# Mathematical Modeling of Virus Spreading

### External Member

## Description

**Motivation:**How does a virus spread in a community? Researchers from a wide spectrum of fields have devoted a substantial amount of effort to address this question. From a theoretical perspective, it is natural to introduce and study mathematical models which mimic the virus spreading process.

**Virus Spreading Model:**Consider a graph G representing a social network, where each node corresponds to an individual and an edge between two nodes indicates that they are friends. Assume that initially all nodes are colored blue, except one randomly chosen node which is colored red. Then in each discrete-time round, all nodes simultaneously update their color according to the following

- For a red node, if it has been red in the last k rounds, then it switches to blue; otherwise, it remains red.
- For a blue node, it chooses one of its neighbors independently at random and picks its color. For example, a blue node which has two blue neighbors and one red neighbor will become red with probability 1/3 and will remain blue with probability 2/3.

*pandemic*if a constant fraction of nodes in the network, say 10% of the nodes, become infected; otherwise, we say that the virus

*vanishes*.

**Contact**

**Supervisor:** Ahad N. Zehmakan**Email:** abdolahad.noorizehmakan@anu.edu.au

## Goals

- What is the probability that the virus results in a pandemic?
- How long does it take in expectation until the process ends (i.e., the pandemic is over)?

## Requirements

- Good understanding of graph theory. (Familiarity with different random graph models and social networks is a plus, but not required.)
- Good knowledge of probability theory. (Familiarity with Markov chains is an advantage.)
- Decent programming skills.

## Keywords

Random Graph Models, Social Networks, Probability Theory, Virus Spreading, Multi-Agent Systems.