Stochastic simulation of a network-based epidemic

1. Authors

Carlos García
Victor Coeto
Yann Le Lorier

2. Context

As of June 2020, the COVID-19 pandemic is rapidly spreading through the world. With this in view, it may be relevant to study how a disease spreads through a population, using the classic Gillespie algorithm found here.

3. Project Description

3. 1 The Gillespie algorithm

The Gillespie algorithm is a methodology that aims to track Markovian processes where objects change status. In this case, we wish to use this algorithm in a given starting graph, see the end graph, and the changes over time. It is based on the SIS model (Susceptible, Infected, Susceptible)

3.1.1 Input/Output

• Input Network graph G, a transmission rate τ, a recovery rate γ, a set of index node(s) of initial_infections, maximum time t_max
  Output The Network graph G after it goes through the Gillespie function.

3.1.2 Variables

• The infected nodes are updated. In the very first iteration, the infected nodes are the initial_infections\
• The at_risk_nodes are updated as the nodes which are direct neighbors of infected nodes\
• The infection rate in the at_risk_nodes is updated as τ*len(infected_neighbors)\
• The total_infection_rate is updated as the sum of all the infection rates in the at_risk_nodes group\
• The total_recovery_rate is updated as γ*len(infected_nodes)
• total_rate is updated as the total_transmission_rate + total_recovery_rate
• time is updated as the exponential variation taking as argument the total_rate

3.1.3 explanation of an iteration

The main loop iterating, until time < t_max and the total_rate < 0

• r is updated as a uniform random distribution taking 0 and total_rate, then

  • if r < total_recovery_rate then remove one node from the infected nodes, and reduce infection rate
  • if not, then add one node in the at_risk_nodes to put it in the infected_nodes group, and update its neghbors to at_risk_nodes group.

• update times, S and I

• update total recovery rate, total infection rate, and total rate

• time is updated as time + exponential_variate(total rate)

4. Topics

1. Pointers
   • To store the graph information in interconnected objects
2. Threads
   • Graph creation
3. Signals
   • To correctly pause or stop the simulation.
4. Dynamic Memory
   • To store the graph information in an vector

5. Use Cases

The programs serves the pupose of being a close representation of a virus outburst in a static network environment. It is quite useful because it allows for different snapshots to be analyzed at different points in time.

6. Dependencies

• C++ environment (gpp 2.0+)
• SFML library
• cmake (3.1 minimum)

7. Running the program

Read the TemplateInstructions file to modify the Nodes file to your liking.

mkdir build && cd build
cmake ..
make
./program-name -i <inputFilename>

References

Stochastic simulations of Epidemics, I.Z. Kiss et al., Mathematics of Epidemics on Networks, Interdisciplinary Applied Mathematics 46, DOI 10.1007/978-3-319-50806-1