AI-generated content

This chapter was automatically generated by an AI agent and has not been reviewed by a human. Read with caution.

Contagion

Susceptible individuals can acquire a virus from any of their infected connections. This page describes the mathematical model governing disease transmission in epiworld.

Transmission Probability

The probability that susceptible individual \(i\) gets virus \(v\) from individual \(j\) depends on three factors:

1. Transmissibility of the virus, \(P_v \in [0,1]\)
2. Contagion reduction factor of individual \(i\), \(C_r \in [0,1]\)
3. Transmission reduction factor of the host \(j\), \(T_r \in [0,1]\)

The last two factors are computed from \(i\) and \(j\)'s tools (e.g., vaccines, masks). The resulting probability of \(i\) getting virus \(v\) from \(j\) is:

\[ P(\text{Virus } v) = P_v \times (1 - C_r) \times (1 - T_r) \]

Single-Virus Constraint

By default, epiworld assumes that individuals can acquire at most one virus at a time. Under this constraint, the actual probability that agent \(i\) acquires virus \(v\) from agent \(j\) is:

\[ P_{ivj} = P(\text{Virus } v \mid \text{at most one virus}) \]

This is calculated using Bayes' rule:

\[ \begin{align*} P_{ivj} &= \frac{P(\text{at most one virus} \mid \text{Virus } v) \times P_v}{P(\text{at most one virus})} \\ &= \frac{P(\text{Only Virus } v)}{P(\text{at most one virus})} \end{align*} \]

Computing the Probabilities

The component probabilities are defined as:

\[ \begin{align*} P(\text{Only Virus } V) &= P_v \times \prod_{m \neq V} (1 - P_m) \\ P(\text{at most one virus}) &= P(\text{None}) + \sum_{k \in \text{viruses}} P_k \times \prod_{m \neq k} (1 - P_m) \\ P(\text{None}) &= \prod_{k \in \text{viruses}} (1 - P_k) \end{align*} \]

This formulation ensures that viruses with higher transmissibility are more likely to be acquired when competing with other variants.