key: cord-0992347-a7vypgby
authors: Jithesh, PK
title: A model based on cellular automata for investigating the impact of lockdown, migration and vaccination on COVID-19 dynamics
date: 2021-09-08
journal: Comput Methods Programs Biomed
DOI: 10.1016/j.cmpb.2021.106402
sha: f8c6aa348dbdd83be8860e76b8025d4a267ba54d
doc_id: 992347
cord_uid: a7vypgby

Background and Objective COVID-19 pandemic continues unabated due to the rapid spread of new mutant strains of the virus. Decentralized cluster containment is an efficient approach to manage the pandemic in the long term, without straining the healthcare system and economy. In this study, the objective is to forecast the peak and duration of COVID-19 spread in a cluster under different conditions, using a probabilistic cellular automata configuration designed to include the observed characteristics of the pandemic with appropriate neighbourhood schemes and transition rules. Methods The cellular automata, initially configured to have only susceptible and exposed states, enlarges and evolves in discrete time steps to different infection states of the COVID-19 pandemic. The transition rules take into account the probability and proximity of contact between infected hosts and susceptible individuals. A transmittable and transition neighbourhoods are defined to identify the most probable individuals infected from a single host in a time step. Results The model with novel neighbourhood schemes and transition rules reproduce the macroscopic behaviour of infection and recovery observed in pandemics. The temporal evolution of the pandemic trajectory is sensitive to lattice size, range, latent and recovery periods but has constraints in capturing the changes in the infectious period. A study of lockdown and migration scenarios shows strict social isolation is crucial in controlling the pandemic. The simulations also indicate that earlier vaccination with a higher capacity and rate is essential to mitigate the pandemic. A comparison of simulated and actual data shows a good match. Conclusions The study concludes that social isolation during movement and interaction of people can limit the spread of new infections. Vaccinating a large proportion of the population reduces new cases in subsequent waves of the pandemic. The model and algorithm with real-world data as input can quickly forecast the trajectory of the pandemic, for effective response in cluster containment.

• Lattice size is varied to accommodate incoming susceptible or infectious population.

• Sensitivity analysis of model parameters is presented.

• Peak and duration of active infections under different scenarios are studied.

healthcare system and economy.

In this study, the objective is to forecast the peak and duration of COVID-19 spread in a cluster under different conditions, using a probabilistic cellular automata configuration designed to include the observed characteristics of the pandemic with appropriate neighbourhood schemes and transition rules.

The cellular automata, initially configured to have only susceptible and exposed states, enlarges and evolves in discrete time steps to different infection states of the COVID-19 pandemic. The transition rules take into account the probability and proximity of contact between infected hosts and susceptible individuals. A transmittable and transition neighbourhoods are defined to identify the most probable individuals infected from a single host in a time step.

The world is facing an unprecedented challenge. The challenge, Coron- Stringent implementation of healthcare advisories from WHO, strategies such as national lockdowns [2, 3] and effective interventions by Governments in testing, isolating and treating the infected aided to control the spread of the virus in its first wave [4] [5] [6] . Nevertheless, the viral pandemic affected human life and the global economy in an adverse manner [7] . The relentless efforts of the scientific community in developing vaccines for the disease is finally a success and vaccination programs are taking place all over the world [8, 9] .

Strategic planning and efficient deployment of resources is crucial in controlling this pandemic without overwhelming the healthcare system or further affecting the economy [10, 11] . This requires information on future demands of medical equipment and additional infrastructure that may be needed [12] .

Computational models can provide valuable insights in this regard by forecasting the growth trends of the pandemic under different circumstances [13] [14] [15] . Possibility for repeated and cost-effective parametric studies in a virtual environment, flexibility to include and adapt to real-time data and simulation in a short duration, are some of the other advantages of modeling.

Mathematical modeling of infectious diseases is a mature area with many well-known methods [16, 17] and there have been several efforts in modeling COVID-19 too [18, 19] . Simple deterministic compartment models are a popular approach in simulating evolving pandemics [20] . In this approach, the entire population is divided into distinct compartments at a particular instant of time, based on infection status. The simplest compartment model is SIR model, which consists of three states, S for susceptible individuals, I for infected and R for recovered individuals [21] . Additional compartments can be added to the model to simulate other infection states observed in a pandemic. These are either formulated and computed as ordinary differential equations or methods such as Cellular Automata (CA) approach are used for the simulation [22] [23] [24] .

COVID-19 exhibits different transmission characteristics compared to other infectious diseases [25] and mathematical models have been successful in identifying relevant parameters and reproducing observed disease dynamics [26] [27] [28] [29] [30] . CA configurations, with their flexibility for defining heterogeneous states and local rules for state transitions based on neighbourhood information, are ideal for simulating epidemics [31] [32] [33] . This method has been successfully used for predicting long term behaviour of COVID-19 [34] and also to understand the effect of lockdown measures in a region [35, 36] .

In a recent study, Ghosh and Bhattacharya [37] , used Probabilistic Cellular Automata (PCA) with simulation parameters optimized using a sequential genetic algorithm to forecast COVID-19 dynamics. This combination is shown to accurately estimate the time trajectory of COVID-19 when physically meaningful parameters are used. In this CA configuration, the initial population with different disease states are distributed in a 2D lattice of fixed size. In another work, the same authors incorporated factors like population density, testing efficiency and movement restriction into a PCA model through appropriate probabilities to explain the variability of disease dynamics observed in different countries [38] .

In another important work, Schimit [39] , studied the impact of social isolation on dynamics of COVID-19 using PCA and ordinary differential equations. The model consists of eight disease states and fifteen parameters and uses an extended neighbourhood to establish random contact networks between cells. Each state transition is based on either probabilities or predefined periods to account for the uncertainties in COVID-19 propagation.

The results suggest that maintaining social isolation is crucial to keep the pandemic spread under control and to avoid the overwhelming of the healthcare system. The model is formulated for a country, with a fixed lattice size of the order of 210 million and the simulation is reported to take over a week to complete.

A spatiotemporal epidemiological forecast model for short and long term local infection risk predictions for smaller regions such as counties is proposed by Zhou et al. [40] . The work shows that, by using local information in the model through a CA which enables interconnectivity and regional variations, the prediction error is significantly reduced compared to that for a larger space. Dai et al. [41] predict the spread patterns of COVID-19 in cities using a CA model that incorporates factors such as sex ratio, age, immunity and various disease characteristics. The movement of the population is simulated by defining moving proportion and a maximum moving step length in a fixed lattice size with occupied and empty cells.

Decentralised response through cluster containment or regional lockdowns is an effective approach to manage the pandemic in a short period [42] . Realtime surveillance of small geographical clusters called micro-containment zones enables error-free data collection and analysis, followed by rapid decision making and interventions with minimum disruption to the overall econ-omy. Local administration can mobilise their resources to support the community during the containment and also to implement the find, test, trace, isolate and support system [4] , successfully carried out to control the spread of the pandemic in its first wave. Additionally, in cluster containment, the size of the susceptible population, which is a crucial parameter in determining the spread of the pandemic [12] is small and can be easily monitored.

The computational model in the present work is developed based on the idea of monitoring a susceptible population in a cluster to mitigate the pandemic. The susceptible population can be people in a geographical region, or a subgroup in the region identified through the process of contact tracing and kept in quarantine, or people in a particular age group or gender [43] . The size of the susceptible population is not constant but can change during the spread of the pandemic or migration of people. In the literature, most studies on modelling COVID-19 using CA have used a fixed lattice size with human-occupied and empty cells. We propose to use a variable size lattice, initially with only susceptible and infected cells in it, which increases in size to accommodate the incoming susceptible or infectious population. The discrete-time step used in the simulation is 1 day. Hence, the state value v of a cell also indicates days passed after that cell became infected. The 

updating of state value is stopped when v = t R , the recovered state. It is assumed that the vaccinated people are immune to infection and all infected people are recovered in due course of time.

A cell, with a particular state assigned to it at time t, is denoted by

There will be a corresponding global state configuration of CA denoted by L t = {x t ij }, which is an array of states of all cells at time t. The most probable S state cells, in L t , to which virus transmission can happen in a particular time step are identified by the procedure described further.

The neighbourhood of a cell is a set of its surrounding cells whose states influence the evolution of that cell. The state x t+1 ij of a cell at time t + 1 is a function of its own state and that of its neighbourhood cells N ⊂ Z 

Transmission of the SARS-CoV-2 virus is reported to happen when there is direct, indirect or close contact between infected persons and others. Testing, identifying and isolating infected people is an important step in breaking the chain of virus transmission [46] . The computational model thus should include parameters accounting for local interaction of people for realistic forecasting of the disease dynamics [25, 47] . Two parameters, namely, social isolation factor for the locality and social isolation factor for the individual, are defined to account for the possibility of movement and contact of people.

These factors represent the probability of virus transmission and bring in the stochasticity of COVID-19 dynamics into the model.

The imposed restrictions to movement in a region is represented by social isolation factor for locality λ l , with λ l = 0 for unrestricted movement and λ l = 1 for total restriction. The value of λ l is the same for the entire lattice and is kept constant for a particular duration of time such as the lockdown period. The movement of an individual in a region depends on the receptiveness of restrictions and regulations by the individual. The social isolation factor for the individual λ p , which is unique and time-dependent accounts for this.

The value of λ p is randomly generated in the time step and has a range of 0 to 1, with 0 representing total compliance and 1 representing no compliance to restrictions. For an infected individual, the condition λ p < λ l , indicates that the person complies with imposed restrictions and virus transmission will not happen from that individual. Whereas the condition λ p ≥ λ l , indicates the possibility of contact between an infected individual and susceptible people, and a high chance of transmission.

If the condition λ p ≥ λ l is satisfied for an infected cell, then the procedure of identifying exposed cells in its neighbourhood involves a two-step process.

Both these steps are sequentially executed in each time step. In the first step, the transmittable neighbourhood of the infected cell is examined and a finite set {C ij } with all potential S state cells is created. The elements of the set are a tuple, (i, j, 8 1 v) ∈ {C ij }, which contains indices i, j of each S state cell and a sum of state values of its transition neighbourhood 8 1 v. The sum of state values is an indication of the proximity and duration of local interaction between potential and all infected individuals in its transition neighbourhood.

{C ij } can be an empty set or can have a maximum cardinality of 4r(r + 1) in time t. An empty set results when no susceptible cells are available in the transmittable neighbourhood. This is the second condition, after λ p < λ l , that leads to a stochastic outcome of the model.

In the second step, R t number of cells in {C ij } are marked as infected by assigning v = 1, starting from the cell with the highest value of 8 1 v and proceeding in descending order. The number R t represents the maximum number of transmission from a single host per time step. The value of R t can be kept as constant or randomly selected from a set of numbers in each time step. Even though R t could range from 1 to cardinality of {C ij }, selecting a higher value will result in a sudden and unrealistic spread of infection in few time steps. Hence, in the present study a constant value of R t = 2 is used.

It may be noted that R t is not the same as the basic reproduction number R o which represents the average number of secondary infections caused by an infected individual introduced to a susceptible population [34, 48] .

The global state configuration L t of the CA is updated in discrete time steps as a sequence of mappings F : L t → a, a ∈ A, where F represents the local transition function. The transition rules can be formulated as follows,

• A cell in an infectious state infects a maximum of R t susceptible cells in its transmittable neighbourhood when social isolation factor λ p ≥ λ l .

• The R t number of susceptible cells are identified in decreasing order of their sum of state values.

• The value of any state other than susceptible is updated sequentially

In terms of assigned state values v, the transition rules can be written as Equation (1) .

Here, {C s ij } ⊂ {C ij } has R t number of elements with maximum values of 8 1 v. It may be noted that only the transition from S to E state takes place as per the procedure explained in Section 2.3. All other mappings (Fig.1) , is used in the simulation to handle evolution at lattice boundaries. The mapping can be mathematically represented as Equation (2). 

The probabilistic cellular automata model presented above is tested with a fixed susceptible population to see if the model can reproduce the macroscopic behaviour of a pandemic propagation. It is assumed that the entire susceptible population is infected and recovered in due course. Further, a parametric study assesses the qualitative response of the model to changes in input parameters. Table. 2 shows the details of parameters used in the simulations.

The simulation is initiated with one infected cell placed at the center of the lattice and the temporal evolution of the PCA is shown in Fig.2a The number of active cases for the 49 th day obtained from all 1000 simulations is shown in Fig.2b to analyse the distribution of stochastic outcome from the model. The data exhibits a normal distribution with a mean of 5049 and has a standard deviation of 24. The corresponding error estimate based on 95% confidence interval is below 1% of the mean value. 

The effect of CA configuration on the time trajectory of the pandemic is studied first. The lattice dimensions (size of the susceptible population) and the range r of the transmittable neighbourhood are two parameters of the CA that influence the peak and duration of the pandemic. attributed to this constraint in the present model. Figure. 4b also shows the effect of average recovery time of the disease, t R , on rate of propagation.

Clearly, when t R = 21, the peak of active cases is high compared to that for t R = 7 or 14. In the model recovery period t R is user-defined, and hence it is easy to adapt the value reported by authorities.

Control measures such as lockdowns, steps such as relaxing restrictions and allowing migration of people to revive the economy and resume daily life and mitigation strategies such as vaccination roll out are some of the vital phases observed in the course of the COVID-19 pandemic. The developed model is used to carry out a parametric study to simulate these phases and the results are discussed in this section.

Lockdowns aim to prevent the movement and interaction of people and thus break the chain of the transmission cycle. As described in Section 2.3, the social isolation factor for the locality λ l , is the parameter that indicates the intensity of lockdown in that region. In the simulations, the value of λ l is 0 at the start, and on the day of lockdown it is changed to a higher value Previous studies show that a high percentage of reduced contacts through social isolation limits the spread of new cases [39, 47] . The timing of enforcing lockdown restrictions is also crucial, with earlier implementations leading to lesser infections [3, 5] . The present study also agrees with these results. In lockdown situations, if movement and interaction of people are there, then the infection will spread gradually with a smaller peak but for a longer duration.

The healthcare system may not collapse but might exhaust the resources and people involved in it. Aggressive lockdown can curb the curve in a short span without infection spreading to a large section of the population.

Different non-pharmaceutical interventions have varying effects on disease transmission [6] , and the present model can provide initial estimates of such interventions by choosing appropriate values of social isolation factors.

Migration changes the size of the population and the dynamics of the pandemic. In a closely monitored and controlled region, the number of people migrating by various modes of transportation is accurately registered. The size of the cellular automata is incremented by a value δ in each time step to account for the migrating population. It is assumed that people are only migrating into the region and not moving out, so the size of the lattice only increases. In the migrating population, susceptible as well as exposed (infected) people will be present. Exposed states are assigned proportionally and distributed randomly in the lattice in each time step to account for this. We studied two factors of importance in allowing migration in a region.

First, the day on which migration can be started in a region with lockdown restrictions and second, the relaxations in lockdown that can be allowed during migration. Figure The results are in line with the observations of Sirakoulis et.al [32] , that movement of people increases the overall infected population. A second surge is unavoidable during migration, but delaying the process can limit the peak of active cases. The spread of infections is also closely linked to the interactions between people as described in Section.4.1. The surge in infections observed in delayed migration is due to this fact. Thus, the results infer that migration should be allowed only under strict lockdown conditions so that the peak of active cases is limited to a manageable level.

Universal immunisation is the scientific solution to control and mitigate R

The logistic function is shown in Equation (3) The present model is also used to investigate the peak and duration of a second wave of infection in a partially vaccinated population. An immune state V is assigned for vaccinated people in the initial configuration of the lattice. As shown in Fig.9 , when the percentage of the immune population is increased from V = 15% to 30% and 45%, the peak of the active infections is seen to reduce significantly. The decline in susceptible population and Initially, people received the COVID-19 vaccination campaign with hesitancy due to various concerns [9] , and now there is a shortage of supply. The model can forecast the overall impact of the vaccination program and prioritise population subgroups for effective inoculation.

To assess the performance of the computational model and to validate the results, a comparison of the simulated and actual pandemic propagation data for the state of Kerala, India, is done. The first case of the COVID-19

pandemic in India is reported in Kerala on 30 January 2020. The data for 237 days from 9 March 2020, when the first wave of infections started, to 31

October 2020 is used for comparison [49] . A statewide lockdown is declared from 23 March 2020, and after lifting of restrictions in phases, notable migration to the state started on 7 May 2020, which are equal to 15 th and 60 th day respectively from 9 March 2020. From day 60, the size of the lattice is proportionally increased (m = m + δ)

per time step to account for the change in susceptible population. The additional exposed cases introduced due to migration is taken as 4% of the population. Figure. 

The Simulations using the present model, with minimum parameters and computational requirements, can provide valuable insights into evolving disease dynamics in a cluster for effective decision making and quick response. Improvements by including fluctuations in migrating population, using scientifically estimated social isolation factors and testing different vaccination scenarios with real data will enable the model to forecast the long-term behaviour of the pandemic.

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