On 30th December 2019 World Health Organization’s Country Office in the People’s Republic of China picked up a media statement by the Wuhan Municipal Health Commission from their website on cases of ‘viral pneumonia’ in Wuhan [1]. In the very first months of 2020 COVID-19 disease spread all around the globe, and on 11th March 2020 WHO characterized it as a pandemic [2].
Europe has been strongly affected by COVID-19 emergency, with severe impact on economy, education, health system and many other sectors.
Also Small and Medium-sized Enterprises (SMEs), which represent 99% of all businesses in the EU, have been deeply affected by the pandemic [3].
Since the very first stages of COVID-19 pandemic, research groups focused their efforts on studying the phenomenon from different points of view. One of the possible strategies is represented by compartmental models based on Kermack–McKendrick theory [4][5][6].
In this project we define an epidemic model to simulate SARS-CoV-2 spread in a fictional and simplified SME named CSNS Group. The epidemic model has different control parameters to mimic several scenarios and allows us to test different population-wide countermeasures, such as personal protective equipment, contact tracing and vaccination.
The simulation can be run on NetLogo environment and we use Gephi platform to analyze contact tracing strategy more in depth.
Here you can find the simulation source code, and here a more refined report of this project.
- Epidemic Model
- Simulation Description
- CSNS group
- Temporal Evolution
- Contact Tracing Program
- Run the Model
Epidemic Model
The compartmental model we define to simulate SARS-CoV-2 epidemic has the following 6 classes:
- Susceptible (S): people who can be infected.
- Infected (I): highly infective people. We refer to the infectivity of this class of people with letter α.
- Hospitalized (H): infected people who have been diagnosed and hospitalized,so they cannot infect anyone. Hospitalized people do not have contacts with other people.
- Exposed (E): people who have been infected, but are not completely infectiveyet. Exposed people are less infective than infected people (the infectivity is reduced by a factor ƐE= 0.6).
- Quarantined (Q): people who have been infected and quarantined, thanks tocontact tracing. The quarantine is not perfect: quarantined people may have contacts and infectother people, but their infectivity is reduced according to Quarantine Efficiency (ƐE). As a result, the infectivity of quarantined class is equal to ƐQ × ( α × ƐE), with 0 ≤ ƐQ ≤ 1 (1).
- Immune (V): people who have been vaccinated and are not susceptible anymore.
The model has 5 control paramters to simulate different scenarios:
- Contact Tracing Efficiency (θ)
- Diagnose Efficiency (ξ)
- Vaccination Efficiency (ν)
- Quarantine Efficiency (ƐQ)
- Face Masks
Parameters θ, ξ and ν can be set to 0%, 10%, 50%, 90% or 100%.
The transition rate I→S – i.e. the recovery chance of the infected people – is computed according to ξ so that, if an infected employee is neither diagnosed nor died, he has a chance of 20% to recover (see the table below).
Quarantine Efficiency goes from 0 (perfect isolation) to 1 (no isolation) to mimic the efficiency of self-quarantine.
Face Masks is a Boolean variable:
False: without face masks the infection chance α is equal to 95%;True: with face masks the infection chance is reduced by a 65% [10].
Simulation Description
CSNS Group
The setting of our simulation is a fictional medium-sized enterprise with 100 employees called CSNS Group.
Our SME has a team-based organizational structure: employees are divided in small teams, which communicate with each other thanks to supervisors. Thus, there are few people with many contacts (connectors), while the vast majority of the employees has very few contacts. Moreover, CSNS Group employees do not have any contacts with external people: the company is perfectly isolated.
To build CSNS Group network of contacts we use the power-law distribution, which is a heavy-tailed distribution that generates a scale-free network with few hubs and many low-degree nodes. In the simulation, we set the power-law exponent to 2, and it is important to keep in mind that the way in which the network is built – that is the way in which people make contacts with each other – strongly affects the simulation.
In the figure below there is an example of the initial network of contacts of the company and the corresponding degree distribution. As we can see, the large majority of the employees has at most 5 contacts, while only few of them have more than 6 contacts, which is consistent with CSNS Group team-based organization.
On the right: an example of initial degree distribution of the network of contacts.
In a disease-free scenario, the network of contacts of CSNS Group does not change over time, since working teams and supervisors are supposed to be the same during all the simulation.
However, there are two epidemic-related factors that can modify the network:
- some employees may die because of COVID-19 and be replaced with new susceptibles who make new contacts;
- some employees may be completely isolated (enter in H class) and, once they recover, they make new working contacts.
Temporal Evolution
The simulation is daily-based and continues until there are only susceptible or immunepeople.
At day 0 (setup procedure):
- 100 employees come to their offices and make working contacts following the power-law distribution described in the previous section. All the employees are susceptible except for an initial number N of infected, which can be set by the observer.
- A fraction of working contacts may be traced thanks to the Contact Tracing Program and the tracing last forever
Each day (go procedure):
- I, E or Q employees infect their susceptible contacts according to the epidemic model parameters.
- Then, following to the transition rates:
- infected people may be diagnosed and isolated; they may also recover, die or remain infected;
- exposed people can recover or move toIclass or even remain exposed;
- quarantined people may either recover or remain quarantined, as well as hospitalized people, who can either recover or remain isolated;
- susceptible people may be vaccinated and enter in V class.
- At the end of the day, missing people are replaced by new susceptible employees, so the headcount remains steady.
Contact Tracing Program
One of the control measures used to slow down the spread of disease or even to eradicate infection is the contact tracing, which aim is to gather information about the network of contacts and identify asymptomatic infected individuals who can then be treated or quarantined. Conversely to other techniques, contact tracing has not been applied as a network evaluation device, but as a control tool. Unfortunately, it strongly relies on people accuracy and correctness, since each person should provide all the information about his relationships to have a perfect contact tracing [11] [12].
We introduce contact tracing into CSNS Group simulation by giving the employees the chance of taking part at the Contact Tracing Program. The degree at which an employee joins the Program is determined by the Contact Tracing Efficiency θ, which simulates employees accuracy on providing complete and precise data about their working relationships.
The Contact Tracing Program works as follows:
- when an employee joins the company, he reports each of his working contacts with a probability equal to θ;
- whenever an employee A is infected by a colleague B and their contact A−B is traced, A self-quarantines.
Self-quarantine is different from isolation: quarantined people do not loose their working-contacts.
Moreover, self-quarantine efficiency is determined by the Quarantine Efficiency (ƐE), thus a quarantined employee may still infect his colleagues.
Here we report an example of a real network of contact (on the left) and the corresponding subgraph of traced contacts (on the right). In the figure gray edges are not traced, while green ones are traced; white nodes refer to susceptible employees, red ones to infected and the size of each node is proportional to its degree.
On the right: the corresponding subgraph of traced contacts.
In the project report you can find a more detailed analysis on the network of contacts shown above.
Run the Model
The model can be easily run on your local machine by downloading the source code and opening it with NetLogo (recommended version >= 6.1.1.).
All the instructions are reported in the Info section of the model.
Here is an overview of the interface:
References
[1] ProMED. Undiagnosed pneumonia – china.
[2] T.A.Ghebreyesus. Who director-general’s opening re-marks at the media briefing on covid-19-3march2020.
[3] European Commission. Internal market, industry, entrepreneurship and smes -sme definition.
[4] W. O. Kermack and A. G. McKendrick. Contributions to the mathematical theoryof epidemics — 1.Bulletin of Mathematical Biology, 53(1-2):33–55, 3 1991.
[5] W. O. Kermack and A. G. McKendrick. Contributions to the mathematical theoryof epidemics – 2. the problem of endemicity.Bulletin of Mathematical Biology,53(1-2):57–87, 3 1991.
[6] W. O. Kermack and A. G. McKendrick. Contributions to the mathematicaltheory of epidemics – 3. further studies of the problem of endemicity.Bulletin ofMathematical Biology, 53(1-2):89–118, 3 1991.
This project is part of my Complex Systems and Network Science exam of the Master’s degree in Artificial Intelligence, University of Bologna.