Agent based simulators for epidemic modelling: Simulating larger models using smaller ones

A susceptible individual $n$ at any time $t$ receives a total infection rate $\lambda_{n}(t)$ which is sum of the infection rates $\lambda_{n}^{h}(t)$ (from home), $\lambda_{n}^{s}(t)$ (from school), $\lambda_{n}^{w}(t)$ (from workplace) and $\lambda_{n}^{c}(t)$ (from community) coming in from infected individuals in respective interaction spaces of individual $n$.

Briefly, the transmission rate ($\beta$) of virus by an infected individual in each interaction space is the expected number of eventful (infection spreading) contact opportunities with all the individuals in that interaction space. It accounts for the combined effect of frequency of meetings and the probability of infection spread during each meeting. An infected individual can transmit the virus in the infective (pre-symptomatic or asymptomatic stage) or in the symptomatic stage. Each individual has two other parameters: a severity variable (individual attendance in school and workplace depends on the severity of disease) and a relative infectiousness variable, virus transmission is related linearly to this. Both bring in heterogeneity to the model.

Specifically, let $e_{n^{\prime}}(t)=1$ when an individual $n^{\prime}$ can transmit the virus at time $t$, $e_{n^{\prime}}(t)=0$ otherwise. To keep notation simple, we avoid describing the details of individual infectiousness, severity, age dependent mobility and community density factor (see [1] for details). Let $\beta_{h}$, $\beta_{s}$, $\beta_{w}$, and $\beta_{c}$ denote the transmission coefficients at home, school, workplace and community spaces, respectively. A susceptible individual $n$ who belongs to home $h(n)$, school $s(n)$, workplace $w(n)$ and community space $c(n)$ sees the following infection rates at time $t$ in different interaction spaces:

$\lambda_{n}^{h}(t)$ is the average transmission rate coming in to individual $n$ from each infected individual in his home.

$\lambda_{n}^{s}(t)$ and $\lambda_{n}^{w}(t)$ are also computed similarly based on infected individuals in respective interaction spaces of the individual $n$.

Different wards in the city constitute different communities. In our numerical experiments for Mumbai, since slums form more than half the population and are extremely dense, we further divide each ward into slum and non-slum community. To compute infection rate to an individual from the community, we first compute the infection rate $h_{c^{\prime}}(t)$ for each community $c^{\prime}$ at a given time $t$. This is set to the sum of transmission rate from all the infected individuals of the community assigned a weight that is proportional to the strictly decreasing function of the distance between the individual and the community centre. Specifically,

where $f(d)$ is a distance kernel that is strictly decreasing in $d$. As in [6], one may select $f(d)=1/(1+(d/a)^{b})$ and $d\ll a$ for appropriately chosen non-negative constants $a$ and $b$. Note that $h_{c^{\prime}}(t)$ is dominated by $\beta_{c}$ and it equals $\beta_{c}$ only when everyone in the community $c^{\prime}$ is infected.

The infection rate $\lambda_{n}^{c}(t)$ seen by an individual $n$ in community $c$ is set to sum of the community infection rates from different communities of the city multiplied with weights that are again proportional to the strictly decreasing function of the distance between the two communities. This quantity is further adjusted for the distance between individual $n$ and its community centre. Specifically,

Again observe that, if $f(d_{n,c(n)})<1$, then $\lambda_{n}^{c}(t)<\beta_{c}$. Further, an age dependent factor determining the mobility of the individuals in the community may also be accounted in $\lambda_{n}^{c}(t)$ (see [1]).

In the numerical experiments, the disease progression model is based on COVID-19 progression as described in [16] and [5]. An individual may have one of the following states: susceptible, exposed, infective (pre-symptomatic or asymptomatic), recovered, symptomatic, hospitalised, critical, or deceased. Briefly, an individual after getting exposed to the virus at some time observes an incubation period which is random with a Gamma distribution. Individuals are infectious for an exponentially distributed period. This covers both presymptomatic transmission and possible asymptomatic transmission. We assume that a third of the patients recover, these are the asymptomatic patients; the remaining develop symptoms. Individuals either recover or move to the hospital after a random duration that is exponentially distributed (see Appendix F for model input data details). The probability that an individual recovers depends on the individual’s age. While hospitalised individuals may continue to be infectious, they are assumed to be sufficiently isolated, and hence do not further contribute to the spread of the infection. Further progression of hospitalised individuals to critical care and further to fatality is also age dependent.

We introduce methodologies to model different PHSMs (or Interventions) such as lockdowns, home quarantine, case isolation, social distancing of elderly population, mobility restrictions, masks etc. Table 7 in Appendix F summarizes some of the key interventions implemented. The PHSM’s mentioned above when implemented put some restrictions on the individual’s mobility. However, it is often the case that when several restrictions are in place, only a fraction of the population comply with these restrictions. Therefore, we restrict the movement of only the individuals in the compliant fraction of households in the city.

In this section, we first define a multi-type branching process and state a key result associated with a super critical multi-type branching process and an assumption outlining a set of sufficient conditions for it to hold. See [2] for details.

Let ${{\mathbf{\tilde{}}{B}}_{t}}$ be a $\tilde{\eta}$ dimension vector denoting the multi-type branching process at time $t$, where $\tilde{\eta}<\infty$. Component $i$ of ${{\mathbf{\tilde{}}{B}}_{t}}$ denotes the number of individuals of type $i$ at time $t$. As is well known, in a multi-type branching process, at the end of each time period an individual may give birth to children of different types and itself dies (it may reincarnate as a child of same or different type). The number of children each individual of type $i$ gives birth to is independent and identically distributed. Therefore, multi-type branching process is a Markov chain $({\mathbf{\tilde{}}{B}}_{t}\in\mathbb{{R^{+}}^{\tilde{\eta}}}:t\geq 0)$. Suppose ${\mathbf{\tilde{}}{B}}_{t}=(b_{1},\ldots,b_{\tilde{\eta}})$, then ${\mathbf{\tilde{}}{B}}_{t+1}$ is sum of independent offsprings of $b_{1}$ type $1$ parents, independent offsprings $b_{2}$ type $2$ parents, and so on. Thus, ${\mathbf{\tilde{}}{B}}_{t+1}$ is sum of $b_{1}+b_{2}+...+b_{\tilde{\eta}}$ independent random vectors in $\mathbb{{R^{+}}^{\tilde{\eta}}}$.

Consider a matrix $\tilde{K}\in{\Re^{+}}^{\tilde{\eta}\times\tilde{\eta}}$ such that $\tilde{K}(i,j)=\mathbb{E}\left({\tilde{B}_{1}(j)|{\mathbf{\tilde{}}{B}}_{0}={\mathbf{e}_{i}}}\right)$, that is expected number of type $j$ offsprings of a single type $i$ individual in one time period. Then,

By $X_{n}\xrightarrow{P}X$ we denote that the sequence of random variables $\{X_{n}\}$ converges to $X$ in probability as $n\rightarrow\infty$. Theorem 4 below is well known (see [2]).

Suppose that Assumption 1 holds. Then, corresponding to eigenvalue $\tilde{\rho}$, there exist strictly positive right and left eigenvectors of $\tilde{K}$, ${\mathbf{\tilde{}}{u}}$ and ${\mathbf{\tilde{}}{v}}$ such that ${\mathbf{\tilde{}}{u}}^{T}{\mathbf{\tilde{}}{v}}=1$ and $\sum_{i=1}^{\tilde{\eta}}{\tilde{u}(i)}=1$.

Given ${{\mathbf{X}}_{t}^{N}}$, ${{\mathbf{X}}_{t+\Delta t}^{N}}$ is arrived at through two mechanisms. (For the ease of notation we will set $\Delta t=1$.) i) Infectious individuals at time $t$ who make Poisson distributed infectious contacts with the rest of the population moving the contacted susceptible population to exposed state, and ii) through population already affected moving further along in their disease state. Specifically,

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  We assume that an infectious individual with characteristic $a\in\mathcal{A}$ spreads the disease in the community with transmission rate $\beta_{a}$. Thus, the total number of infectious contacts it makes with all the individuals (both susceptible and affected) in one time step is Poisson distributed with rate ${\beta_{a}}$. The individuals contacted are selected randomly and an individual with characteristic $\tilde{a}\in\mathcal{A}$ is selected with probability proportional to ${\psi_{a,\tilde{a}}}$ (independent of $N$). $\psi_{a,\tilde{a}}$ helps model biases such as an individuals living in a dense region are more likely to infect other individuals living in the same dense region. As a normalisation, set $\sum_{\tilde{a}}\pi_{\tilde{a}}\psi_{a,\tilde{a}}=1$.

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  Once the number of infectious contacts for a particular characteristic $\tilde{a}\in\mathcal{A}$ have been generated, each contact is made with an individual selected uniformly at random from all the individuals with characteristic $\tilde{a}$.

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  If an already affected individual has one or more contact from an infectious individual, its type remains unchanged. On the other hand, if a susceptible individual has at least one contact from any infectious individual, it gets exposed at time $t+1$. Each susceptible individual with characteristic $\tilde{a}\in\mathcal{A}$ who gets exposed at time $t$, increments ${{\mathbf{X}}_{t+1}^{N}}(\tilde{a},exposed)$ by 1. A susceptible individual that has no contact with an infectious individual remains susceptible in the next time period.

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  Once an individual gets exposed, its disease progression is independent of all the other individuals and depends only on its characteristics, that is, the disease progression profile of the characteristic class the individual belongs to. The waiting time in each state (except susceptible, dead and recovered) is assumed to be geometrically distributed. Transition to symptomatic, hospitalised, critical, dead or recovered state happens with respective characteristic (disease progression profile) dependent transition probabilities. Thus, an individual of type ${\mathbf{s}}\in\mathcal{S}\setminus\mathcal{U}$, some disease state other than susceptible, at time $t$ transitions to some other state ${\mathbf{q}}$ at time $t+1$ with the transition probability $P({\mathbf{s}},{\mathbf{q}})$ in one time step. The probability $P({\mathbf{s}},{\mathbf{q}})$ is independent of time $t$ and $N$. If the transition happens, $X_{t+1}^{N}({\mathbf{s}})$ is decreased by 1 and $X_{t+1}^{N}({\mathbf{q}})$ is increased by 1. Observe that $P({\mathbf{s}},{\mathbf{q}})=0$ if characteristic of types ${\mathbf{s}}$ and ${\mathbf{q}}$ is different.

For each $t$, let ${{{\mathbf{B}}_{t}}\in\mathbbm{Z^{+}}^{\eta}}$, ${{\mathbf{B}}_{t}}=(B_{t}({\mathbf{s}}):{\mathbf{s}}\in\mathcal{S}\setminus\mathcal{U})$ where $B_{t}({\mathbf{s}})$ denote the number of individuals of ${\mathbf{s}}$ at time $t$ in the branching process.

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  At time $t$, every infectious individual of type $a$, for all $a$, gives birth to independent Poisson distributed offspring of type $(\tilde{a},exposed)$ at time $t+1$ with rate $\pi_{\tilde{a}}\psi_{a,\tilde{a}}\beta_{a}$ for each $\tilde{a}$. $B_{t+1}(\tilde{a},exposed)$ is increased accordingly.

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  Once an individual gets exposed, disease progression of the individual has same transition probabilities as in each epidemic process. An individual of type ${\mathbf{s}}\in\mathcal{S}\setminus\mathcal{U}$, that is, in disease state other than susceptible at time $t$, transitions to some other disease state ${\mathbf{q}}$ at time $t+1$ with probability $P({\mathbf{s}},{\mathbf{q}})$. If the transition happens, $B_{t+1}({\mathbf{s}})$ is decreased by 1 and $B_{t+1}({\mathbf{q}})$ is increased by 1.

Let $A_{t}^{B}=\sum_{{\mathbf{s}}\in\mathcal{S}\setminus\mathcal{U}}B_{t}({\mathbf{s}})$ denote the total number of offsprings generated by time $t$ in the branching process.

As described above, an individual of type ${\mathbf{s}}\in\mathcal{S}\setminus\mathcal{U}$, may give birth to other exposed individuals if it is infectious, and/or may itself transition to some other type in one time step. Then, $K$ can be written as,

It follows that

Theorem 5.2 for multi-type branching processes holds under a standard assumption that the expected offspring matrix ${K}$ is irreducible (for more details, see Appendix 5.1). However, $K$ as defined in (5) is not irreducible. Lemma 5.1 below sheds further light on the structure of $K$. Theorem 5.2 observes that standard conclusions continue to hold for the branching process $\{{{\mathbf{B}}_{t}}\}$ associated with epidemic processes $\{{{\mathbf{X}}_{t}^{N}}\}$.

For any matrix $M$, let $\rho(M)$ denote its spectral radius, that is, the maximum of the absolute values of all its eigenvalues. Henceforth, we assume that $\rho=\rho(K)>1$ in all subsequent analysis.

As $K_{1}$ defined in Lemma 5.1 is irreducible, its Perron Frobenius eigenvalue is equal to its spectral radius $\rho(K_{1})$.

Observe that $\mathbb{E}\left({{B}_{1}(j)\log{B}_{1}(j)|{\mathbf{B}}_{0}={\mathbf{e}_{i}}}\right)<\infty$ for all $1\leq i,j\leq{\eta}$ holds because the offsprings for our branching process have a Poisson distribution.

Proof sketch for Lemma 5.1 and Theorem 5.2: Lemma 5.1 follows by observing that: 1) $K_{1}$ correspond to types of individuals with disease states that are infectious or may become infectious. These types may give birth to exposed individuals of each characteristic in subsequent time steps and $K_{1}$ is therefore irreducible. 2) As any individual in hospitalised, critical, dead or recovered disease states neither transitions nor gives birth to offspring in exposed, infective or symptomatic disease states, therefore, we have $0$ submatrix in $K$. 3) $C$ corresponds to type of individuals that do not contribute to any infections in the city. In addition, these individuals either transition to the next disease state or remain in the same disease state with positive probability. This results in all the eigenvalues of C being less than or equal to 1. Theorem 5.2 then follows easily by combining the Lemma 5.1 and the standard results for branching processes. For detailed proof, see Appendix C.1 and C.2.

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  Recall that at time zero, ${\mathbf{B}}_{0}={\mathbf{X}}_{0}$. We couple each exposed individual of each type in the epidemic process to an exposed individual of the same type in the branching process at time zero.

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  The coupled individuals in each of these processes follow the same disease progression (using same randomness) and stay coupled throughout the simulation.

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  Further, when infectious, they generate identical Poisson number of contacts (in epidemic process) and offsprings (in branching process). Specifically, when a coupled individual with characteristic $a\in\mathcal{A}$ is infectious, the number of contacts it makes in a time step with all the individuals with characteristic $\tilde{a}\in\mathcal{A}$ in epidemic process is Poisson distributed with rate $\pi_{\tilde{a}}\psi_{a,\tilde{a}}\beta_{a}$. This equals the offsprings generated by the corresponding coupled individual in the branching process where the offsprings are with characteristic $\tilde{a}\in\mathcal{A}$ and are of type $(\tilde{a},exposed)$.

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  In epidemic process, each contact is made with an individual randomly selected from the population with the same characteristic. If a contact is with a susceptible person, then that person is marked exposed in the next time period and is coupled with the corresponding person in the branching process.

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  On the other hand, if in epidemic process, a new contact is with an already affected individual, then this does not result in any person getting exposed so that the corresponding offspring in the branching process is uncoupled. The descendants of uncoupled individuals in the branching process are also uncoupled. In our analysis, we will show that such uncoupled individuals in the branching process vis-a-vis epidemic process are a negligible fraction of the coupled individuals till time $\log_{\rho}N^{*}$ for large $N$, where recall that $N^{*}=N/i\log N$ and $\log_{\rho}N^{*}=\log N^{*}/\log\rho$ and $\rho$ denotes the exponential growth rate of the branching process.

Denote the new uncoupled individuals born from the coupled individuals in branching process at each time $t$ by $G_{t}^{N}$ (referred to as “ghost individuals”). As mentioned earlier, these and their descendants remain uncoupled to epidemic system $N$. Let $D_{i,t-i}^{N}$ denote all the descendants of $G_{i}^{N}$ (ghost individuals born at time $i$) after $t-i$ time steps, i.e. at time $t$.

Following result shows that epidemic process is close to the multi-type branching process till time $\log_{\rho}N^{*}$.

Result (7) was earlier shown for SIR setting in [3]. We extend this result to general models and also prove the result (8) in this more general setting.

Proof sketch for Theorem 8 and Lemma 5.2: For all $\bm{s}\in\mathcal{S}\setminus\mathcal{U}$, $\left\lvert{{{B}}_{t}(\bm{s})-{{X}}^{N}_{t}(\bm{s})}\right\rvert$ is bounded from above by $\sum_{i=1}^{t}D_{i,t-i}^{N}$ (total number of uncoupled individuals till time $t$). Recall that the ghost individuals ($G_{i}^{N}$) arise as a result of the infectious individuals in epidemic process making contact with already affected individuals and is related to their product. Further, infectious individuals are upper bounded in no. by affected individuals and hence it can easily be shown that $\mathbb{E}\left({{G}^{N}_{i}}\right)\leq\hat{\beta}\mathbb{E}\left({\frac{(A_{i-1}^{N})^{2}}{N}}\right)$ for some $\hat{\beta}>0$ and any $i\geq 1$. Further, from coupling we have $A_{t}^{N}\leq A_{t}^{B}$ for all $t\in\mathbb{N}$. Inequality (9) then follows by bounding from above the no. of descendants of $G^{N}_{i}$ individuals at time $t-i$ in the branching process. ${\mathbf{1}}\in{\Re^{+}}^{\eta}$ is a vector with each entry equal to 1.

Lemma 5.2 follows from (9) and a standard branching process result that $\mathbb{E}\left({\left({A^{B}_{i}}\right)^{2}}\right)\leq\tilde{c}\rho^{2i}$, for some $\tilde{c}>0$. Theorem 8 follows from Lemma 5.2 using Markov’s inequality. Details are given in Appendix C.3 and C.4.

Recall from Theorem 5.2 that, ${{\mathbf{B}}_{t}}/{\rho^{t}}\xrightarrow{P}W{\mathbf{v}}$ as $t\to\infty,$ where $W$ is a non-negative random variable representing the intensity of branching process and ${{\mathbf{v}}}\in{\Re^{+}}^{\eta}$ is the left eigenvector corresponding to eigenvalue $\rho$ of matrix $K$. Therefore, initially (till time $\log_{\rho}N^{*}$) epidemic process grows exponentially at rate $\rho$, with sample path dependent intensity being determined by $W$.

The following proposition follows directly from Theorem 5.2 and Theorem 8 and justifies the fact that the proportions across different types stabilize quickly in the epidemic process, and thus early on, till time $\log_{\rho}N^{*}$, paths can be patched from one time period to the other with negligible error due to change in the proportions.

Compartmental models are widely used to model epidemics. Usually, in these models we start with infected population which is a positive fraction of the overall population. However, little is known about the dynamics if we start with a small, constant number of infections. In Appendix B we describe the results for some popular compartmental models using Theorem 8 and 5.1.

As Theorem 8 and Proposition 5.1 note, early in the infection growth, till time $\log_{\rho}N^{*}$ for large $N$, while the number affected grows exponentially, the proportion of individuals across different types stabilizes. Here, the types corresponding to the susceptible population are not considered because at this stage, the affected are a negligible fraction of the total susceptible population. The growth in the affected population in this phase is sample path dependent and depends upon non-negative random variable $W$.

However, at time $\log_{\rho}(\epsilon N)$ for any $\epsilon>0$ and large $N$, this changes as the affected population equals $\Theta(N)$. Hereafter, the population growth closely approximates its mean field limit whose initial state depends on random $W$ and where the proportions across types may change as the time progresses. Our key result in this setting is Theorem 5.4. To this end we need Assumption 2.

Let $t_{N}=log_{\rho}(\epsilon N)$. Let ${\mathbf{\mu}}^{N}_{t}$ denote the empirical distribution across types at time $t_{N}+t$. This corresponds to augmenting the vector ${\mathbf{X}}_{t_{N}+t}^{N}$ with the types associated with the susceptible population at time $t_{N}+t$ and scaling the resultant vector with factor $N^{-1}$.

Observe that ${\mathbf{\bar{}}{\mu}}_{0}(W)$ above is path dependent in that it depends on the random variable $W$. The above assumption is seen to hold empirically. While, we do not have a proof for it (this appears to be a difficult and open problem), Corollary 5.1 below supports this assumption. This corollary follows from Lemma 5.2.

Also note that Assumption 2 holds along some subsequence $\{N_{i}\}$ since ${\mathbf{\mu}}^{N}_{0}$ is a bounded sequence.

Let $c^{N}_{t}({a})$ denote the total incoming-infection rate from the community as seen by an individual with characteristic ${a}\in\mathcal{A}$ at time $t$. It is determined from the disease state of all the individuals at time $t$. In our setup, $c^{N}_{t}({a})$ equals

where $\mathbbm{1}$ denotes the indicator function. For each individual $n\leq N$, let $S^{t}_{n}$ denote its type at time $t$. Then, for ${\mathbf{s}}=(a,d)\in\mathcal{S}$, $\mathbb{P}\left(S^{t+1}_{n}={\mathbf{s}}|\mathcal{\mu}_{t}^{N}\right)={p}(S^{t}_{n},{\mathbf{s}},c^{N}_{t}({a})),$ for a continuous function $p$. In particular, the transition probability only depends on the disease-state of the individual at the previous time, the disease-state to which it is transitioning, and the infection rate incoming to the individual at that time.

Recall that from Assumption 2 we have defined ${\mathbf{\bar{}}{\mu}}_{0}(W)\in{\Re^{+}}^{(\eta+1)}$ such that ${\mathbf{\mu}}^{N}_{0}\xrightarrow{P}{\mathbf{\bar{}}{\mu}}_{0}(W)$ as $N\rightarrow\infty$. Define ${\mathbf{\bar{}}{\mu}}_{t}(W)\in{\Re^{+}}^{(\eta+1)}$ such that for all $t\in\mathbbm{N}$, ${\mathbf{s}}=(a,d)\in\mathcal{S}$,

where

where $\mathcal{M}^{N}_{t+1}(\bm{s})=\frac{1}{N}\sum\limits_{n=1}^{N}\mathbbm{1}\left(S^{t+1}_{n}=\bm{s}\right)-{p}(S^{t}_{n},\bm{s},c^{N}_{t}({a})).$ Observe that $\mathcal{M}^{N}_{t+1}(\bm{s})$ is a sequence of $0$-mean random variables whose variance converges to $0$ as $N\rightarrow\infty$. Also, $c^{N}_{t}({a})$ is a continuous and bounded function of $\mu^{N}_{t}$ and ${h}$ is uniformly continuous in its arguments since it is a continuous function defined on a compact set. Therefore, ${\mathbf{\mu}}^{N}_{t}\xrightarrow{P}{\mathbf{\bar{}}{\mu}}_{t}(W)$ follows by inducting on $t$. Further details are given in Appendix C.5.

In particular, if ${\mathbf{\bar{}}{\mu}}_{t}$ denotes the mean field limit of the normalised process at time $t+\log_{\rho}(\epsilon N)$, then, the number of infections observed in a smaller model with population $N$ is approximately $N*{\mathbf{\bar{}}{\mu}}_{t}$ and that of a larger model is approximately $kN*{\mathbf{\bar{}}{\mu}}_{t}$. Thus, the larger model infection process can be approximated by the smaller model infection process by scaling it by $k$.

In our earlier experiments the interaction spaces included homes, workplaces, schools and communities. In practice, there maybe subspaces within these interaction spaces where individuals have more frequent interaction. In the following experiments, we capture these subspaces by introducing smaller project networks in workspaces, classes within each school, and neighbourhood clusters within communities. The exact details of these constructions are given in [1]. Roughly, speaking a project network within a workplace is a cluster of size that is uniformly chosen within 3 to 10. In schools, students of same age are kept in the same class. Each neighbourhood consists of approximately 200 families. The $\beta$’s within these clusters are increased while those outside the cluster are reduced.

Theses clusters are identically created in the larger as well as the smaller city (in the larger city, their numbers are scaled up along with the population size). Figure 9 shows that the numerical results using the SSR algorithm on the smaller 1 million city match the larger 12.8 million city under no-intervention scenario. Figure 9 shows analogous results under the intervention - home quarantine day 40 onwards.

While our experiments are designed for Mumbai, a very dense city, it is reasonable to wonder if the results would hold for less dense cities or regions. In this section we report the experimental results for the sparser cities that are created by reducing the contact and hence the transmission rates between individual in different interaction spaces. Specifically, we consider two reasonable scenarios:

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  Scenario 1: The community beta is kept at one tenth the level of the community beta for Mumbai considered in our earlier experiments in Section 2. Figure 11 shows that the result using SSR algorithm matches the larger model in the no-intervention scenario. Figure 11 shows that the result using SSR algorithm matches the larger model in the scenario with an intervention of home quarantine after 40 days. Although we did not conduct elaborate experimentation, similar results should be true under more general and extensive interventions.

  

  

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  Scenario 2: Beta community is reduced by a factor of 20, beta home, school and workplace by a factor 4. Again as seen in Figure 13, the result using SSR algorithm matches the larger model in the no-intervention scenario. It can be seen that even in the no-intervention scenario, the number of infections are very low after 200 days. Under home quarantine after 40 days, the exposed cases from the smaller model are within 20% of the larger model although both are more-or-less negligible numbers, the highest less than 250 (see Figure 13).

In the earlier experiments in Section 2, we had compared the results of a 12.8 million city to the results from using SSR on a smaller 1 million city. A natural question to ask is how small the city can be for SSR to continue to be accurate. Below we consider cities of size 500,000 and 100,000.

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  500,000 city: Figure 15 shows that the result using SSR algorithm matches the larger model in the no-intervention scenario. Figure 15 shows that the result using SSR algorithm matches the larger model in the scenario with illustrative single intervention - home quarantine day 40 onwards.

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  100,000 city: Figure 17 shows that in this case using SSR algorithm poorly matches the larger model in the no-intervention scenario. The reason for this mismatch is that 12.8 million city and 100,000 city are essentially identical for only 24 days (see Figure 17). However, the number of exposed at day 24 are around 8,000. After scaling 8,000 by 128, we get 62.5, we need to take the simulation output from the smaller model when the number of exposed equalled 62.5 and stitch that to day 24 by scaling it by 128. This is not feasible as we start with 100 persons exposed. Ad-hoc adjustments, that allow us to start with 100 or more exposed under SSR lead to much noisier output. For this reason, we are also unable to implement SSR in this setting when there are interventions.