Incidence from Recent Infection Prevalence

Transcription

1 (with Norman Ives and Tom McWalter) School of Computational and Applied Mathematics, University of the Witwatersrand and DST/NRF Centre of Excellence in Epidemiological Modelling and Analysis (SACEMA) DIMACS/SACEMA Workshop 25 July 2007

2 Outline of Talk 1 Introduction

3 Motivation Challenges 1 Introduction Motivation Challenges

4 Motivation Introduction Motivation Challenges Determination of epidemiological trends requires knowledge of both incidence and prevalence. This is required for: Allocating resources Evaluating interventions Planning cohort based studies

5 Motivation Challenges Measuring Incidence is Problematic Gold Standard of cohort follow up is: Expensive Slow Not free from bias Intervention Loss to follow up

6 Alternatives to Gold Standard Motivation Challenges Incidence from snapshots of recent infection prevalence Many pasts can produce one present Problem of calibrating recent Forever recent? Sample sizes

7 Disease Progression and Survival Relating History to Present 1 Introduction 2 Disease Progression and Survival Relating History to Present

8 Stages of Infection Introduction Disease Progression and Survival Relating History to Present As observed through two assays of different sensitivity, infected individuals experience the following events: 0 The moment of infection, at t 1 Infection becomes detectable by assay one, at t + t 1 2 Infection becomes detectable by assay two, at t + t 2 3 Death, from whatever cause, at t + t 3

9 Counting Infectees Introduction Disease Progression and Survival Relating History to Present All historically infected individuals N i (0) = 0 λ(t) dt = 0 I(t)N s (t) dt After infection at time t < 0 the probability of a being alive and classified as recently infected at time zero is given by P { (t 1 < t) AND (t 2 > t) AND (t 3 > t) } = t 0 t t ρ(t 1, t 2, t 3 ) dt 3 dt 2 dt 1 The probability of a being alive and classified as not recently infected at time zero is given by P { (t 2 < t) AND (t 3 > t) }

10 Disease Progression and Survival Relating History to Present Recently and Long Infected Individuals Counts of recently and long infected individuals observed in a cross-sectional survey R = 0 I(t)N s (t)p { (t 1 < t) AND (t 2 > t) AND (t 3 > t) } dt L = 0 I(t)N s (t)p { (t 2 < t) AND (t 3 > t) } dt

11 What Can We Hope to Know? Modelling the Forever Recent Infection Equilibrium Population Dynamics Beyond Equilibrium Population Dynamics 1 Introduction 2 3 What Can We Hope to Know? Modelling the Forever Recent Infection Equilibrium Population Dynamics Beyond Equilibrium Population Dynamics 4 5 6

12 Simplifications Introduction What Can We Hope to Know? Modelling the Forever Recent Infection Equilibrium Population Dynamics Beyond Equilibrium Population Dynamics In order to infer incidence from observables, we need to make numerous simplifications of the general model. Redefine incidence to start at t 1. Factorise ρ(t 2, t 3 ) into ρ r (t 2 )ρ a (t 3 ) (independent events). Use only a few moments of ρ s (preferably just the means). Now define the survival times S r (t) = 1 S a (t) = 1 t 0 t 0 ρ r (s) ds ρ a (s) ds

13 What Can We Hope to Know? Modelling the Forever Recent Infection Equilibrium Population Dynamics Beyond Equilibrium Population Dynamics Which Incidence Can Possibly Be Measured? If I(t) is not constant then define a weighted average I w = 0 I(t)W (t) dt 0 W (t) dt where W (t) is a statistical weight implied by the sampling methodology and formal assumptions. It will be useful to consider in which case 0 W (t) = S a ( t)s r ( t) = S ar ( t) W (t) dt = 0 S ar ( t) dt = E [τ r ]

14 When Recent is Forever What Can We Hope to Know? Modelling the Forever Recent Infection Equilibrium Population Dynamics Beyond Equilibrium Population Dynamics BED assay may be best bet for a practical definition of recent. But: not everyone leaves this category. Note symmetry!

15 General Practical Model What Can We Hope to Know? Modelling the Forever Recent Infection Equilibrium Population Dynamics Beyond Equilibrium Population Dynamics Counts of infection categories R t = 0 L = (1 P ) R f = I(t)N s (t)s r ( t)s a ( t) dt 0 P (1 P ) L Unbiased estimation of R t is possible! R t = R o I(t)N s (t)(1 S r ( t))s a ( t) dt P (1 P ) L

16 Constant Number of Susceptibles What Can We Hope to Know? Modelling the Forever Recent Infection Equilibrium Population Dynamics Beyond Equilibrium Population Dynamics If N s (t) = n 0 then R t = n 0 0 I(t)S ar( t) dt E [τ r ] E [τ r ] = n 0 I ar E [τ r ] Hence I ar = R t n 0 E [τ r ] is an unbiased estimator.

17 When N s (t) is Non-Constant What Can We Hope to Know? Modelling the Forever Recent Infection Equilibrium Population Dynamics Beyond Equilibrium Population Dynamics It is not possible to obtain unbiased estimator for I w with non-constant I(t) and non-constant N s (t). Need to assume specific functional forms for N s (t) and I(t). Can infer up to two parameters in I(t), N s (t). In the case where I(t) = i 0 and N s (t) = n 0 + n 1 t we recover I w = i 0 = R t n 0 E [τ r ] n 1 2 E [τ 2 r ]

18 What Can We Hope to Know? Modelling the Forever Recent Infection Equilibrium Population Dynamics Beyond Equilibrium Population Dynamics When N s (t) and I(t) are Both Non-constant Even the simple case of N s (t) = n 0 + n 1 t and I(t) = i 0 + i 1 t requires calibration for E [ τr 2 ] [ ], E τ 3 r and E [τa ], which seems impossible. But: we can use a simple formula and then see how wrong it is when we violate the assumptions under which it would be correct.

19 Analytical Simulations 1 Introduction Analytical Simulations 5 6

20 Closed-Form Calculation of Bias Analytical Simulations Declare I(t), N s (t), ρ a (t) and ρ r (t). Evaluate I ar from knowledge of declared inputs. Evaluate naive formula to estimate I: Compare I ar and I est. I est = R t E [τ r ] N s (0) Note bias as a function of parameters which break assumption in naive formula.

21 Analytical Simulations 0.35 I(t) = t Relative error in Iar n 1 /n 0, where Ns(t) = n 0 +n 1 t

22 Analytical Simulations 0.06 I(t) = I 0 2 t 0.05 Relative error in Iar r, where Ns(t)=n 0 e rt

23 Matlab Simulations Introduction Analytical Simulations Produce infection events according to λ(t) = N s (t)i(t). Obtain samples after burn-in time. Confirms analytical results. Can explore sample-path specific I w. Can combine sample size effects with sample path effects. Future implementation of hierarchical Bayesian inference.

24 Analytical Simulations x Population Long infected True recents 7 False recents Naive incidence (total population) Naive incidence (sample population) Realised incidence Platonic incidence Incidence

25 Sample Sizes Calibration 1 Introduction Sample Sizes Calibration 6

26 How Big Must Surveys Be? Sample Sizes Calibration Precision limiting step in a survey is the count of recent infections. Need to find tens of recent infections for reasonable precision. Example: suppose we wish to distinguish I = 0.05 confidently from I = 0.04, then: BED definition of recent leads to N RNA+/AB- definition of recent leads to N

27 Sample Sizes Calibration Determining S ar (t) and P Most important parameter is P. Need to observe this failure to progress several tens of times in a calibration study. Need to estimate the mean of S ar (t) in a cohort censored for progression. Non equilibrium formulas require estimation of long term survival times or short term standard deviations.

28 1 Introduction

29 Long Term Incidence Surveillance Can t run from the need for incidence surveillance. There is no magic bullet. BED type definition is perhaps the best short term option. Just discussed one-shot methods. Need to set up open ended calibration. Need to manage multi-survey meta analysis to: construct informative priors ; calibrate demographic/epidemiological parameters.