2.1 Synthetic population

BharatSim’s synthetic population models those details of the actual population that are relevant to the interactions between agents. We use a variety of data sources to generate a population of individuals and households with demographic attributes that are statistically similar to survey data, using statistical methods and machine learning algorithms. These can be used to generate data at various scales, ranging from small communities to entire states [45].

2.1.1 Datasets and data preparation.

The primary sources of data for our algorithms include the Census of India, the India Human Development Survey (IHDS), the National Sample Survey (NSS), and the Gridded Population of the World (GPW). We use sample survey data (microdata) from India Human Development Survey-II [46], marginals from the 2011 census [47], data for employment from NSS [48], and population density from GPW grid population density dataset [49]. A hybrid process is used to scale up these limited datasets to the size of a population. The data is initially curated to remove obvious inconsistencies. Then, the customized hybrid of statistical and machine learning techniques detailed below is used to generate new data on the required scale. Table 1 lists the different datasets and the role they play in the generation of the synthetic population.

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Table 1. Source datasets for generating the synthetic population.

The different data sources used are specified, along with the role they play in the generation of the synthetic population.

https://doi.org/10.1371/journal.pcbi.1012682.t001

The IHDS-II (2011–12) [46] is a nationally representative, multi-topic survey of 42,152 households in 1,503 villages and 971 urban neighbourhoods across India. We use microdata from two datasets from this survey, Individual and Household, in the machine-learning model described below. The 2011 census was performed across 28 states and 8 union territories, covering 640 districts, 5,924 sub-districts, 7,935 towns and more than 600,000 villages. We use marginal data from this census as input to the Iterative Proportional Updating (IPU) method described below, since the census microdata is not publicly available. The NSS collects data through nationwide sample household surveys on various socio-economic subjects. We use employment data from this survey. GPW provides spatial population maps for all countries and their sub-divisions—we use population density data for India to place the generated synthetic population in a geographical context. The next section describes the use of these different datasets in our hybrid model to generate a synthetic population for a district of India.

2.1.2 Population generation.

Our objective is to generate an accurate synthetic population for India with joint distributions of relevant variables that statistically match those of the actual population. The synthetic population should capture relevant attributes of individuals in the real population, such as the distributions of age, height, weight, and sex. The synthetic population must also incorporate social network structures appropriate to households and workplaces. We might also require the distribution of family sizes, the joint distribution of age and sex of individuals within a family, the geographical distribution of households, workplaces, and schools within administrative units (cities, districts, and states), and the number of people who are associated with a workplace or school. The population generation pipeline described below is illustrated in Fig 1.

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Fig 1. Pipeline for synthetic population generation.

Census, gridded population, and survey data are first processed and subsequently used as the input to generate the final synthetic population.

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Iterative Proportional Updating (IPU) [50] is a statistical sampling technique that accepts both information regarding marginals from the real population, and individual-level data from a sample of the real population. IPU is used to iteratively scale up the sample to generate a large-scale population, ensuring that the marginal distributions for both households and individuals in the synthetic population match with the known marginal distributions in the real population.

While IPU generates a large-scale population in which households have a realistic distribution of the number of family members and their age and sex, the joint distribution of various attributes in the synthetic population should also resemble those of the real population. To achieve this, we use a machine learning technique, Conditional Tabular Generative Adversarial Network (CTGAN) [51], to add individual-level attributes for which marginals are not available. CTGAN is trained using data from a sample survey and used to generate highly correlated individual attributes like heights, weights, and comorbities. This provides a reasonable joint distribution of various individual-level attributes. Additionally, different trained instances of our CTGAN model can also be used to combine different survey datasets with an overlapping set of attributes.

A realistic geographical distribution of the population must describe the spatial distribution of households, workplaces, schools, and public places, and the number of individuals that frequent them. We use grid population density data to assign individuals to specific locations. A row in the grid population density data consists of latitude (X), longitude (Y) and the number of individuals (Z) who live within a square of side 1 km, centred at (X, Y). We use rejection sampling to assign geo-location points within a geographical boundary, faithful to population density distributions [49]. An individual is then assigned locations such as workplaces, schools and public places, based on a probability inversely proportional to their distances from the individual’s household [52].

The sampling method determines the geographical boundaries of the region for which data is being generated using a polygon provided by a geojson file. We filter the population density dataset by only retaining those rows for which the latitude X and longitude Y are within the boundary polygon of the region. We sample, with replacement, a large number of latitude-longitude pairs from this filtered subset, using the number of individuals (Z) as the weight for this sampling. For a given row in the sample, we then add uniformly distributed random noise to both the latitude X and longitude Y. We reject those latitude-longitude pairs that are not within the polygon, until the required number of points is reached. Given that we do not have access to workplace distribution data, we make the natural assumption that it follows the population density distribution as well. Our technique can use actual workplace density data to obtain distributions more consistent with reality.

Individuals in our synthetic population are associated with external locations which include workplaces, schools, and public places. These locations define their contact networks. All individuals are assigned job descriptions. This job description is sampled with replacement from the list provided in the IHDS-II dataset. Individuals below the age of 3 are “Homebound”. For individuals above the age of 3 and below the age of 18, “Student” is assigned as the job description. The number of workplaces, schools and public places varies for each district. Each workplace has three attributes: workplace type, latitude and longitude. Just as for the job description, the attribute “workplace type” is generated by sampling (with replacement) from the list of jobs in the IHDS-II subset. Specific workplaces are assigned as schools by looking at a subset of workplaces of individuals with job descriptions set to “Teacher”. Students are assigned to schools and adults to workplaces, provided they are not homebound. We assume that agents are more likely to visit public places which are closer to their homes.

Given an individual with home latitude X_h and longitude Y_h and a list of k possible external locations {E_k} with latitude and longitude pairs , we calculate the Euclidean distance r between the home of the individual and the external location. Among those who visit external locations, a fixed fraction is labelled as taking public transport. (Although we do not explore this in this paper, such a label would allow us to examine the possibility of increased transmission in such crowded locations. This has been examined in prior work [39]). We then choose a real-valued function f(r) which is strictly decreasing and non-negative in the positive real domain, and weight the probability of an individual being assigned the external location by the value f(r). For our analysis, we choose f(r) ∝ 1/r² so that the probability of an external location being assigned is inversely proportional to the square of Euclidean distance between the individual’s house and the external location. A geodesic distance metric can also be used, but the difference is insignificant at these scales.

As an example, in S2 Appendix we describe the creation of a synthetic population for Mumbai, along with a visualization of the geographic distribution of synthetic households, workplaces and schools in the combined districts of Mumbai and Mumbai Suburban. We also include a statistical analysis of the synthetic population vis-a-vis the survey population.

The code that generates the synthetic populations used in this paper available on the BharatSim GitHub page (www.github.com/bharatsim). In addition, the synthetic populations used in our simulations can be found on the BharatSim website (bharatsim.ashoka.edu.in).

One of the challenges in making a synthetic population is the need for up-to-date survey data. The most recent iteration of our main source of microdata, the IHDS-III survey, was expected to be released in 2023, but has still not yet made public. Additionally, the last India-wide census was conducted in 2011. The absence of up-to-date census data has made the task of extrapolating to the current date far more difficult. Following most, if not all such modelling efforts, we have used data from the 2011 census to generate the populations used in this paper. However, we understand that some modellers may prefer to use a more up-to-date population. To facilitate this, we have used state-wide population projections from the Niti Ayog, India’s apex planning body, extrapolating to 2021, and assuming that distributions remain similar in form. This projected population is also available from the BharatSim website.

2.1.3 Population benchmarks.

We evaluate the synthetic population based on the following three criteria: (i) does the distribution of individual features in synthetic population match those in sample surveys? (ii) do the joint distributions of selected features in synthetic population match those in sample surveys? and, (iii) can our synthetic population replace survey data in a machine learning regression task?

S2 Appendix compares the marginal distributions of attributes of age, height, and weight across samples of the real and the synthetic population for the combined districts of Mumbai and Mumbai Suburban. We work with a randomly chosen subset comprising 10,000 individuals to compare with the survey data, although our synthetic population has 12 million individuals.

To benchmark the population, we compared the marginals of individual features (age, height, weight, sex) in the synthetic population with those in the survey data to quantify the similarities of their underlying statistical distributions. The results of our tests are shown in S2 Appendix.

To examine whether the synthetic data could reproduce correlations present in the survey data, we use a machine-learning regression task. We first attempt to predict the weight based on age, sex and height. Similarly, we attempt to predict the height based on age, sex, and weight. We use two machine-learning models: Linear Regression and Multi-layer Perceptron (MLP) Regression [53–55]. The results of these regression tasks are reported in Table 2. We begin by training each model on a subset of the survey data. We choose this subset to be 80% of the entire sample. We then report the accuracy of the model on the remaining 20% of the sample. Next, we instead train our models with the entire synthetic population, and again report the accuracy of the model on the same 20% of the survey data. The results of these comparisons are shown in Table 2.

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Table 2. Benchmarking synthetic population quality.

To assess the viability of using synthetic populations as substitutes for real populations, we analyse the predictive performance of various machine-learning models using both real and synthetic data. Each model is tasked with predicting weight (or height) based on age, sex, and height (or weight). Case 1: the model undergoes training and testing using survey data. Case 2: an identical model is trained on synthetic population data and tested on the survey data. This table lists the accuracy of each model in predicting outcomes for linear regression and MLP regression.

https://doi.org/10.1371/journal.pcbi.1012682.t002

2.2 The simulation engine

The simulation engine framework allows modellers to directly specify their models using a domain-specific, high-level language. This domain-specific language (DSL) is based on the Scala programming language [56], allowing modellers to extend their knowledge of Scala when creating their models. The DSL provides abstractions and constructs that make it easier for modellers to define general agent-based models of arbitrary complexity. It simplifies the creation of customized data structures, the execution of scenarios and experiments, as well as the integration with a synthetic population. Modellers can thus focus on the specific problem they wish to address even without being highly computationally skilled.

1. Agents The Agent class encapsulates an individual agent’s attributes, behaviours, and relations with other agents or entities. Modellers can extend the framework’s inbuilt Agent class as required, including behaviours specific to the context of the problem they are addressing. For example, the StatefulAgent described below is one such inbuilt extension of the Agent class.
2. Behaviour An agent’s behaviour is defined as an action that is executed at every simulation timestep (or “tick”). For example, an agent might have some probability of getting themselves vaccinated at each time-step. This probability can depend on their attributes like their age, socio-economic status, and prior vaccination history.
3. Stateful Agents A StatefulAgent is a specialized extension of the Agent class that supports the framework’s Finite State Machine (FSM). This is particularly useful in models where agents can exist at any given time in only one “state”, but where transitions exist between these states. The StatefulAgent interface can define and control these transitions between states. These include actions that must be performed when agents enter or exit these states, as well as actions that are performed in every simulation tick.
4. Networks BharatSim is a network-based simulation model, where networks determine agent interactions. Every Agent or a StatefulAgent is a Node and a Network is composed of such Nodes, with Relations between them. The framework allows the modeller to define relations between agents that mimic real life network locations like houses, offices, restaurants, or other public places. These networks can, in principle, be dynamic.
5. Relations Different Nodes can be related to each other using framework-defined as well as extendable Relations. For example, an agent can be linked to his or her house, as well as his or her workplace. Two agents or workplaces can also be related to each other in this fashion.
6. Schedules Schedules define the Network locations of Agents at a given instant in the simulation. For example, the Schedule can specify that a specific Agent is part of her “House” network for 0 to 8 simulation ticks and “Office” network for 9 to 18 ticks. The BharatSim DSL allows for the definition of a custom Schedule using predefined units (Day, Week, Month), but also allows modellers to define custom units of time, as per their requirement.
7. Interventions Interventions are a set of externally defined rules imposed on the agents, such as lockdowns or vaccination drives. BharatSim allows for the incorporation of such interventions in order to study a range of counterfactual scenarios. Modellers can define multiple types of interventions, with different activation, deactivation, and reactivation conditions over the course of the simulation.
8. Simulation In BharatSim, model developers have to define a Simulation in order to be able to specify input data, register agents, define schedules, and generate output. Along with monitoring clock ticks, the Simulation stores all common data in a Context, which is shared globally and from which other objects like agents can query information.

2.2.1 Creating Agents and Networks.

The simulation engine accepts the synthetic population as a CSV file. This data is stored in a graph data structure, a network of nodes related to each other through user-defined relations. These nodes represent both individual agents as well as network locations such as households or offices. The framework defines a Node class which allows for relations to be established between other nodes. Using such a data structure makes the framework flexible and domain-independent. This can be implemented in one of two ways, either by using Neo4j, a graph database, or using the Scala programming language’s scalable map implementation, TrieMap. Both these structures optimize data operations, allowing the simulation to scale efficiently to larger populations.

Scala allows programmers to define their own datatypes, justifying its choice for the BharatSim simulation framework. In particular, a user can define a special “class”, a user-defined blueprint with a set of properties or functions that are common to all objects of this datatype. The user could then create different objects as “instances” of this datatype, just as a user might create different variables to be instances of the Integer datatype. Additionally, these classes can then be “extended” to define new datatypes. These new datatypes “inherit” the attributes and methods of their parent class, but may also possess new attributes and methods of their own.

The framework uses the Node class to define the Agent and Network classes. The Agent and Network classes can be further extended by the modeller to define the individual agents and locations of their own model. The Agent class possesses a addBehaviour function which can be used to define agent behaviours.

In Scala, case classes are particularly useful for modelling immutable data. In the models specified below, the agents and network locations are thus defined as case classes that extend these inbuilt framework-defined classes. Additionally, BharatSim implements a Finite State Machine (FSM); with this implementation, a modeller may define classes that extend the inbuilt State trait to distinguish agents that exist in different disease states. In order to use this FSM, the framework extends the Agent class to define StatefulAgent, an agent that can be in only one State at any given instant of time. The State class possesses two inbuilt functions:

1. a. An enterAction: a function that defines an action that is to be performed only whenever an agent enters this particular state, and
2. b. A perTickAction: a function that defines an action that is repeated by an agent at every time-step (or “tick”) for as long as they are within this state.

Additionally, every extension of the State class also possesses an addTransition function which allows the modeller to define (i) which State a StatefulAgent can transition to from the current State, and (ii) under what conditions such a transition can occur. Very much like the addBehaviour function, the addTransition function is run for every agent at every time-step, to check whether a transition between states should occur.

Agents are assumed to move between different network locations to interact with each other. These locations can be modelled by extending the inbuilt Network class. Just as with the Agent class, these classes too may possess an arbitrary number of attributes. Additionally, the Network class has an inbuilt getContactProbability function which can be used to define, for example, a location dependent relative risk of infection.

Every individual agent follows a schedule that is defined by the modeller. Such schedules specify agent locations across time. These schedules can be dynamic, can depend on the current state of the agent, and can be affected by interventions that are imposed. In addition to the predefined units of time (Hour, Day, Week, etc). schedules may also be linked to user-defined units of time (instances of the ScheduleUnit class). In BharatSim, these schedules can be registered using the registerSchedules function. When schedules are registered, a priority can also be assigned to them. This may be used to give a lockdown-schedule a higher priority than the routine daily schedule of any agent. In Fig 2 we show a schematic of some of the different components of the simulation engine described above.

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Fig 2. Components of the simulation engine.

(A) A schematic of a graph between nodes representing agents, homes, and work locations. (B) These nodes are user-defined extensions of the in-built Node class. (C) The graph is created by creating bi-directional relationships between these user-defined classes. (D) Agents can have user-defined schedules using user-defined units of time. Additionally, the agent class may posses custom user-defined functions. Some of these functions, called “behaviours”, can be set so that they are executed by every agent at every time-step. (E) Agents’ schedules can be customized based on attributes from the synthetic population like their age, job label, and socioeconomic status.

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2.3 The visualization engine

Using the simulation engine, the modeller can write simulation data to an output file. Such a file could contain, for example, the number of individuals in different disease states for every time step, or more specific geo-spatial information. The visualization engine can then read this output and create multiple dashboards with different types of graphs that can help to analyse the results and visualize them.

The visualisation engine is self-contained, and can accept and visualize any CSV data file. Each dashboard provides the user with a combination of different graphs and charts, including line-graphs, histograms, and pie-charts. Additionally, the visualization engine can also represent GIS data in the geoJSON format to plot heatmaps and choropleths. These can either be static, or can change with time based on the data provided in the input CSV file.

The visualization engine provides the following features:

1. Data import The visualization engine can import data in ZIP, CSV, and GeoJSON formats. The user can add, delete, and link files from different dashboards and can also add columns with custom formulae.
2. Charts The user can represent time-series data as a line-graph. Additionally, bar charts, histograms, and pie charts can be used to study aggregated data. Geographical information and data can also be visualized. Spatial variations in data can be visualized using either heatmaps or choropleths. Heatmaps visualize data in the form of “hot” or “cold” spots, with a warm-to-cool colour scheme. Choropleth maps can also be used to visualize how a quantities vary across a fixed geographic area like a district or state, while simultaneously showing the extent of variation within a region.
3. Project & dashboard management Users can create a Project which can have multiple individual dashboards. Users can add, edit, and delete projects and dashboards for better management. Dashboards are auto-saved to avoid data loss. Users can easily add, edit, duplicate, and delete widgets and charts in each dashboards. Dashboards can be duplicated to allow users to duplicate certain design choices and widget configurations. Each widget also allows the users to export the output either as raster or vector data using the PNG and SVG file formats respectively.

2.4 Implementing a disease progression

The BharatSim framework allows a modeller to implement a variety of agent-based models. The disease states in these models can be labelled according to the compartments in associated compartmental models. In this paper, we restrict ourselves to the compartments described in the INDSCI-SIM model of Ref [42]. This model uses the disease progression shown in Fig 3. It has been used to describe the disease trajectory across the first wave of COVID-19 in India. However, BharatSim allows users to easily use different disease progressions. Indeed, in our study of mpox, described in detail in S9 Appendix, we have modelled a different set of disease compartments.

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Fig 3. Schematic of the epidemiological model.

Boxes show the 9 compartments of the model, i.e., the set of states that characterise each individual in the population. The compartments are (S)usceptible, (E)xposed, (A)symptomatic, (P)resymptomatic, Mildly Infected (MI), Severely Infected (SI), (H)ospitalised, (R)ecovered, and (D)ead. Black arrows indicate possible transitions between states. The dotted lines indicate that the rate of infection of Susceptible individuals is an increasing function of the number of infected individuals they are in contact with. Recovered individuals are assumed to be immune to further infection in this model, however the possibility of including a non-zero recovery rate has been implemented.

https://doi.org/10.1371/journal.pcbi.1012682.g003

The INDSCI-SIM model assumes 9 compartments: Susceptible (S), Exposed (E), Asymptomatic (A), Presymptomatic (P), Mildly Infected (MI), Severely Infected (SI), Hospitalised (H), Recovered (R), and Dead (D). Those in the Susceptible compartment can be infected by all individuals in any infectious compartment, whereupon they transition to the exposed compartment. Exposed individuals can either become asymptomatic, exhibiting no symptoms and eventually recovering, or presymptomatic, wherein they transition into either the mildly or severely infected compartments. Mildly infected individuals recover, while severely infected ones are eventually hospitalised. They go on to either recover or die. Extensions of the INDSCI-SIM model that incorporate vaccinations have been used to describe subsequent waves in the country [57].

The synthetic population described in Section 2.1 provides us with individual-level information about the age distribution, household size distribution and composition, and comorbidities. This information provides further granularity to the description of disease states and the transitions between them. In the model discussed here, for example, the individual agent’s probability of branching between the asymptomatic and presymptomatic states is age-dependent, with older individuals more likely to develop an acute case of the illness. We do the same for the other branching probabilities, as shown in Table 3. In addition to the agent’s age, the synthetic population also provides us with the coordinates for agents’ homes and workplaces. No other columns from the population are currently used, although it is straightforward to extend our simulation to incorporate, for example, the effect of comorbidities on disease spread.

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Table 3. Branching ratios for different compartments.

In our model, we assume that an individual agent’s probability of branching between different compartments is age-stratified. Individuals are divided into age-bands of one decade each. When an individual agent exits a compartment that branches into multiple possible compartments, they are assigned a compartment based on this age-stratified probability. These rates have been taken from Ref [59].

https://doi.org/10.1371/journal.pcbi.1012682.t003

The residence times that different individuals spend in different compartments can, in principle, also be age-stratified. For simplicity here, we assume these parameters to be the same across all age-groups. BharatSim allows the modeller the possibility of choosing these residence times from different probability distributions including the exponential distribution, as simple compartmental models assume, or the lognormal distribution which we use in this paper on grounds of its wider applicability. A set of convenient parameters for these different distributions can be obtained from Ref [58], and can be found in Table 4. These can be updated as more information becomes available.

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Table 4. Residence times in different compartments.

When an individual agent enters a disease state, they remain in this compartment for a period of time drawn from the probability distributions given above [58].

https://doi.org/10.1371/journal.pcbi.1012682.t004

We also incorporate the possibility of vaccination drives in our model. Vaccines are assumed to reduce both the relative risk of infection as well as the probability of symptomatic disease. We also assume that vaccinated individuals are less likely to transmit the disease, as shown in Ref [60]. The vaccine’s efficacy is assumed to ramp-up to its maximum value over a period of 14 days. Additionally, the dose-interval is currently set to 90 days between doses; see S5 Appendix for more information.

All individuals above the age of 20 are assumed to be eligible to receive vaccines, provided they are not exhibiting symptoms. We can begin our simulation with either a vaccine-naive population, or a population where a certain fraction has already received partial protection through one or two shots of the vaccine. For the simulations discussed here, these numbers are chosen to be 20% and 10% respectively, to match the situation in India as of August 1, 2021. These initial vaccines are distributed as follows: first, 80% of the individuals above the age of 60 are assumed to be vaccinated by the start of the simulation. Next, any remaining vaccines are distributed randomly among the full (age-stratified) eligible population.

Additionally, vaccines can be rolled out in different phases, targeting eligible individuals of different age-groups. In the simulations described below, we include two phases. In the first phase, individuals above the age of 40 are given first preference to receive vaccines. This phase spans one month. In case there are no eligible agents above the age of 40, the vaccines are randomly distributed among the entire population to avoid allotment-related wastage. In the second phase, all agents above the age of 20 (not currently exhibiting symptoms) are targeted for vaccination.

2.5 Network dynamics

An individual’s movements in a day are defined through a “schedule” (see Section 2.2), specific to each individual. This schedule dictates which network location (home or workplace) the individual participates in at any given time. The infection spreads stochastically between individuals who share a location at any given time, at rates determined by the infection parameters as well as the vaccination status of the different individuals present in that location. In our simulations, we take a single time step to be 12 hours, defining the probabilities accordingly. Individuals move back and forth between their homes and workplaces or schools twice a day.

Schools are a specific subset of workplaces to which all school-going individuals are assigned. Schools consist of classrooms, and students are assigned specific classrooms and do not mix with students in other classrooms. Multiple network and school structures can be examined. For example, one may allow for students and/or teachers to move between the classrooms of the school. Classrooms may also be defined according to age brackets.

The force of infection at any location l, the rate at which a single susceptible person becomes infected, is proportional to the product of the number of infectious people at that location and the infectivity parameter β. This is divided by a normalization factor : (1) where V_tr(i) is the factor by which the agent i’s probability of transmission is changed due to vaccination, C_q(i) is an “isolation factor” (chosen in the range 0⁺ to 1). In our simulations below, this factor is 0.1 if the agent is hospitalised or quarantined and 1 if not. For density-dependent transmission, as in this paper, we can chose to be the effective number of individuals in location l, not counting dead individuals and only a fraction of hospitalised individuals, as in [61]. However, this can easily be generalised to other types of transmission, including frequency-dependent transmission.

The force of infection at each location controls the probability p_j that a susceptible individual j in that location contracts the disease and therefore transitions to the exposed state in some time-step Δt: (2) where V_inf(j) is the relative risk of infection, given by the vaccination status of the individual j.

When an individual enters the exposed compartment or any of the subsequent compartments, a “sojourn-time” is drawn and stored from the distributions shown in Table 4. The individual remains in this compartment for this period. Once this time has elapsed, the individual transitions to the next state in the disease progression. If two possible outcomes exist—for example, suppose they can transition to both the asymptomatic or presymptomatic states—then one of the outcomes is chosen randomly, weighted by the appropriate (age-stratified) branching probability for that transition. For the new state, a new “sojourn-time” is drawn and stored.

Following an update of all agents, the time is incremented by Δt, all individuals are moved along the network according to their specified schedules, and the process repeated for the next time-step.

2.6 The models and underlying paradigms

There are two ways to model the disease progression in BharatSim and the modeller may choose between then. Both work with the same disease progression and synthetic population and implement the same interventions. In the first, the framework’s inbuilt Finite-State Machine (FSM) controls the movement of agents between the different disease states. The framework ensures that an individual StatefulAgent can only possess one activeState at any given instant of time. Such an implementation is computationally efficient and simple to debug.

When we wish to model situations where multiple strains of a disease could coexist, such an FSM cannot be implemented without a proliferation of states. Indeed, an individual could have recovered from one strain, only to be infected with another, or might in principle be simultaneously infected with both strains. In the second implementation, we do not use the StatefulAgent class, but instead work with the Agent class. In this case, the individual’s state is defined by a set of two attributes: s1_disease_state and s2_disease_state, which contain information of which states they are currently in with respect to the disease dynamics of each individual strain. In the model described here, being infected by one strain reduces an individual’s relative risk of infection to the second strain. We vary this relative protection from the second strain to see how varying levels of immunity provided by the first strain change the disease trajectory. (Reinfections for each individual strain are currently not implemented in the examples discussed below, although it is easy to include them.)

2.7 Interventions and complex dynamics

3.1 Simulating disease spread in a model Indian city

We use our model to simulate the city of Pune, with a population of 3.13 million agents, exploring the different disease trajectories induced by pharmaceutical and non-pharmaceutical interventions.

We consider first the effect of a lockdown on the disease trajectory. Fig 4A shows the fraction of recovered over all age-groups with time in different scenarios. In these scenarios, the daily vaccination rate is varied, and 15-day lockdowns are introduced when the total number of infections crosses a threshold of either 1% of 5% of the total population. If left unchecked the pandemic would have led to 70% of the population having contracted the disease during the epidemic. With the introduction of a 15-day lockdown that begins when the number of active cases is 5% of the total population, this number reduces to close to 60%. The number further reduces if the lockdown began even earlier (when the number of active cases is 1% of the population).

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Fig 4. Simulating disease trajectories with different interventions.

The plots show the disease trajectory for the city of Pune in five different scenarios: without any interventions, with 15-day lockdowns beginning when the number of active infections is 1% and 5% of the population respectively, with a lockdown starting with 1% active infections and concurrent vaccination drive at two different vaccination rates of 0.2% and 0.5%. (A) shows the number of recovered as a function of time, (B) the number of active cases as a function of time, and (C) the number of active severe cases and hospitalizations. We see that imposing a lockdown reduces the total number of people who contract the disease, an effect that is enhanced by starting the lockdown earlier. A concurrent vaccination drive with a high vaccination rate can further augment this. In all scenarios with a vaccination drive, the drive begins on day 0. The curves are averages over 25 runs, with the error bars corresponding to 1.96σ.

https://doi.org/10.1371/journal.pcbi.1012682.g004

We next consider the role of vaccinations. We begin a lockdown when the number of active infections is 1% of the population, and vary the daily vaccination rate of the vaccination drive. By comparing these scenarios with the curve representing the case with the lockdown but no vaccination drive in Fig 4A and 4B, we see that an increased daily vaccination rate reduces peak of the infection. The number of severe and hospitalised, shown in Fig 4C, follows the active-infections curve, although lagging slightly behind it in all scenarios.

Additionally, we also study what effect the initial spatial configuration of the disease has on its trajectory. In Fig 5, we compare two such scenarios: in the first, a total of 3000 individuals (roughly 0.1% of the total population of Pune) are seeded in a single ward, and the disease is allowed to spread. In the second scenario, the same number of individuals are chosen from a much smaller geographical region (a collection of 750 homes), and the disease is again allowed to spread. The figure compares the spread of the disease due to these two initial conditions.

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Fig 5. Effect of initial infection seed on disease spread.

The heatmaps represent the density of the infected at different instants of time for two scenarios. In each case, a single simulation with an initial infection seed of 0.1% of the total population was seeded in (A) uniformly across a single ward, and (B) a geographical cluster of 750 families in a single ward. This illustrates how BharatSim can be used to model the spatial hetereogeneity in disease spread. The central ellipse that remains unchanged across time represents the Army Cantonment region for which no publicly-available data exists. The map image is produced by the BharatSim visualization engine with base maps obtained from OpenStreetMap (https://www.openstreetmap.org/).

https://doi.org/10.1371/journal.pcbi.1012682.g005

Finally, we also compare the effects of different lockdown strategies on disease dynamics. In particular, we study the effect that locking down the most populous wards of the city has on the spread of the disease. In order to do this, the following procedure is used: the entire city is divided into a grid whose unit cell is 200 m × 200 m. Every five days, we count the number of infected individuals in each of these grid boxes. An individual is assumed to count towards the number of infected in a grid box if their household falls within its geographical boundaries. We place the most populous central wards under a lockdown for (i) 15 days and (ii) 30 days when the number of cases is 1% of the total population, prohibiting all entry and exit from these wards, and observe the effect this has on the disease dynamics. In Fig 6 we show the results for one such run in which we see that locking down the most populous wards can slow down the spread of the disease in the city, thereby flattening the curve of infections.

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Fig 6. Effects of different lockdown strategies.

The city of Pune is divided into a grid of 200 m × 200 m, and the number of infected individuals with households in each grid box is plotted as a heatmap. In scenario (A) no interventions are applied and the disease is allowed to progress freely through the population. In order to test the effect that locking down the most populous wards has, these wards are locked down for a period of (B) 15 days (C) and 30 days when the number of active infections is 1% of the total population, which corresponds to day 50. From these graphs, we conclude that locking down the most populous wards can slow down the spread of the disease in the city, thereby flattening the curve of infections. The underlying map of Pune is provided by the Spatial Data of Municipalities (Maps) Project by Data{Meet} [71], made available under the Creative Commons Attribution 4.0 International license. As before, the central region that remains unchanged represents the Army Cantonment area for which we have no data.

https://doi.org/10.1371/journal.pcbi.1012682.g006

In reality, the south-west of Pune saw a larger number of cases than other regions. This has been attributed to the relative importance of public transportation, variations in adherence to non-pharmaceutical interventions and inhomogeneities in case-records, among other explanations.

We recognise the utility of at least an approximate comparison to real data. We have chosen to use our model to describe the onset of the COVID-19 pandemic in the city of Pune, comparing model results to real data. We choose a time period between the 1st of March and the 15th of July, 2020 for this comparison for the following reasons: (i) the simulations are initiated when the lockdown started so spread would primarily have been driven by in-house transmission, (ii) the virus dynamics was largely dominated by a single variant, and (iii) we can study the impact of the relaxation of the nation-wide lockdown on cases.

This direct comparison is shown in S8 Appendix. Briefly, our results are consistent with (i) a relaxation of the first lockdown leading to an increase in cases, (ii) a large background of undetected infection, arising mainly from asymptomatic cases, (iii) an indication that the epidemic arising from the initial strain had run its course by the end of the second lockdown and that (iv) that a complex mix of more transmissible variants would have determined the later-time course of the disease. These agree with a retrospective analysis [59, 72].

Extending our study beyond this window of study would have involved accounting for the emergence and dominance of multiple such variants, as well as an ever-changing suite of non-pharmaceutical interventions. We will leave these to later work. Our model here is not intended to precisely reproduce these complexities, but to describe a framework within which such questions can even be addressed.

3.2 Effects of school reopening on the disease trajectory

To study the impact of school reopening on the disease trajectory, we simulate a number of different counterfactual scenarios. In these simulations, we assume schools are initially closed (i.e., children stay at home) as was the case in India for a close-to two-year period since March 2020. We then examine the effect of opening schools at various times during the epidemic. We assume 20% of our population is initially vaccinated, with 10% having received the second-dose. We further assume that vaccinations proceed at a constant daily vaccination rate with an appropriately chosen prioritization of first and second doses.

We consider two different initial conditions. In the first, 30% of the population is assumed to have already recovered from the disease, and is therefore immune to it. In the second case, this number is set to 50% [73]. In both cases, we observe the effect of varying vaccination rates (including no vaccination), as well as the times at which schools are reopened.

Fig 7 shows our results for infected children (ages 0–18) and adults (ages 18+) as a function of time. We find that for children, reopening schools could lead to a substantial increase in cases with an initial recovered fraction of 30%. In this case, opening schools earlier leads to a greater increase in cases among children. However, with an initial recovered fraction of 50%, this increase is substantially lower. In all simulations with these initial conditions, reopening schools does not lead to any significant increase in infections among adults. As a consequence, the total number of severe infections, hospitalizations, and deaths remains largely unaffected when schools are reopened under any of these scenarios.

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Fig 7. Simulating different school reopening timelines.

The left-panels show results for children (≤18 years) while the right-panels show the results for adults (>18 years). We begin with an initial recovered fraction of 30% of the population, and with 20% of the population vaccinated (10% with the first dose and 10% with both doses). We compare two different scenarios: in (A) we simulate the population in the absence of a vaccination drive, and in (B) with a daily vaccination rate of 0.4%. The initial plateau in active cases for children is due to the fact that a small fraction of children are infected as part of the initial infection seed, but are not able to infect each other since schools are closed. In (C) and (D) we compare the same scenarios (no vaccination drive and 0.4% daily vaccination rate respectively), but with an initial recovered fraction of 50%. In all cases, opening schools earlier leads to a rise in cases among children, but no significant increase in infections among adults. The curves are all averaged over 100 simulation runs, with the error bars corresponding to 1.96σ.

https://doi.org/10.1371/journal.pcbi.1012682.g007

However, any increase of cases among children, even if asymptomatic or mild, may also be a matter of concern. Thus, we also studied the peak number of cases in children as a function of the time at which schools are reopened. This is shown in Fig 8 for initial recovered fractions of 30% and 50%. From these plots, we see that after a certain time, keeping schools closed makes only a marginal difference to the peak number of infections amongst children. This threshold time depends on both the daily vaccination rate and the initial seropositivity. In all cases, this threshold lies well past the date of peak of the infection that would have occurred if schools had been closed throughout. Thus, if cases in children have already reduced significantly from the peak value, reopening schools would only make a marginal difference.

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Fig 8. Effect of schools reopening on the epidemic peak in children.

The peak number of children infected (normalized to their relative population) under different reopening times as indicated by the x−axis and different rates of immunization as indicated by the colour of the plots. The two panels show the results (A) assuming 30% initial recovered (B) assuming 50% initial recovered. The values are all averaged over 100 simulation runs, with the error bars corresponding to 1.96σ.

https://doi.org/10.1371/journal.pcbi.1012682.g008

Another useful criterion for this threshold is obtained by examining the vaccine coverage and net seropositivity as a function of time (see Table 5). Here we define vaccine coverage as the fraction of the total population (i.e., both adults and children) that has received at least one dose. We define net seropositivity as the fraction of the total population that has either been infected and recovered, or has been vaccinated by at least one dose. For this to hold true, the net infection-induced seropositivity among children must be comparable to that among adults.

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Table 5. Vaccine coverage and net seropositivity when schools are reopened.

The fraction of the population that have received at least one vaccine dose (vaccine coverage) and the net seropositivity (both infection and vaccination derived) are tabulated at the moment schools are reopened, for simulations with daily vaccination rates (DVR) of 0% (no vaccination), 0.2% and 0.4%. The top table shows results for an initial recovered fraction of 30% and the bottom corresponds to an initial recovered fraction of 50%.

https://doi.org/10.1371/journal.pcbi.1012682.t005

Comparing Fig 8 with Table 5, we conclude that in order to minimize the peak number of infections in children, schools should be unlocked only when the net seropositivity reaches the range 75–85%. This net seropositivity will be achieved by a combination of natural infections and vaccination. Therefore, vaccinating the population at a higher rate will allow one to open earlier, as will an initially higher seropositivity. Our model allows us to compute these thresholds for any arbitrary vaccination schedule and prior history of infections.

Our results differ in some aspects from those in Ref [74], a modelling study that explores scenarios of partial, progressive, or full school reopening in France, finding somewhat larger severe disease than reported here. We note, however, that the fraction of the population in the 65+ category is over 21% in France, while it is less than 7% in India. Given the strong dependence of the IFR on age, we would expect mortality arising from school reopenings to impact the population far more substantially in France in comparison to India [75]. A different modelling study using a compartmental model with India-specific assumptions yields results very similar to those described here [57].

3.3 Modeling multiple strains with variable immunity

Finally, we study how the trajectory of a disease is influenced by the introduction of a new and more transmissible strain. The two strains are modelled as identical, except for the parameter β. We wish to examine how levels of immunity granted by both infection- and vaccination-induced immunity alter the disease trajectory of the second strain.

We consider a community of 10,000 unvaccinated individuals. We initially seed the population with 10 individuals infected by the first strain. The infection is then allowed to propagate without intervention. Once the infections due to this strain have died out, we then introduce the more transmissible second strain by seeding 100 individuals at random.

Individuals who recover from infections with the first strain have their disease trajectories following a second infection altered in two ways: (i) they have a lower relative risk of infection when they come in contact with infected individuals (modeled by introducing a “reinfection probability” parameter), and (ii) prior infection reduces an individual’s risk of severe illness, with a protection equal to a single vaccine dose [57].

The trajectory of the second-wave of the disease, induced by the more transmissible variant, is shown in Fig 9A. The worst case occurs when the probability of reinfection is 100%. In this case, since the variant is more transmissible and can infect the entire population, the peak in infections is substantially higher than that of the first-wave. As the reinfection probability is reduced, this peak is diminished.

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Fig 9. Effect of reinfection probability on disease progression.

A second more virulent strain is introduced once the first strain has died out. Individuals who contracted the first strain have a reduced relative risk of infection to the second strain, representing a “reinfection probability”. In (A) the reinfection probability is varied, producing different disease trajectories. As the reinfection probability reduces, so does the peak of the second wave. In (B) we study the effect of a delayed vaccination drive on the disease trajectory. A constant daily vaccination rate of 0.5% is used. For simplicity, only the curve for 100% reinfection is shown, although the trend is similar for other values of the reinfection probabilities. Starting the vaccination drive later causes the more virulent strain to spread much faster, reaching a much higher peak earlier.

https://doi.org/10.1371/journal.pcbi.1012682.g009

We can further study the effect that vaccination has on the disease trajectory by delaying the start of the vaccination drive. In Fig 9A, no vaccination drive was included, but we can modify this. Fig 9B shows what happens when the start of the vaccination drive is delayed by successive intervals of 50 days. For simplicity, only the curve for 100% reinfection is shown.

As can be expected, delaying the start of the vaccination drive causes the second, more virulent, strain to spread much faster, and reach a higher peak earlier. The trend is identical for lower daily vaccination rates, although the relative difference between the peaks is less noticeable. Because our model can keep track of each agent’s infection history, this study could also be extended to simulate any complex sequence of variants with given epidemiological characteristics, as discussed in [76–78], for example.