Heterogeneity (next 3 classes) 2 case studies –Age –Sexual activity –Others: Day care vs. not, Genetic heterog. Use data Understand behavior of complicated models

Age-Structured Serological Data (next 2 classes) Biological Basis Modeling context (multicompartment ODEs, PDEs) Estimate Age-Specific FOI Problems with interpretation Other interpretations Mixing Matrix (WAIFW) Elimination Threshold

Biology: Humoral Immune Response

Humoral immunity Protection from subsequent infection Standing antibody levels Long-lived antibody-secreting cells Ability to mount anamnestic (memory, 2 o ) response Record of history of infection Presence of antibody

Serological testing Assays: –ELISA: Count antibodies (usually IgG) –Functional assays: biological function Arbitrary cutoff for positivity must be specified: Continuous measurement [Ab] with cutoff SHOW Figs 1 and 2

Seroprevalence: involves making cutoffs in continuous data Rotavirus in Sao Paolo: MJ Cox et al Epidemiol Infect IgG IgMIgA

Same: RSV in Sao Paolo MJ Cox et al J Med Virol

Cellular immunity Important for protection (long-term immunity) vs. viruses, intracellular bacteria Less frequently assayed for history of infection Exception: TB skin test (tuberculin, PPD) –DTH: CD4+ T-cell mediated monocyte response –Ambiguous indicator of history of infection, protection

Antibody decline with age Significance for immunity to reinfection? Significance for record of infection? 3 weeks 70 years time ln [IgG] infection

Ideal Serological Marker of Infection permanent highly reproducible sensitive (everyone who has had the infection is positive on the test) specific (only people who have had the infection are positive on the test) biologically relevant (i.e., history of infection = immunity = assay positivity) -- DEPENDS ON BIOLOGY

Often ideals are approximately met, but problems include: waning immunity with age differences among individuals in their immune responses background ("natural" false positives, false positives induced by infection with related agents) ignorance of appropriate cutoffs (both for "record keeping" and for functional immunity)

Poor resolution at older ages for highly transmissible diseases MJ Cox et al J Med Virol FT Cutts et al J Med Virol

Hepatitis A in NL Termorshuizen et al Epidemiol Infect

And more.... Measles: US Measles: India, NigeriaMumps: E/W, NLPolio: US Malaria: NigeriaHBV: Senegal Rubella: GambiaRubella: E/W Abbreviation: E/W England & WalesAnderson & May Nature 1985

Uses of Age-SP Data Infer characteristics of transmission process –age-specific rates of infection –predictors of infection at various ages –changes in transmission over time Estimate age-specific: –force of infection ( ): (time-to-event analysis, same as survival analysis in other contexts) –incidence –transmission coefficients given a mixing matrix structure

From: OL Chason, DPH Thesis, HSPH, Diphtheria Immunity in Rural Alabama: A survey of social conditions, environment, and Schick Test Results

End of Biology Back to Models

Why to use an age-specific model? 1. To capture more accurately the dynamics of the population –Especially important when infectiousness is long relative to the lifespan (HSV, HIV) –Because death is then a likely end to infection, so modeling it correctly is important

Exponential stay in each compartment without age structure Non age-structured model S1S1 I1I1 R1R1 SS II RR No age component. Stay in any compartment has an exponential distribution Bad approximation for lifespan in developed countries

Why to use an age-specific model? 2. To keep track of when people get infected: important when –Different ages contribute to transmission differently –Age at infection matters to impact Cost Severity (chickenpox, rubella, flu) –Vaccines are introduced at ages other than near-birth

Idealized Population for Analyzing Age-Specific Data Everyone lives exactly L years and then dies Population is in demographic equilibrium: –N(0) born each year in our unit area –N(0) in each year’s age cohort in unit area Total population density is LN(0)

Multi Age-Group Model: compartments S1S1 S2S2 S3S I1I1 I2I2 I3I3 R1R1 R2R2 R3R (1/30)S 2 (1/15)S 1 (1/55)S 3

More compartments better approximate the ideal Ideal

PDE Model: Limiting Case ( ∞ age classes)

Using Age-Stratified Seroprevalence Data

Seropositives Incident infections Infectious cases (sources) CausalityInference Immunity

Calculations Seronegatives reduced by force of infection Incidence = force of infection * prop. Susceptible Prevalence ~ incidence * duration Note: lower case letters = proportions; capitals = densities)

Formal analysis of AS Data Start with AS profile Calculate Age-specific Force of Infection (a) Calculate Age-Specific Prevalence Y(a) Calculate Mixing Matrix Coefficients  ij given a structure

Calculating Force of Infection, Incidence and Prevalence from Serologic Data

Diphtheria Immunity in Rural Alabama, (whites) AgeS+S- IncPrev E E E E E E E E E E P S+ I

Diphtheria Immunity in Rural Alabama, (whites) P S+ I For age classes wider than 1 yr

Caveats

Data Available What we usually have: Cross-sectional age-seroprevalence What we usually want: Longitudinal (cohort) age-seroprevalence ?? WHEN ARE THESE THE SAME?

Looking at Cross-Sectional AS Data

HTLV-1

Other examples (EXCEL) STD that has been present for a long time New STD New zoonosis (e.g. tick-borne disease) Recently eradicated disease Disease transmitted primarily in childhood with fast waning of immunity

Genetic Heterogeneity in Susceptibility: Can Masquerade as Age-Heterogeneity Younger age classes: mix of genetically S & genetically R individuals, relatively high incidence Older age classes: S individuals will have been infected already: incidence will be rate relevant to R individuals Frailty model