SIR Model
The foundational epidemiological model (Susceptible, Infectious,
Recovered). It assumes that individuals who recover from the disease develop lifelong immunity.
Example: Widely used for modeling fast-spreading diseases like
Measles or Rubella, where recovery confers long-term immunity.
SEIR Model
Adds an "Exposed" state to account for the incubation period during
which individuals have been infected but are not yet infectious themselves.
Example: Suitable for diseases with significant incubation periods like
COVID-19 or Ebola.
SIS Model
Assumes that infection does not confer long-lasting immunity. Upon
recovery, individuals immediately become susceptible to the disease again.
Example: Often used for modeling bacterial infections like
Gonorrhea or common colds caused by rhinoviruses.
Network Model
Simulates disease spread through a social contact network. This model
assumes that individuals are nodes in a graph, and transmission occurs only between connected nodes.
Example: Modeling a School Outbreak, where children only interact
with their classmates and teachers (localized clusters).
Differential Equations
$\frac{dS}{dt} = -\frac{\beta S I}{N}$
$\frac{dI}{dt} = \frac{\beta S I}{N} - \gamma I$
$\frac{dR}{dt} = \gamma I$
$\frac{dS}{dt} = -\frac{\beta S I}{N}$
$\frac{dE}{dt} = \frac{\beta S I}{N} - \sigma E$
$\frac{dI}{dt} = \sigma E - \gamma I$
$\frac{dR}{dt} = \gamma I$
$\frac{dS}{dt} = -\frac{\beta S I}{N} + \gamma I$
$\frac{dI}{dt} = \frac{\beta S I}{N} - \gamma I$
$P(\text{inf} | k ) = 1 - (1 - T)^k$
$\frac{dI}{dt} \approx (\text{Contacts}) \cdot T \cdot \frac{S}{N} \cdot I - R \cdot I$
Where $T$ is Trans. Prob, $R$ is Recovery Prob.