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Infectivity, Basic Reproduction Number, R-Naught, R0, Herd Immunity, Population Immunity, Serial Interval, Incubation Period, Effective Reproductive Number, SIR Model, Susceptible-Infected-Removed Model

  • Definitions
  1. R0 (pronounced "R naught", or Basic Reproduction Number)
    1. R0 was derived by epidemiologist from its original use in organism reproduction (Basic Reproduction Number)
    2. R0 refers to the number of people that a single infected individual spreads to others
      1. R0 <1 suggests the infection will not spread broadly
      2. R0 >1 risks epidemic spread
    3. R0 = transmissibility * contactRate * infectiousnessDuration
      1. Where transmissibility = infection per contact
        1. Probability of infection in a susceptible host exposed to an infected host
      2. Where contactRate = mean contact per time period (cMean)
        1. Rate of contact between those susceptible and those infected
      3. Where infectiousnessDuration = Duration that an infected patient may infect others on contact
  2. R (Effective Reproductive Number)
    1. R = R0 * X
    2. Where X is the percentage of the population that is susceptible to the infection
    3. Example: if 50% of the population is immune to the pathogen, X=0.5
    4. As with R0, when R <1, the infection will not continue to significantly spread
  3. Herd Immunity Threshold (HIT)
    1. Threshold over which a large percentage of the population's specific pathogen Immunity protects the non-immune
    2. In general, approximately, HIT = 1-1/R0
    3. Example: If R0 = 5, HIT = 80%
  4. Serial Interval
    1. Time duration between symptom onset in a source patient (primary case) and symptom onset in a secondary case
  5. Incubation Period
    1. Duration between the time of infection (transmission) and the onset of disease symptoms
  • Precautions
  1. R0 is significantly modified by various environmental factors
    1. Immunization and Herd Immunity
      1. See Effective Reproductive Number (R) as above
    2. Crowding in large gatherings tends to increase R0 (depending on modes of transmission)
      1. Includes super spreading
  • Approach
  • Susceptible-Infected-Removed Model (SIR Model)
  1. Calculate three compartments
    1. Susceptible (s) = S/N
      1. ds/dt = -Bsi
    2. Infected (i) = I/N
      1. di/dt = Bsi - vi
    3. Removed (r) = R/N
      1. dr/dt = vi
  2. Where
    1. B = Effective Contact Rate = transmissibility * contactRate
      1. transmissibility = infection/contact
      2. contactRate = contact/time
    2. v = Removal Rate
    3. d = Infection Duration = 1/v
  3. Derive R0 > 1 is consistent with epidemic
    1. Given a closed, well-mixed population size (N) that does not change (including no births or deaths)
    2. Given constant rates of transmission
    3. When di/dt >0, those infected exceed those removed, and an epidemic occurs
      1. Bsi - vi > 0
      2. Bsi/v >i and given 100% are susceptible at the start of an epidemic, s=1
      3. B/v >1, and since R0 = B/v
      4. R0 > 1 implies an epidemic
  4. References
    1. Notes on R0 (James Holland Jones)
      1. https://web.stanford.edu/~jhj1/teachingdocs/Jones-on-R0.pdf
  • Interpretation
  • R0 and Herd Immunity Threshold (HIT) for Serious Infections
  1. Pandemic Influenza
    1. R0: 1.5-1.8
    2. HIT: 33 to 44%
    3. Serial Interval: 2-4 days
  2. Ebola
    1. R0: 1.5-2.5
    2. HIT: 33 to 60%
    3. Serial Interval: 10-16 days
  3. Covid19
    1. R0: 2-3
    2. HIT: 50-66%
    3. Serial Interval: 4-8 days
  4. SARS
    1. R0: 2-5
    2. HIT: 50-80%
    3. Serial Interval: 7-8 days
  5. Mumps
    1. R0: 4-7
    2. HIT: 75 to 86%
    3. Serial Interval: 18-20 days
  6. Smallpox
    1. R0: 5-7
    2. HIT: 80 to 85%
    3. Serial Interval: 18 days
  7. Polio
    1. R0: 5-7
    2. HIT: 80 to 86%
    3. Serial Interval: 2-45 days
  8. Rubella
    1. R0: 6-7
    2. HIT: 83 to 85%
    3. Serial Interval: 15-23 days
  9. Measles
    1. R0: 12-18
    2. HIT: 92 to 95%
    3. Serial Interval: 9-13 days
  • Resources
  1. Herd Immunity (Wikipedia)
    1. https://en.wikipedia.org/wiki/Herd_immunity
  2. Epidemic Theory
    1. https://www.healthknowledge.org.uk/public-health-textbook/research-methods/1a-epidemiology/epidemic-theory
  • References
  1. Vink (2014) Am J Epid 9(1):864-75
    1. https://academic.oup.com/aje/article/180/9/865/2739204
  2. Delamater (2019) Emerg Infect Dis 25(1): 1-4 [PubMed]
  3. Fine (1993) Epidemiol Rev 15(2): 265-302 [PubMed]