term data support well-parameterized transmission models.Microparasites include viruses, bacteria, and protozoans, whereas macroparasites include ticks, mites, helminths, and flukes.Source: §2 · p. 3

Transmission biology

• Transmission may be direct through close contact or indirect through vectors and environments, with urbanization, habitat loss, travel, and trade increasing opportunities for spread.Transmission route, infectiousness timing, and infectious-period duration influence disease dynamics and intervention effectiveness.Source: §2 · p. 3
• Latent, incubation, and infectious periods must be distinguished because symptomatic surveillance is generally ineffective when infectiousness begins before symptoms.The proportion of transmission occurring before symptom onset is an important measure of outbreak controllability.Source: §2 · p. 4

Interventions and modeling

• Infectivity, virulence, and pathogenicity describe distinct pathogen properties shaped by pathogen–host interactions, while antibiotics, distancing, hygiene, and vaccination provide complementary control strategies.Antibiotic overuse can select resistance, and social distancing can reduce epidemic attack rates and help explain multiple infection waves.Source: §2 · pp. 4, 6, Figure 1 · p. 5

Population immunity and vaccination

• Herd immunity reduces susceptible hosts’ infection risk, depends on disease infectivity and population mixing, and is commonly induced through mass vaccination, although vaccine opposition can undermine protection.Immunity may arise naturally, through previous exposure, or through vaccination.Source: §2 · p. 4
• Vaccines provide direct individual protection and indirect population protection, but licensed vaccines can have primary failure rates of 2–50% even under ideal clinical-trial conditions.Effectiveness comparisons require vaccinated and unvaccinated groups to be comparable in exposure, infection risk, access, and diagnosis opportunity.Source: §2 · p. 5

3.1. Mathematical epidemiology: a story shaped by statistical physics

• McKendrick and Kermack transformed contagion modeling by importing statistical-physics mass action, representing infection as collisions between susceptible and infective individuals.For infections without immunity, the resulting logistic dynamics drive prevalence toward I = 1 through a sigmoid trajectory.Source: §3.1 · p. 7

3.2. Basic concepts: homogenous social mixing, infection incidence, force of infection

• Compartmental models represent infection life courses with transitions among susceptible, exposed, infective, and removed classes, accommodating lifelong immunity or reinfection.SEIR describes infections such as measles, mumps, and rubella, whereas SEIRS allows return to susceptibility for pertussis and varicella.Source: §3.2 · p. 8
• Under homogeneous mixing, the force of infection is λ = βI, where β combines contact behavior and biological transmission, and incidence follows the standard mass-action law.The framework assumes similar social activity and transmission probability across individuals, with compartment populations represented by dynamic variables.Source: §3.2 · pp. 8–9
• Mean-field SEIRS formulations assume exponentially distributed compartment durations, large nearly closed populations, demographic balance, and no infection-related mortality or vertical transmission.Restricting parameters yields SIRS, SIR, SIS, and outbreak models including the Kermack–McKendrick SIR model.Source: §3.2 · pp. 9–10

3.3. Reproduction numbers of epidemic spread

• The basic reproduction number R0 is the expected secondary cases from one typical infective in a wholly susceptible population and determines invasion when R0 > 1.During initial spread, R0 also gives the per-generation growth factor; real-time growth satisfies R0 = e^(rD) when generation time equals infective duration D.Source: §3.3 · p. 11
• The effective reproduction number RE declines as susceptible individuals are depleted, reaches 1 when incidence peaks, and falls below 1 during the epidemic’s declining phase.With R0 = 2, immunizing 25% reduces RE to 1.5, while immunizing 50% reduces RE to 1.Source: §3.3 · p. 11
• For a perfect vaccine, critical coverage x_c satisfying R0(1 − x) < 1 prevents outbreaks or eliminates endemic infection through herd immunity; measles requires x_c = 93.3% when R0 ≈ 15.R0 and RE depend on the chosen mean-field model, while next-generation operators generalize reproduction-number calculation through their dominant eigenvalue.Source: §3.3 · pp. 11–12

3.4. SIR and SEIR models for epidemic outbreaks

• The SIR formulation assumes a short, fixed-population outbreak introduced into an initially wholly susceptible population, neglecting demographic change and the exposed class.These assumptions make the model purely transient and apply the classical Kermack–McKendrick framework.Source: §3.4 · p. 12
• R0 determines whether an outbreak occurs: with initial susceptible fraction S(0), invasion requires R0S(0) > 1, equivalently S(0) > 1/R0.If R0 < 1/S(0), prevalence does not increase; for a largely susceptible population, the threshold is R0 = 1.Source: §3.4 · pp. 12–13
• After initial exponential growth, prevalence peaks when S(t) = 1/R0 and then declines as the susceptible pool and effective reproduction number decrease.For a wholly susceptible population, maximum prevalence increases with R0; examples are approximately 45% for Spanish flu and 80% for measles-like R0 ≈ 15.Source: §3.4 · p. 13
• The final attack rate increases concavely with R0 but remains below complete infection for immunity-conferring infections; for Spanish flu with R0 ≈ 2, it is close to 80%.Measles-like R0 values can produce attack rates only slightly below 100%.Source: §3.4 · p. 13

3.4.1. The SEIR model for outbreaks

• The SEIR model adds a latent, non-infective phase, yet the same threshold R0 < 1/S(0) implies die-out, while R0 > 1/S(0) produces initial exponential growth.The exposed–infective subsystem is cooperative, and the BRN governing the transient SIR model also governs SEIR dynamics in a vectorial sense.Source: §3.4.1 · pp. 14–15
• Although SEIR dynamics can be increasing in vectorial terms, individual components may temporarily decrease; with E(0) = 0, infective prevalence initially satisfies I′(0) = −γI(0) < 0.When the latent phase is much longer than the infective phase, one initial infectious subject can produce no new infectious subjects during the first infective-period interval.Source: §3.4.1 · p. 15

3.5. Emergence and features of endemicity: the SIR and SEIR models with vital dynamics.

• When R0 ≤1 infection dies out and the disease-free equilibrium is globally attractive, whereas R0 >1 creates a globally attractive endemic state with prevalence increasing in R0.The threshold corresponds to a transcritical bifurcation; for childhood infections with γ >> µ, endemic prevalence satisfies Ie < (µ/γ) << 1.Source: §3.5 · p. 16
• Births rebuild the susceptible pool after outbreaks, producing alternating epidemic and demographic-replenishment phases that converge toward endemicity and can generate biennial oscillations.For measles-like parameters, 1/γ ≈1 week, R0 ≈15, and life expectancy 1/µ =75 years yield an inter-epidemic period T ≈2.01 years.Source: §3.5 · pp. 15–16, Figure 6 · p. 17

3.5.1. Periodic contact rates

• For periodic contact rates in SIR models, the disease-free equilibrium is globally attractive when average R0(t) ≤1 and unstable when average R0(t) >1, independent of β(t)’s shape.In SEIR models, the disease-free equilibrium instead depends on the whole shape of β(t), with Floquet eigenvalues determining stability; yearly forcing can also produce nonlinear resonances.Source: §3.5.1 · pp. 17–18

3.5.2. Nonlinear contact rates as a phenomenological route to behavior change

• Prevalence-dependent behavior can reduce contacts or increase protection, destabilize endemic equilibria through delayed responses, and interact with annual forcing to produce sustained or chaotic oscillations.This extends models beyond constant or exogenously periodic transmission rates and includes movement between inactive and core active groups with prevalence-dependent transfer.Source: §3.5.2 · p. 19

3.6. Mass vaccination and herd immunity for vaccine preventable infections: the SIR model with vaccination at birth

• The model assumes vaccination of a constant proportion x of newborn susceptible individuals using a perfect vaccine with 100% take and lifelong protection.Successfully vaccinated individuals are fully protected and behave as removed individuals; waning immunity or residual susceptibility requires additional compartments.Source: §3.6 · pp. 19–20
• Coverage x ≥ 1 − 1/R0 eliminates infection; below this threshold, infection persists endemically with lower prevalence and longer inter-epidemic periods.For measles-like parameters, x = 0.90 raises the long-term inter-epidemic period above 10 years, compared with about 2 years before vaccination.Source: §3.6 · p. 20
• In the measles-like example, x = 0.95 eliminates infection, whereas x = 0.80 and x = 0.90 retain endemic infection with much lower prevalence and markedly longer inter-epidemic periods.For x = 0.90, the inter-epidemic period initially exceeds 20 years before converging toward approximately 10 years; these scenarios are illustrated in Fig. 7.Source: §3.6 · p. 20, Figure 7 · p. 21
• Vaccination reduces the force of infection, shifting the endemic level from µ(R0 − 1) to µ(R0(1 − x) − 1) and increasing the average age at infection.Because unvaccinated individuals may remain susceptible longer, delayed infection can occur at ages associated with greater risk of serious disease, especially for measles and rubella.Source: §3.6 · p. 20

3.7. Return to susceptibility: SIS and SIRS models

• Loss of immunity itself can generate endemicity by rebuilding the susceptible pool, and globally attractive endemic equilibria can occur even without vital dynamics.SIS assumes direct return from infection to susceptibility, whereas SIRS inserts a prolonged recovered period before susceptibility returns.Source: §3.7 · pp. 21–22
• SIS and SIRS models predict extinction or endemicity according to whether ⟨R0(t)⟩ is below or above one, with constant transmission yielding globally attractive endemic states when R0 > 1.In SIS models with periodic transmission, infection approaches a one-year periodic endemic solution; with constant transmission, the equilibrium is constant and globally attractive.Source: §3.7 · pp. 21–22
• Imperfect vaccination can leave recipients fully susceptible, temporarily immune, or partially susceptible to breakthrough infection, making elimination more difficult; these possibilities occur with varicella vaccination.Reduced susceptibility can be represented by λ = σβI with σ < 1, while vaccine-derived immunity may wane faster than natural immunity.Source: §3.7 · p. 22
• Backward bifurcation creates a vaccination range with a stable disease-free equilibrium alongside unstable and stable endemic equilibria, so elimination requires x > x∗c.Below x0c, only one endemic steady state coexists with an unstable disease-free equilibrium.Source: §3.7 · p. 22

3.8. Remarks on the wise use of the mean-field principle in theoretical epidemiology

• Deterministic epidemic equations arise as large-population mean-field limits of stochastic birth-death processes, which neglect correlations by approximating ⟨Y^2(t)⟩ with (⟨Y(t)⟩)^2.Exact stochastic dynamics can be represented by Gillespie simulations and master equations, while system-size expansions yield Fokker-Planck equations and stochastic differential equations.Source: §3.8 · pp. 22–23
• Mean-field and system-size approximations require sufficiently large state variables and can fail when any state variable becomes small, so deterministic models need condition-specific validation.For small populations, stochastic birth-death models are required; approximation accuracy also varies over time because event rates depend on the state.Source: §3.8 · pp. 23–24
• Stochastic extinction can occur between epidemics when prevalence is low, including under vaccination, whereas deterministic periodically forced models may maintain positive infection indefinitely.The critical community size for prevaccination measles was estimated at 250000-400000 and depends strongly on infectious and latent-period distributions and spatial structure.Source: §3.8 · p. 24
• Stochastic amplification can convert deterministic damped endemic oscillations into sustained oscillations, and periodic vaccination may theoretically induce eradication through fluctuation-driven resonance.These stochastic effects demonstrate phenomena absent from deterministic counterparts, including resonance-based population extinction.Source: §3.8 · p. 24

3.9. Space and beyond: mean-field metapopulations

• Metapopulation models divide populations into locally modeled patches linked by commuting or other population fluxes, enabling analysis of spatial epidemic spread and persistence.The approach accommodates patch-specific dynamics and inter-patch connections, including reduced mobility of infectious individuals and transmission through patch networks.Source: §3.9 · p. 25
• Multi-patch systems reproduce traveling waves, spatial instabilities, and synchronized or asynchronous patterns, while vaccination can explain the observed post-vaccination de-correlation of measles time series across UK cities.Unforced SIR and SEIR oscillations synchronize at very low connectivity, whereas seasonal forcing preserves some phase differences and yields more plausible inter-epidemic prevalence levels.Source: §3.9 · pp. 25–26

3.9.1. Large scale meta-population networks

• Global epidemic spread requires a second threshold beyond R0 > 1, determined by population mobility and inter-patch network topology; heterogeneous networks can make mobility restrictions less effective.In very large heterogeneous networks, the second threshold approaches the classical one and becomes nearly independent of population mobility.Source: §3.9.1 · p. 26

3.9.2. Metapopulation models beyond space: social and age-specific heterogeneities

• Beyond geography, multigroup models use transmission matrices to represent social and age-specific contact heterogeneity, supporting analyses of sexually transmitted infections and vaccination-driven changes in age at infection.Age-since-infection and age-since-vaccination variables modulate contact, disease duration, mortality risk, immunity, and vaccine-related side effects; continuous versions produce integro-differential models.Source: §3.9.2 · pp. 26–27

4. Basic concepts and methods in (non-behavioral) epidemiological modeling and vaccination: network models

• Real-world contact networks are heterogeneous complex systems with hubs, communities, clustering, correlations, and heavy-tailed connectivity distributions, so regular lattices rarely suffice for epidemic modeling.These structural features motivate mathematical approaches that capture large-scale fluctuations and correlated organization in networks.Source: §4 · p. 28

4.1.1. Definitions and Notations

• Network definitions formalize nodes and links through adjacency matrices and characterize structure using degree distributions, paths, correlations, centrality, clustering, and community structure.Degree correlations range from −1 ≤r ≤1; social networks are described as assortative, whereas technological and biological networks are disassortative.Source: §4.1.1 · pp. 28–30

4.1.2. Type of networks

• The section distinguishes random, small-world, scale-free, evolving, weighted, spatial, temporal, adaptive, and multilayer networks for representing different connectivity patterns.Random graphs yield binomial or Poisson degree distributions, small-world rewiring adds long-range shortcuts while preserving clustering, and Barabási–Albert networks combine growth with preferential attachment.Source: §4.1.2 · pp. 30–31

4.2. Epidemic spread over networks

• Network topology replaces homogeneous mixing by restricting transmission to contacts with heterogeneous probabilities, while percolation theory maps outbreaks to clusters connected by occupied edges.Transmissibility T is the probability of transmission before recovery, and outbreak size corresponds to the size of the reachable occupied-edge cluster.Source: §4.2 · pp. 33–34

4.2.1. Percolation theory and spreading process

• For uncorrelated networks, epidemics emerge when T exceeds the critical transmissibility Tc; for scale-free networks with ζ < 3, Tc tends to zero as N →∞.The percolation framework identifies the epidemic onset at TG′1(1, T) = 1; above Tc, a giant component forms, while heterogeneous networks with ζ < 3 can sustain spread for arbitrarily small infection probabilities in the infinite-size limit.Source: §4.2.1 · p. 35

4.2.2. Disease spreading in multilayer networks

• Multilayer networks support same-node inter-layer, other-node inter-layer, and other-node intra-layer transmission, with survival and absolute-dominance thresholds for interacting processes.Different layers and inter-layer links may have distinct transmissibilities, motivating generalized epidemic mean-field models with more complex states than SIS or SIR.Source: §4.2.2 · p. 36

4.2.3. Simulation-based models

• Simulation-based models construct large, demographically and socially structured contact networks with household, school, workplace, hospital, and public-place interactions to estimate epidemic variability.Microsimulation incorporates stochasticity and observed census or demographic characteristics, including time-varying daily activities.Source: §4.2.3 · p. 36
• Scale-free networks can sustain disease spread at vanishingly small transmission probabilities, eliminating an epidemic threshold in the infinite-network limit.This persistence reflects the joint roles of infection processes and heterogeneous network topology.Source: §4.2.3 · p. 37
• Simulation models are difficult to assess for sensitivity to network representativeness, between-community variation, and missing rare but important contact structures.Idealized networks and analytical tools help identify which structural elements determine epidemic dynamics.Source: §4.2.3 · p. 37

4.3. Vaccinations over networks

• Network vaccination is formulated as site-bond percolation: removing immune nodes reduces transmissibility, and eradication occurs when the network falls below its percolation threshold.In heterogeneous networks, optimal connectivity-dependent vaccination can eradicate disease with fewer vaccinated individuals than uniform coverage.Source: §4.3 · pp. 37–38
• Targeted vaccination of highly connected nodes strongly improves infection tolerance in scale-free networks, requiring an exponentially small critical fraction across broad spreading rates.By contrast, uniform vaccination requires complete coverage when <k^2>→∞, whereas targeting hubs can eradicate infection with a tiny immune fraction.Source: §4.3 · p. 38
• Targeted vaccination is effective in highly heterogeneous scale-free networks but offers little advantage over uniform vaccination in small-world networks with limited heterogeneity.Acquaintance vaccination approximates hub targeting without requiring global degree information and consistently uses fewer vaccinated nodes.Source: §4.3 · p. 39

5.1. Uniform (or random) vaccination

• Uniform vaccination protects homogeneous networks when x exceeds xc, but in scale-free networks with vanishing epidemic thresholds it requires x = 1 in the thermodynamic limit.For scale-free networks with degree exponent ζ < 3, ⟨k2⟩ diverges and xc →1 for both SIS and SIR models; measles vaccination requires 95% coverage.Source: §5.1 · p. 40

5.2.1. Degree-based targeted vaccination: theoretical prediction of the vaccination threshold

• Vaccinating high-degree nodes reduces degree variance, raises the epidemic threshold, and can make βIc exceed βI, while the theoretical vaccination threshold is exponentially small across broad spreading-rate ranges [275].For 2 < ζ < 3, progressively lowering the cutoff kc by vaccinating more large-degree nodes increases the epidemic threshold.Source: §5.2.1 · pp. 40–42
• Targeted vaccination sharply outperforms random vaccination on scale-free networks, yielding an estimated threshold xc ≃0.16 while random vaccination suppresses prevalence only as x →1.At βI = 0.25, targeted vaccination produces a sharp prevalence drop above its threshold, whereas random vaccination decays slowly.Source: §5.2.1 · p. 42, Figure 9 · p. 43

5.2.2. Centrality-based targeted vaccination: definitions and simulation-based models

• Centrality-based strategies exploit network bridges, distances, influential neighbors, or spectral importance; random-walk centrality minimizes infected cases at low coverage, while PageRank performs best among compared indexes.Adaptive recalculation can improve efficiency but requires more information and may be impractical beyond very small populations.Source: §5.2.2 · pp. 43–45

5.3. Vaccination without global knowledge

• Because global network structure is seldom known, local-information strategies seek comparable efficiency without identifying globally influential nodes, addressing ambiguity in how social-network links are defined.Targeted vaccination otherwise requires complete knowledge of network structure to identify and immunize influential nodes.Source: §5.3 · p. 45

5.3.1. Acquaintance vaccination: theoretical prediction of the vaccination threshold

• Acquaintance vaccination randomly selects nodes but vaccinates a neighbor, thereby preferentially targeting higher-degree nodes without global knowledge; percolation theory gives pc when branching becomes subcritical and the epidemic is contained.Under the uncorrelated-network assumption, the branching process continues when the relevant sum exceeds 1 and is contained when it is below 1.Source: §5.3.1 · pp. 45–46
• In scale-free networks, acquaintance vaccination requires a lower critical eradication threshold than uniform vaccination across both homogeneous and heterogeneous degree distributions.The comparison is shown for SIS disease spreading as a function of degree exponent ζ.Source: §5.3.1 · p. 47, Figure 11 · p. 47

5.3.2. Other strategies: definitions and simulation-based models

• Local network strategies include random walks, common acquaintance vaccination, degree-biased exploration, and D-steps targeting of the largest-degree node within distance D.Common acquaintance vaccination is somewhat more effective than acquaintance vaccination, while using neighbor-degree information significantly improves acquaintance-based efficiency.Source: §5.3.2 · p. 47
• Targeted vaccination performs best when global network structure is known, whereas D-steps vaccination is the practical compromise when only local structure is available.D-steps vaccination is more effective than acquaintance vaccination, is outperformed only by targeted vaccination across tested D and βI values, and has robustness close to targeted vaccination.Source: §5.3.2 · pp. 48–49

5.4.1. Vaccination on community networks

• Community-network vaccination should target bridge nodes or bridge hubs because weak inter-community ties carry disease between communities, even when bridge importance is not reflected by degree centrality.The community bridge finder uses local self-avoiding walks and is more efficient than hub-targeting strategies, while bridge-hub detection further improves prevention over acquaintance vaccination and CBF.Source: §5.4.1 · pp. 49, 51

5.4.2. Vaccination on adaptive networks

• Adaptive rewiring increases epidemic thresholds and reduces infections, while vaccination targeting infected bridge nodes on SI links outperforms random infected-node vaccination and need not be implemented as early as possible.The ISSI strategy is more effective than ISR because it cuts invasion pathways into susceptible clusters; optimal vaccination can follow the community structure induced by early rewiring.Source: §5.4.2 · pp. 51, 53

5.4.3. Vaccination on multiplex networks

• On multiplex networks, random vaccination works better on homogeneous ER networks, whereas targeted vaccination is needed for heterogeneous SF networks; layer correlations and cross-layer protection alter these outcomes.Multiplex-node and layer-node vaccination provide different protection frameworks, and vaccinating one layer does not efficiently protect individuals on other layers.Source: §5.4.3 · pp. 54, 56

• Vaccines have prevented substantial infectious-disease morbidity and mortality, including millions of children’s lives saved annually and major reductions in measles deaths.Smallpox eradication relied substantially on ring vaccination, while large-scale immunization reduced estimated annual measles deaths from 766,000 in 2000 to 164,000 in 2008.Source: §6 · pp. 56–57
• Vaccine hesitancy and refusal can reduce coverage and cause resurgent infections even where vaccination is accessible, as illustrated by measles and polio outbreaks.These barriers may become increasingly important as logistical, financial, technological, and administrative obstacles recede and eradication thresholds approach.Source: §6 · p. 57
• Herd immunity can reduce perceived infection risk and vaccination motivation, creating complacency and conditions for rumors, regardless of whether non-vaccinators consciously free-ride.Observed increases in pertussis and MMR vaccination during and after outbreaks demonstrate feedback between disease prevalence and vaccinating choices.Source: §6 · p. 57
• Vaccinating behavior responds to disease prevalence, including during the 2014 Disneyland measles outbreak, demonstrating strong coupling particularly near the elimination threshold.Surveyed motives may vary between outbreaks and elimination periods, but observed behavioral responses provide direct evidence of the coupling.Source: §6 · pp. 57–58
• Peer groups, medical professionals, and contagious social influence shape vaccine decisions, making contact networks and statistical-physics methods natural tools for modeling behavior-disease dynamics.Bluetooth and related technologies can help map social contact networks.Source: §6 · p. 58
• Behavior-inclusive models explain infectious-disease prevalence more effectively than behavior-free models with little or no parsimony penalty, show retrospective predictive power, and can use surveys, games, and social-media data.These models also exhibit dynamics analogous to physical systems, including phase transitions, oscillations, and spatio-temporal patterns.Source: §6 · pp. 58–59

7. Basic concepts in behavioral modeling

• Behavioral models are grouped into phenomenological, game theoretical, and psychological categories, although the distinctions are not always well defined.The classification concerns ways of modeling behavior rather than classifications of coupled behavior–disease interactions.Source: §7 · p. 59

7.1. Phenomenological models

• Phenomenological models describe behavioral effects without specifying underlying mechanisms, including prevalence-dependent transmission saturation and contagion metaphors for ideas or behaviors.Their simplicity can establish that behavior affects disease spread and clarify how information transmission produces emergent infection control, but contagion metaphors may overlook distinctions among social norms, social learning, and cognition.Source: §7.1 · p. 60

7.2. Game theoretical models

• Game-theoretical vaccination models define vaccinator and non-vaccinator strategies, assign payoffs, and analyze Nash equilibria or evolutionarily stable states, but classical rationality can misrepresent decisions shaped by social learning, bounded rationality, and imperfect information.Network and lattice extensions incorporate social influences, norms, rules of thumb, and more realistic cognitive processes, blurring the boundary with psychological models.Source: §7.2 · p. 61

7.3. Psychological models

• Psychological models formulate vaccination mechanisms through social learning and imitation, subjective expected utility, prospect theory, or Dempster–Shafer theory.These approaches model imperfectly rational strategy switching, coupled vaccination and contact-precaution decisions, distorted perceptions of rare risks, and awareness spreading through social connections.Source: §7.3 · p. 62

8.1. Phenomenological SIR models with information-dependent vaccine uptake

• Information-dependent models represent uptake as increasing with current or remembered disease trends, while their structure can also encompass delayed vaccination, adverse-event responses, and smoothed game-theoretic behavior.Exponential fading memory has characteristic length T = 1/a, and fast imitation dynamics recover the current-information phenomenological model.Source: §8.1 · pp. 63–64, §8.1.4 · pp. 65–66, §8.2.4 · p. 69

8.1.1. Implications of information-dependent vaccination: elimination “mission impossible”

• Prevalence-dependent vaccination generally makes infection elimination impossible because low prevalence reduces perceived vaccination benefits and drives uptake toward baseline levels.Elimination requires baseline uptake above the critical threshold, effectively reproducing mandatory-like coverage conditions.Source: §8.1.1 · p. 64

8.1.2. Memory-triggered oscillations

• Delayed behavioral feedback can generate sustained oscillations through Hopf bifurcations when vaccine uptake responds steeply to perceived risk within an intermediate memory-delay window.The result persists across more realistic memory kernels, generic prevalence responses, disease mortality, nonconstant populations, adverse-event information, and intrinsic SEIR delays.Source: §8.1.2 · pp. 64–65

8.1.3. Dynamic pattern for a measles-like infection

• With a 4-month average delay, the measles-like model converges to a limit cycle with an approximately 12-year inter-epidemic period, while uptake peaks at 96% but averages below 80%, under the critical coverage xc ≈0.93.Increasing prevalence raises vaccine demand after a delay, which lowers prevalence and subsequently feeds back to reduce uptake.Source: §8.1.3 · p. 65

8.2.1. Imitation dynamics

• Behavior-explicit models treat vaccination as a strategy that spreads through imitation, with switching determined by perceived payoff differences that may depend on infection and vaccination risks.If the payoff advantage has a constant sign, one strategy takes over; state-dependent payoff differences produce richer vaccination dynamics.Source: §8.2 · p. 66, §8.2.1 · p. 67

8.2.2. Basic SIR models with imitation-based vaccinating behavior

• Imitation-based SIR models always contain no-vaccinator disease-free, pure-vaccinator disease-free, and pre-vaccination endemic equilibria, while additional equilibria and their stability depend on perceived payoff structure.The vaccination fraction becomes a dynamic state variable representing parents’ current willingness to vaccinate.Source: §8.2.2 · p. 67

8.2.3. The baseline SIR model with imitation-based vaccinating behavior

• In the baseline imitation model, low relative disease cost prevents vaccine adoption, whereas high relative disease cost supports a partially vaccinated endemic state and sufficiently rapid imitation can cause Hopf-bifurcation oscillations.The fully vaccinated disease-free equilibrium is unstable because zero prevalence removes the perceived payoff of vaccination.Source: §8.2.3 · p. 68

8.2.4. An alternative model with myopic perception of the cost of vaccine adverse events

• When vaccination cost is perceived from current adverse-event burden, elimination is possible only if perceived disease-reemergence costs raise uptake above the critical threshold.The partial-vaccinator endemic state can oscillate through Hopf bifurcations when perceived disease cost is sufficiently steep and imitation speed lies within a finite interval.Source: §8.2.4 · pp. 69–70
• The vaccine-scare model fits pertussis and MMR data best when combining disease-related feedback with social learning, predicting pertussis trends for up to 10 years.Its full model outperformed simplified variants, while oscillations occurred for intermediate imitation rates; large rates could generate long inter-epidemic periods and uptake above the elimination threshold [465].Source: §8.2.4 · pp. 70–71

8.2.5. Delaying vaccination as a further strategy

• Adding delayed vaccination creates multiple stable endemic states and anti-phase switching between timely vaccination and delay, while the third strategy stabilizes dynamics relative to the timely-versus-nonvaccination model.The model represents timely vaccinators, delayers, and nonvaccinators, with prevalence-dependent age-specific infection risks and vaccination costs.Source: §8.2.5 · pp. 71–72

8.2.6. Dynamical effects of social norms

• Injunctive social norms generate bistability and oscillations but can stabilize the pure-vaccinator equilibrium at sufficiently strong social pressure, mitigating the prevalence-dependent impossibility of elimination.The model adds group-pressure terms to vaccinator and nonvaccinator payoffs; social norms can therefore either support or hinder vaccination goals and fit UK pertussis coverage and prevalence data.Source: §8.2.6 · pp. 73–74

8.2.7. Implications of bounded rationality in vaccination decisions

• Bounded rationality, vaccine imperfection, and asymmetric social pressure produce nonlinear stability regions, steady oscillations, and a pure-vaccinator endemic equilibrium, making infection elimination harder.The model uses prospect-theoretic utilities and probability-weighted perceived rewards; the pure-vaccinator endemic state is stable when vaccinator pressure is large and vaccine efficacy is low [466].Source: §8.2.7 · p. 75

8.2.8. Modeling public communication on disease and vaccine risks

• Public communication creates effort thresholds: high effort makes full vaccination globally stable, intermediate effort makes above-threshold partial vaccination globally stable, and low effort permits endemicity or oscillations.Increasing public effort can raise coverage, eliminate the endemic state through a stability exchange, and eventually collapse the partial-vaccination equilibrium into full vaccination; calibration to Italian measles data outperformed imitation-only models.Source: §8.2.8 · pp. 76–77

8.3. Behavior-explicit game-theoretic models of vaccinating behavior

• Behavior-explicit game-theoretic models frame voluntary vaccination as a conflict between individual self-interest and community interest, focusing on Nash equilibria and socially optimal uptake.These approaches combine SIR epidemiology with classic game-theoretic questions rather than primarily analyzing dynamic behavior.Source: §8.3 · p. 77

8.3.1. The baseline equilibrium model

• The baseline equilibrium model represents parents’ vaccination probability x and seeks a convergently stable Nash equilibrium x* determined by perceived vaccine-to-disease risk and infection probability at endemic equilibrium.The framework integrates an endemic SIR model with vaccination at birth, with each parent’s payoff depending on other parents’ strategies.Source: §8.3.1 · p. 77
• Self-interested vaccination cannot eliminate infection: r > x_c yields no vaccinators, while a vaccine scare can sharply reduce uptake and make restoring coverage difficult.The baseline model has a unique CSNE solution: x* = 0 when r > x_c, but strictly positive vaccination when r < x_c; payoff gains from increasing perceived vaccination risk have no plateau.Source: §8.3.1 · pp. 77–78, Figure 25 · p. 78

8.3.2. Best individual vs community vaccination response in a heterogeneous population

• In heterogeneous populations, Nash equilibrium systematically produces less vaccination than community optimization unless universal vaccination is already optimal, while uptake increases with prevalence-linked information signals.Individuals choose between vaccination and refusal using utilities affected by a shared infection-risk signal and average uptake; the resulting equilibrium response is self-consistent and unique at the population-average level.Source: §8.3.2 · pp. 78–79

8.3.3. Markov decision processes show that the interaction between vaccine-believers and vaccine-skeptics yields sub-optimal measles vaccine uptake

• Among vaccine-believers and skeptics, Nash uptake is below the community optimum and can fall sharply below critical measles coverage as skeptic prevalence or vaccination cost rises [476].Each group’s best response is zero below a group-specific force-of-infection threshold, one above it, and intermediate at the threshold; skeptics have the higher threshold, producing abrupt uptake collapse as relative vaccination cost increases.Source: Figure 26 · p. 81, §8.3.3 · p. 82

8.4. Other contributions to mean-field coupled disease-behavior models

• Mean-field behavior-disease research expanded from vaccination free-riding and HIV/AIDS behavior change to Markov, utility-based, game-theoretic, and bounded-rational models for multiple infections.This literature established inefficiently low free-market uptake and motivated increasingly structured models before the transition toward network-based approaches.Source: §8.4 · pp. 82–83

9. Behavior-vaccination dynamics in structured, networked populations

• Network models are needed because homogeneous-mixing theory loses effectiveness and predictive value for close-contact epidemics; nodes represent individuals and edges represent disease-transmitting contacts.The review applies this framework to the interplay between vaccination behavior and disease dynamics in structured populations.Source: §9 · p. 83

9.1. Behavior-vaccination dynamics on static networks

• Static-network vaccination decisions depend on neighbors’ strategies, perceived infection risk, vaccine safety and efficacy, and vaccination or infection costs, producing collective phenomena analogous to statistical-physics phase transitions.The reviewed networks include lattices, small-world networks, Erdös-Rényi graphs, and Barabási-Albert scale-free networks.Source: §9.1 · p. 83

9.1.1. Impact of imitation dynamics: The universal “double-edged sword” effect

• Networks create a “double-edged sword”: they promote vaccination below a cost threshold but suppress it above that threshold, with stronger effects in heterogeneous networks and wider coexistence phases under stronger selection.Peer pressure reproduces this pattern, while memory and perceived-risk thresholds preserve the qualitative result through vaccination-cluster formation and decomposition.Source: §9.1.1 · pp. 85, 87

9.1.2. Impact of interaction network topology

• On scale-free networks, vaccination can better constrain epidemics than on Erdös-Rényi graphs because middle- and high-degree nodes are more inclined to vaccinate, but imperfect vaccines can reverse this advantage when infection rates are high.For βI = 0.2, c1 = 1, and c2 = 0.7, vaccination is favored when kinf ≥6; with imperfect vaccines, ER graphs can outperform SF networks because disease spread can outpace vaccination uptake on SF networks.Source: §9.1.2 · pp. 88–89

9.1.3. External incentive programs

• Under imitation dynamics, free subsidies produce higher vaccination coverage and stronger outbreak control than partial-offset subsidies by making recipients role models that seed vaccination clusters.Under myopic updating, the result reverses: partial-offset subsidies better support herd immunity, and a moderate subsidy rate maximizes socioeconomic benefit.Source: §9.1.3 · pp. 90–91
• Committed vaccinators increase uptake by acting as steadfast role models, but committed non-vaccinators can worsen spread; effects depend on their fraction, network structure, and vaccine cost.On lattices, committed vaccinators and non-vaccinators have positive and negative effects, respectively, while on scale-free networks committed vaccinators may not prevent epidemics at low or intermediate vaccine cost because hubs reject vaccination.Source: §9.1.3 · p. 91

9.1.4. Multiple-strategy dilemma framework: A counter-intuitive observation

• Intermediate self-protection effectiveness increases epidemic size and collective cost, whereas sufficiently effective self-protection suppresses epidemics and makes vaccination nonviable, producing a Braess-like paradox.Higher individual payoffs from self-protection can coincide with lower population payoff, and collective payoff is negatively correlated with epidemic size.Source: §9.1.4 · pp. 92–93

9.2.1. How does feedback influence well-known passive and pro-active policies?

• Voluntary ring vaccination contains outbreaks below a neighborhood-size threshold of 57, but larger neighborhoods allow infection to percolate through imperfect vaccination rings and produce substantial epidemics.Smallpox-specific simulations showed similar results, indicating that eradication depends on social-contact numbers and local population structure.Source: §9.2.1 · pp. 94–95
• Behavioral feedback largely eliminates the advantage of superspreader-targeted vaccination by reducing coverage among non-superspreaders, although substantial economic incentives can partly restore effectiveness.Targeting connection heterogeneity alone is insufficient because infectiousness and infectious-period duration may not coincide with high connectivity.Source: §9.2.1 · p. 96
• In mixed-strategy models, vaccination likelihood increases with degree, while vaccination thresholds depend on network heterogeneity and disease transmissibility; recalling more prior seasons can stabilize otherwise oscillatory behavior.Low transmissibility produces no epidemic risk or vaccination, intermediate conditions generate oscillations, and high transmissibility returns behavior toward a steady state.Source: §9.2.1 · pp. 96–97, Figure 34 · p. 97

9.2.2. Social clustering of behavior and opinions – assortativity of (non-)vaccination

• Positive assortativity among non-vaccinators forms susceptible clusters that increase outbreak likelihood and can undermine herd immunity even when overall vaccination coverage is high.This effect appears in models, opinion networks, and high-school contact networks, and becomes stronger with longer or combined contact durations.Source: §9.2.2 · pp. 98–99

9.2.3. Belief-decision models and dual-perspective systems

• Belief-decision models represent vaccination as acceptance, rejection, or indecision determined by beliefs about disease severity and vaccine safety within a social network.The framework explicitly couples awareness with vaccination choices.Source: §9.2.3 · p. 100, Figure 37 · p. 100
• Reporting rates mainly affect vaccination coverage, whereas the awareness fading coefficient determines the timing of vaccine administration.Beliefs are updated using extended Dempster-Shafer theory incorporating evidence from connected neighbors.Source: §9.2.3 · pp. 100–101
• Social influence and cost jointly determine vaccination outcomes: low conformity makes coverage decline with vaccine cost, while full conformity makes coverage depend on initial willingness.At intermediate conformity, vaccination coverage and epidemic incidence are determined by competition between vaccine cost and initial vaccination willingness [573].Source: §9.2.3 · p. 101

9.2.4. Competing controlling measures (going beyond traditional binary selection)

• Vaccination and non-pharmaceutical interventions trade off: expanding non-pharmaceutical interventions reduces vaccine coverage, while expanding vaccination is only weakly affected by the other measure.Vaccination interferes more strongly than non-pharmaceutical interventions as the primary determinant of influenza incidence, so policies should communicate that the measures are complementary.Source: §9.2.4 · pp. 101–102

9.3. Behavior-vaccination dynamics on temporal networks

• Temporal networks alter vaccination and epidemic outcomes, motivating models that move beyond static interaction structures.The review identifies temporal networks as an important framework because social interactions change over time and can affect both vaccination behavior and intervention effectiveness.Source: §9.3 · p. 102

9.3.1. Outcome inelasticity and outcome variability on adaptive networks

• Adaptive-network edge turnover produces successful ring vaccination, widespread vaccination, or no vaccination, with stochastic realizations yielding highly variable outcomes under identical parameters.Changing interaction structure creates nonlinear behavior-disease feedback, policy resistance, and outcome inelasticity that static traditional models do not capture.Source: §9.3.1 · p. 102

9.3.2. The impact of closeness driven time-varying topology

• Greater acquaintance-driven connectivity weakens vaccination enthusiasm because social links facilitate herd immunity, while vaccination makes epidemic incidence peak before declining with crisis awareness.Without vaccination, epidemic size increases monotonically; temporal-network dynamics also depend on the relative time scales of network change, transmission, and vaccination uptake.Source: §9.3.2 · pp. 102–103

9.4.1. The general framework and an example

• Multilayer systems couple social opinion spreading with biological transmission, creating feedback dynamics beyond either isolated layer and generally suppressing vaccination coverage.The resulting lower coverage produces larger and more frequent epidemics, while negative opinions and many small-degree nodes further impede vaccination propagation.Source: §9.4 · p. 103, §9.4.1 · p. 104

9.4.2. Parallel, interacting dynamics on different networks – The impact of opinion clusters

• Opinion clustering among parents increases outbreak likelihood by creating susceptible offspring clusters, and this effect grows with overlap between parent and offspring networks.The effect persists even with low overlap, while non-overlapping intra-layer connections can help determine vaccination coverage needed for disease mitigation.Source: §9.4.2 · pp. 104–105

10. Measuring and assessing vaccination levels: traditional approaches

• Vaccination assessment must capture coverage, its distribution, and vaccine effectiveness because all three shape population disease dynamics and inform modeling.Source: §10 · p. 105

10.1. Measuring vaccination coverage

• Administrative coverage estimates are distorted by registration gaps, migration, duplicate records, inaccurate denominators, and especially weak registries in low- and middle-income countries.Source: §10.1 · pp. 105, 107
• Surveys such as DHS, MICS, EPI, and LQAS complement administrative data but remain vulnerable to selection and information biases.These biases arise from non-representative sampling, poor field practices, and inaccurate recording on household-retained paper cards.Source: §10.1 · p. 107
• Biomarkers rarely distinguish vaccine-induced antibodies from natural infection, while waning immunity, immune boosting, and pathogen-specific exceptions limit inference about vaccination and dose history.Source: §10.1 · p. 107

10.2. Vaccine efficacy and effectiveness

• Vaccine efficacy varies substantially across vaccines and contexts, ranging from 8% to 93% for influenza, 78%–90% for HPV, and 92% for acellular pertussis.Efficacy is typically estimated in double-blind randomized controlled trials, whereas many models incorrectly assume 100% efficacy.Source: §10.2 · p. 108
• Under random mixing, observed disease among vaccinated and unvaccinated people can jointly estimate coverage or effectiveness, and more cases among the unvaccinated may indicate higher coverage rather than worsening risk.The relationship uses susceptible fractions 1 − x + x(1 − τ) = 1 − τx and can also signal increasing vaccination hesitancy when the pattern reverses.Source: §10.2 · pp. 108–109

11.1.1. Participatory platforms for digital surveillance

• Digital surveillance platforms estimate influenza-like illness incidence highly correlated with classic surveillance while providing inexpensive, scalable, near-real-time, geographically resolved data with fewer reporting delays.FluNearYou and InfluenzaNet use weekly user-submitted geographic, behavioral, and medical information to complement traditional systems and inform model initialization.Source: §11.1.1 · p. 111
• Participatory surveillance can estimate geographically resolved vaccination rates and vaccine efficacy near real time, but it complements rather than replaces clinical systems needed to characterize circulating strains.Questionnaire design can mitigate, but not eliminate, selection bias and other observational-study limitations.Source: §11.1.1 · p. 111

11.1.2. Measuring and understanding close proximity interactions

• Wearable measurements show that traditional surveys miss most short contacts, overestimate interaction duration, and omit weak ties that bridge clusters and enable large-scale outbreaks.Wearables provide more objective and scalable contact measurements, while traditional methods still reproduce strong local ties with reasonable accuracy.Source: §11.1.2 · p. 112
• Wearable data reveal heterogeneous, time-varying contact activity, durations, and intercontact intervals, so transmission depends on disease-matched interaction timescales and burstiness can either slow or accelerate spread.Activity-driven models encode this heterogeneity and yield epidemic thresholds determined by the first and second moments of the activity distribution rather than time-aggregated topology.Source: §11.1.2 · pp. 112, 114
• Empirical contact networks contain clustering and vaccination assortativity, and simulations suggest these correlations can increase average outbreak size while dense local ties promote spread and sparse intercommunity links slow it.A high-school contact study found vaccination assortativity significantly more positive than expected by chance, increasing with contact duration and aggregation across days.Source: §11.1.2 · pp. 114–115
• Vaccinating just the top 0.04% most active nodes is enough to halt contagion, whereas random vaccination requires more than 50% coverage to reduce endemic infection below 20%.These comparisons come from activity-driven-network simulations of random and targeted vaccination strategies.Source: §11.1.2 · p. 116, Figure 42 · p. 117
• Wearable and RFID data characterize close-proximity contacts and hospital transmission pathways, supporting the development and testing of infection-prevention and containment measures.A local exploration strategy reaches active nodes more efficiently than random sampling, although it remains less effective than targeted sampling.Source: §11.1.2 · p. 117

11.2. Non participatory digital epidemiology

• Nonparticipatory digital epidemiology mines data generated for everyday technology use rather than explicit epidemiological studies, extracting proxies of relevant behaviors and indicators.The approximation for vaccination-probe selection is reliable for small w and T, when repeated selection is unlikely.Source: §11.2 · p. 117

11.2.1. Social media based methods

• GFT initially achieved about 0.97 mean correlation with surveillance data and predicted influenza one or two weeks ahead, but failed to predict the 2009 pandemic’s unfolding.It overestimated 92% of weeks in the 2011 and 2012 seasons and more than doubled the 2013 peak; the service was later discontinued.Source: §11.2.1 · pp. 118–119
• Social media can amplify vaccine misinformation and polarized echo chambers, while also supporting vaccine communication and coverage through texts, calls, apps, emails, and web services.Negative vaccine sentiment appears contagious whereas positive sentiment is not, and highly connected opinionated users may suppress their own expression.Source: §11.2.1 · p. 119
• Social-media proxies from searches, tweets, Wikipedia views, and other online activity can estimate epidemic indicators, forecast disease activity, and inform models where surveillance data is unavailable.These approaches have been applied across diseases and geographic scales, but simple correlations, changing platforms, and context-insensitive keywords can create spurious signals [727].Source: §11.2.1 · pp. 118–120

11.2.2. Mobile phone based methods

• CDRs reveal large-scale contact and mobility networks whose structure informs infectious-disease models and enabled malaria strategies targeting high-risk human-movement flows.Data from about 15 million people across 12,000 cell towers and 692 Kenyan settlements identified parasite source and sink regions and importation routes.Source: §11.2.2 · pp. 120–121

11.3. Mixed approaches

• Mixed systems combine online streams, expert reports, official data, and user participation to monitor, forecast, classify, and communicate disease threats at population level.Health Map provides continuous global processing, while Crowdbreaks crowdsources tweet classification so machine-learning performance improves through user engagement.Source: §11.3 · p. 121

• Coupled mean-field models generate emergent dynamics, explain observed data better when available, and can inform vaccination policy and responses to declining coverage.They can produce oscillations in outbreaks and vaccine coverage and support questions about communication modes, resilience, and vaccine introduction.Source: §12 · p. 122
• Future mean-field research should incorporate behavioral theories, cross-country and cross-infection datasets, stochasticity under disease rarity, and behavioral responses in age-structured intervention models.These directions aim to preserve parsimony while improving behavioral explanation, empirical testing, and prediction during elimination programs such as measles control.Source: §12 · p. 122
• Network structure is essential for predicting population dynamics from individual observations and modeling interventions such as contact tracing, ring vaccination, and time-varying contact control.Network-based coupled models additionally show behavior-dependent phenomena such as difficulty eradicating infection and large outbreak probabilities within social clusters despite high vaccination coverage.Source: §12 · pp. 122–123
• Network vaccination research should address mobility, multilayer processes, heterogeneous updating criteria, and the integration of local and global risk information.Metapopulation networks can represent human movement and information exchange, while future models should cover targeted vaccination, disease-awareness layers, individual differences, and public-media dissemination.Source: §12 · p. 123
• The field needs stronger interdisciplinary collaboration, shared terminology, and new analytical approaches from statistical physics to connect individual behavior with population-level disease dynamics.Physicists, mathematicians, economists, epidemiologists, public-health researchers, statisticians, and computer scientists can reduce duplication, improve model application, and exploit methods including Ising models and percolation theory.Source: §12 · pp. 123–125
• Testing coupled behavior-disease hypotheses against data remains difficult because vaccination coverage, vaccine efficacy, and effectiveness are uncertain, especially in low-income countries.Digital epidemiology can provide high-resolution contact, mobility, and behavioral data, but requires scalable analytical methods, privacy and access frameworks, and theories of temporal, high-order, and social contagion dynamics.Source: §12 · pp. 123–124

Appendix

• The appendix lists abbreviations for terminology used in the main text and report.Table A1 presents the abbreviation summary.Source: Appendix · p. 126, Table A1 · p. 126