# Contact Tracing. Johannes Müller. Technical University Munich Center for Mathematics

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1 Contact Tracing Johannes Müller Technical University Munich Center for Mathematics

2 Contact Tracing - Aim Concept: If we find an infected individual (so-called index case ), we ask for (potentially) infectious contacts. Contact Partners are called in to a doctor. In this way, it is possible to find more infectious persons. Related Methods: Clustering ( who has a similar behaviour like you instead of with whom did you have contact ), Screening Diseases: HIV, SARS etc., Tuberculosis, sexually transmitted diseases (Gonorrhea, Chlamydia)...

3 Contact Tracing - Questions What s the effect of a contact tracing program? How is an efficient contact tracing procedure designed? Do we trace as much as we can? Do we trace only the contacts of the primary index case? How hard do we search for contacts? How does contact tracing differs in the onset of an epidemic and an endemic state? Estimate Parameters? Can we find monitoring tools for contact tracing programs?

4 Contact Tracing - Some Models (a) Hethcote/York, 1984: Gonorrhea Deterministic model. Nonlinear correction of the incidence(!) function. No connection to stochastic models. (b) Lourdes/Azoza, 1995: HIV Deterministic model. Linear (!) incidence function. No connection to nonlinear stochastic models. (c) Kretzschmar/Duynhjoven/Severijnen, 1996: Gonorrhea, Chlamydia Detailed simulation model. No attempt for analysis. (d) Huerta, Tsimring 2002 Stochastic network ( small world network ) and mean field (e) Fraser, Riley, Anderson, Ferguson 2004 Deterministic PDE-Model; Tracing process is handled in a linear way. (f) Eames 2006 Pair approximation in stochastic network.

5 Basic Epidemic Model Simple stochastic SIS model with random mixing. Consider a population of N individuals. Define the standard epidemic model on this population: contact rate: β (random mixing) probability q for the transmission of the infection per S/I-contact recovery rate µ

6 Contact Tracing - Different Situations Onset of the epidemic / initial phase of CT Onset of the epidemic / equilibrium phase of CT Epidemic equilibrium / initial phase of CT Epidemic equilibrium / equilibrium phase of CT Disease is subcritical / equilibrium phase of CT

7 Contact Tracing - Onset Onset of the diseases / initial phase of CT Onset of the diseases / equilibrium phase of CT Epidemic equilibrium / initial phase of CT Epidemic equilibrium / equilibrium phase of CT Disease is subcritical / equilibrium phase of CT Graph of Infecteds: Nodes: Infected and Infectious individuals. Edges: Directed edge from Infector to Infectee. Tracing Process: Define a screening rate: with rate σ an individual is detected (the primary index case) and removed. Edges have the probability p to become detected and to be removed. Neglected: (many things, especially) contact tracing is no instantaneous process. no test is perfect. diseases are not Markovian processes. diseases are not homogeneous. p p p Time of Infection

8 Contact Tracing - Onset Onset of the epidemic / initial phase of CT Onset of the epidemic / equilibrium phase of CT Epidemic equilibrium / initial phase of CT Epidemic equilibrium / equilibrium phase of CT Disease is subcritical / equilibrium phase of CT Results in the Onset of the diseases / equilibrium phase of CT: The process tends to a stable exponentially growing phase, where distributions (e.g. size of connected components) stabilizes Formula for the effective reproduction number R e (though not nice in the general case) E(Number of detected cases per index case) = p R 0 1+R 0 + O(p 2 ) where p is the tracing probability E(size of a connected component in the infection graph) 1 + R 0 Structure of a connected component in the infection graph (without contact tracing) (M,Kretzschmar, Dietz 2000; M, Möhle 2003; M, Hösel 2007)

9 Contact Tracing - Endemic Case Onset of the epidemic / initial phase of CT Onset of the epidemic / equilibrium phase of CT Epidemic equilibrium / initial phase of CT Epidemic equilibrium / equilibrium phase of CT Disease is subcritical / equilibrium phase of CT S S Suscept. S I Infect. I I S I Infect. No Inf.

10 Contact Tracing - Endemic Case Onset of the epidemic / initial phase of CT Onset of the epidemic / equilibrium phase of CT Epidemic equilibrium / initial phase of CT Epidemic equilibrium / equilibrium phase of CT Disease is subcritical / equilibrium phase of CT S S Suscept. S I Infect. I I S I Infect. No Inf. Index case Detected case

11 Contact Tracing - Tracing Model in the Endemic State Two different type of traced contacts: Infection Tracing Tracing of infectee or infector Random Tracing Tracing of contacts via that no infection has been transmitted. Tracing Model: All contacts have to be considered. In a finite, randomly mixing populations, any person has had contact to all other persons (perhaps long ago). Declining of tracing probability with the time since contact is necessary to incorporate. Detection/screening rate σ. P (tracing of a contact a time units ago) = p 1 e p 2a

12 Contact Tracing in the Endemic state - Analysis Central tool: κ(a) = P ( individual has been infected a time units ago and is infectious since then) d κ(a) = κ(a){µ + σ + Removal rate via a traced contact}, κ(0) = 1. da

13 Contact Tracing in the Endemic state - Analysis Central tool: κ(a) = P ( individual has been infected a time units ago and is infectious since then) d κ(a) = κ(a){µ + σ + Removal via a traced contact}, κ(0) = 1. da Assumption: approximately constant prevalence. Ten cases (blue: infectious, black: susceptible): Infector Infectee + Random a=0 a Individual Random time

14 Contact Tracing in the Endemic state - Analysis Good news: Up to the first order in p 1, the differential equation for κ is possible to work out. Bad news: The result is terribly lengthy. Attempt: Approximation. (a) neglect all contacts that took place before the last infectious period of an individual started (b) Approximate terms in exponent by quadratic terms in time since infection a Let τ 0 = qβ/(µ + σ) (Fraction of susceptibles without contact tracing) then κ(a) e (µ+σ)a }{{} No Screening e σp 1a }{{} Infector e βqτ 0 p 1 σ a 2 /2 }{{} Infectees e β(1 τ 0) p 1 σ a 2 /2 }{{} +O(p 2 1) Random tracing: I-I

15 Contact Tracing in the Endemic state - Results Endemic Equilibrium: 1 = R e = qβs 0 κ(a) da prev. of diseases (β/(µ + σ) {3, 4, 5}, q = 0.4, σ/(µ + σ) = 0.2) p1 Effect even for small values like p 1 0.1

16 Contact Tracing in the Endemic state - Results Contact Tracing: Rough Estimation of scanned persons per index case (do not through away these data!): E(No of traced/scanned contacts per index case) p 1 R 0 /q Rough Estimation of scanned persons per index case: Relative number of hits: E(Detected Persons per Index Case) p 1 ( 1 }{{} Infector + 1 }{{} Infectee detected cases scanned cases 2q/R 0 + (1 τ 0 ) Screening: Compare with the case of pure screening: detected cases screened cases = (1 τ 0) + (1 τ 0 )R 0 /q ) }{{} Random Tracing For R 0 1, the relative success (number of hits versus number of tests) is in the same range for contact tracing and screening; the latter is cheaper!

17 Contact Tracing in the Endemic state - Questions Are there (endemic) situations where contact tracing my be of use? Core groups - Contact tracing may reveal a core group structure (a lot of detailed data is necessary - unrealistically?) - Contact tracing may reveal core group members (clustering may be more efficient in this) Asymptomatic cases - Contact tracing may reveal asymptomatic cases

18 Summary: Contact Tracing Different situations (onset/endemic state) require different mathematical methods and may lead to different conclusions! Contact tracing in the Endemic State and a Random Mixing Population For large populations, we can analyse models for contact tracing in the endemic case. Contact tracing is able to reduce the prevalence of diseases even for small tracing probabilities. In a random mixing population, screening may be preferable. In a core group situation, clustering may be preferable If you have many asymptomatic cases, contact tracing may be of use

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