In order to observe the effectiveness of the proposed methods, we use two different ABMs: the first one is a simple toy example without any practical value, while the other is a complex model with a real application, so we can see how the behaviour of the proposed methods changes with increased complexity.
4.1. Agent-Based Model of the Spread of Contagious Disease (ABM-SIR)
As a toy model, we use the ABM that simulates the spread of contagious diseases. Each agent has three states: susceptible to the disease, infectious, and recovered (SIR), similar to the SIR model [
26]. The model has discrete time steps, and on each step, agents interact with each other with the number of contacts following Poisson distribution. If an infectious agent interacts with a susceptible one, the latter can be infected with a certain probability. Infectious agents recover after some time and become immune to the disease. The population consists of 100 thousand agents. The model simulates 40 days with a step of 2.4 h.
The step of the model is as follows: we iterate over the agents, and if we find a susceptible agent, we randomly choose agents with whom it would make contact. If we choose an infectious agent, there is a defined probability that the virus will be transmitted to the susceptible agent. If we find an infectious agent, it can recover with a defined probability.
The model has four adjustable parameters:
is the probability of the disease transmission from an infectious agent to a susceptible one,
c is the contact rate between agents,
is the probability of the recovery, and
is the initial number of infectious agents. Since it is just a simple model without real data behind it, we manually set the reference parameter values and get the curves that we want to reproduce (
Figure 1).
4.2. Model of the Circulation of Respiratory Viruses in a City (ABM-ARI)
In order to analyse the behaviour of different parameter tuning methods on more complex models with a real application, we use the ABM, called ABM-ARI, of the co-circulation of seven respiratory viruses (influenza A and B, rhinoviruses, respiratory syncytial viruses, adenoviruses, parainfluenza and common human coronaviruses) in a city [
27]. In the model, viruses differ in their properties, such as the mean viral load, duration of incubation period, duration of symptoms, and the probability of developing symptoms. Moscow was chosen as the model city. The model consists of 10 million agents representing city residents who are characterised by a set of properties. Properties such as age, sex and social status were based on the demographic and socio-economic data for various municipalities of the city according to the 2010 Russian census. Epidemiological properties include total immunoglobulin levels dependent on the age and health status of the agent. Each agent may be fully or partially susceptible to different viruses, exposed to the virus for one day, infectious with or without symptoms, or resistant to all viruses, depending on the stage of infection (
Figure 2).
Agents are assigned to households, as well as other social groups, which include groups of educational institutions and workplaces. Interactions of agents with each other are modelled using contact networks in the form of complete graphs for households and Barabasi-Albert graphs [
28] for groups of educational institutions and workplaces with connectivity parameters equal to 10 and 5, respectively (
Figure 3). The model simulates one year, starting from August 1, and has a discrete time step of one day.
Transmission of viruses occurs through contact between fully or partially susceptible agents and infectious agents. The risk
of infection of the agent
i includes the risks of infection by each simulated virus
v in each simulated social setting
c that agent
i attends at the model step
t from each infectious agent
j with whom agent
i makes contact:
where
is the set of modelled viruses,
is the set of social settings to which the agent
i belongs,
is the set of infectious agents with whom agent
i makes a contact,
is the state of the agent’s health,
I and
S are infectious and susceptible states, respectively.
The risk of virus transmission is determined by the product of five independent risks:
where
is the risk of transmitting the virus
v from infectious agent
j infected
days ago, and
and
are the risks of infecting the susceptible agent
i with the virus
v for a given total level of immunoglobulins and a number of days
that have passed since its last recovery from an illness caused by the virus
v, respectively.
h is the risk of infection transmission between two agents for a given contact duration, and
is the risk of transmitting the virus
v for a given average daily air temperature.
The risk of transmitting the virus from an infectious agent depends on the viral load, which in turn depends on the age of the agent, the number of days passed since becoming infectious, the virus with which the agent was infected, and the presence or absence of symptoms:
where
and
are the maximum and average viral load for the virus
v and the age group of agent
j, respectively.
and
are durations of the incubation period and symptoms, respectively.
and
are infectious states with symptoms and without, respectively.
The risk of infecting the susceptible agent depends on the total level of immunoglobulins, which in turn depends on the age and sex of the agent:
where
is the normalised total immunoglobulin level of agent
i, and
is the adjustable parameter for the virus
v.
The risk of infecting the susceptible agent also depends on the level of virus-specific antibodies, which in turn depends on the number of days passed since the recovery and on the duration of the immunity to the virus:
where
is the adjustable parameter denoting the duration of specific immunity of agent
i to the virus.
The risk of infection transmission during the contact between two agents depends on the duration of contact, which in turn depends on the social setting where it was made:
where
is the duration of contact (in hours) between agents
i and
j in the social setting
c on the step
t,
is the adjustable parameter.
The risk of transmitting the virus for a given average daily air temperature depends on the time step of the model:
where
is normalised average air temperature for the step
t, and
is the adjustable parameter for the virus
v.
The model consists of initialisation and simulation stages (
Figure 4). Simulation stage consists of multiple model steps where each step goes as follows:
First, we determine whether the current day is a holiday or a day off for different social settings. National holidays and Sundays are days off for everyone. Children have summer, autumn, spring, and winter vacations. Workers do not work on Saturdays.
We iterate over the agents, and if we find an infectious agent, we iterate over the agents with whom the agent makes contact with the current step. In the case of finding an agent, susceptible or partially susceptible to the virus, we sample the duration of their contact from a given distribution [
29] and evaluate the risk of infection transmission. If the transmission is successful, the agent becomes exposed.
Next, we update the agents’ properties based on their health state:
- (a)
Susceptible. There is a small chance that the agent may have been exposed to a virus from an unknown source at the current step. The virus is selected randomly, and if it is able to overcome the level of specific immunity, the agent becomes exposed.
- (b)
Immune to at least one of the viruses. We decrease the level of specific immunity to the virus.
- (c)
Infected at the current step. We find the duration of the incubation period and the duration of symptoms or the duration of the asymptomatic period from the given distribution [
30,
31,
32].
- (d)
Infectious. The agent can exhibit symptoms or progress to the asymptomatic stage after the end of the incubation period, self-isolate and become diagnosed [
33], or recover.
- (e)
Recovered. We check if it is able to be infected with viruses again.
In the end, we update the model date and air temperature.
The model has 26 adjustable parameters divided into 5 groups: a group corresponding to the dependence of the risk of infection on the duration of contact (), a group corresponding to the dependence of the risk of infection on the total level of immunoglobulins (), a group corresponding to the dependence of the risk of infection on the air temperature (), a group corresponding to the average durations of immunity to each virus (), where is the virus, and a group corresponding to the probabilities of infection from an unknown source (), where is the age group.
The model is used to reproduce the average weekly incidence of acute respiratory infections in Moscow for different age groups and viruses for a single year based on the data from 1997 to 2002. It consists of three groups of incidence curves: an overall incidence curve (
Figure 5), incidence curves for age groups 0–2, 3–6, 7–14 and 15+ years, and incidence curves for seven modelled viruses.