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## Infectivity

*Aka: 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
- R0 (pronounced "R naught", or Basic Reproduction Number)
- R0 was derived by epidemiologist from its original use in organism reproduction (Basic Reproduction Number)
- R0 refers to the number of people that a single infected individual spreads to others
- R0 <1 suggests the infection will not spread broadly
- R0 >1 risks epidemic spread

- R0 = transmissibility * contactRate * infectiousnessDuration
- Where transmissibility = infection per contact
- Probability of infection in a susceptible host exposed to an infected host

- Where contactRate = mean contact per time period (cMean)
- Rate of contact between those susceptible and those infected

- Where infectiousnessDuration = Duration that an infected patient may infect others on contact

- Where transmissibility = infection per contact

- R (Effective Reproductive Number)
- R = R0 * X
- Where X is the percentage of the population that is susceptible to the infection
- Example: if 50% of the population is immune to the pathogen, X=0.5
- As with R0, when R <1, the infection will not continue to significantly spread

- Herd Immunity Threshold (HIT)
- Threshold over which a large percentage of the population's specific pathogen Immunity protects the non-immune
- In general, approximately, HIT = 1-1/R0
- Example: If R0 = 5, HIT = 80%

- Serial Interval
- Time duration between symptom onset in a source patient (primary case) and symptom onset in a secondary case

- Incubation Period
- Duration between the time of infection (transmission) and the onset of disease symptoms

- R0 (pronounced "R naught", or Basic Reproduction Number)
- Precautions
- R0 is significantly modified by various environmental factors
- Immunization and Herd Immunity
- See Effective Reproductive Number (R) as above

- Crowding in large gatherings tends to increase R0 (depending on modes of transmission)
- Includes super spreading

- Immunization and Herd Immunity

- R0 is significantly modified by various environmental factors
- Approach: Susceptible-Infected-Removed Model (SIR Model)
- Calculate three compartments
- Susceptible (s) = S/N
- ds/dt = -Bsi

- Infected (i) = I/N
- di/dt = Bsi - vi

- Removed (r) = R/N
- dr/dt = vi

- Susceptible (s) = S/N
- Where
- B = Effective Contact Rate = transmissibility * contactRate
- transmissibility = infection/contact
- contactRate = contact/time

- v = Removal Rate
- d = Infection Duration = 1/v

- B = Effective Contact Rate = transmissibility * contactRate
- Derive R0 > 1 is consistent with epidemic
- Given a closed, well-mixed population size (N) that does not change (including no births or deaths)
- Given constant rates of transmission
- When di/dt >0, those infected exceed those removed, and an epidemic occurs
- Bsi - vi > 0
- Bsi/v >i and given 100% are susceptible at the start of an epidemic, s=1
- B/v >1, and since R0 = B/v
- R0 > 1 implies an epidemic

- References
- Notes on R0 (James Holland Jones)

- Calculate three compartments
- Interpretation: R0 and Herd Immunity Threshold (HIT) for Serious Infections
- Pandemic Influenza
- R0: 1.5-1.8
- HIT: 33 to 44%
- Serial Interval: 2-4 days

- Ebola
- R0: 1.5-2.5
- HIT: 33 to 60%
- Serial Interval: 10-16 days

- Covid19
- R0: 2-3
- HIT: 50-66%
- Serial Interval: 4-8 days

- SARS
- R0: 2-5
- HIT: 50-80%
- Serial Interval: 7-8 days

- Mumps
- R0: 4-7
- HIT: 75 to 86%
- Serial Interval: 18-20 days

- Smallpox
- R0: 5-7
- HIT: 80 to 85%
- Serial Interval: 18 days

- Polio
- R0: 5-7
- HIT: 80 to 86%
- Serial Interval: 2-45 days

- Rubella
- R0: 6-7
- HIT: 83 to 85%
- Serial Interval: 15-23 days

- Measles
- R0: 12-18
- HIT: 92 to 95%
- Serial Interval: 9-13 days

- Pandemic Influenza
- Resources
- Herd Immunity (Wikipedia)
- Epidemic Theory

- References