Comparison of pandemic intervention policies in several building types using heterogeneous population model

1. Introduction

Humanity has experienced multiple epidemics over the centuries which caused significant mortality, economic losses, and political shifting [1]. In the last fifty years, the world has experienced four large-scale outbreaks of pandemics: HIV/AIDS, Seventh Cholera, 2009-flu, and Coronavirus [2]. The question of how policymakers can control a pandemic spread is becoming increasingly more relevant as urbanization in the developing world is bringing more people into denser neighborhoods, which increases the speed at which new infections are spread [3]. Moreover, globalization has facilitated pathogen spread among countries through the growth of trade and travel [4]. These socioeconomic processes keep the infectious disease outbreaks nearly constant, even with modern medicine, technology, and government awareness [2].

Thus, the preparedness of policymakers is a necessary step to ensure the ability of a country in handling the next pandemic. A useful tool to obtain data-driven decisions such as lockdowns [5], artificial job separation [6], school-work duration [7], mask-wearing [8] and others [8] pandemic intervention policies (PIPs) are epidemiological-mathematical models which allows to investigate the influence of these PIPs in multiple scenarios. A large portion of the epidemiological models are based on the SIR model [9] with different extensions related to biological, economical, and spatial properties of the disease that one is aiming to model [10–12]. Nevertheless, these models treat individuals homogeneously, assuming everybody has the same biological properties, located in the same places during the day, and meet each other randomly or by a static network of connections [7,13]. To address this shortcoming, we developed a spatio-temporal social-epidemiological model which treats each individual heterogeneously by allowing a unique walk between the locations (rooms) based on the individual’s social role and ’’personality". The proposed model describes the way populations move around a building using a graph-based spatial model and the pandemic spread using an extended SIR model. In addition, we developed a computer simulation that provides an in silico tool for evaluating the performance of four PIPs over four different types of buildings: home, office, school, and mall. The proposed model allows a more accurate investigation of the epidemiological dynamics in relatively small population sizes. Using the model, we show that temporal-based PIPs behave identically if the population is dense enough while spatial-based PIPs behave differently in these building types. Moreover, we show that the PIPs’ dynamics in the home building type is chaotic due to the low density of population except for the social distancing PIP which significantly reduces the pandemic spread.

This paper is organized as follows: In Section 2, we provide background about mathematical models for pandemic spread followed by graph-based population interaction models and a review of PIPs. In Section 3, we introduce our spatio-temporal social-epidemiological model. In addition, we provide an agent-based approach [14] to simulate the model’s dynamics. In Section 4, we present four PIPs used to control the pandemic spread. In Section 5, we present the implementation of the model for the COVID-19 pandemic and data of buildings from Israel, followed by sensitivity analysis of the PIPs for the different building types. In Section 6, we discuss the main advantages and limitations of the model. In Section 7, we briefly describe the possible usages of the model for policymakers in future pandemics.

2. Background

Historical records from the last few hundred years show that pandemics caused significant political shifting, economical losses, and mortality [1]. In particular, in the 20th century (1918) the influenza pandemic struck the world and killed an estimated 50 to 100 million individuals worldwide [1] and in 2020 the coronavirus (COVID-19) pandemic was declared by the World Health Organization (WHO) as a public health emergency of international concern and killed millions [15].

One approach to allow policymakers to better manage pandemics is using mathematical models and computer simulations. Indeed, multiple models are shown to be efficient in obtaining data-driven decisions such as artificial job separation [6], lockdowns [5], masks wearing [16], school-work duration [7], and others PIPs [8]. These mathematical models can be divided into two main groups. First, models aim to predict the different parameters such as the total death and peak in hospitalized individuals, given historical data [17,18]. Second, models that aim to analyze and optimize PIPs [19,20].

Models related to the second group are usually focusing on large populations such as cities [20], countries [17,21], and even the entire world population [18]. As such, they ignore the small, daily interactions of the individuals in the population that influence the overall pandemic spread dynamics [22]. The world’s population is growing [23] and is concentrated in cities in general and megalopolis cities in particular [24]. As a result, most of the individuals spend most of their time indoors (22–23 h a day on average) [25].

We present an overview of epidemiological models, in which, general and spatio-temporal epidemiological models are shown as a method for the heterogeneous indoor epidemiological model. Afterward, several graph-based population interactions models are presented, where using this representation allowed us to investigate the emerged behavior of the population from the individual level. Finally, several studies outline possible PIPs.

2.1. Epidemiological models

Mathematical models of epidemiological dynamics usually represent the transformations of individuals in the population between several epidemiological states [7,10,12,26]. These models can be roughly divided into two main groups: diseases with short-term and long-term immunity memory when more often than not an immunity memory is considered long-term if it is longer than the average duration of two infection waves in the population. For example, influenza is a short-term immunity memory disease since it occurs on average once a year when the immunity memory is shorter [27].

The group of short-term immunity memory diseases can be represented by the Susceptible–Infected–Susceptible (SIS) model, where susceptible (S) individuals are infected on average at a rate β relative to the size of the infected individuals sub-population, and infected individuals (I) recover at a rate γ and become susceptible (S) again [28]. The SIS model is represented by the following non-linear ODEs:

In the context of indoor dynamics with a relatively short time frame, individuals develop and retain the immune system needed to be safe from re-infection [29]. Therefore, the epidemiological dynamics are a better fit for the long-term immunity memory diseases regardless of the type of the disease itself. Therefore, the model takes the form of the Susceptible–Infected–Recovered (SIR) model [9], represented as a system of ODEs where susceptible (S) individuals are infected on average at a rate β relative to the size of the infected individuals sub-population and infected individuals (I) recover at a rate γ and become recovered individuals that cannot be infected again. The SIR model is represented by the following non-linear ODEs:

While the SIR model is widely used due to its simplicity, which is commonly used to obtain an initial estimation of the pandemic behavior, its simplicity is also its shortcoming as it fails to represent the complexity of the pandemic spread dynamics [18]. Therefore, several extensions of the SIR model are proposed to better represent the spread dynamics:

First, the mortality due to the pandemic defines a state D. This state plays a meaningful role in the dynamics as it removes individuals from the interaction, besides allowing to investigate and predict the portion of individuals that die due to the pandemic [19].

Second, adding an exposure state (E) which represents the period between the time a susceptible individual is infected and until it starts to be infectious to others [17]. The exposed state better represents the biological settings as many diseases go through a phase of incubation [30,31].

Third, individuals experience diseases in a range of severity [17]. Therefore, by dividing the infection group, I, into two degrees of infection severity: symptomatic, I^s, and asymptomatic, I^a, it is possible to obtain a more accurate representation of the social and epidemiological dynamics as well as to fine-tune the infection rates as different severity degrees have corresponding infection rate.

Fourth, data from several epidemiological studies show that children and adults transmit diseases at different rates and have different recovery duration [32,33]. As a result, it is possible to divide the population into two age groups, adults, and children, such that each of the values of the transformation between epidemiological states within and between groups is different [34].

Fifth, the places where individuals spend their time during the day affect the pandemic dynamics by changing the rate of infection. Indeed, in the case of COVID-19, Viguerie et al. [35] showed that Spatio-temporal SIR-based models better predicted the COVID-19 spread in the Italian region of Lombardy. Their version of the spatial dynamics assumes the static distribution of the population over the course of the day and does not take into consideration the unique dynamics of a different location as is possible by using a graph-based spatial model [7].

Moreover, several attempts to obtain a more accurate pandemic spread model resulted in a mathematical description that incorporates some level of randomness in the formulation of the model parameters and state transformations [36,37]. For example, Li et al. [38] proposed a SIRS-based model with complex heterogeneous networks. In their analysis, the authors show the influence of community connectivity on the pandemic spread. In particular, individuals in the population have a random variable of individuals they interact with and therefore can be infected from. This randomness in the SIRS model’s infection rate parameter resulted in complex dynamics that in some cases will result in the disease-free equilibrium state while in others cases one obtains an endemic equilibrium state [38]. In addition, Yuan et al. [39] proposed a multi-group SIR model with random perturbation, showing that the endemic equilibrium of the model is stochastic asymptotically stable. Notably, the authors show the stability condition is obtained by the construction of the Lyapunov function and graph theory [39]. As such, generalizing the work proposed by [40,41]. Nonetheless, the numerical evaluation has shown worse results for short-medium periods of time compared to the asymptotic state, which points to the under-representation of the model to realistic epidemiological scenarios. Furthermore, Liu et al. [42] proposed a two-group stochastic SEIR (E-exposed) with infinite delays, showing it well fits the deterministic prediction if it is stable for sufficiently small noise. Therefore, providing an important mathematical connection between stochastic and deterministic epidemiological SIR-based models [43].

2.2. Graph-based population interactions models

Individuals’ movement around in buildings, occupancy, and interactions with devices are influenced by variables in three main categories: environmentally-related, time-related, and random. The environmentally-related variables include a physical aspect related to the building’s characteristics and location. Solar orientation, envelope, building layout, and local climate are some examples of environmentally related factors. The time-related variables comprehend the occupants’ routine. In that manner, occupancy and interactions with devices and individuals in buildings are influenced by time of day and day of week [44].

Noakes et al. [45] modified the SIR model, taking into consideration several physical properties of airborne diseases in a close space based on the physical airborne infection model proposed by Riley et al. [46]. The model is able to evaluate the effects of room size, occupancy and ventilation conditions on the number of new infections which makes it more accurate than the general-proposed SIR model. Nevertheless, the model assumes the population is located in the room during the entire dynamics which is a poor approximation of the real movement dynamics of individuals over relatively long periods (days, weeks, and months) in which usually pandemic prediction takes place. Therefore, we assume individuals move between rooms over time, which changes the infection rate in rooms accordingly.

Several more studies investigated the epidemiological dynamics of people as a network of interaction represented using a graph-based model. Hau et al. [11] proposed an SEIR-based model (E-exposed) for sexually transmitted diseases where the interactions between individuals happened randomly on a bipartite (male and female nodes) static graph. Similarly, Wang et al. [13] proposed an SIS model where each individual is represented as a node in a static graph that has between 1 and k ≪ N edges randomly set (where N is the size of the population). The assumption of a static graph over time introduces an error to the simulation as unplanned interactions (for example, using an elevator with unfamiliar people), changes in contacts (for instance, when replacing a workplace or when meeting new members in a sports team), and other factors make edges of the graph dynamic. In the proposed model, we consider these dynamics using the walk between rooms of a building which in its turn define the interactions between individuals at any point in time.

2.3. Pandemic intervention policies

Governments across the world are facing the task of selecting suitable intervention strategies to cope with the effects of pandemics. This is a highly challenging task, since harsh measures may result in an economic collapse while a relaxed strategy might lead to a high death toll. Pandemic intervention policies (PIPs) are aimed to optimize the trade-off between the number of infections (and therefore deaths) and the socioeconomic costs.

Non-pharmaceutical intervention (NPI) policies are actions, apart from getting vaccinated and taking medicine, that individuals and communities can take to slow the spread of a pandemic. There are multiple NPI policies such as school-work duration [7], lockdown [6], mask-wearing [12] and others. Zhao et al. [19] proposed an extension to the SEIR model where the susceptible population is separated into two groups: individuals not taking infection-prevention actions and individuals taking infection-prevention actions as an NPI policy. The authors introduce a stochastic element to the SEIR model making it more robust for social changes that happened during the epidemic. Di Domenico et al. [20] used data from March 17 to May 11 (2020) in Ile-de-France with a stochastic age-structured transmission extension of the SEIR model integrating data on the age profile and social contacts of four age-based classes. The authors investigated the influence of average social distancing on the duration of the pandemic and the total number of infected individuals.

On the other hand, pharmaceutical intervention policies are clinical actions such as vaccination and taking medication. Moore et al. [47] investigated the influence of age-based vaccination on the changing levels of infection based on epidemiological data from the UK using the SIR model for the case of COVID-19.

3. Model definition

The model can be mathematically described using an interaction between two sub-models: epidemiological (temporal) ODE-based and social (spatial) graph-based sub-models. It is difficult to numerically solve this representation due to the noncontinuous accrual as a result of the spatial dynamics (population mobility in the building). In addition, it is also difficult to use this representation to obtain analytical results, due to the nontrivial integration of ODE and graph theories and the large-scale representation of the dynamics it yields. Therefore, we proposed an agent-based approach to simulate the proposed model. The examined PIPs are treated as an additional layer to the model by modifying several attributes of the model.

3.1. Epidemiological (temporal) sub-model

The model considers a constant population with a fixed number of individuals N. In the context of an indoor pandemic, the time horizon of interest is relatively short and therefore the native population growth can be neglected. Each individual belongs to one of seven groups: susceptible (S), asymptomatic exposed (E^a), symptomatic infected (E^s), asymptomatic infected (I^a), symptomatic infected (I^s), recovered (R), and dead (D) such that N = S + E^a + E^s + I^a + I^s + R + D. Individuals in the first group have no immunity and are susceptible to infection. When an individual in the susceptible group (S) is exposed to the pathogen, the individual is transferred to either the asymptomatic exposed group (E^a) or symptomatic exposed group (E^s) at a rate corresponding to the average interaction between infected individuals and susceptible individuals. The individuals stay in the symptomatic, asymptomatic exposed group on average ξ ^s, ξ ^a days, respectively, after which the individual is transferred to the symptomatic, asymptomatic infected group. Afterward, the individual stays in the symptomatic infected group on average γ ^s days, after which the individual is transferred to the recovered group (R) or the dead group (D). Therefore, a rate of (1−ψ) of symptomatic infected individuals remain seriously ill or die while others recover. All asymptomatic infected individuals stay in the infected I^a group on average γ ^a days, after which the individual is transferred to the recovered group (R). The recovered individuals are again healthy, no longer contagious, and immune from future infection.

The population is further divided into two age classes: adults and children, because these groups experience the disease in varying degrees of severity, have different infection rates, and have different social roles. Individuals below age A are associated with the children age-class while individuals in the complementary group are associated with the adult age-class.

By expanding the designation to two age-classes, we let S_c, E^a _c , E^s _c, I^a _c , I^s _c, R_c, D_c, S_a, E^a _a, E^s _a, I^a _a, I^s _a, R_a, and D_a represent susceptible, asymptomatic exposed, symptomatic exposed, asymptomatic infected, symptomatic infected, recovered, and death groups for children and adults, respectively such that

The epidemiological dynamics (e.i., the SEEIIRD model) are described in detail in Eqs. (S1–S14) in the supplementary material. A summary of Eqs. (S1–S14) is shown in Eq. (3).

Since the coefficients of the model embed terms that represent probabilities, the model is stochastic. Therefore, the coefficients represent the probabilities for state transfer in the individual level [10]. As a result, Eq. (3) is a non-linear, first order, stochastic system of ODEs with 14 states. A schematic view of the transformation between the epidemiological model’s state is shown in Fig. 1.

3.2. Social (spatial) sub-model

The spatial sub-model is a graph-based model G = (V, E). The population N from the temporal dynamics is allocated in some distribution to the nodes of an undirected, connected graph (G). The graph is defined according to the rooms V (nodes) of a building and the connections E (edges) between them which are defined if there is a door between two rooms. For each step in time, each agent in the population is moving to one of the nodes in the graph or staying in the same node, according to the inner moving policy this agent has, in addition to a global policy which overtakes the decision in the case of a conflict between the two policies. The transition between any two nodes is assumed to be immediate and that everybody is following the same clock. Between each population movement on the graph, the temporal sub-model is performed simultaneously on all the nodes of the graph.

Therefore, based on the Wells–Riley equation [46] and the extension proposed by Noakes et al. [45], in each room the infection probability is computed using the formula:

where C is the probability a susceptible individual will be infected by spending time in the same room as infected individuals, Ω is the set of all sub-groups of susceptible S and infected (P) sub-populations. For example,

for the proposed epidemiological model. β(ω) is the infection probability of the pair of sub-populations ω in general, p is the average pulmonary ventilation rate of the susceptibles (m³/t), q is the quanta production rate per infector (t⁻¹), A is the ventilation rate in air changes per hour (t⁻¹), and V is the room volume (m³).

Eq. (4) holds in the case of airborne disease. In the case where the disease is not airborne (like sexual diseases), the probability C equals the coefficient of the SI term in the epidemiological (temporal) model (see Section 3.1). This is because non-airborne diseases do not fulfill the assumptions required for Eq. (4) [45]. For example, Wang and Cao [48] proposed a SI-based model with a waterborne disease where the infection rate between individuals is equal to the coefficient of the SI term in the model. In a similar manner, Kibona and Yang [49] proposed a SIR model for the Zika virus which is a non-airborne disease where the infection rate is again equal to the coefficient of the SI term in the model.

3.3. Numerical simulation

Due to the non-linear and large-scale (14V equations) nature of the system in addition to the noncontinuous behavior as a result of the population mobility between the rooms — it is hard to numerically solve the system in both a stable and fast way as current ODE solvers struggle in such a case even on small-scale systems [50].

Therefore, we take advantage of the P-system model [51] to simulate epidemiological and social dynamics as an agent-based simulation is a powerful tool to simulate complex social systems [14]. Specifically, we extend the model proposed by [52]. First, let us define the system as an instance of a P-system model. A P-system model PM is defined as a tuple of three elements P_s := (P, M, I) where P is a set of finite state machine agents, M is a set of locations (originally, membranes) over which the population is distributed, and I : (p_i, p_j) → (p_i, p_j) is a pair-wised interaction protocol as a function between two agents p_i, p_j ∈ P such that i/ = j changes their states. The location m ∈ M that the agent is located at a given time is part of the state definition of the agent. At each point in time, the population is randomly divided into pairs such that agents in a pair are constrained to be located in the same location m ∈ M. Afterward, the interaction protocol I is performed on each pair.

We introduce an extended P-system definition. An EP-system is defined as a tuple EP_s := (P, G, I_p, S_p, M_p) where P is a set of timed finite state machine agents where the inner clock measure the time pass from the last state change, G = (M, E) is an underacted, connected graph over which the population is distributed, I_p : (p_i, p_j) → (p_i, p_j) is a pair-wised interaction protocol as a function between two agents p_i, p_j ∈ P such that i/ = j changes their states, S_p(p) → p is a mono-wised spontaneous protocol as a function that changes the state of an agent based on the inner clock and current state, and M_p(G, P) → P is a movement protocol as a function from the graph G and the population P and return a new distribution of the population P in the graph G. At each point in time, the population is randomly divided into pairs such that agents in a pair are constrained to be located in the same location m ∈ M. Afterward, the interaction protocol I is performed on each pair. Then, the spontaneous protocol S_p and the movement protocol M_p are performed on each agent in the population.

Therefore, let M be an EP-system which represents the model defined in Sections 3.1, 3.2. Namely, P is a set of individuals, G is the graph of rooms in the building, connected by edges, which is a physical way to go from one room to another (usually, via a door). The interaction protocol I_p implements the infection dynamics of the SEEIIRD model stochastically:

y where θ ∼ U[0, 1] is the chance that the current infection succeeds, x, z ∈{c, a}, y ∈{s, a}, β_xz is the average probability that a susceptible x age group individual would be infected from a z age group individual with y infection severity, and α ∈{s, a} is set to s at a probability φ_x and a otherwise. The spontaneous protocol S_p implements the infection and recovery dynamics using the inner clock (T):

Finally, the movement protocol M_p is a general-purpose function that is defined as part of a specific policy.

4. Pandemic intervention policies

Pandemics have negatively impacted many aspects of humanity’s existence, causing massive unrest around the world with significant loss of life. Policymakers are forced to aim to execute PIPs in the form of non-pharmaceutical intervention (NPI) policies such as social distancing and masks and pharmaceutical intervention (PI) policies such as vaccination to control the epidemic. These PIPs can be divided into two main groups: temporal and spatial. The temporal PIPs modify the individual or population’s properties related to the transformations between the epidemiological stages (see Section 3.1) while spatial PIPs modify the individual or population’s properties related to the walks of the population in the topology.

4.1. Spatial-based policies

In the model, a spatial-based policy fully or partially overrides the walk on the topology dynamics each individual has. Specifically, in each step in time, the policy modifies the walk dynamics of all individuals in the population at once followed by regular heterogeneous walk behavior of each individual independently, according to these modifications.

4.1.1. Isolation of symptomatic infected individuals

Symptomatic (exposed and infected) individuals produce clinical signs which are relatively easy to measure (for example, increased body temperature), and it is recommended to isolate these individuals from the rest of the population to reduce the infection rate. Since the measuring process requires effort in the form of manpower and technical means (for instance, a distance thermometer), it is limited by the availability of these factors.

Therefore, the isolation of symptomatic infected individuals (ISII) policy is realized as follows. Every τ ∈[0, ∞) steps in time, all individuals at this point in a portion σ ∈[0, 1] of the rooms of the building are tested. Individuals that belong to the symptomatic exposed (E^s) symptomatic infected (I^s) are isolated out of the building. In addition, at each check, individuals that were isolated in a previous check are re-tested and if they are recovered (R) they are allowed to return to the building and continue their original behavior.

4.1.2. Social distancing

During an airborne type pandemic, the instruction for individuals is to keep social distancing to reduce the infection rate. Social distancing reduces the infection rate as the sum of possible infections in each room Σ_ω∈Ωβ(ω) is decreasing (see Eq. (4)).

Therefore, the social distancing (SD) policy was realized as follows. There is a probability χ ∈[0, 1] that an individual overrides its original walk dynamics with a walk to a neighboring room that optimizes the SD, where SD in a room i is defined as follows:

where V_i is i’s room volume.

4.2. Temporal-based policies

In the model, a temporal-based policy modifies the spontaneous and interaction protocols of the individuals of the population and does not change their behavior.

4.2.1. Mask wearing

During an airborne type pandemic, wearing masks reduces the rate of infection in the event of an encounter between individuals [53]. However, masks have several levels of protection that differ according to their materials [54].

Therefore, the mask-wearing (MW) policy was realized as follows. When two individuals such that one of them is infected and the other is suspicious interact, there is a probability C (see Eq. (4)) that the suspicious individual would be infected. If one of the sides wears a mask with quality α ∈[0, 1], the infection probability becomes αC. If both sides wear a mask with quality α ∈[0, 1], the infection probability becomes α²C. The portion of individuals that are wearing masks all the time is marked by Γ ∈[0, 1].

4.2.2. Vaccination

Vaccination is known to be the golden pandemic intervention policy [55]. Vaccination of the entire population for large populations is a complex, time and resource-consuming task [55]. As a result, it is common that only a portion of the population is vaccinated. In addition, a vaccine is not a silver bullet since it is effective only for a portion of the time and vaccinated individuals may be infected anyway.

Therefore, the vaccination (V) policy was realized as follows. A portion of the population ζ ∈[0, 1] is vaccinated and has a probability Λ ∈[0, 1] to be infected.

4.3. Policy simulation

A policy is defined on the dynamics that emerge from both the spatial and temporal sub-models, as shown in Fig. 2. First, we define the state of the model as follows.

Definition 1. The Model’s state at time t_i is defined by the set S_i such that:

A policy P is a function P : {t_j ∈[t₀, t_i] | S_j} → Ψ such that S ∈ Ψ that gets the states of the model from the beginning t₀ and up to the current point in time t_i and returns a model’s state.

5. Results

Based on the proposed model, we examined the performance of the simulation on four building types (home, office, school, and mall). First, we gathered data regarding each building’s type topology, population, and its walk in the topology. Second, we evaluated the spread of the pandemic on all buildings, divided by type without any intervention which is defined as the baseline dynamics. Afterward, we examined the influence of the four proposed PIPs on each building type separately.

5.1. Experiment setup

We implemented the proposed model as a computer simulation. The building’s topology was obtained using maps of the building (either in physical or electronic form) or by physically mapping the building ourselves. All the buildings are located in Israel.

5.1.1. Epidemiological values

We define the model’s parameters used in the following experiments. The parameters reflect the SARS-CoV-2 (COVID- 19) pandemic. The parameters obtained according to biological and clinical data are presented in Table 1. The exposed to infection rate ξ in both asymptomatic and symptomatic and in adults and children parameters are estimated by fitting the SEEIIRD model (see Section 3.1) on the Israeli COVID-19 data [56], where N = 8· 10⁶, N_c = 2.24· 10⁶, and N_a = 5.76· 10⁶ using the method proposed by [6], as no relevant clinical data was found.

The model uses abstract discrete time steps. To calibrate the simulation’s abstract time step into a real one, we define each time step to be the duration of the shortest meaningful interaction between two or more individuals in a room. For example, in a school, the shortest event of the day is a 15-min break so ∆t = 0.25 h. On the other hand, if in an office each meeting or task is assumed to be in quantities of half an hour, therefore ∆t = 0.5 h. The parameters in Table 1 are linearly scaled according to the chosen time step.

5.1.2. Topology of the building and the population’s social behavior

For each building, we obtained the schematic walk of the population in one of two ways. First, the building’s operative managers (schoolmaster for the schools, chief executive officer (CEO) of the operating company of the mall, the companies’ chief operating officer (CEOs), and family members (for the home-type buildings) were interviewed. Second, offices had doors that open (both inside and outside) and the building’s population used a personal card that recorded each entrance and exit (per room). A log of five days was obtained and analyzed. A summary of the building types with a qualitative description of the population and topology of the building is shown in Table 2.

Parameter definition	Symbol	Value	Source
Susceptible contacts in children which become infected due to direct disease transmission from an symptomatic/asymptomatic adult in an hour [t−1]	βs ca, βa ca	0.0110	[32]
Susceptible contacts in adults which become infected due to direct disease transmission from symptomatic/asymptomatic children in an hour [t−1]	βs ac, βa ac	0.0010	[33]
Susceptible contacts in children which become infected due to direct disease transmission from symptomatic/asymptomatic children in an hour [t−1]	βs cc, βa cc	0.0128	[57]
Susceptible contacts in adults which become infected due to direct disease transmission from symptomatic/asymptomatic adult in an hour [t−1]	βs aa, βa aa	0.0128	[57]
The probability that a child will be asymptomatic [1]	ψc	0.998	[58]
The probability that an adult will be asymptomatic [1]	ψa	0.078	[59]
Asymptomatic infected to recover average duration for children in hours [t−1]	γ a c	0.025	[58]
Symptomatic infected to recover average duration for children in hours [t−1]	γ s c	0.02083	[58]
Asymptomatic infected to recover average duration for adults in hours [t−1]	γ a a	0.0075	[60]
Symptomatic infected to recover average duration for adults in hours [t−1]	γ s a	0.002975	[60]
The probability an infected adult will recover from the disease [1]	ρa	0.942	[61]
The probability an infected child will recover from the disease [1]	ρc	0.99	[58]
The rate a symptomatic exposed child becomes infected in hours [1]	ξ s c	0.0018	estimated
The rate a asymptomatic exposed child becomes infected in hours [1]	ξ a a	0.0104	estimated
The rate a symptomatic exposed adult becomes infected in hours [1]	ξ s c	0.0056	estimated
The rate a asymptomatic exposed adult becomes infected in hours [1]	ξ a a	0.0065	estimated

Name	Population	Topology
Home 1	Sa(0) = 1, Sc(0) = 2, Ea a(0) = 1	|V| = 7, |E| = 6
Home 2	Sa(0) = 1, Ea a(0) = 1	|V| = 7, |E| = 6
Home 3	Sa(0) = 2, Sc(0) = 3, Ea a(0) = 1	|V| = 6, |E| = 5
Home 4	Sa(0) = 2, Sc(0) = 2, Ea c (0) = 1	|V| = 7, |E| = 6
Home 5	Sa(0) = 2, Ea c (0) = 1	|V| = 11, |E| = 10
Office 1	Sa(0) = 23, Ea a(0) = 1	|V| = 10, |E| = 10
Office 2	Sa(0) = 46, Ea a(0) = 1	|V| = 15, |E| = 15
Office 3	Sa(0) = 26, Ea a(0) = 1	|V| = 10, |E| = 9
Office 4	Sa(0) = 11, Ea a(0) = 1	|V| = 9, |E| = 8
Office 5	Sa(0) = 6, Ia a (0) = 1	|V| = 5, |E| = 4
Office 6	Sa(0) = 17, Ea a(0) = 1	|V| = 11, |E| = 10
Office 7	Sa(0) = 64, Ea a(0) = 1	|V| = 25, |E| = 25
School 1	Sa(0) = 77, Sc(0) = 896, Ea c (0) = 1	|V| = 57, |E| = 56
School 2	Sa(0) = 48, Sc(0) = 450, Ea a(0) = 1	|V| = 26, |E| = 25
Mall 1	Sa(0) = 400, Sc(0) = 70, Ia a (0) = 1	|V| = 47, |E| = 46

A schematic description of the population’s walk in each building type is provided below. The description aims to generalize the underline dynamics proposed by the representatives of the buildings shown in Table 2.

We define a home-type building as a house with one family. More often than not, there are both adults and children at home. While the behavior of a family changes according to its cultural, social, and economic characteristics, it is possible to draw general guidelines for the schematic behavior that a family has in its home. Specifically, both adults and children get up and prepare for the day as they spend time in the bathroom after sleeping at night in their bedroom. Afterward, part of all the individuals in the house prepare breakfast in the kitchen and eat it together in one room or separately in multiple rooms. Later, some of them spend time focusing on their tasks mostly independently in one of the rooms of the house until noon where they repeat the preparation and eating process for lunch. After lunch, some of the family members continue with their tasks until dinner where the process repeats itself for the third time. Eventually, the children repeat the morning actions as they prepare for sleep and then spend the night in their rooms. At the same time or later, the adults do the same. During the day, individuals may spend short periods in the bathroom and kitchen.

In addition, we define a office-type building as an office used by a single company, where all the employees of the company are adults. The employees arrive at the office during the morning at a range of times and go directly to their desks. Sometimes, the employees spend some time in the office’s kitchen if it exists. Afterward, during the day, the employees take small breaks in the kitchen or go to the bathroom. Around noon, the employees eat lunch in the kitchen or at their desks. In addition, subsets of the employees gather together at different meetings in the meeting room or an available room in the office and sometimes even in the corridor. Finally, between afternoon and the evening employees leave the office at a range of times.

Furthermore, we define a school-type building as a single school which may include several physical buildings and an outdoor area in one location occupied by both adults and children. Firstly, the school’s administrative staff arrive at the school in the morning and shortly after both pupils and teachers arrive. Children have a planned program over the day where they spend the time in a classroom (either the same one or a different one according to the subject they have in the program). In between classes, pupils are allowed to move freely (and usually randomly) in the school. Teachers, on the other hand, start their day in the teachers’ lounge until the first class of the day. During a lesson, one teacher and a group of pupils are in each classroom. Teachers that do not have a class stay in the teachers’ lounge. During a break, part of the teachers is in the teachers’ lounge while the other part moves randomly and supervises the pupils. Every now and then, both pupils and teachers go to the restroom. Moreover, the administrators either spend time in their offices or perform tasks in random locations in the school.

Finally, we define a mall-type building as an indoor shopping center with multiple shops, that serve both adults and children. One can divide the population of the mall into working individuals such as the salespeople in the shops including the administrative staff of the mall itself and the shoppers visiting the mall. Usually, during the morning until the afternoon, most of the shoppers are adults. Later on in the day, children visit the mall as well. The working individuals are all adults. The shoppers’ sub-population can be further divided into individuals that randomly visit shops and individuals that visit a targeted list of shops to find desired items. At the end of the day, all shoppers leave the mall and only afterward the employed individuals.

5.2. Baseline model dynamics

Figs. 3(a)–3(d) presents the model baseline dynamics. The x-axis shows the time (in hours) from the beginning of the simulation and the y-axis shows the distribution of the population to S(t), E^s(t), E^a(t), I^s(t), I^a(t), R(t), and D(t). The initial condition and topology of the building are taken from Table 2 such that the results are shown as the average of the same building type for n = 100 repetitions each.

5.3. Sensitivity analysis

Figs. 4–7 present the model sensitivity for each one of the PIPs (see Section 4) for each building type. The figures are divided into four plots as follows: First, the ISII policy sensitivity graphs, where the x-axis shows the rate of tests (τ) and the y-axis shows the portion of rooms (σ) that are included in each test. Second, the SD policy sensitivity graphs, where the x-axis shows the probability (ξ) that an individual overrides its original walk dynamics with one optimization of social distancing and the y-axis shows the number of infected individuals from the population. Third, the MW policy sensitivity graphs, where the x-axis shows the average quality of mask-wearing in reducing infection rate (α) and the y-axis shows the portion of the population that wears masks (Γ ). Finally, the vaccination policy sensitivity graphs, where the x-axis shows the efficiency of vaccination in reducing infection (Λ) and the y-axis shows the portion of the population that is vaccinated (ζ). In all plots of PIPs with two parameters, the color indicates the portion of infected individuals from the population.

In order to evaluate the average influence of each parameter of the PIPs on the pandemic spread in the context of each building type, we fitted the results from the simulation. The fitting function is calculated using the least mean square (LMS) method [62]. To use the LMS method, one needs to define the family function approximating a function. The one dimensional and two dimensional linear family function (e.g., f (x, y) = p₁ + p₂x + p₃y) are chosen to obtain the linear influence of each parameter — in the form of the coefficient of the respected parameter in the fitted parameter. The results are shown in Table 3.

The motivation of using the linear approximation to the connection between each PIP and its parameters is rooted in the explainability of these functions. Linear functions are considered easy to interpret and commonly used to learn the connection between the parameters of models [63]. However, the linear connection is not necessarily appropriate as for the case of Vaccination PIP and the Mall where the coefficient of determination R² = 0.32, as shown in Table 3. On the contrary, the linear approximation provides a fine approximation for the ISII PIP for the School (R² = 0.86) and Mall (R² = 0.95) building types. Similarly, the SD PIP with the Office and Mall building types obtain relatively well-fitting with R² = 0.81 and R² = 0.97, respectively. Nonetheless, nonlinear functions can provide better results for some cases as one can see from the Home building type but are not as easy to interpret compared to the linear function which makes them harder to analyze and use. Of note, this analysis does not provide an explanation on how the PIPs’ parameters influence the pandemic spread pandemic which is nonlinear, as one can see from Eq. (3) and Section 4.1, rather propose an approximation to the influence of each parameter to the pandemic spread.

	Home	Office	School	Mall
ISII	Not converged	−0.002τ −0.006σ +	0.003τ −0.003σ +	0.001τ −0.002σ +
		0.777 (R2 = 0.73)	0.220 (R2 = 0.86)	0.304 (R2 = 0.95)
SD	−0.296χ +	−0.169χ +	−0.045χ +	−0.473χ +
	0.064 (R2 = 0.96)∗	0.198 (R2 = 0.81)	0.261 (R2 = 0.52)	0.325 (R2 = 0.97)
MW	Not converged	−0.001α −0.001Γ +	−0.001α −0.002Γ +	−0.001α −0.002Γ +
		0.2261 (R2 = 0.73)	0.387 (R2 = 0.57)	0.489 (R2 = 0.78)
Vaccination	Not converged	−0.001ζ −0.001Λ +	−0.001ζ −0.002Λ +	−0.001ζ −0.001Λ +
		0.229 (R2 = 0.62)	0.407 (R2 = 0.80)	0.401 (R2 = 0.32)

6. Discussion

In future pandemics, policymakers will be able to use our model to investigate the consequences of several PIPs on the pandemic spread. The model has seven extensions to the traditional SIR model: separation of the population into two age groups (children and adults); separation of the infected group into asymptomatic and symptomatic infected groups;

introduction of a dead group (D); introduction of asymptomatic and symptomatic exposed groups, including graph-based spatial dynamics; use of the Wells–Riley [46] in-room pandemic spread; and heterogeneous walks.

Using these extensions, the model shows that pandemic spread highly differs in homes, offices, schools, and malls as shown in Fig. 3. The model shows the pandemic spread in homes is more chaotic than the other building types, as shown in Fig. 3(a), which can be associated with the low density of the population (average number of individuals in a room) which is 0.53 compared to 4.06, 17.75, 10.02 in the office, school, and mall, respectively. On the other hand, due to low density, a large portion (38% on average) is kept uninfected even without any PIPs. The (high-tech) office is the most pandemic spread resistant, as shown in Fig. 3(b), and can be explained by the fact that individuals are staying in their rooms most of the time and therefore produce only a local spread. On the other hand, the model shows that schools are the building type with the highest infection rate (with around 90% infection), as shown in Fig. 3(c). There are several reasons for this. First, pupils spend long periods in the classrooms which results in infection between classmates, as shown by the drops in the susceptible (blue) population size. Second, during breaks, the pupils move around the building randomly which operates as a bridge of infection between classrooms. Third, infected adults (teachers, maintenance employees, etc.) operate as an infection vector as they move between large groups of pupils. In the case of a mall, the pandemic spread is similar to the standard two-age SIR [34] model with one location. This can be explained by the large portion of the population that performs a somewhat random walk which is known to approximate the temporal SIR model [11]. However, some portion of the population performs a targeted buying which makes it less likely to be infected and therefore not (almost) the entire population being infected at some point.

Based on the proposed model, we evaluated four PIPS by the portion of the total number of infected individuals from the population. The four PIPs are isolation of symptomatic infected individuals, social distancing, mask-wearing, and vaccination. The model shows that all four PIPs are random for the home building type, as shown in Figs. 4(a), 4(c), and 4(d). This behavior is associated with the low density of the population together with the small population size which results in large differences even by changes in a single individual’s state. As a result, the social distancing PIP is very useful, as individuals can find empty rooms to stay in and thereby avoid infections, as shown in Fig. 4(b).

Furthermore, we evaluated four pandemic intervention policies (PIPs) — isolated symptomatic infected individuals (ISII), social distancing (SD), mask-wearing (MW), and vaccination. Each PIP is evaluated on each of the four building types, showing the influence of the pandemic spread on the range of the parameters that define each PIP as shown in Figs. 4–7. Linear regression has been performed on each simulation result for a specific PIP and building type to obtain the linear influence of each PIPs’ parameters on the pandemic spread for each building type as shown in Table 3.

Similar to the baseline dynamics of the home-type building, the ISII, MV, and vaccination PIPs do not have a clear behavior that can be associated with small population size and low density together. On the other hand, the SD PIP is able to reduce the pandemic spread such that on average if 20% of the time individuals were to avoid being in the same room without other individuals, the pandemic would spread after the first infected individual which exploits the low density. As a result, the SD PIP is the most suitable for homes while other PIPs (even vaccination) have a chaotic behavior which is an issue for policymakers.

For the office, school, and mall building types, the MV and vaccination PIPs have similar average behavior as shown in Table 3. Indeed, temporal PIPs are less sensitive to spatial dynamics (e.g., individuals’ walks in the building). In addition, by definition, the MW and vaccination are similar, the results show that indeed over several building types the influence of these PIPs on the pandemic spread is equal while the baseline (the free coefficient) differs between building types. On the other hand, the ISII PIP is spatial-based and differs between the building types such that in the office the testing rate is more important than the coverage of testing, in the school both have the same influence on the pandemic spread, and in the mall, it is the opposite to the office, as shown in Table 3. Similarly, the SD PIP differ between the building types, showing a quick decrease to zero in the mall (70%) and a slower one in the office (15% from the case without SD) building types while less than 20% influence in the case of a school building type. A possible reason for the inefficiency of the SD in the school is the relatively high density compared to the office and mall building types which makes the SD less effective in practice.

Therefore, while there is a similarity between the temporal-based PIP between building types, spatial-based PIPs highly differ in their effectiveness to control the pandemic spread.

One limitation of the proposed agent-based simulator is that it is linear (O(|P|)) to the size of the population (P) which is significantly worse than solving the ODE at each point in time (which is independent of the size of the population and therefore computes in O(1)). Therefore, for large size populations such as in the case of cities, countries, or the entire world’s population the computation increases. However, most of the computation steps in the simulations are independent in the scope of a single individual in the population which allows relatively easy parallelization of the computation which can reduce the computing time to make it feasible even for large populations.

7. Conclusion

The model developed in this study allows us to examine the impact of non-pharmaceutical and pharmaceutical PIPs on the course of a pandemic spread in the context of a single building. The model is implemented for the COVID-19 outbreak with data of buildings from Israel. It extends the traditional SIR model by introducing asymptomatic and symptomatic exposed groups and a dead group, splitting the infected group into asymptomatic and symptomatic infected two age-groups, including graph-based locations where individuals can be present during the day, and introducing a heterogeneous walk on the graph for each individual in the population. These spatio-temporal interactions allow us to explore the reciprocal effects of both spatial and temporal PIP on the spread of the pandemic in different buildings and social contexts such as the effect of mask-wearing in school as shown in Fig. 6(c). The inclusion of these interactions improves the accuracy of the model’s forecasts and allows for multidimensional analysis of the impact of PIPs and the dynamics of the crisis.

Our results indicate that policymakers need to take into consideration the unique social properties that individuals in the population have in different buildings (in which they spend most of their time) to find the optimal PIP as differential mask-wearing among different age groups or social roles (such as teachers and pupils in schools), or varying symptomatic infection testing, that significantly improves policy trade-offs, enabling considerable reductions in the pandemic spread and excess deaths.

The results in this study were obtained given the values in Table 1, buildings data from Table 2, and walks description from Section 5.1.2 which can vary significantly across countries and time as it depends on several hyper-parameters such as the local culture, architectural style, and population density. However, the proposed model does not take into consideration the movement of individuals between buildings during the day and the differences between the days of the week (such as weekends and working days). Assuming such dynamics, the results shown in Fig. 3 could be significantly altered. Therefore, we intend to include these dynamics in future studies.

Abbreviations

Pandemic intervention policy — PIP.

Funding

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

Availability of data and material

The buildings’ raw topological data that has been used is not available due to security reasons but the graph representation of the topology used in the simulation is available upon request from the authors.

Code availability

Upon acceptance, we will publish all the source code used in a GitHub repository with technical documentation for easy usage.

CRediT authorship contribution statement

Teddy Lazebnik: Conceptualization, Methodology, Formal analysis, Investigation, Data curation, Software, Validation, Writing – original draft, Writing – review & editing, Supervision. Ariel Alexi: Formal analysis, Investigation, Data curation, Software, Visualization, Manuscript editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

All the data that has been used is available online. In the manuscript, we provide links and cite the works that originally presented the data sets.

Acknowledgment

The authors wish to thank Gal A. Kaminka for his insightful suggestions regarding the research question.

Appendix A. Supplementary data

Supplementary material related to this article can be found online at https://doi.org/10.1016/j.cnsns.2021.106176.

Article notes

Publication history

Received 29 July 2021 · Accepted 1 December 2021 · Published 11 December 2021

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