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# Applying Network Theory to Epidemics: Control Measures for Mycoplasma pneumoniae Outbreaks

Lauren Ancel Meyers*† , M.E.J. Newman*‡, Michael Martin§, and Stephanie Schrag§
Author affiliations: *Santa Fe Institute, Santa Fe, New Mexico, USA; †University of Texas at Austin, Austin, Texas, USA; ‡University of Michigan, Ann Arbor, Michigan, USA; §Centers for Disease Control and Prevention, Atlanta, Georgia, USA

Main Article

## Table

Notation for epidemiologic interaction network model

Notation Definition
W Number of wards in the facility
C Number of caregivers working in the facility
μw Average no. of caregivers working in a ward
μc Average no. of wards in which a caregiver works
r Probability that a given caregiver works in a given ward
Ρχ Probability that a caregiver works in k wards
qχ Probability that a ward has k caregivers working in it
ƒ0(x) Probability generating function (pgf) for the degree distribution of caregivers
g0(x) pgf for the degree distribution of wards
ƒ1(x) First select a random ward, and then select a random caregiver working there. This expression represents the pgf for the number of other wards in which that caregiver works.
g1(x) First select a random caregiver, and then select a random ward associated with that caregiver. This expression represents the pgf for the number of other caregivers working in that ward.
τw Probability of transmission from a ward to a caregiver
τc Probability of transmission from a caregiver to a ward
Ф0(x) pgf for the number of wards affected by transmission from a random caregiver
Ф1(x) First select a random ward and assume that it is affected by the bacterium, then select a random caregiver working there. This expression represents the pgf for the number of other wards affected by that caregiver.
Γ0(x) pgf for the number of caregivers affected by transmission from a random ward
Γ1(x) First select a random caregiver and assume he/she is infected, then select a random ward in which that caregiver works. This expression represents the pgf for the number of other caregivers infected by individuals working/living in that ward.
Average number of wards affected in an outbreak
1 - Ѕc The size of the caregiver giant component—the largest set of infected caregivers that are all connected through work in common wards
2- Ѕw The size of the ward giant component—the largest set of affected wards that are all connected through common caregivers
βw(x) pgf for the number of patients in affected ward w who contract the bacterium
Β(x) pgf for the total number of patients in the facility who are infected during an epidemic

Main Article

1We calculate these rates by averaging the fraction of infected patients per ward across the 15 wards and compute the error by taking the standard deviation of these fractions, divided by the square root of the sample size.

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