Effect of lockdown interventions to control the COVID-19 epidemic in India
Ankit Sharma, Shreyash Arya, Shashee Kumari, Arnab Chatterjee
EEffect of lockdown interventions to control theCOVID-19 epidemic in India
Ankit Sharma , Shreyash Arya , Shashee Kumari , and Arnab Chatterjee TCS Research, New Delhi, India
ABSTRACT
The pandemic caused by the novel Coronavirus SARS-CoV2 has been responsible for life threatening health complications,and extreme pressure on healthcare systems. While preventive and definite curative medical interventions are yet to arrive,Non-Pharmaceutical Interventions (NPIs) like physical isolation, quarantine and drastic social measures imposed by governingagencies are effective in arresting the spread of infections in a population. In densely populated countries like India, lockdowninterventions are partially effective due to social and administrative complexities. Using detailed demographic data, wepresent an agent based model to imitate the behavior of the population and its mobility features, even under intervention.We demonstrate the effectiveness of contact tracing policies and how our model efficiently relates to empirical findings ontesting efficiency. We also present various lockdown intervention strategies for mitigation – using the bare number of infections,the effective reproduction rate, as well as using reinforcement learning. Our analysis can help assess the socio-economicconsequences of such interventions, and provide useful ideas and insights to policy makers for better decision making.
The rapidly spreading infectious disease COVID-19 is caused by the novel coronavirus SARS-CoV-2, has become a globalpandemic . Spreading fast, with short doubling times there are usually long delays in showing effects from interventions .With increasing number of complex, life threatening infections , it has overwhelmed healthcare systems around the worldand caused thousands of deaths worldwide. A majority of infected individuals need medical treatment, some of them criticalattention. The current case fatality ratio (CFR) stands around 7%, although countries across the world show variations between1% and 15% . While the World Health Organization (WHO) currently maintains that there are no reports of reinfection yet,they were cautious in stating that there is “currently no evidence that people who have recovered from COVID-19 and haveantibodies are protected from a second infection” .The propagation of a transmitting infection is usually well understood by the basic reproduction number ρ which is theaverage number of infections caused by a single individual in a fully susceptible population, or in other words, a populationwithout any immunity. When immunity develops in a population, or otherwise, when other mitigating factors are in play, theeffective reproduction number ρ t (reproduction number when both susceptible and non-susceptible are present) decreases tobelow unity, thereby rapidly decreasing the proportion of infected population. Suppressing infections is an alternative way ofstopping the spreading, since the population still remains susceptible. True immunity is only achieved using pharmaceuticalintervention like vaccination. In a well mixed population, the herd immunity is either achieved by naturally recovering frominfections or by vaccinations, when 1 − ρ fraction of population becomes immune.In absence of vaccines and consensus on effective medication for treatment, drastic Non-Pharmaceutical Interventions (NPIs) are the only possible ways to control the contact mixing of the population that is the basis of the spreading of anycontact mediated infectious disease. The effect of such interventions only delay the spread of the infections in the susceptiblepopulation, thereby reducing the pressure on the healthcare system, and buying time for viable, effective pharmaceuticalanswers.One of the basic and widely used intervention is to impose strict restriction on population contact, mixing and movement,usually regulated by governing authorities, and are commonly termed as lockdowns . When the duration of lockdowns keepsincreasing, the public costs can be immense, both in terms of its economic and financial fall-outs, as well as from the social andpsychological perspective. Recent proposals of isolation and other restrictions have barely been able to flatten the curve andkeep the critical cases below the healthcare capacity. Flattening the curve along with managing social and economic costs is notpossible unless NPIs are introduced and relaxed to re-start the social and economic activities in a civil society.One of the effective ways of monitoring and controlling the spread of infections is through the efficient use of instantaneouscontact tracing data acquired through smartphone based application ( app ), and administer regulated interventions accordingly.The app keeps track of proximity contacts and notifies individuals at risk, who can isolate, quarantine or get tested and treated.Such targeted recommendations can serve as a better alternative for infection control, compared to mass quarantines, which can a r X i v : . [ phy s i c s . s o c - ph ] S e p ave heavy socio-economic consequences. However, this can only be possible if the application can be used by a large fractionof the population. A recent study argued that while almost 87% of Indian population has access to a mobile phone , thenumber of smartphone users is still well below 40%. If there is a mechanism to augment the location data from non-smartphoneusers using basic feature phones, the Indian population can achieve digital herd immunity , thereby facilitating the governmentauthorities to track individuals, trace their contacts and impose targeted interventions like isolation, testing and quarantine.However, such a theoretical possibility is yet to mature to a practical reality.Modeling epidemics on graphs has been an extremely important and fertile area of research. The natural reason is tounderstand the mechanisms of disease propagation in closely connected and mixing population in human societies, using avariety of frameworks from paradigmatic toy models to detailed, data driven, agent based models. The huge amount of datagathered through a multitude of sources to create a multi-scale data of demographics, mobility and other essential componentshave opened up the possibility to perform large scale data-driven simulations. This helps to answer important, specific anddetailed questions related to epidemics, and particularly for prediction and forecasting purposes.In this paper, we introduce an agent based model of the Indian population, with demographic and mobility features modeledusing publicly available data sources. The COVID-19 epidemic is studied using a microscopic model that takes into account theavailable data on transmission, infections and mortality. Under this detailed framework, we study the consequences of a fewlockdown strategies that can be effective in containing the spread of infections as well as possibly minimize the socio-economicfootprints. In the following section, we describe our model framework in detail by introducing the structured population andthen the epidemic process. Next, we discuss our results and conclude with discussions. We separate our model into two distinct parts. First of all, we model the population and its contacts as a hierarchically structuredgraph, in order to facilitate the effect of lockdown intervention at different spatial scales by rearranging and removing linkssystematically. The movement related behavior of individuals depend on several demographic features. The second part of themodel deals with the epidemic dynamics and its details.
We build a hierarchically modular network to simulate a structured population (See, e.g. Ref. ) where the basic unitsare individuals. While the individual agents are the nodes, the links represent the close physical proximity or contact andassociations over a short period of time, which can be dynamic in nature. We will interchangeably use the term agents to referto individuals in the rest of the paper, since they will eventually have several static and dynamic features depending on spatialand temporal properties. This can be imagined to be the aggregated network of all contact graphs of agents over a short periodof time. The modular graph is conceived and constructed using the following steps: Figure 1.
The hierarchical structure of the population: the basic units are individuals, forming families residing in residentialzones which also contain retail. The contact networks between families are governed by Barabási-Albert type networks. Thereare also zones for education and entertainment, as well as employment (public or private) opportunities. Combinations of thesezones create district. Travel between various areas create contact networks between individuals. Several such districtsconstitute a state. Contact networks between districts are created using gravity law. • Individuals (agents) form the basic units of a society. However, at the basic level, several of them constitute a family. It isfair to assume that individuals in a family form a complete graph or a clique, because they come in close contact in theirresidential space. • In a residential neighborhood, several families live in close proximity. It is fair to assume that number of contacts withothers across different families have a broad distribution. In fact, there are strong empirical evidences that the degreedistribution of physical social networks have power law tails (see, e.g. ). We construct a neighborhood using different amilies as units. We use the Barabási-Albert algorithm for preferential attachment to connect a new family of size n k to an existing family i , with probability n i / ∑ j n j , where n j are the sizes of existing families. Once a target familyis selected, we create a single link between any two members of the two families to keep the connectivity sparse. Thefamily size distribution is sampled from the Census data . See Appendix for details in Algorithm 1. • Next, we introduce the concept of zones as a spatial feature. Each zone is tagged to some special amenities like residentialneighborhoods (defined in the previous paragraph), retail outlets, entertainment centers, educational institutions, officespaces, etc., which provide distinguishing characteristic to it. In a simplistic picture, we assume that there are 4 differenttypes of zones in a district: (i) residential and retail (RR), (ii) education and entertainment (EE), (iii) private employment(EPV), and (iv) public employment (EPB). A district usually comprises of several of the above zones. Zones in eachdistrict are assumed to be distributed uniformly for sake of simplicity. For instance, Mumbai Suburban district has a totalof 25 zones divided in ratio 8:5:5:7 for RR:EE:EPV:EPB respectively. • Agents are embellished with various features from the demographic data available from Census of India 2011 . We useage, gender (male or female) and employment status. We consider 3 broad age classes – (i) 0-19 years, (ii) 20-59 yearsand (iii) 60 years and above. We also have employment status as – (i) employed or (ii) unemployed. The unemployed donot move to the employment zones. The employed in turn also fall in two categories – (i) essential and (ii) non-essentialworkers. The essential workers have the liberty to move around different zones even during the lockdown. For instance,48.73% of Mumbai Suburban’s population is employed out of which 1% of agents are tagged as essential workersrandomly (See Appendix Table 1 for parameter details, F essential ). • We consider 4 time bins – (i) morning, (ii) afternoon, (iii) evening and (iv) night, which sets the time granularity of ourmodel. Thus, 4 time steps correspond to one day in our subsequent simulations.Inspired by models of intra-urban mobility (see, e.g., Ref ), we simulate the movement of agents across different zonesas a function of their age and time of the day, using probabilities for transiting from one zone to another. We alsoconstruct a similar set of conditional probability tables for the lockdown periods taking ideas from , when movementis rarely allowed outside residential zones except for obtaining essential commodities.At this point it is important to note that the mobility data at the level of individuals is useful, to compute origin-destinationmatrices, dwell estimates at hotspots, amount of time spent at various locations, and also contact matrices . Comparedto other countries like Italy , the data at this level of detail is not available for India. However, aggregate mobilitytrends give a rough idea of the time aggregated mobility across a variety of locations like retail and recreation, groceriesand pharmacy, parks, transit stations, workplaces and residential areas. See Appendix Table 6, Table 7, and Table 8 fordetails. • Restructuring the network due to mobility: We model the movement of agents across different zones and the resultingcontact structure with others, using a process of removing and adding links in the existing graph. If an agent does notmove out of its own zone, it breaks one of its existing links with probability P re and creates a new link inside the zonewith probability P ce . In case if it moves out of its own zone to a target zone (can be within or outside its own district), alllinks from its previous zone are broken with a probability P re and a new link is created in the target zone with probability P ce . An agent moves to another district as per the inter-district mobility matrix M DD . This dynamics of links effectivelycaptures the properties of temporal contact graphs at the granularity of 4 time steps in a single day. At night, agentsreturn back to their residence locations with a probability C H , restores their family links with probability P ce , and theirexisting links are removed with a probability P f e . As soon as an agent is tested positive, its existing links are broken withprobabilities C I and C E depending upon the current state of agent i.e., Infected (symptomatic) or Exposed (asymptomatic)respectively (See Sec. 2.2 for epidemic states). If an agent dies, all its links are broken and the agent is removed from thesystem. To restrict/limit inter-district mobility, inter-district links are broken with a probability C BS when lockdown isimposed. During lockdown, parameters are changed to bring in the effect of social distancing. Refer to Table 3, Table 4,and Algorithm 3 in Appendix for details. • Connections across districts: Empirical evidences regarding population movement across space is a well studied anddebated problem. While earliest research pointed the flux between two population agglomerations to be proportional tothe respective population sizes and a inverse dependence on the distance separating them , recent studies presentstrong evidence in favour of a generalized gravity law , which holds reasonably well depending on the scale underconsideration. In absence of empirical data for population flux in India, we assume that the gravity law holds, as in thecase of Korean highways . e assume that the number of links across districts is directly proportional to the population flux between two districts ofpopulation sizes P i and P j , separated by distance d i j , and is given by F i j = C P i P j d i j . (1)The normalization factor C is set to unity. In each district, nodes are chosen randomly to connect with randomly chosennodes from other districts. We use data from Census data of 2011 to create population samples for districts. • Mobility during lockdown: When lockdown interventions are imposed at different stages, the corresponding links arebroken and mobilty is restricted inside different zones, along with a compliance factor. Severity of lockdown can bemanyfold – from containment zones, district lockdown and sealing of state border. See Table 3, Table 4, and Table 8 inappendix for details.
In the standard mathematical treatments of epidemics, there is a concept of the basic reproduction number ρ which is theexpected number of cases generated by an infected one in a population where there is no immunity and all are susceptible toinfection. This is not a fundamental constant, and depends on behavioral properties of the given population, and hence can differacross countries for the same disease. In fact, recent reports on dependence of weather suggest that the effective reproductionnumber ρ for COVID-19 decreases by around 3% per degree of increase of temperature beyond 25C . The generation andincubation times are rather more fundamental, as is the serial interval of infections. In case a fraction f attains immunity to thedisease due to some reason, an infected individual will, on the average, infect ρ = ρ ( − f ) . For the epidemic to die, needs tosatisfy the condition ρ <
1, i.e., f > − ρ . f c = − ρ is known as the herd immunity threshold . In this simplistic picture,this infection spreading is a branching process , and is basically a percolation problem , in the language of statistical physics.There is a critical phenomenon associated with the phase transition, with the critical point at f c , separating the epidemic phase( f < f c ) and the immune phase ( f > f c ) (Fig. 2). f c (cid:16) = 1 − ρ (cid:17) herd immunitythreshold 1epidemic immune Figure 2.
The epidemic phase diagram with the herd immunity threshold f c in terms of the basic reproduction number ρ .While the early findings from COVID-19 cases in China suggested a value of 2 . − .
7, the latest findings suggest a muchhigher average value of ρ around 5 . , which pushes the herd immunity threshold from around 60% to above 80%. Thisimplies that more than 80% of the population needs to have immunity, naturally by recovering or through vaccination, for theinfections to stop spreading further.The transmission of the virus is known to happen through various routes, mostly through exhaled droplets, but surfacesand fecal-oral contamination have also been reported. The transmission are either through the symptomatic, pre-symptomatic,asymptomatic or environmental in nature. In what follows, we do not take the environment into account, explicitly. As ina macroscopic mathematical description of the epidemic , we model the states of an individual in a population as thefollowing: (i) Susceptible S , (ii) Exposed E , (iii) Infected I , (iv) Recovered R and (v) Dead D . We will denote the correspondingfractions of these states in the population as S , E , I , R and D respectively. This is similar to the SEIR model with an additionalstate D corresponding to the dead. The microscopic dynamics of the states of the model are described in the following.The initial state for the entire population is set to S . Epidemic spread is initialised with a few infected agents I init which represent imported cases. With each passing time-step, agents move and interact with other agents, and the mobilitycharacteristics are dictated by the agent level features as described in Sec. 2.1. The infected I coming in contact with S canconvert them to E with rate β I . Once the state E is produced in the population, they in turn can convert the susceptible S into E with rate β E . F sym of the total exposed agents E can mature into infected I . 1 − F sym of the total exposed agents E may stayasymptomatic and eventually recover to state R . An early report from ICMR claimed that the asymptomatic cases constituted80% of the total COVID cases . However, we have used 60% asymptomatic cases in our epidemic modelling (refer to Table 2in Appendix for the value of the fraction F sym ). The infected agents may decease to D dictated by probability P µ or I caneventually recover to R by probability 1 − P µ . Transitions to I , R and D are bounded by their corresponding time periods – the Indian Council of Medical Research, , (2020), [accessed 20 April 2020]
E I DR β E , β I F sym P µ − P µ − F sym Figure 3.
The basic SEIRD model: Susceptible S can come in contact with the exposed E or the infected I to transition toexposed E with rates β E and β I respectively. F sym of the total exposed agents E can mature into infected I . 1 − F sym of the totalexposed agents E may stay asymptomatic and eventually recover to state R . Infected agents may die to D with a probability P µ or can recover to R by probability 1 − P µ .time to infection T I , recovery from infected T RI , asymptomatic recovery from exposed T RE , and mortality time T M . Lockdownintervention mechanism is also triggered which locks out districts depending upon the deployed strategy (for details referto Sec. 3.2). This controls the contact process and hence the rapid spread of the infections. Fig. 9 in Appendix shows anillustration of epidemic spread for Mumbai suburban district at different temporal snapshots.In order to keep track of the individuals in our agent based model, we carry out the microscopic simulation of the abovedynamics using the Gillespie algorithm . We use the model described in the previous section to simulate the behavior of agents in a population, with the epidemic processrunning on top of it, spreading through the process of contacts between individuals and spatially propagating due to mobility ofexposed and infected individuals.We initiate our simulations with susceptible agents in the population, while we seed the epidemic process by assuming afew infected agents. The first quantities to calculate are the fractions S , E , I , R and D and their evolution over time. Using amodel of their choice, the usual studies compute the infected fraction and try to match the numbers from reported data. Inreality, reported data does not give us the correct number of infected population, specially for a socially complex, and populouscountry like India. This is because, it is practically impossible to test each individual and trace their contacts for possible furthertesting. Usually, the rate of testing is low and is unable to capture the real numbers. Moreover, there is a lag between testing andreporting of results, and this can be a major factor which makes tracing difficult. Trying to attempt to match reported number ofcases with simulations will be a futile exercise.In our work, we do not attempt to match the number or fraction of infected individuals to real data. Our model, on the otherhand, keeps information regarding each individual’s movements in space and contact with others, enabling us to theoreticallymatch the tracing and testing scenario, within the limits of practical error in real data. For a disease like COVID-19, where the incubation period is long and the fraction of asymptomatic population is quite large, astandard method of relying on people’s reporting accuracy to trace who they came in contact with, is insufficient. In that case,the tracing starts with an individual showing symptoms and its contacts can possibly be traced both forward and backward intime, the latter being more important because it can enable tracing the other possible paths the infection has branched into,and thus recursively trace the parent of the transmission sub-tree. In this digital era of smartphones, contact tracing can beenhanced, not only in terms of the number of contacts but also in real time. The theoretically zero time lag between occurrenceof a contact and it being reported is a matter of technology.
The most important part of creating a microscopic agent based model with a variety of agent features, mobility behavior anddetailed epidemic parameters, is to be able to correspond to the scenario where each agent can be traced spatio-temporallyfor its contacts. In real data, the testing rate is defined as the number of individuals tested per unit population. Anotherimportant quantity of our interest is the test positive rate , which is given by the number of individuals testing positive perunit tested population. In reality, individuals are rarely tested completely randomly because that is economically inefficient.Usual procedures include testing someone who reports a symptom, then test a list of its contacts, and maybe more – secondarycontacts, and so on. Often individuals who have been in vicinity of an area which reports a lot of infected cases, are also tested. -5 -4 -3 -2 -1 popu l a t i on f r a c t i on i n ep i de m i c s t a t e s days MaharashtraEIRD f r a c t i on o f po s i t i v e c a s e s f r o m t e s t s days Maharashtramodelreal -5 -4 -3 -2 -1 popu l a t i on f r a c t i on i n ep i de m i c s t a t e s days Tamil NaduEIRD f r a c t i on o f po s i t i v e c a s e s f r o m t e s t s days Tamil Nadumodelreal
Figure 4. (Left Panel) The fractions E , I , R and D over time as observed in numerical simulations. (Right panel) Thecorresponding test positive rates compared with real data over time (right panels) for the states of Maharashtra and Tamil Nadu.The results are shown for 10 networks and 10 runs for each network for both the states. Maharashtra has 35 districts and TamilNadu has 32 districts (as per 2011 census) and our simulation results are shown for a sample population of 19944 agents and19727 for Maharashtra and Tamil Nadu respectively. The vertical lines correspond to the change in intervention (lockdown)rules in India (25/03, 15/04, 04/05, 18/05, 01/06).We use the real data of testing and test positive rate in a given state and try to fit that from our simulation results. In doingso, a set a parameters are used for fitting. When an agent gets infected, all the contacts of the agent for the last T LB days alongwith N RT number of random agents are tested for the infection. The real scenario corresponds to the infected agent being ableto recollect (or through a tracing app) all other agents who came in contact in that span of time T LB . N RT can be interpreted asan offset term taking care of the error and delay in tracing. In Fig. 4 we show the correspondence between the real data for thefraction of positive cases over time, and tried to match our simulation results by tuning several parameters related to mobilityand lockdown compliance (see Table 3 in Appendix for details). We find the best fits for T LB = N RT is either 2 or 3depending on the lockdown regime.In our simulations, we constructed 10 different networks for the sample populations and simulated 10 runs of the epidemicprocess on each of them. The mean and standard deviations of the test positive rates are shown, along with the real data. Inthe Indian context, the strictness of the lockdown had varied through the timeline and thus the parameters have been alteredaccordingly (see Table 3 in Appendix for details). These are also taken into consideration during the simulation. However,there are irregular spikes in the empirical data since the reporting is known to irregular, with data missing for few days andbeing accumulated for a particular day when they are actually available.The corresponding density of different epidemic states in the population is also shown (left panels of Fig. 4) for the samplepopulations from Indian states of Maharashtra and Tamil Nadu. The growth of the infected state I is found to be slow, with aslow growth rate consistent what is observed in empirically reported data. Containing infections being the main objective, the primary task is restricting the contacts of the population by arresting theirmovement at different scales. This leads to asking the question that what could be a viable strategy which is both operationallyeasy and functional, as well as effective in containment. Of course, the brute global lockdown can slow down the spreading butis not the best option for several reasons, mainly because certain economic sectors need to mobilize to facilitate the restart ofthe economy, which otherwise during the restriction period comes to a standstill.In the following, we discuss a few different strategies that can be realized. We try to use the information about the barefraction of infections in a given area and subsequently use the instantaneous reproduction number for the infections as triggersfor deciding when to impose and relax restrictions in the different constituent districts of a given state. We also try a more ophisticated reinforcement learning based protocol to make the same decisions. In all the results that follow, we assume thatno pharmaceutical interventions are being used, and the population is left to itself to acquire immunity through recovering fromthe infections. Our simulated time horizons are long for demonstration, as a consequence of the above assumption.
The natural trigger for imposing lockdown interventions will be by looking at the fraction of infections in a particular area.We use districts as the geographical entity where the fraction of infections is calculated and depending on thresholds I u and I d , the interventions are turned on or off. In our simulations, shown in Fig. 5, we demonstrate the case when the interventionswitch is I u = . × − , i.e. 0.175%, i.e. interventions are effective when I ( t ) > I u and relaxed when I ( t ) < I d with I d = . × − , i.e. 0 . -5 -4 -3 -2 -1
0 200 400 600 800 1000 1200 1400 popu l a t i on f r a c t i on i n ep i de m i c s t a t e s days MaharashtraEIRD d i s t r i c t s unde r l o ck do w n days Maharashtraaveragesingle run
Figure 5.
The simulation results for lockdown on districts depending on the fraction of infections. Lockdown interventionsare activated when infections are greater than I u and deactivated when less than I d . The result shown is the average for 50 runs(10 runs each on 5 networks ). (Left panel) The fractions E , I , R and D over time. (Right panel) The average number of districtsunder lockdown at a certain day for the state of Maharashtra. The grey shaded region denotes one standard deviation. Theaveraged data is compared with a single run on a particular network realisation. Maharashtra has 35 districts (as per 2011census) and our simulation results are shown for a sample population of 19943 agents. Compared to using the fraction of infected cases, one can imagine a rather sophisticated way of designing a lockdown triggerby using the instantaneous effective reproduction number ρ ( t ) . The values of ρ ( t ) can be calculated at the level of differentstates of India (following the recipe of Ref. and is shown in Fig. 8 in Appendix). In our model simulations, we computethe values of ρ ( t ) for each district. We prescribe a rule where a district goes under lockdown if the ρ ( t ) exceeds a particularthreshold, ρ ( t ) > ρ u and is relaxed when is below another low threshold, ρ ( t ) < ρ d . In Fig. 6, we show the case when theinterventions are in place while ρ u = ρ d = . -5 -4 -3 -2 -1
0 200 400 600 800 1000 1200 1400 popu l a t i on f r a c t i on i n ep i de m i c s t a t e s days MaharashtraEIRD d i s t r i c t s unde r l o ck do w n days Maharashtra
Figure 6.
The simulation results for lockdown on districts depending on the reproduction rate ρ ( t ) . Lockdown interventionsare activated when reproduction rate ρ ( t ) is greater than 1 and deactivated when less than 0.5. The results are averaged over 25runs (5 runs each on 5 networks). (Left panel) The fractions E , I , R and D over time. (Right panel) The average number ofdistricts under lockdown at a certain day for the state of Maharashtra. The grey shaded region denotes one standard deviation.Maharashtra has 35 districts (as per 2011 census) and our simulation results are shown for a sample population of 19943 agents. .2.3 Using reinforcement learning Critical factors in decision making while imposing lockdown include infections, recoveries, deaths and economic loss. Inprevious sub-sections, we focused on infection and recovery as triggers whereas death and economic impact were not considered.To address this, we used reinforcement learning (RL). Reinforcement Learning is a technique in which an agent learns andimproves using its experience and performs actions while interacting with environment with a goal to maximize reward.Observable characteristics of the environment is referred to as state and its representation is given as context to the agent. -6 -5 -4 -3 -2 -1
0 100 200 300 400 500 600 popu l a t i on f r a c t i on i n ep i de m i c s t a t e s days MaharashtraEIRD d i s t r i c t s unde r l o ck do w n days Maharashtra
Figure 7.
The simulation results for lockdown on districts as proposed by reinforcement learning. The results are shown for 5networks and 3 runs each of 100 episodes for each network. (Left panel) The fractions E , I , R and D over time. (Right panel)The average number of districts under lockdown at a certain day for the state of Maharashtra.The grey shaded region denotesone standard deviation. Maharashtra has 35 districts (as per 2011 census) and our simulation results are shown for a samplepopulation of 19943 agents.We trained a Deep Q Network (DQN) agent to learn an optimal intervention strategy. State representation presented tothe RL agent in order to choose an action (imposing lockdown or not for every district) consisted of district-wise population,infections, recoveries, deaths, infection rate, recovery rate and the death rate at every time-step. An action with a higher Qvalue is chosen greedily by the RL agent during exploitation phase. Whereas, in the exploration phase, actions are chosenrandomly. Rewards are calculated on the basis of number of districts under lockdown, infection count and death count for everydistrict. Training was carried out using 2 hidden layers with learning rate = .
001 and momentum = . ) using the mean squared error as the loss function. Fig. 7 shows the essentials of epidemic spread using trained DQNagent for the state of Maharashtra. The COVID-19 pandemic has become one of the threats to humanity, by spreading rapidly, causing life threatening clinicalcomplications by itself and somewhat fatal for individuals with other co-morbidities . With less information about theepidemic in the first few months, healthcare systems were overwhelmed with critical cases. With time, some reliable scientificinformation has helped healthcare systems across different countries to cope with the pandemic. However, the number of caseshave multiplied by then. Researchers have explored various classical and improvised modeling techniques to understand andpredict the rising number of cases, with little success.Modeling epidemics in real populations for studying various possible scenarios and enable prediction is an extremelychallenging task. While basic mathematical models established over time provide the necessary framework for realizing thepopulation and the processes of spreading, relating to real data is tricky. This is mainly because the data captured throughsurveys, reporting and testing are not at par with the actual information on the infections in the population. For new pathogens,whose transmission and infectious behavior (e.g., symptoms, detection, etc.) are not well established or known, the ambiguitycontributes to the error in estimating the actual number of infections and its temporal characteristics. In most countries, the testprotocol and reporting of results are quite erratic and thus the gap between the actual infections and the measured, is beyond thescope of modeling. Under these circumstances, almost all of the modeling efforts for COVID-19 are thus, not accurately able toestimate or predict the actual number of infections. In our study, we do not attempt to match the actual number of reportedcases as well.In our approach, we have constructed a framework that takes into account the spatio-temporal complexity of the populationand its contact characteristics. Our study, for the case of India, makes use of the available Census data for extracting social,demographics and the spatial population features, along with hypothesis driven modeling of spatially hierarchical organizationof population movement and contact. This enables us to monitor the dynamics of the epidemic through contact at variousspatial scales. Using our microscopic, agent based model of the COVID-19 epidemic for Indian population, we demonstrate our esults for sample populations at the level of ‘states’, the basic federal administrative units of India. However, the frameworkcan be used to focus small populations in municipalities. On the other hand, the model can be scaled up to the level of thecountry using appropriate modifications due to realization of multiple ‘states’. The simple, toy model approaches help inunderstanding the gross behavior of the epidemic and its states, but mostly fail to answer detailed questions related to thepopulation, demographic and spatial features. Although our detailed model contains many parameters, most of them can betuned and validated with real data, whenever and wherever they are available. In that way, our approach provides a possibilityof answering several fine grained questions.We demonstrated how the framework can be used to compute the fraction of cases tested positive in a sample populationwhose contact behavior is regulated by the restrictions in movement due to imposed lockdowns. The proposed frameworkalso allows to test hypothetical lockdown protocols to study the time evolution of densities of epidemic states. In fact, wedemonstrated the use of three different lockdown strategies for a particular state of India. To study this, we utilized the fractionof infections I ( t ) , the reproduction rate ρ ( t ) and also reinforcement learning to decide which of the districts should come underlockdown restrictions. Our results show that these three methods can be utilized to dynamically decide lockdown in variousdistricts and thereby restrict the spread of infections. However, from the point of view of socio-economic impact of lockdowns,one can compare these protocols as well as new ones to decide which can be deployed in a real population. Regarding thefeasibility as well as practicality of switching lockdowns on and off at a rapid rate, our results merely demostrate a possibility,which can be modified according to administrative capabilities. In a realistic scenario, lockdowns may not be implementedwith a possibility of changing state at the scale of a single day. In addition, our model formulation is flexible in answeringfeature related queries, as well as adapting to new features which may be added for the population, spatial information as wellas disease characteristics. References Pellis, L. et al.
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Centers for Dis. Control. Prev. , DOI: 10.3201/eid2606.200320 (2020). ppendix Transition rates in the epidemic model
Rate parameters α , δ , γ , µ are respectively used in Algorithm 2 and Algorithm 4 for calculating the total infection, recoveryfrom exposure, recovery from infection and death rates respectively. The corresponding time periods are already defined as T I , T ER , T IR and T M respectively in Sec. 2.2. These rates are used to decide the transition, whereas parameters F sym and P µ are usedfor selecting eligible nodes for transition. Tuning and changing parameters during the lockdown periods
Isolation compliance for infected C I has been kept a constant throughout the simulation period as symptomatic agents havebeen following the same norms throughout. As testing rate has increased gradually in different phases of lockdowns, isolationcompliance for exposed agents C E increases with the increasing phases of lockdowns. Border seal compliance C BS decreaseswith lockdown phases as border seal policies become less stringent with passing time and inter-district mobility is allowed withvarying degrees in different phases of the lockdown. The probability of returning to home C H is least when there is no lockdownas people are free to move and stay away from their homes when there is no intervention. It increases in the first phase of thelockdown due to curbing of mobility and then decreases in the subsequent phases (constant in last three phases) as restrictionsare lifted gradually to some extent. The probability of removing an edge P re is the probability of breaking an existing edge ofan agent when it moves to the same or a different zone. This probability is very low when there is no lockdown and is highduring lockdowns as agents tend to break contacts and keep it to minimum during lockdowns. The probability of creating anedge P ce is kept high when there is no lockdown as agents move and make contacts freely. However, this probability is very lowin the first phase of lockdown due to strict social distancing norms and increases with small values in the subsequent phasesas restrictions on mobility are lifted gradually. As inter-family interactions are limited during lockdowns, the probability ofremoving edges between families is quite high during lockdowns and decreases with lockdown phases as families start movingand interacting with each other with more liberty. The probability of moving to a different district M DD is comparatively lowerduring lockdowns as agents move to other districts only if it is very urgent. Refer to Table 3 for the values of these parametersduring different lockdown phases. able 1. Parameters of network part of the model, with notations for the variables and parameters
Network Parameter Notation Remarks
Population network N Hierarchical network formed using Algorithm 1Household size distribution for a district P household India Census 2011 data Age distribution for a district P age India Census 2011 data Gender distribution for a district P gender India Census 2011 data Age-wise probability of employment P emp India Census 2011 data Fraction of essential workers F essential N zones Min families
Backtracked from state’s population size to get 20k nodes for each networkName of the district district _ name India Census 2011 data Name of the state state _ name India Census 2011 data District’s Latitude-Longitude district _ latlong Mapped from GeoPy State’s Latitude-Longitude state _ latlong Mapped from GeoPyEpidemic state of a node ep _ st Susceptible(S), Exposed(E), Infected(I), Recovered(R), Dead(D)Last update time of a node time
Day(relative to start of the simulation) when the state of agent changed lastNode’s free mobility flag is _ f ree Parameter to control node’s movement
Table 2.
Parameters of epidemic part of the model and notations for variables and parameters
Epidemic Model Parameter Notation Remarks
Initial infections I init β E Sampled on daily basis for each node from Gamma distribution, shape=8.0, scale=0.6, constant=1.2 Contact rate for infected population β I Sampled on daily bases for each node from Gamma distribution, shape=18.0, scale=0.4, constant=0.85 Adjusted case fatality ratio P µ Specific to each Indian state (Code , Data )Fraction of symptomatic nodes F sym Incubation time T I Gamma distribution, shape=5.8, scale=0.95
Recovery time from infection T RI Gamma distribution, shape=1400.0, scale=0.01 Recovery time from exposure T RE Gamma distribution, shape=25.0, scale=0.9 Mortality time from infection T M
13 days Maximum simulation days T max able 3. Parameters of model and notations for variables and parameters: Fixed day intervention
Epidemic Model Parameter Notation No Intervention Intervention 1 Intervention 2 Intervention 3 Intervention 4 Intervention 5
Intervention application dates (DD/MM) Dt Int C I C E C BS C H P re P ce P fe M DD T LB N RT Table 4.
Parameters of model and notations for variables and parameters: Interventions based on (a) Number of infections in adistrict I(t), (b) Effective Reproduction Number ρ ( t ) , and (c) Reinforcement Learning. Epidemic Model Parameter Notation No Intervention Intervention
Isolation compliance for infected C I C E C BS C H P re P ce P f e M DD N RT ρ ( t ) T max T max Table 5.
Threshold for intervention
Epidemic Model Parameter Upper threshold Lower threshold
I(t) I u = . I d = . ρ ( t ) ρ u = ρ d = . Table 6.
Mobility Matrix ( M ZZ ) for weekdays in the absence of intervention (lockdown) for all source zones Age group Time bin Destination zoneRR EE EPB EPV NM
Age Bracket 1 Morning 0.4 0.45 0.01 0.04 0.1Afternoon 0.4 0.45 0.01 0.04 0.1Evening 0.47 0.34 0.07 0.02 0.1Night 0.46 0.29 0.13 0.02 0.1Age Bracket 2 Morning 0.05 0.5 0.25 0.1 0.1Afternoon 0.2 0.45 0.15 0.1 0.1Evening 0.225 0.375 0.175 0.125 0.1Night 0.29 0.13 0.24 0.24 0.1Age Bracket 3 Morning 0.2 0.6 0.05 0.05 0.1Afternoon 0.25 0.55 0.05 0.05 0.1Evening 0.3 0.5 0.05 0.05 0.1Night 0.55 0.25 0.05 0.05 0.1 able 7.
Mobility Matrix ( M ZZ ) for weekends in the absence of intervention (lockdown) for all age brackets Source zone Time bin Destination zoneRR EE EPB EPV NMRR
Morning 0.22 0.52 0.1 0.06 0.1Afternoon 0.29 0.48 0.07 0.06 0.1Evening 0.33 0.41 0.1 0.06 0.1Night 0.43 0.19 0.16 0.12 0.1 EE Morning 0.22 0.66 0.01 0.01 0.1Afternoon 0.29 0.59 0.01 0.01 0.1Evening 0.33 0.55 0.01 0.01 0.1Night 0.43 0.45 0.01 0.01 0.1
EPB
Morning 0.225 0.05 0.62 0.005 0.1Afternoon 0.295 0.05 0.55 0.005 0.1Evening 0.335 0.41 0.15 0.005 0.1Night 0.655 0.19 0.05 0.005 0.1
EPV
Morning 0.225 0.05 0.005 0.62 0.1Afternoon 0.295 0.05 0.005 0.55 0.1Evening 0.335 0.41 0.005 0.15 0.1Night 0.655 0.19 0.005 0.05 0.1
Table 8.
Mobility Matrix ( M ZZ ) during intervention (lockdown) for all zones and all age brackets. Time bin Destination zoneRR EE EPB EPV NM
Morning 0.3 0.001 0.02 0.01 0.669Afternoon 0.1 0.001 0.02 0.01 0.869Evening 0.3 0.001 0.02 0.01 0.669Night 0.5 0.001 0.02 0.01 0.469 lgorithm 1
Network Model create _ zones _ grid ( . ) : Assigns random zones to the grid within a district. get _ distributions ( . ) : Returns the distribution of household size, age, gender and employment according to age for each districtfrom census data. get _ complete _ graph ( . ) : Create complete graph according to P household . assign _ attributes ( . ) : Assigns state, district, zone, grid and global (district) coordinates, age bucket, employment, gender,essential traveller flag (1 = essential, 0 = non-essential), epidemic state, last update time (used in epidemic modeling), mobilityflag (1 = movable, 0 = non-movable) attributes to each node of a complete family graph. connect _ f amilies ( . ) : Connect families based on preferential attachment rule. get _ gravity _ attributes ( . ) : Returns district’s size and latitude-longitude. gravity _ law ( . ) : Returns edges based on formula: size _ district i × size _ district j d H ( latlong _ district i , latlong _ district j ) ; where d H is haversine distance. get _ dis joint _ union ( . ) : Returns a disjoint union graph of all districts without edges between them. sample _ nodes ( . ) : Returns sampled nodes from two input district graphs based on the edges (calculated from gavity law)between them (with replacement). connect _ edges ( . ) : Connect the sampled nodes in the disjoint union graph. procedure create _ network ( Min f amilies , F essential , N zones , state _ name , district _ name , district _ latlong , ep _ st , time , is _ f ree ) position _ zone _ mapping = create _ zones _ grid ( N zones ) P household , P age , P gender , P emp = get _ distributions ( Min f amilies ) for complete _ graph in get _ complete _ graphs ( P household ) do f amilies = assign _ attributes ( complete _ graph , P household , P age , P gender , P emp , F essential , state _ name , district _ name , district _ latlong , position _ zone _ mapping , ep _ st , time , is _ f ree ) end for districts = connect _ f amilies ( f amilies ) state _ graph = connect _ districts ( districts ) return state _ graph end procedure procedure connect _ districts ( districts ) for district i in districts do for district j in districts do size i , size j , latlong i , latlong j = get _ gravity _ attributes ( district i , district j ) edges _ btw _ districts i , j = gravity _ law ( size i , size j , latlong i , latlong j ) end for end for dis joint _ union _ graph = get _ dis joint _ union ( districts ) for district i in districts do for district j in districts do nodes i , nodes j = sample _ nodes ( district i , district j , edges _ btw _ districts i , j ) dis joint _ union _ graph = connect _ edges ( nodes i , nodes j , dis joint _ union _ graph ) end for end for return dis joint _ union _ graph end procedure lgorithm 2 Epidemic spread model num _ time _ o f _ day : 4 divisions within a day: morning, noon, evening and night. set _ time ( . ) : Initializes incubation, recovery and mortality time for every node. rand ( a , b ) : Returns a random number between a and b . get _ exp _ in f _ nodes ( . ) : Returns exposed and infected nodes eligible for recovery. get _ exp _ nodes ( . ) : Returns exposed nodes eligible for infection. get _ scp _ nodes ( . ) : Returns susceptible nodes eligible for exposure. get _ in f _ nodes ( . ) : Returns infected nodes eligible for death. get _ lockdown _ f lags ( . ) : Returns district-wise lockdown flags on the basis of the current network state. α (infection rate) = 1 / T I γ (recovery rate from infection) = 1 / T RI δ (recovery rate from exposure) = 1 / T RE procedure Gillespie
SEIRD ( N , I init , β E , β I , α , δ , γ , µ , T I , T RI , T RE , T M , C I , C E , C BS , P re , P ce „ M DD , M ZZ , T max ) num _ time _ o f _ day = N = set _ time ( N , T I , T RI , T RE , T M ) N , in f ected _ nodes = set _ initial _ in f ections ( N , I init ) exposed _ nodes = get _ exposed _ nodes ( N ) nw _ params = get _ nw _ params ( N , exposed _ nodes , in f ected _ nodes , β E , β I , α , δ , γ , µ ) day _ count = 0 nw _ tra jectory = [] while day _ count < T max and nw _ params . tot _ rt > do day _ count + = time _ o f _ day = while time _ o f _ day < num _ time _ o f _ day do time _ o f _ day + = ct = get _ current _ time ( day _ count , time _ o f _ day ) if not ( day _ count == 1 and time _ o f _ day == 1) then N = mobilize ( N , day _ count , time _ o f _ day , C I , C E , C BS , P re , P ce , M DD , M ZZ , lockdown _ f lags ) nw _ params = get _ nw _ params ( N , exposed _ nodes , in f ected _ nodes , β E , β I , α , δ , γ , µ ) end if r = rand ( , ) P R = nw _ params . tot _ rec _ rtnw _ params . tot _ rt P I = nw _ params . tot _ in f _ rtnw _ params . tot _ rt P E = nw _ params . tot _ exp _ rtnw _ params . tot _ rt P D = nw _ params . tot _ mor _ rtnw _ params . tot _ rt if r < P R then exp _ in f _ nodes = get _ exp _ in f _ nodes ( P µ , F sym , exposed _ nodes , in f ected _ nodes ) for ei in exp _ in f _ nodes do if ei . ep _ st == I and ct − ei . time > = T RI ( ei ) then N . ei . ep _ st = R N . ei . time = ct else if ei . ep _ st == E and ct − ei . time > = T RE ( ei ) then N . ei . ep _ st = R N . ei . time = ct end if end for else if P R < = r < P R + P I then exp _ nodes = get _ exp _ nodes ( F sym , nw _ params . in f _ prob _ mapping ) for e in exp _ nodes do if ct − e . time > = T I ( e ) then exposed _ nodes . remove ( e ) in f ected _ nodes . append ( e ) N . e . ep _ st = I N . e . time = ct end if end for else if P R + P I < = r < P R + P I + P E then scp _ nodes = get _ scp _ nodes ( nw _ params . exp _ prob _ mapping ) for s in scp _ nodes do exposed _ nodes . append ( s ) N . s . ep _ st = E N . s . time = ct end for else in f _ nodes = get _ in f _ nodes ( P µ , in f ected _ nodes ) for i in in f _ nodes do if ct − i . time > = T M ( i ) then in f ected _ nodes . remove ( i ) N . i . ep _ st = D N . i . time = ct end if end for end if lockdown _ f lags = get _ lockdown _ f lags ( N ) nw _ tra jectory . append ( N ) if nw _ params . tot _ rt < = then break end if end while if nw _ params . tot _ rt < = then break end if end while return nw _ tra jectory end procedure lgorithm 3 Mobility model isolate _ in f ected ( . ) : Isolates infected nodes with a compliance probability C I and sets is _ f ree attribute to 0. isolate _ exposed ( . ) : Isolates exposed nodes with a compliance probability C E and sets is _ f ree attribute to 0. isolate _ dead ( . ) : Isolates dead nodes and sets is _ f ree attribute to 0. seal _ borders ( . ) : Disconnects districts on the basis of lockdown _ f lags with a compliance probability C BS . contact _ tracing _ and _ isolation ( . ) : Traverses and tests the neighbourhood of infected agents and isolates them if tested positive. get _ lckdwn _ f lag ( . ) : Returns lockdown flag for the current district. get _ mat ( . ) : Returns the appropriate matrix with movement probabilities on the basis of day of week, current time and lockdown. get _ target ( . ) : Returns the target district and zone on the basis of mvmnt _ mat and agent features like employment status etc. Italso ensures that at night, agents return to their respective residential districts with probability C H and inter family edges areremoved with a probability P f e . update _ edges ( . ) : Updates network interactions on the basis of target district and target zone. In case of short range move-ment(within the same zone of the same district), single edge is removed with P re and another is created with P ce . Otherwise, alledges are removed with P re and a single edge is created in the target area with P ce . No changes are made if the node does notmove. update _ spatial _ con f ig ( . ) : Updates latitude-longitude and zonal coordinates as per the movement. procedure mobilize ( N , day _ count , time _ o f _ day , C I , C E , C BS , P re , P ce , M DD , M ZZ , lockdown _ f lags ) ct = get _ current _ time ( day _ count , time _ o f _ day ) N = isolate _ in f ected ( N , C I ) N = isolate _ exposed ( N , C E ) N = isolate _ dead ( N ) N = seal _ borders ( N , C BS , lockdown _ f lags ) N = contact _ tracing _ and _ isolation ( N , T LB , N RT ) for node in N . nodes do if node . is _ f ree then if node . is _ essential _ service _ provider then mvmnt _ mat = get _ mat ( node , ct , , M DD , M ZZ ) else lckdwn _ f lg = get _ lckdwn _ f lag ( lockdown _ f lags , node . curr _ dist ) mvmnt _ mat = get _ mat ( node , ct , lckdwn _ f lg , M DD , M ZZ ) end if target _ dist , target _ zone = get _ target ( node , mvmnt _ mat , ct , C H , P f e ) N = update _ edges ( node , target _ dist , target _ zone , N , P re , P ce ) N = update _ spatial _ con f ig ( node , target _ dist , target _ zone , N ) end if end for return N end procedure lgorithm 4 Get Network Parameters num _ neighbors ( . ) : Returns number of neighbors of a node in a specific epidemic state. procedure get _ nw _ params ( N , exposed _ nodes , in f ected _ nodes , β E , β I , α , δ , γ , µ ) nodes _ at _ risk _ o f _ exposure = [] neighbors _ o f _ exposed = N . neighbors ( exposed _ nodes ) neighbors _ o f _ in f ected = N . neighbors ( in f ected _ nodes ) nodes _ at _ risk _ o f _ exposure . append ( neighbors _ o f _ exposed ) nodes _ at _ risk _ o f _ exposure . append ( neighbors _ o f _ in f ected ) exp _ prob _ mapping = dict () tot _ exp _ rt = 0 for node in nodes _ at _ risk _ o f _ exposure do er = β E × num _ neighbors ( node , E ) + β I × num _ neighbors ( node , I ) exp _ prob _ mapping [ node ] = ertot _ exp _ rt + = er end for in f _ prob _ mapping = dict () tot _ in f _ rt = 0 for node in exposed _ nodes do in f _ prob _ mapping [ node ] = α tot _ in f _ rt + = α end for tot _ rec _ rt = γ × len ( in f ected _ nodes ) + δ × len ( exposed _ nodes ) tot _ mor _ rt = µ × len ( in f ected _ nodes ) tot _ rt = tot _ exp _ rt + tot _ in f _ rt + tot _ rec _ rt + tot _ mor _ rt return nw _ params ( exp _ prob _ mapping , in f _ prob _ mapping , tot _ exp _ rt , tot _ in f _ rt , tot _ rec _ rt , tot _ mor _ rt , tot _ rt ) end procedure TG TN UP WB MH MP OR RJ GJ HR JK KA AP BR CH DL INDIA
Figure 8.
The evolution of the effective reproduction rate ρ ( t ) with time for some states of India as well as the whole country(shown until 31/07/2020). ρ ( t ) calculated using code from https://github.com/k-sys/covid-19/blob/master/Realtime%20R0.ipynb and statewise India datafrom Covid19india API. igure 9. The simulated connected network of families in a single residential zone of
Mumbai suburban district inMaharashtra during the epidemic on (a) day 6 (b) day 26 and (c) day 75 at night time. Some nodes are absent and new nodesare present due to mobility across zones and districts. The colors depict the epidemic states of the individuals. Susceptible S(cyan), Exposed E (blue), Infected I (red), Recovered R (green), Dead D (grey). This is one of the simulated networks for thedata presented in Fig. 4.district inMaharashtra during the epidemic on (a) day 6 (b) day 26 and (c) day 75 at night time. Some nodes are absent and new nodesare present due to mobility across zones and districts. The colors depict the epidemic states of the individuals. Susceptible S(cyan), Exposed E (blue), Infected I (red), Recovered R (green), Dead D (grey). This is one of the simulated networks for thedata presented in Fig. 4.