An agent-based model of COVID-19 transmission and mitigations
Modelling the effectiveness of mitigation strategies requires that we simulate them in parallel with the spread of the disease itself. Building on the principles of the SEIR approach, an agent-based model (ABM) was developed, simulating the spread of COVID-19 within the population of an urban area (full source code at http://github.com/harrykipper/covid). Since the virus is transmitted through contacts between infected and susceptible individuals, in order to understand the dynamics of the spread it is essential to represent the multiple social networks that connect individuals within a population and determine patterns of contacts. A major weakness of traditional compartmental infection models is their aggregated nature. The models divide the population into homogenous groups (compartments) in accordance with the state of the disease (SEIR); and disease transmission dynamics is assumed to occur as the infected group mixes with the susceptible group at certain rates16. These model assumptions do not account for the heterogeneity that exists between individuals within the groups, and simplifies the complexity of contact patterns in social networks that are important to understanding the course of an epidemic17. Our ABM intends to bridge this gap by modelling more realistic contact patterns that take place among heterogeneous agents interacting within a social network. Central to the agent-based approach is that each agent is represented in the model individually, with their specific characteristics (such as age and sex) and behaviour. Furthermore, interactions between agents are explicitly simulated. In the proposed model, agents are linked in a multi-layered social network that determines potential contacts. A contact is defined as an encounter between two agents where the virus may be transmitted. The social network includes several types of social ties: household, relations, friends, workplace colleagues (for adults) and classmates (for school children). Contacts outside of the social network also take place and represent everyday “random” encounters with strangers in public places. In addition, a proportion of the working age population is assumed to work in customer-facing employment, which entails additional random contacts.
Agents and social networks
We generate a synthetic population of circa 103,000 agents, derived from the 2011 UK Census18, including household type, gender and age within geographical zones (Detailed Characteristic Sector 2011). The population represents the city of Glasgow, Scotland. A multi-layered social network links agents within the following social structures:
- A household structure is created as follows: individuals who belong to households classed with the same type in the Census, and reside in the same locale, are linked together on the basis of age difference; single people below the age of 20 are assumed to live at home with one or two parents and siblings. Single people above the age of 20 are assumed to live independently, with a certain proportion co-habiting. Links of type ‘household’ are built among these
- Family relatives who don’t live in the same household (i.e. grandparents) are linked
- Several workplace sites are created, based on the distribution of workplace sizes in the city of Glasgow19. Active working-age agents are distributed among workplaces and linked to all co-workers at the same site as well as to a subset of colleagues who are assumed to be in closer, more frequent Out of the working-age population, 13% of agents are assigned to customer-facing employment20, experiencing frequent contact with random agents of the population during work.
- A friendship network links agents over 14 years of age, generated following the Barabasi-Albert model21so that a scale-free network is produced, characterised by variation in number of friends per agent, with a median of 14 friends per agent, skewed towards similar
- Children between 6-17 years of age also belong to classes of maximum 30 children of the same age from the same zone and are linked together as
The subset of network contacts that an agent meets on a given day are determined based on the frequency of encounters and the number of contacts per encounter that characterise each type of social network (Table 1). For example, we assume that agents meet their household members every day and have contact with all of them, whereas they meet other relatives only twice a week, each time having contact with only one of them.
Type of contact
|
Frequency of encounters
|
No. of contacts per encounter
|
Transmission
probability (β ) per contact
|
Household
|
Daily
|
All household members
|
β
|
School
|
5 days per week
|
50% of the class
|
β × 0.5
|
Friendship /
Acquaintance
|
Daily (age < 65 years)
2.5 days per week (age > 65 years)
|
1-10% of their friends
|
β
|
Relations
|
2 days per week
|
One relative per household
|
β
|
Workplace
|
5 days per week
|
All close colleagues and one from other
colleagues
|
β
|
Workplace (public facing)
|
5 days per week
|
Random contacts are drawn from a
Poisson distribution Pois(λ )
λ = 1.5% × zone− population
|
β × 0.1
|
Random
|
Daily (age < 65 years);
2.5 days per week (age > 65 years)
|
Random contacts are drawn from a
Poisson distribution Pois(λ )
λ = 0.5% × zone− population
|
β × 0.1
|
Table 1. Contact type and transmission probability for social network based and random encounters
Viral transmission
During each simulated day, all infected agents come into contact with a subset of agents from their social network and with a proportion of the population residing in their area (as defined in Table 1). For each contact with a susceptible agent the virus may be transmitted with a certain probability (Table 1). The probability of transmission during a contact with an agent in the network (β ) is higher than during a random contact with a stranger (0.1 β ), as we assume that a contact with a stranger is of shorter duration and reduced closeness, translating in a reduced likelihood of transmission (Table 1). In accordance with evidence that children are less susceptible to COVID-1922, we reduce the probability of infection by 50% for agents under the age of 16.
Progression of the disease
Once a susceptible agent is infected, she progresses through the various states of the disease (Figure 1). The progression between disease states and the duration of each state are based on probabilities and durations which are age and gender dependent, as estimated in recent research on COVID-19 patients (Supplementary Table S1). Initially, the disease is in the incubation phase, a stage in which the agent is not infectious. A fraction of the infected agents becomes infectious 1-3 days before the end of the incubation period (pre-symptomatic infection), while others are infectious only at the end of incubation6, 23, 24. To reflect that, during each of the last 3 days of incubation agents have a 25% probability of becoming infectious. Following the incubation period, agents are either asymptomatic or symptomatic. Asymptomatic agents are able to infect others, but do not feel symptoms, and we assume that after the 3rd day of being infectious, infectiousness declines by 10% each day that follows25. Symptomatic agents with a mild disease are assumed to feel the symptoms, but do not require hospitalisation; severely ill symptomatic agents initially stay at home and are then admitted to the hospital. Once severely ill agents are admitted to the hospital, we assumed they do not come into contact with any other agent. The model does not simulate nosocomial infection and always assumes the availability of hospital beds. In accordance with findings, agents stop infecting others after 7-11 days from the onset of symptoms.26.
Mitigation strategies: testing, contact tracing app
The model includes two types of mitigation tools to track and trace infected agents: the contact tracing app (CTA) and COVID-19 detection tests. The CTA is distributed among a fraction of the population aged over 14 years old. It stores in memory the ID of all other CTAs it came into contact with over the course of the previous 10 days. Infected agents who are aware of their illness (by either feeling symptoms or following a positive test) can use the CTA to notify their contacts of possible exposure. Tests are administered between 1 and 3 days after the onset of symptoms in an agent, and results are assumed to be determined within a day. We assume that a fixed number of tests are available; as agents are tested the stocks decrease and restocking takes place daily. Agents seek testing when: (a) they feel symptoms, (b) they are notified of possible exposure by the CTA or directly by an infected relative. We assume that over the course of any given week, 3.5% of the population has contracted influenza27, of which 30% will seek COVID-19 testing. These agents test negative and contribute to the depletion of tests.
Agents’ compliance to self-isolation
When agents self-isolate all their social ties are removed, except for household ties, as they are assumed to self-isolate at home. However, it is likely that some precautions are put in place between the infected and her household members; therefore, we assume a 30% reduction in the probability of transmission.
Without the certainty that testing provides, surveys suggest that not all agents will comply with self-isolation guidelines, both when feeling symptoms or when notified by the CTA28. We denote a parameter ωi representing the probability of agent i to self-isolate when feeling symptoms of COVID-19. We also assume that agents who are notified by the CTA, but do not feel any symptoms, are less likely to self-isolate (than if they had symptoms) without testing. Therefore, their probability to
self-isolate is reduced by factor Ω, where 0 < Ω < 1. In the model, the probability of self-isolation varies between agents with mean ω = 70%. Figure 2 presents the algorithm triggered once an agent becomes aware of her symptoms. The procedure
triggers a chain of actions performed by symptomatic agents that involves testing (if available), decision to self-isolate and notifying relatives and CTA contacts. Following that, exposed agents who were notified preforms similar actions.
Model calibration and baseline scenario
The initial scenario reproduces a ‘business as usual’ situation with no mitigation in place, with contact frequencies as specified in Table 1. To verify the contact patterns generated by the ABM, we compared the properties of the distribution of agents’ daily contacts generated by the model to a distribution of contacts derived from a survey conducted in the UK29. Like the survey results, the distribution is characterized by a lognormal body and a power-law tail with an exponent of -2.1 (Figure 3a). The median and mean number of daily contacts are 17 and 12, respectively.
To calibrate the model, we tested a range of transmission probability (β ) values to generate the basic reproduction number of R0 2.8 in the initial three weeks of the epidemic, as estimated for the UK30. The best fit was achieved for β = 0.08.
After establishing the initial scenario, we simulate the post-lockdown situation expected in several countries, in which most restrictions are lifted but citizens are still encouraged to work from home when possible, limit social interactions, maintain physical distancing and wear face masks in public. Therefore, in this scenario we assume 3 days attendance per week at workplaces and schools; and a reduction of 30% in contacts in schools, with strangers, as well as the frequency of social meetings (within the ‘friendship’ network). In addition, β is reduced by 30% to reflect measures such as face mask usage, social distancing and increased hygiene, all of which reduce the likelihood of viral transmission. We refer to this situation as our baseline scenario. The reproduction number in this scenario comes down to 1.5. Comparing the scenarios, when social distancing is practised, the proportion of infected agents at the peak of the epidemic is significantly reduced from 34% to 10% (Fig 3c). The distribution of the sources of infection also varies, as the proportion of infection originating in workplaces and schools is reduced and the household becomes the predominant locus of transmission (Figure 3b).
Experimental design
The core of our study explores the introduction of the CTA and the availability of testing into the baseline scenario of social distancing. We simulate the impact on viral spread of various combinations of: (1) proportion of CTA users in the population; (2) levels of testing capacity; (3) levels of compliance with self-isolation on the part of CTA users; (4) testing Table 2 summarises the parameter combinations explored in the model. Overall, we simulated 140 scenarios, each repeated 20 times to account for uncertainty in the results due to the stochasticity embedded in the model.