#biology #disease #epidemic


An epidemic is the rapid spread of a disease.

SIR model

Epidemic is modeled and measured using several variables. The most common model is SIR[1], with the following three compartments:

graph LR;
  Susceptible --> Infectious;
  Infectious --> Recovered;

Where susceptible represents the number of people who can contract the disease, infectious represents the number of people of can spread it, and recovered represents the number of people who don't spread it anymore. The SIR model assumes immunity for recovered people.

These vary as a function of time. A general pattern is: $S(t)$ goes down as the number of people who contract the disease move to further compartments. $I(t)$ goes up as the epidemic ramps up, then down as the proportion of susceptible people goes down. $R(t)$ goes up as the epidemic spreads.

The infection is characterized by $\beta$ representing the transmission of a disease (number of contacts divided by probability of transmission), and $\gamma$ representing the rate of recovery. The resulting disease characteristic is $R = \frac{\beta}{\gamma}$, representing the difference between transmission and recovery.

$R_0$ is the basic reproduction ratio[2]: the transmission rate at $t_0$ for a specific disease and population when outbreak is starting. It materially corresponds to the number of people an infectious person is infecting on average. If $R$ is greater than 1, then the disease is epidemic and spreads exponentially. If $R$ is lower than 1, then the outbreak is in remission - the number of diseased individuals is going down. If $R$ is 1 the outbreak increases linearly.

epidemic control

Epidemic can be controlled by acting on each of the transition variables:

Reducing transmissibility, by limiting the number of contacts or the probability of transmission.

Number of contacts: achieved by isolating people through confinement (reducing contacts for everyone, including both susceptible and infectious), and quarantining (reducing contacts for infectious or apparent susceptible with a higher chance of being infectious, e.g. when having been in contact with an infectious).

Specific implements include: random or risk-based testing, voluntary isolation and social distancing, quarantining people who have been in contact with infectious (and apps to track such contacts), limiting the number of friends physically visited.

Probability of transmission: achieved through increased hygiene, such as washing hands more than usual, and protective measures against transmission vectors, such as masks, gloves or glasses.

Since for $R > 1$ evolution is exponential, benefits coming from representative application of these measures are exponential.

The goal is to keep $R$ under 1[4], so even partial reductions are sufficient to achieve recovery.

Increasing recovery through medical treatment ; which is improved by reducing transmission, thusly reducing pressure on medical resources. This is also virtually improved by vaccination, which transits susceptible people to the recovered compartment without being infectious (or suffering from the disease itself...).

Since the transmissibility can have a direct impact on availability of medical treatment (e.g. overcrowded emergency departments, production capacity for cures and treatments), corresponding implements can be used as well to control the recovery factor. This can be for example intermittent isolation, where isolation is set when the $I$ population grows beyond a certain threshold such as hospital capacity.

other models

Another such model is the SIS model, where recovered people get back into the pool of susceptible people, such as the flu or the common cold.

graph LR;
  Susceptible --> Infectious;
  Infectious --> Susceptible;


Covid-19 is an example of epidemic. $R_0$ was somewhere between 1.5 and 5, located at around 2.5 in the US.



transition rates details

[3]: The transition between $S$ and $I$ is $\beta\frac{I}{N}$, where $\beta$ is the average number of contacts per person over time multiplied by the probability of the disease being transmitted. $\frac{I}{N}$ is the fraction of contact occurrences involving an infectious person.

The transition between $I$ and $R$ is $\gamma I$ which is the rate of recovery or death. If recovery or death duration is $D$ then $\gamma = \frac{1}{D}$.


$$ \frac{d S}{d t} = - S \times \beta \frac{I}{N} \frac{d I}{d t} = S \times \beta \frac{I}{N} - \gamma I \frac{d R}{d t} = \gamma I $$