Epidemiology and Disease Dynamics

Epidemiology

• The study of epidemics using statistics and probability to study disease.
• John Snow is considered the father of epidemiology due to his work tracing a cholera outbreak to a contaminated water pump.

Epidemics, Endemics, and Crowd Diseases

• Epidemic: A sudden, short-term outbreak in a specific location or population over a short time.
• Endemic: A long-term, consistent presence of a disease at a baseline level within a specific region.
• Crowd Diseases: Spread rapidly and are sustained in large populations, typically in dense communities. Examples include smallpox and measles.

Disease Transmission and Cycles

• Critical Community Size (CCS): The minimum population size needed for a disease to sustain itself without dying out.
• Zoonotic: A disease that is transmitted from animals to humans.
• Sylvatic Cycle: The transmission cycle of a pathogen that primarily circulates in wild animal populations without immediate human involvement.
• Enzootic: A long-term consistent presence of a disease within an animal population in a specific region.
• Epizootic: A disease outbreak that briefly affects an animal population in a short period of time.

Plague

• Three clinical forms:
  • Bubonic: Inflamed lymph nodes, transmitted by fleas. Mortality rate of 30-90%, death occurs within 10 days.
  • Pneumonic: Lung infection, transmitted human-to-human via droplets. Mortality rate of 90-100%.
  • Septicemic: Flea-human transmission within the blood.

SIR Model

• Answers the question: "Will an epidemic occur?"
• A compartment model that measures the rate of individuals moving across compartments.
• β: Transmission rate or rate of an infected person getting someone sick.
• ν: Clearance rate or rate of recovery/death.

SIR Model Assumptions

• Population size is constant (N = S(t) + I(t) + R(t)).
• No births/deaths.
• Well-mixed populations: everyone has an equal chance of coming into contact with one another.
• Mass action: The rate of change of new infections (β) is proportional to S x I, assuming well-mixing.
• Everyone starts susceptible, except for one infected.
• Once removed, individuals cannot become infected again.
• Rates are uniform across individuals.

SIR Model Equations

• Susceptible Population:
  • dS(t)/dt = -β * (I(t) * S(t) / N)
• Infected Population:
  • dI(t)/dt = β * (I(t) * S(t) / N) - ν * I(t)
• Recovered Population:
  • dR(t)/dt = ν * I(t)

Reproductive Numbers

• Basic Reproductive Number (R0):
  • R0 > 1: More infected than recovered, resulting in an epidemic.
  • R0 < 1: More recovered than infected.
• Effective Reproductive Number (Re):
  • Accounts for changes in the susceptible population over time.
  • Re = R0 * (S(t) / N)
  • Re > 1: Each infected person causes more than one new infection, leading to exponential growth.
  • Re < 1: Epidemic decline.

Slowing Epidemic Spread

• Increasing ν: Reducing the duration of infection through antibiotics and vaccines.
• Reducing β: Public health measures such as masking and quarantine.

Breaking SIR Model Assumptions

• N = constant: Assumes a constant population size, which is a reasonable assumption.
• Everyone starts susceptible except for one infected:
  • This assumption is challenged by herd immunity.
• Rates are uniform between individuals:
  • Superspreaders can significantly impact transmission.
• Mass action:
  • Social distancing and quarantine decrease contact frequency, challenging the mass-action principle.
• Once removed cannot get sick:
  • The reality is more complex, with multiple compartments.

Herd Immunity and R0

• Herd Immunity: The protection of a population from a disease when a sufficiently high percentage of individuals are immune.
• Herd Immunity Threshold: The percentage of the population that needs to be immune to achieve herd immunity.
  • R(t) / N > 1 - (1 / R0)
• ν and Infection Time:
  • ν = 1 / Infection Time
• S(0): At the beginning of an epidemic (t = 0), the entire population is susceptible.
  • S(0) = N
• Estimating R0 after an Epidemic:
  • R0 = ln(S(∞) / N) / [(S(∞) / N) - 1]
    • S(∞)/N represents the fraction of the population that remains susceptible at the end of the epidemic.

SIR Model Limitations

• The SIR model is a good model for the initial exponential rise in cases.

Non-Pharmaceutical Interventions (NPI)

• NPIs, such as social distancing and quarantine, can "flatten the curve" of an epidemic.
  • Examples include:
    • The 1918 flu outbreak in St. Louis (appropriate NPI) versus Philadelphia (inadequate NPI).
    • Early COVID-19 mortality in China decreased due to the availability of resources.

Contact Tracing and Novel Strategies

• Stochastic Models: Account for certain levels of unpredictability or randomness.
• Backward Tracing: Identifies the source of infection by tracing who infected the patient.
  • Pros: Useful for identifying superspreaders.
  • Cons: Difficult to keep up.
• Forward Tracing: Identifies people who were in contact with an infected person.
  • Pros: Effective for stopping future transmissions.
  • Cons: May miss superspreading events.
• Contact Tracing Impact: Contact tracing reduces transmission rates (β) by decreasing the number of infected people.
• Novel Contact Tracing Strategies:
  • Wastewater Analysis: 50% of asymptomatic/symptomatic patients shed the virus in feces.
    • Helps identify novel variants.
  • Genetic Epidemiology:
    • Everyone with the same strain indicates a hotspot.
    • Two people with different strains suggests they did not infect each other.
    • It can supplement contact tracing.