Mathematical Modeling of Virus Spreading

External Member

Ahad N. Zehmakan


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
updating rules:
  • 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.
This process is supposed to mimic the spread of a virus in a community, where red and blue represent infected and uninfected respectively and k is the number of days an individual needs to recover from the
We say that the virus results in a pandemic if a constant fraction of nodes in the network, say 10% of the nodes, become infected; otherwise, we say that the virus vanishes.
Note that in this process eventually all nodes become blue and the pandemic is over. (Why?)



Supervisor: Ahad N. Zehmakan

Please feel free to contact me if you have any questions. I would be glad to have a discussion on whether this project suits you.


The goal of this project is to study the behavior of the above virus spreading model. This includes both theoretical and experimental analysis of the model. The exact objectives of the project can be decided based on your interests and background.
Some interesting questions to study, from both a theoretical and an experimental perspective, are:
  • 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)?
Of course, the answer to these questions very much depends on the underlying graph structure. One can investigate the aforementioned questions on various random graph models (which mimic the real-world social networks) or the available real-world graph data.


  • 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.


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

Updated:  10 August 2021/Responsible Officer:  Dean, CECS/Page Contact:  CECS Marketing