The standard SEIR equation-based models represent the state-of-the-art approach in epidemiological modelling. Their drawbacks include unrealistic infectionrelated contact estimates and difficulties in modelling nonpharmaceutical interventions, such as contact reductions or partial closures. In this paper, we present our agent-based model that addresses the above-mentioned issues. It works with a population of individuals (agents) and their contacts are modelled as a multi-graph social network according to real data based on a Czech county. Custom algorithmic procedures simulating testing, quarantine and partial closures of various contact types are implemented. The model can serve as a tool for relative comparison of the efficacy of various policies. It was also used for a study comparing various interventions in Czech primary and secondary schools, using a graph based on real data from a selected Czech school.
Mathematical models play an important role in a variety of scientific fields, including epidemiology. Mathematical modelling has been an integral part of epidemiology for more than 100 years. Epidemiological models serve many different purposes, namely as a tool for hypothesis verification, explanation of observed data, understanding basic principles of infectious dynamics, prediction of the future development and understanding the current one, calculation of fundamental epidemiological metrics.
During the current world-wide pandemic the interest in epidemiological modelling significantly increased not only in the scientific community, but also in the general public. The ability to model and predict the development of the epidemic became important from day to day.
There exists a variety of epidemic models, including
well established S(E)IR models[
In this paper we present our agent based model that was
designed for the purpose of comparing various
interventions, measures and policies. We focus on the simulation
of non-pharmaceutical interventions, such as partial
closures, contact restrictions and quarantines. More details
on the model itself can be found in our preprint [
The interventions we simulate cover protective measures and contact restrictions. By protective measures we understand masks, stronger hygiene, and general cautiousness. Protective measures decrease the probability of transmission of the disease when the infectious contact happens. On the other hand, the contact restrictions decrease the probability that such contacts are realised. Contact restrictions can be either flat, i. e. whole public places are closed (school closures, shop closures, etc.), or individual. Individual contact restrictions cover isolation or quarantine of single individuals. The effective contact restriction on individual level requires to actively search for individuals that were in contact with infectious ones. This process is known as contact tracing.
All mentioned interventions play an important role in the epidemic and the model should be able to reflect them.
Some of the available epidemiological models posses
mechanisms for modelling some of the mentioned
interventions including contact tracing [
The paper is organised as follows. In the next section we briefly explain the framework of our model and its main properties. In Section 3 the principles of simulation of individual interventions are described. Section 4 brings discussion of the usage of the model, its capabilities and Our model can be classified as a multi agent model. A key property of agent models is that they are built in a bottomup manner. We learn about the behaviour of the system as a whole from the detailed description of its components by means of agents and their interactions by computer simulation. This is completely different approach than in classical compartment models, where the whole system is described by set of equations.
The core of our model framework is formed by three modules. They are depicted in the Fig. 1. Namely, they are the SEIR model, the contact graph and the policy module. Let us describe them briefly.
The SEIR model module is the base part of our model. It is
a standard epidemiological model of the S(E)IR type [
One iteration of the model corresponds to one day, i. e.
an individual can change its state once a day. The
transitions between the states, in other words times for which
the individual stays in each state, are given by the
parameters of the disease (in this case COVID-19). For details on
model parameters used for COVID-19 simulations see [
The exception is the transition S ! E that depends both on the infectiousness of the disease modelled (a global model parameter b ; note that b has a different meaning than in typical SEIR models) and the contact graph. This transition is also the one influenced by non-pharmaceutical interventions.
While the majority of epidemiological models use
synthetic population (if any), our model uses a multi-graph
that is based on real data. The graph is built as a model
of a Czech county (a town together with its surrounding
villages). Great attention was given to the graph
construction so as it is as realistic as possible. Main data
sources used include Czech Statistical Office [
The resulting graph has about 56 thousands nodes (representing agents) and 2.8 millions edges (contacts between agents). Nodes have a list of attributes, including sex, age and economic activity. An example of an individual (and a node in a graph) can be a 43 years old teacher, who lives with his wife and three children in a family house. He has 12 friends, who he regularly meets, most of them are of the similar age. Every day he commutes to work and regularly visits his parents in a close village. All these activities have to be reflected by contact edges in a graph.
The detailed explanation of the graph construction is out
of the scope of this paper and can be found in [
Formally, the contact graph is a multi-graph G = (V; E), where V is a set of nodes, each node corresponding to one individual in a population of the concerned agent model. E is a set of undirected edges representing contacts between individuals. The prefix multi refers to the fact that two nodes can be connected with arbitrary number of edges (see Fig. 2).
Edges are divided into 30 layers, each layer refers to a particular type of contact (such as family, work, leisure time, etc.). Layers are associated with weights w that control the activity on the layer as a whole.
Each edge e stores three parameters pe, ie and le. pe is the probability that a contact represented by the edge e is realised in the current iteration, ie is the intensity of that contact, and le is the type of the layer.
Each day the edge is activated with the probability pactive(e) = wle pe:
If the edge is activated and one of its nodes is in an infectious state and the second node is in the state S, the second node is moved to the state E with the probability pS!E (e) = b ie; where b is a global parameter of the model (however may differ for each node depending on its symptomaticity) and corresponds to the disease modelled. (1) (2)
The Policy Module is an optional, yet important part of the model. It is responsible for a simulation of policies and interventions as will be described in the next section. It communicates both with the model and with the contact graph.
It has access to all parameters of the model and can modify them. In addition it can ask the model to move a particular node to a required state (besides normal state transitions). It has the right to modify the graph, change weights of layers or parameters of individual edges.
It is called between each two iterations of the SEIR model. 3
The main purpose of the model presented in the previous section is the simulation of various interventions, their study and comparison.
One family of interventions covers protective measures
(masks, distancing, hygiene) that make the infection
transmission during the contact with an infectious individual
less probable. This is modelled simply by reduction of
the model parameter b . The parameter b is a vector
over all individuals, so individual-based approach is
possible. Also asymptomatic individuals have different b than
symptomatic ones. In addition, we distinguish between
a contact in a family (based on the type of the layer in
the graph), where less reduction is applied (no masks in
households), and other contacts. The reduction is
controlled by the rate parameter ranging from 0:0 to 1:0 that
reflects the compliance with protective measures. For the
reconstruction of a past progression of the epidemic we
use rates based on data from a sociological survey [
The other type of interventions covers contact
restrictions. The global contact restrictions cover closures of
public places, such as school, shops or restaurants. In the
context of our model, they refer to restrictions of whole
layers. They are modelled by modifying the layer weights
wl (see Eq. (1)). For example, complete closing of schools
is realised by setting wl to zero for all l corresponding to
schools. Setting wl to 0:5 for all l work layers reduces
the probability of all work contacts (for example because
people switched to home-office). Again, for past epidemic
progression reconstruction data from [
Individual contact restrictions are the most challenging ones from the modelling point of view. Isolation of individual nodes requires to reduce probabilities of individual edges adjacent to these nodes. This means local temporal modifications of the graph.
If a node v is isolated, we modify the probabilities of all his edges.
8e 2 v : penew peql ; where ql are quarantine coefficients of individual layers and v is a set of edges adjacent to v.
An example of simple quarantine coefficient setup follows: ql = (1 l is a family edge 0 otherwise. (3) (4)
We implemented three types of policies that use isolation of individual nodes. Namely, they are self-isolation, testing and contact tracing.
Self-isolation simply reflects the fact that a portion of individuals that exhibit symptoms or do not feel well decides to stay home on their own. In the model, as the node starts to exhibit symptoms, it is isolated with a certain probability and it stays in its isolation as long as the symptoms are present.
Second basic policy is the testing. The individuals are tested with a probability q . If the test is positive, they are marked as detected. We typically use non-zero q only for symptomatic individuals (for the case of simplicity).
As soon as the individual is detected, it is sent to isolation for a given number of days. The isolation ends with a stop condition, which is optional. It can be two consequent negative tests or just a number of days past from the positive test. As soon as the stop condition is fulfilled, the isolation stops, i.e. the edges of the node are returned to their normal values (unless the second node of the edge is isolated as well).
The last policy is contact tracing. Contact tracing in the real life can be done in an individual manner, algorithmic simulation on the other hand requires a certain level of simplification. The quarantine-detection-isolation life cycle is depicted in the Fig. 4.
As soon as a node is detected it is sent into isolation, i.e. the probabilities of its edges are modified as in (3), the original values are backed up. After a given time, the contact tracing takes place. All edges that were active in the X days before the first symptoms of the node or Y days before its detection are collected. (X , Y are parameters of the contact tracing.) The model is stochastic, extensively using random numbers, and the variance of results is typically quite high. Therefore in our experiments we always use 1000 simulations for 1000 fixed unique random seeds. Median and mean values are then observed. The time needed for one simulation is approx. 20 seconds on a common CPU.
An example of a possible experiment is depicted in the Fig. 5. There the experiment compares five contact tracing strategies of different strength in the scenario where no other interventions are present.
The strength is defined only with probabilities 1:0 or 0:0 for simplicity, so all contacts from the particular group are always collected. The following strategies were tested: no contact tracing (0; 0; 0; 0), tracing family contacts only (1; 0; 0; 0), tracing family, work and school (1; 1; 0; 0), tracing everything except “others” (1; 1; 1; 0) and ideal tracing (1; 1; 1; 1). We can see that the ideal tracing does not bring much improvement as compared with (1; 1; 1; 0), while the difference between no tracing and tracing family only is more evident.
More experiments on contact tracing strategies
comparison and their detailed results can be found in the
preprint [
The advantage of the model is its modularity. One
module can be easily replaced by its alternative. For example,
one can replace a contact graph by a contact graph specific
to a given environment. Such a graph was a graph built
for a school environment, based on real data collected in
one Czech school. The results of the simulation study of
spread of COVID-19 in a secondary school environment
can be found in [
In the paper we presented a novel agent-based epidemic model. The main difference compared to other models is that it uses a realistic contact graph instead of a synthetic population. Another key feature is the simulation of various non-pharmaceutical interventions that enables to make relative comparisons between them and study their efficacy.
The model is implemented in Python and is publicly
available on GitHub [
As a future work we plan to include export nodes that will simulate the possibility of infection import from the surrounding world.
The work has been supported by the "City for People, Not for Virus" project No. TL04000282 of the Technology Agency of the Czech Republic. [2] Openstreetmap. htps://www.openstreetmap.org. [3] State administration of land surveying and cadastre.
https://cuzk.cz. [4] Réka Albert and Albert-László Barabási. Statistical mechanics of complex networks. Reviews of Modern Physics, 74(1):47–97, Jan 2002.