††thanks: Authors contributed equally to this research

# TinyFedTL: Federated Transfer Learning on Tiny Devices

Kavya Kopparapu *
Department of Computer Science
Harvard University
Cambridge, MA, USA
<EMAIL_ADDRESS>
&Eric Lin *
Department of Computer Science
Harvard University
Cambridge, MA, USA
<EMAIL_ADDRESS>

###### Abstract

TinyML has rose to popularity in an era where data is everywhere. However, the
data that is in most demand is subject to strict privacy and security
guarantees. In addition, the deployment of TinyML hardware in the real world
has significant memory and communication constraints that traditional ML fails
to address. In light of these challenges, we present TinyFedTL, the first
implementation of federated transfer learning on a resource-constrained
microcontroller.

_Keywords_ TinyML $\cdot$ Federated Learning $\cdot$ Microcontrollers $\cdot$
Transfer Learning $\cdot$ Privacy $\cdot$ Machine Learning

##  1 Introduction

Our presentation is available at this link.

In recent years, the emphasis on data privacy has grown in the wake of several
privacy scandals and information leaks. More than ever, individuals are
concerned with who has access to their personal data and where it is being
shared. Most current successful machine learning methods, for both the Tiny
domain and not, benefit from large and diverse datasets, so this rise in
concern regarding digital privacy appears to come at the cost of progress.

In addition, in many applications there is a significant need for distributed
learning agents that operate in an environment with significant communication
costs and minimal on-device storage. These distributed computing setups have
even taken hold in consumer-facing applications such as the Amazon Go store.
Tiny Machine Learning is a rapidly growing field at the intersection of
embedded systems and machine learning, allowing significant insights, data
collection, and algorithmic development that was not previously possible.

Put together, the opportunity is twofold: data from several sources can no
longer be consolidated for a singular learner to access due to privacy
concerns and low-cost learning devices on the edge need a new method to
aggregate shared insights that doesn’t require large on-device memory and
constant communication to a central server. Thus, we see the field of TinyML
is ripe for applications of privacy-preserving machine learning. However,
frameworks like Tensorflow Lite do not currently support model training on-
device, but rather enable the deployment of static models for inference [1].

In this work, we contribute:

  * •

The first implementation of federated learning on a microcontroller.

  * •

A method of deploying transfer learning on a resource-constrained
microconroller without growing storage costs as the number of training
examples increases.

  * •

The method of federated transfer learning for the CIFAR-10 benchmark doesn’t
necessitate a dense layer with pre-trained weights, meaning devices can be re-
trained to different types of classification problems.

  * •

Implemmentation of federated transfer learning on an Arduino microcontroller.

  * •

Identification of challenges and limitations encountered in the process of
training on-device with a federated learning framework.

##  2 Related Work

###  2.1 TinyML

Previous work in TinyML is focused on optimizations to compress models and
reduce the inference latency but does not aim to allow continuous learning,
especially in a privacy-preserving manner. Prior research in deploying
learning on-device in the TinyML domain has included simple NN classifiers
like k-Nearest Neighbors to transfer learn on-device [2]. However, such a
classifier scales poorly with more training examples and does not follow
federated learning privacy guarantees, as information about the input to a
network can be generated from the embedding generated without the final dense
layer. Therefore, such an implementation is essentially infeasible on a
hardware platform such as the Arduino Nano.

###  2.2 Federated Learning

The domain of Federated Learning has been a hot research topic in recent
years. The topic first emerged in late 2015, with the seminal paper by McMahan
et al. of Google AI detailing their novel approach and proposing a concrete
“Federated averaging” algorithm [3]. The introduction of this algorithm
presented a solution to several bottlenecks concerning mobile devices and
privacy constraints. By distributing the data across edge devices, user
privacy can be respected while learning models can be conducted on the
aggregate updates on the collection of devices [4]. There has been continued
work in addressing problems in Federated Learning, including non-iid data
distribution, attacks, and communication cost [5, 6, 7].

###  2.3 FL for TinyML

Due to the fact that both fields are relatively new, there has been little
work conducted in the intersection of tinyML and Federated Learning. There has
been some work in the implementation of Federated Learning in mobile edge
network and in selectively updating parts of large networks to make transfer
learning more feasible but none have integrated into a significantly
constrained device, but instead onto a Raspberry Pi [8, 9].

##  3 Methods and Experiments

###  3.1 Transfer Learning Task

In order to turn a CNN into a feature extractor, we removed the final fully
connected (dense) layer, so the output of the CNN was a feature vector $n$
units long. We took the weights for this feature vector off-the-shelf, meaning
we did not train the model on any specific task but rather kept the weights
from a large dataset that were used to generate meaningful features (in this
case, the dataset was either ImageNet or Visual Wake Words) [10, 11]. We then
used the features extracted by the CNN extractor to be the input into a fully
connected layer, whose outputs were then put through a softmax to get class
probabilities. Training the weights of this FC on-device later but not the
feature extractor (which is set aside as constant) is the task.

A typical benchmark for Federated Learning is the CIFAR-10 dataset, which
consists of $32$x$32$ images in 10 different classes. These images were
upscaled to $96$x$96$ and turned into two binary classification problems: dog
versus no-dog and cat versus no-cat. The dog classification problem was
treated as the original problem (for optionally pre-training the the dense
layer of the off-the-shelf model trained on ImageNet or Visual Wake Words) and
the cat classification problem was treated as the transfer problem.

###  3.2 Hardware

The microcontroller of choice for these experiments was the Arduino Nano 33
BLE Sense. The BLE Sense is the hardware of choice for many TinyML
applications due to its variety of sensors (including temperature, pressure,
humidity, light, color, and more) and interface with the Arduino IDE.

The Arduino federated learning implementation also used a 5MP Arducam to
collect real-world images as local training data. A macbook simulated a global
server by communicating with the Arduino through a serial port. Finally,
unlike other approaches for on-device learning, no SD card or external storage
was used in simulations. This meant that all training and data storage had to
take place in the 1MB of flash memory and 256KB of SRAM, further constraining
the memory capacity.

###  3.3 Model Transfer Learning On-Device

Due to the incompatibility of the current version of Tensorflow Lite Micro
with on-device training and update techniques, we implemented our own fully
connected (FC) layer inference and backpropogation update in C++.

###  3.4 Arduino Federated Learning Implementation

The Federated Averaging algorithm, the gold standard for enabling federated
learning across distributed devices, is shown in Algorithm 1. This algorithm
is the basis for our TinyFedTL system.

Algorithm 1 FedAvg Algorithm [3]

Input: Devices $i=1,...,N$

for epoch $t=1,2,\dots,T$ do

for $i\in\textit{N}$ do

Device $i$ trains all models $m$ on its local data for $E$ local episodes

end for

for $m=1,2,\dots,M$ do

$\textit{w\\_avg}=\text{AverageWeights}(i\text{ s.t. }c_{m}^{(i)}\neq 0)$

Learner updates model $m$ with w_avg

end for

Evaluate models with global validation data

end for

Federated Learning in the real-world likely only sees a given example or data
point once. This is notably different than normal federated (or not) training
schemes, in which training examples are reused over epochs of training.
Furthermore, theoretical federated learning schemes push and send weight
updates on a carefully scheduled and consistent basis. This is simply
infeasible in many applications of tinyML edge devices where unstable network
connection may prevent comunication for long stretches of time. Moreover, data
input is not uniform – changing environments may result in some days where a
certain edge device sees many pieces of new data, while other devices see
none.

Our setup mirrors real-world scenarios as described above. Data available to
the model is different for every epoch of training and collected through the
Arducam. Furthermore, the arduino continuously sees and trains on new data
until it is contacted by the global server to update its weights. When the
server does initiate contact, model weight and bias updates have to be sent
and read byte by byte via the serial port between the arduino and the server.
The batch size on our real-world implementation is 1 (instantaneous inference
and model update).

##  4 Results

###  4.1 Models

We implemented two versions of MobileNetV2 for our task: the Tensorflow built-
in MobileNet compressed with a factor of $alpha=0.35$ (hereby referenced as
tf-mobilenet) pretrained on ImageNet and the MobileNet from the TinyML Perf
Benchmark (hereby referenced as perf-mobilenet) [10, 12]. The compressed tf-
mobilenet has more parameters than perf-mobilenet has x parameters. Both
models were frozen and the input and outputs were quantized using post-
training 8-bit quantization to interface correctly with the microcontroller.

Table 1: Deployed Model Descriptions Model | Model Parameters
---|---
Name | Total | Trainable | Trainable | Trained on
tf-mobilenet | 412,770 | 2,562 | 410,208 | ImageNet
perf-mobilenet | 221,794 | 514 | 221,280 | Visual Wake Words

We ran into issues with transfer learning on the perf-mobilenet model.
Specifically, the embeddings generated from the network were incredibly
sparse, where only 16 of the 256 outputs were ever non-zero. This may be as a
result of encouraging sparsity during training via methods such as L2 weight
regularization so the embeddings generated are highly specific to the task it
was initially trained on (person detection through the visual wake words
dataset). If given more time, we would have re-trained the perf-mobilenet to
get more meaningful embeddings.

In our simulations, tf-mobilenet performed much better and successfully
learned the task of cat identification. However, it was too large to fit on-
device for our microcontroller. Since we were using an Arduino Nano 33 BLE
Sense with 1MB of flash memory and 256KB of SRAM, we believe that other
microcontrollers could have fit the tf-mobilenet model. Therefore, we used the
perf-mobilenet for memory and time benchmarking on our arduino and the tf-
mobilenet model in our simulations to understand performance.

###  4.2 Transfer Learning from Scratch versus Pretrained

Figure 1: Train and validation accuracy across 10 experiments comparing the
training performance of the tf-mobilenet and perf-mobilenet models. The
results represent the mean $\pm$ standard deviation across 10 experiments with
5 local episodes and a batch size of 20. The FL implementation has 2 devices.

There are two methods of transfer learning under our setup: using a FC layer
trained on a different task and changing the weights, or starting with a
randomly-initialized FC layer. Ideally, the differences would be negligible or
the randomly-initialized layer would perform better, since then a CNN-based
feature extractor could be developed and deployed once and then the size of
the dense layers (embedding size x class number) can be varied for various
types of applications with different class sizes. As we can see in Figure 1,
the differences between the randomly initialized model and the model pre-
trained on the Dog classification problem have equivalent performance in the
Cat classification problem. This is a great result, because it means that
whenever we want to switch our feature extractor to a different task to
transfer learn, we can just initialize a random FC layer, and the features
that were lost in the replacement are not useful for transfer learning.

###  4.3 Federated Learning

Just to note on the following graphs, the measure training examples seen is
across all the devices. Therefore, both devices in a federated context and
devices not in federated context are trained and benchmarked against each
other. In addition, these results were collected through simulation of the
same C++ code that was deployed on the Arudino platform: so the results are
identical to performance on-device without having to wait for training and
transfer of weighs to occur.

It’s interesting to see that accross all our graphs, models did not benefit
from additional data after $\sim 3000$ training examples likely because they
were stuck in local optima. This is likely due to the simplicity in the
optimizer we used in training our model and presents significant future
opportunity.

####  4.3.1 Number of Devices

Figure 2 shows the effectiveness of federated learning. As we can see, the
validation accuracy drops as the number of devices increases, as with more
distributed data model updates may cancel each other out or progress may be
lost during the averaging step. This is as compared to the baseline of regular
(non-FL) transfer learning, shown on the graph as device number of 1.

Figure 2: The validation accuracy across epochs of training across different
number of devices participating in federated learning. The results represent
the mean $\pm$ standard deviation across 10 experiments with 5 local episodes
and a batch size of 20.

####  4.3.2 Batch Size

As we can see in Figure 3 that batch size has a significant impact on the
variability in epoch-to-epoch performance of FL. This makes sense even in a
non-FL scenario, as more data during an update better approximates the true
gradients in stochastic gradient descent and prevents over-indexing on a
certain piece of data. Ideally in a TinyML application, we would use a batch
size of 1 since inference would be instantaneous rather than having to
allocate storage and produce results once sufficient data has been collected.

Figure 3: The validation accuracy across epochs of training across different
batch sizes. The results represent the mean $\pm$ standard deviation across 10
experiments with 5 local episodes and 2 devices.

####  4.3.3 Local Episodes

The number of local episodes is defined as the number of times the model was
trained on a given batch of data. In Figure 4, it’s clear to see that the
number of local episodes leads to smaller variability in the epoch-to-epoch
validation accuracy. On the other hand, extra local episodes may lead to
overfitting to the specific epoch and has greater associated computational
costs. For this task, the increase in the number of local episodes from 5 to 6
doesn’t present a significant increase in validation accuracy or a decrease in
variability.

Figure 4: The validation accuracy across epochs of training across different
number of local episodes. The results represent the mean $\pm$ standard
deviation across 20 experiments with a batch 20 and 2 devices.

###  4.4 Performance on-Device

As stated above, the perf-mobilenet model was used to measure on-device memory
and communication cost time. Images were collected through the 5MP arducam,
converted to 96x96x3 RGB data, and trained on for 20 local epochs.

####  4.4.1 Memory

Deploying the model utilized 210 KB (80%) of dynamic memory and 657 KB (66%)
of program storage space. This excludes further memory costs associated with
communicating with the global server and encoding captured images.

An important advantage of federated learning is the ability for the device to
continuously take in new input data then discard it after it’s done training.
Since data is not transferred from the edge device to a central server, the
device only has to retain its model weights in the federated learning scheme.
Therefore, memory footprint does not increase with the number of training
samples per device. This is particularly useful in scenarios where external
data storage (even SD cards) are not available. Compared to implementations in
previous work with KNN and other similar models, we clearly see a vast
improvement in memory footprint.

####  4.4.2 Time

By picking a sparse model and writing implementation from scratch without use
of external libraries, we were able to optimize our deployment runtime. Image
capture and inference took between 8-10 seconds as a buffer from Arducam data
was processed as input to the perf-mobilenet model. The actual training
process was quite fast as it took only 214ms to go through 20 episodes of
local epochs on-device. Then, weights and bias data had to be sent between the
arduino and the global server in both directions. These were encoded as bytes,
then sent through the serial port in packets of 32 bits. The 514 floats (256x2
weights and 2 bias) often took up on average $\geq$ 6000 bytes, and thus took
over 30 seconds (one way) to upload and download.

##  5 Discussion

###  5.1 Responsible AI and Privacy

Our work has shown the possibility of utilizing a federated learning schema to
learn tasks in a tinyML domain without the need to share data with a central
server. Since data never leaves the device, this opens up avenues for a
multitude of applications that have been thus far hesitant to adopt machine
learning techniques due to privacy concerns. For instance, in the healthcare
context, patient wearables and sensors can continuously assess situations and
learn without the risk of privacy infringement for the user. This opens up
data in many new fields to be utilized to improve quality of life and user
experience without sacrificing privacy.

Moreover, we show that privacy-centric on-device transfer learning is not only
possible but also effective. In our results above, we see that tinyFedTL
performed on par with our simulations with regular learning techniques. The
federated averaging method effectively captures learnings from edge devices
through only sharing of the weights without the need for any attributes of the
data.

###  5.2 Future opportunities for On-Device Learning

There are a myriad of future directions this work can go. The first is to
implement support for better weight decay and a better optimizer to solve the
local optima issue discussed earlier. The second is more algorithmic: in order
to further compress the amount of memory needed to run the FC layer training
on-device, we could implement feature reduction methods between the embedding
from the CNN and the FC layer trained on-device. This would significantly
reduce the number of weights stored and trained, therefore also increasing
latency and allowing the storage of embeddings to form batches.

Communication cost is also an area of improvement. Although a 1-minute round-
trip communication time per update is not infeasible, this presents problems
for edge devices in remote areas with unstable wifi. There, satellites may
only have clear connection at short intervals, and packets may be dropped
without notice. Thus, possible future work will be to modify our communication
protocol for greater resiliency. Other work may take advantage of the sparsity
of the model to develop a more efficient message encoding rather than directly
translating all weight parameters into bytes. However, there are some costs
and decreases in accuracy as the number of devices grow in the federated
learning scheme, as shown above. This can be mitigated with capturing more
data (our data was limited by the number of images per class in CIFAR-10) and
also employing other federated learning techniques like hierarchical training.
In those schemas, instead of only having one global server, several
intermediary nodes are used to break up edge devices into different groups,
meaning there are fewer devices per server node. This would decrease the
likelihood of conflicting updates and help sustain accuracy.

##  6 Conclusion

We have shown a successful first-attempt at deploying on-device federated
learning. We have demonstrated the efficacy of tinyFedTL with one-shot
examples such that storage costs don’t increase as the number of training
examples increase. We have also shown that pre-trained weights aren’t
necessary for transfer learning, meaning that edge devices can be re-trained
for multiple different classification problems. Finally, our work has also
identified several challenges and future avenues of research. The lack of
precedent in this task means there is ample room for exploration in terms of
designing better optimizers and hypertuning parameters.

## References

  * [1] Robert David, Jared Duke, Advait Jain, Vijay Janapa Reddi, Nat Jeffries, Jian Li, Nick Kreeger, Ian Nappier, Meghna Natraj, Shlomi Regev, et al.  Tensorflow lite micro: Embedded machine learning on tinyml systems.  arXiv preprint arXiv:2010.08678, 2020.
  * [2] Simone Disabato and Manuel Roveri.  Incremental on-device tiny machine learning.  In Proceedings of the 2nd International Workshop on Challenges in Artificial Intelligence and Machine Learning for Internet of Things, pages 7–13, 2020.
  * [3] Brendan McMahan, Eider Moore, Daniel Ramage, Seth Hampson, and Blaise Aguera y Arcas.  Communication-efficient learning of deep networks from decentralized data.  In Artificial Intelligence and Statistics, pages 1273–1282. PMLR, 2017.
  * [4] Peter Kairouz, H Brendan McMahan, Brendan Avent, Aurélien Bellet, Mehdi Bennis, Arjun Nitin Bhagoji, Keith Bonawitz, Zachary Charles, Graham Cormode, Rachel Cummings, et al.  Advances and open problems in federated learning.  arXiv preprint arXiv:1912.04977, 2019.
  * [5] Maria Florina Balcan, Avrim Blum, Shai Fine, and Yishay Mansour.  Distributed learning, communication complexity and privacy.  In Conference on Learning Theory, pages 26–1, 2012.
  * [6] Neta Shoham, Tomer Avidor, Aviv Keren, Nadav Israel, Daniel Benditkis, Liron Mor-Yosef, and Itai Zeitak.  Overcoming forgetting in federated learning on non-iid data.  arXiv preprint arXiv:1910.07796, 2019.
  * [7] Kavya Kopparapu, Eric Lin, and Jessica Zhao.  Fedcd: Improving performance in non-iid federated learning.  arXiv preprint arXiv:2006.09637, 2020.
  * [8] Wei Yang Bryan Lim, Nguyen Cong Luong, Dinh Thai Hoang, Yutao Jiao, Ying-Chang Liang, Qiang Yang, Dusit Niyato, and Chunyan Miao.  Federated learning in mobile edge networks: A comprehensive survey.  IEEE Communications Surveys & Tutorials, 2020.
  * [9] Han Cai, Chuang Gan, Ligeng Zhu, and Song Han.  Tinytl: Reduce memory, not parameters for efficient on-device learning.  Advances in Neural Information Processing Systems, 33, 2020.
  * [10] Jia Deng, Wei Dong, Richard Socher, Li-Jia Li, Kai Li, and Li Fei-Fei.  Imagenet: A large-scale hierarchical image database.  In 2009 IEEE conference on computer vision and pattern recognition, pages 248–255. Ieee, 2009.
  * [11] Aakanksha Chowdhery, Pete Warden, Jonathon Shlens, Andrew Howard, and Rocky Rhodes.  Visual wake words dataset.  arXiv preprint arXiv:1906.05721, 2019.
  * [12] Colby R Banbury, Vijay Janapa Reddi, Max Lam, William Fu, Amin Fazel, Jeremy Holleman, Xinyuan Huang, Robert Hurtado, David Kanter, Anton Lokhmotov, et al.  Benchmarking tinyml systems: Challenges and direction.  arXiv preprint arXiv:2003.04821, 2020.

## Appendix A: High Level Description of Code

All our code, including model modifications, C++ NN and FL implementations,
Arduino modifications, and more is available at
https://github.com/kavyakvk/TinyFederatedLearning. Below is information on our
file structure, graph and result generation code, and the important files for
training (marked with a ***).

    dl
    source
    ----arduino_training_final_v3
    ------->***the .ino file has the implementation
            of our FL code for the Arduino IDE to
            compile
    ------->***python_final_script.py acts as the
            "central server" for the arduino
    ----simulation
    ------->***NeuralNetwork.cpp has our FC
            implementation and the FL implementation
    ------->***simulation.cc is the file with the
            code necessary to run our simulations.
    ------->simulation-xxx the executable that can
            be run with ./ for each of our
            experiments
    -------> the .txt files are the output from
            terminal when running the experiments
    ------->fl_simulation_analysis generates the
            .csv from the .txt
    ------->graphing.ipynb has the information to
            graph our figures from the paper
            from the .csv files
    tensorflow (no changes)
    third_party (no changes)


# Spatiotemporal patterns of Covid-19 pandemic in India: Inferences of
pandemic dynamics from data analysis.

Preet Mishra and R. K. Brojen Singh<EMAIL_ADDRESS>School of Computational
& Integrative Sciences, Jawaharlal Nehru University, New Delhi-110067, India.

###### Abstract

Modeling and analysis of the large scale Covid-19 pandemic data can yield
inferences about it’s dynamics and characteristics of disease propagation.
These inferences can then be correlated with contextual factors like
population density, effects of strategic interventions, heterogeneous disease
propagation etc, and such set of validated inferences can serve as precedents
for designing of subsequent mitigation strategies. In this work, we present
the analysis of Covid-19 pandemic data in Indian context using growth
functions fitting procedure and harmonic analysis method. Our results of
growth function fitting to the data indicate that the growth function
parameters are quite sensitive to the growth of the infected population
indicating positive impact of lockdown strategy, identification of inflection
point and nearly synchronous statistical features of disease spreading. The
harmonic analysis of the data shows the countrywide synchronous incident
features due to simultaneous implementation of control strategies. However, if
one analyzes the data from each state of the India, one can see various forms
of travelling waves in the countrywide wave pattern. Hence, one needs to do
these analysis from time to time to understand the effectiveness of any
control strategy and to closely look at the disease propagation to devise the
required type of mitigation strategies.

Keywords: Covid-19; Growth functions; Harmonic analysis; Empirical Mode
Decomposition(EMD); Control strategy.

##  I Introduction

The ongoing Covid-19 pandemic is a major health threat to the human population
across the world Wang . The threat is quite dangerous just like arms conflict
Marciano , and can cause serious problems in various perspectives, such as,
unstoppable infections due to genetically driven continuous emergence of
multiple variants Bano , more dangerous forms of Covid-19 disease when it
aligns with co-morbidities namely, malaria Mohanan , tuberculosis Maia ,
monkeypox Leon , cancer Yang , diabetes Steenblock , cardiovascular diseases
Amankwaah , other viral diseases Wilder  etc to cause major harm in global
economic crisis Morgan , collapse of healthcare systems Christopher , risk due
to global environmental change Barouki , harm in the tourism policies LiZ ,
major loss in agricultural products and policies Lioutas , instabilities in
socio-psychological equations Jaspal  etc. These problems will persist until
ongoing pandemic is controlled or the disease is cured. Even though few
vaccines are available Ndwandwe , it is still hard to control the pandemic due
to the emergence of new variants, and the psychological fear still remains due
to the possibilities of emergence of new variants in future.

The problem that we are still facing is how to optimally control the pandemic
or cure the disease. One straightforward way to control this pandemic is to
hunt for drug/s to kill the virus, SARS-CoV-2 and its variants Otto  which is
still not successful. To control the pandemic, besides the large scale
vaccination Dinleyici , there have been various strategies, such as lockdown,
social distancing, wearing mask etc at various social gatherings, but the
pandemic is still hard to control Perra . One of the reasons for inability to
control this pandemic could be problems in proper understanding of the
dynamics of the pandemic and its causative agent. In this perspective,
modeling of the disease spreading scenario data, their patterns and
spatiotemporal attributes could probably give more concrete answers. Modeling
the nature and growth of numbers of an epidemic/pandemic outbreak is important
for optimizing the tangible solutions needed to contain and mitigate the
epidemic/pandemic. In India, a multi-step approach to design a comprehensive
strategic response to the risk posed by COVID-19 caused by SARS-CoV-2 is in
progress and implemented time to time as government policy by looking at the
patterns of disease spreading Rohith2020 ; Biswas2020 ; Tiwari2020 . With the
current wide-scale biotechnology-aided designing of strategies and solutions
(e.g. vaccine, anti-viral drugs), and by systematic analysis of spatiotemporal
disease spreading patterns, there is a high probability that the risk will be
lowered significantly in near future.

Working under constraints of economic, health-system’s carrying capacities,
socio-psychotic fear and drastically affected food products supply, one of the
major aims of a strategic optimal Zadeh1958  response to the pandemic is to
mitigate the mortality or infection risks. In this context, the majority of
response design decisions are based on sequentially or hierarchically
structured “if-then” or “cost-benefit” analysis drawing inferences from
“cause-effect” scenarios. One way to reduce errors associated with the
probabilities in calculations of efficiency or pay-offs from response
strategiesShea2020 ; Bjornstad2020a ; Drake2019  is by supplanting it with
scientific evidence through accumulating decision-support information of
causal inferences from epidemiological data analysis and models.

Figure 1: A hypothetical but probable scenario in which multiple recurrences
in the incidence data of the country occurs due to state-levels dynamics.

Modeling is a process of inclusion and exclusion of parameters measuring real
observables within a given context and there are many tacit assumptions,
contexts, and trade-offs involved in such a model Iwasaki1994 . When using the
model inferences to design any mitigation strategies, the feedback from
convolution between contexts and parameters must be carefully analyzed to
prevent spurious correlation or false causations entering the subsequent
strategy designs. Modeling the ongoing epidemic/pandemic as a dynamical system
is a drastically simple but useful level of abstraction which allowed us to
quantify the dynamics of the trajectories of this system from multiple
perspectives. The well-established epidemiological models based on
compartmental subdivisions of population, in both stochastic and deterministic
form, and taking into account the demographic diversity have been done
rigorously. Emerging attributes like recurrences Whitman2020 ;
Cacciapaglia2020 ; Hitz2020 ; Mollison1986 ; bartlett1956 ; Anderson1984 ;
Schwartz1983 ; Gomes2014 ; Krylova2013 ; Korevaar2020 , synchrony in infection
occurrences Cui2010 ; Cummings2004 ; Grenfell2001 ; Bjornstad1999 ; He2003 ,
super spreading Li2020 ; Wang2020 , technology-based contact tracing, etc.
also have been studied and contextualized. However, still we are facing
problems to control this ongoing pandemic.

In this paper, we have modeled statistically and theoretically the development
and evolution of the COVID-19 scenario in some of the individual states of
India. One of the central motivations for the mode of analysis taken in the
paper is the fact that multiple agents are influencing the pandemic dynamics
as inferred from the incidence data. Figure 1 presents one of such scenarios
in which there are small recurrences of the outbreak after the pandemic has
been controlled to a certain minimal level. These recurrences may correspond
to multiple wave like features which are seen in ongoing Covid-19 pandemic
Hale . One inference which could directly obtained is that the dynamics of the
pandemic is necessarily stochastic, where, fluctuations in the dynamics may
come from different factors including problems in the control policies, random
interaction of individuals, environmental fluctuations, random climate change,
change in the food habits etc Bittihn ; Gardner  which may trigger the
dynamics far from equilibrium Chanu  and may drive the system to various
recurrences. Hence, fluctuation is one of the contextual factors to be taken
into consideration when modeling the data of the pandemic incidence. Another
example of causative inference could be that the manual interventions like
lockdowns and time-dependent behavior of individuals’ mobility could also act
as contextual factors. Thus, the central theme is set to analyze these
causations or correlations between contextual factors and the model variables
obtained from pandemic data.

The paper is organized to describe inferences from two conceptual data-based
modes of analysis. The repertoire of statistical tools and methods, to draw
inferences about the pandemic dynamics from data is huge and each tool is
optimized for a certain class of constraints. The choice of the tools
presented here is supplanted by their power to extract information about the
latent factors which can then be interpreted safely with proper tests and
checks. Firstly, parametrized inferences and their respective critiques are
given based on fitting the data to growth functions like the Gompertz’s,
Richard’s function, and renormalization group theory functions. Secondly,
analysis of the spatiotemporal data using Empirical Mode Decomposition (EMD)
algorithm is performed to untangle the qualitative-quantitative effects of
various latent factors on the dynamics of the pandemic. The data we have used
might contain errors or biases that cannot be removed therefore the results
are completely dynamic and may change with scenarios.

Figure 2: FLow chart used for data analysis

##  II Methods

We describe the Covid-19 data acquisition from various data repositories, then
to clean the data for analysis. We then explain various fitting procedures to
the acquired data for possible analysis and prediction. Then we explain the
method of harmonic analysis we used in this work.

###  II.1 Data acquisition and cleaning

The epidemic/pandemic growth can be transformed to obtain inferences through
several methods like fitting, visualizations, filtering etc of the data Lega .
The flowchart in Fig. 2 shows the approach we used in this work. The cleaning
and transforming processes of the retrieved data are done to minimize the
errors in subsequent operations of the methods on the data. The two main data
sets used are: (a) the accumulated number of infected individuals denoted as
$I(t)$, and (b) the new number of infected individuals denoted as ’$n(t)$’.
The data used in the work has been obtained from the dedicated Covid-19
database PRS  and then cross verified with the data present in the other
government provided websites. The data used may contain many spurious human
input errors thus inferences from the causative statements should be used
taking these factors into account.

###  II.2 Growth curves for fitting procedure

Fitting of the growth functions to the accumulated number of infected
individuals data $I(t)$ RICHARDS1959 ; Winsor1932 ; Bertalanffy1957 ;
Tjorve2017 ; Somers1988  and then tracking the fit statistics in time may
reveal growth-phase transitions of the pandemic which is both important in
designing preemptive response programs and will provide a correlative-measure
of the qualitative and quantitative effects of decisions taken to control the
epidemic/pandemic. In this work, we analysed the the growth of the pandemic in
Indian states by modelling the data through the fitting of the following
growth functions:

Gompertz function
This function has a rich history Winsor1932 ; Tjorve2017 ; Consolini2020  of
being used to fit multi-phase growth processes’ data. It is a three-parameter
function given by,

$\displaystyle I(t)=a\mathrm{e}^{-\mathrm{e}^{b-{\gamma}t}}$ (1)

where, $a$, $b$ and $\gamma$ are the three parameters in the function (1). To
analyse the nature of the data and growth of the pandemic, this function is
fitted to the curve of the accumulated number of infected individuals with
time. The three parameters control the following features of the curve: ’$a$’
controls the value of saturation plateau, ’$b$’ controls the inflection point
and ’$\gamma$’ controls deviations from the logistic growth curves. The
inflection point $T_{i}$ (stationary point) can be calculated by taking the
derivative of the function (1) and equating it to zero, which is given by,
$\displaystyle T_{i}=\frac{b}{\gamma}$. This inflection point in the fit
parameters correlates with the turning point of the growth curve indicating
the plateau of the growth curve is reached. If there are multiple inflections
corresponding to multiple pandemic waves at $t_{1}$, $t_{2}$,…,$t_{m}$ ($m$
multiple waves), then $\displaystyle T_{i}(t)=\frac{b(t)}{\gamma(t)}$, where,
$t_{1}\leq t\leq t_{m}$.

Epidemic-Renormalization Group (e-RG ) theory growth function
Renormalization group theory, which is used in various branches of Physics and
Mathematics Fisher ; Jona , can be applied to epidemiology Cacciapaglia2020a ;
Morte2020  to model and analyse the growth of the epidemic. The growth
function given by this theory is given by,

$\displaystyle
ln[I(t)]=\dfrac{a\mathrm{e}^{{\gamma}t}}{\mathrm{e}^{{\gamma}t}+b}$ (2)

where, $a$, $b$ and $\gamma$ are the three parameters involved in this growth
function. In the same way as is done in Gompertz function case, the one wave
inflection point is given by, $\displaystyle
T_{i}=\frac{1}{\gamma}ln\left[\frac{b}{2}\left(a+\sqrt{4+a^{2}}\right)\right]$.
For $m$-multiple wave scenario, the multiple inflection points are given by,
$\displaystyle
T_{i}(t)=\frac{1}{\gamma(t)}ln\left[\frac{b(t)}{2}\left(a(t)+\sqrt{4+a(t)^{2}}\right)\right]$,
where, $t_{1}\leq t\leq t_{m}$.

Richard growth function:
Richard’s function RICHARDS1959 ; Somers1988  is a four-parameter growth
function which can be used to study the characteristics and growth properties
of an epidemic/pandemic by fitting the epidemic/pandemic data. The function is
given by,

$\displaystyle
I(t)=\frac{a}{\left[1+{{e}^{-b(t-d)}}\right]^{\frac{1}{\gamma}}}$ (3)

where, the growth parameters are $a$, $b$, $d$ and $\gamma$. The main
underlying assumption of this function is that there is a single maxima peak
in the daily new cases data i.e. $n(t)$. For single wave epidemic/pandemic,
the inflection point is given by, $\displaystyle T_{i}=d-\frac{ln\gamma}{b}$.
However, for $m-$multiple epidemic waves, there will be $m$ such inflection
points given by, $T_{i}(t)=d(t)-\frac{ln\gamma(t)}{b(t)}$ for $t_{1}\leq t\leq
t_{m}$.

###  II.3 Harmonic analysis of spatiotemporal data

The theory of drawing causal inferences from a time series data through
spectral methods is a powerful methodology of extracting patterns and
characteristics of the data based on the decomposition of the signal into
inherent modes Huang1998 . In the context of a pandemic, the data of daily new
cases $n(t)$ (i.e. a time-series data) can be analyzed using harmonic analysis
to understand the nature of the epidemic growth patterns Anderson1984 ;
Huang1998 .

Geo-encoded temporal data of incidence Bjornstad1999 ; Bjornstad2001 ; He2003
; Cacciapaglia2020  gives a powerful way to visualize the physical spread of
the pandemic through the states. Coupled with the information provided from
statistical analysis of the spatiotemporal (time-series) data, the harmonic
analysis can provide both correlative and causative insights into the
relational mappings of the contextual factors and the data-derived inferences.
Correlations among this geo-coded spatiotemporal data can be used to visualize
the measurable population attributes like synchrony in spreading of the
disease with time and spatial regions Cummings2004 . These features have been
observed in various diseases spreadings Hitz2020 ; Grenfell2001  and their
quantification could help in risk mitigation and design of response by the
appropriate administrative authorities.

Empirical mode decomposition(EMD) Huang1998 ; Stallone2020  is a robust method
by which we decompose the given time series into inherent signals with
different frequencies known as intrinsic mode functions (IMF). The method is
done by a process of sifting (Please refer to the Supplementary Information
section). The process can be thought of as analogous to a filtering technique
where multiple frequency components in a time-series can be extracted step-by-
step in a data-adaptive manner. One of the major advantages of this method is
the fact that all analyses are done a posteriori and based on data solely.

##  III Results

The retrieved data from the Covid-19 dedicated database in Indian context PRS
is cleaned and applied the methods mentioned in the Methods to analyse the
data. The results are discussed as in the following.

####  III.0.1 Mapping of growth phases of Covid-19 on a country-wide scale

The data of the cumulative number of infected individuals I(t) for the whole
country for 260 days from the period of March 2020 to November 2020 was fitted
to three growth curves as shown in the Figure 3. The fitting of the three
growth curves mentioned in Methods has been done using Python codepy . All the
fitted curves on the data as shown in the Figure 3 provide a picture of a
smooth growth with a saturation reached after some time. This fitting results
provide a good basis for comparative modeling of the pandemic with
compartmental models.

Figure 3: Fitting of countrywide accumulated no. of infected individuals data
( I(t) ) to growth functions and tracking the temporal changes in fit
parameters to measure the reaction of the disease spreading to interventions.
(a-b)Gompertz growth function and temporal changes in fit parameters, (c-d)
epidemic Renormalisation Group theory growth function and temporal changes in
fit parameters, (e-f) Richards growth function and temporal changes in fit
parameters.

The fit parameters are in the table provided in the supplementary information.
The first panel on top of the Figure 3 shows the data fit with a Gompertz
growth function. The fit parameters show a time-dependent behavior as shown in
the three panels in the row of Figure 3, (b). The parameter $a(t)$ encoding
the number of infected individuals increases rapidly but as strict lockdowns
are enforced, it peaks and then saturates to a lower value correlating with
the fact that the intervention measures were working to contain the spread of
the disease. A major transition in the growth phase of the pandemic is
prominently shown by the existence of inflection points in the plots of the
time-dependent parameters. This feature is seen in the behavior of all the
growth models. The correlation between the parameters also changes its sign
after the inflection point has occurred e.g. the correlation between $b(t)$
and $\gamma(t)$ goes from negative to positive after an inflection point
occurring around 150-160th day. If one had analyzed the data from the start of
the pandemic till the 110th day, one could easily notice that it could predict
the time when there would be a peak in the number of cases with an error of
$\pm$ 7 days.

In case of fitting with the RG function as shown in the second panel of Figure
3 (c-d) the parameter a(t) shows an increase correlating with the facts that
increase in the number of next-neighbor infections resulting from the increase
in diffusion mobility amongst the population due to uplifting of strict
lockdowns on 80th day. The parameters b(t) encodes information about the
offsetting time in the occurrence of the turning point of the growth curve.
The parameter $\gamma(t)$ encodes information about the flattening of the
growth curve. Thus lowering of $\gamma(t)$ indicates the lockdown measures
were effective in the control of the pandemic spread. The parameters a(t) and
$\gamma(t)$ are correlated. Again a similar change in behavior of the
correlation between parameters is seen after the occurrence of an inflection
point.

The bottom panel of the Figure 3 (e-f) shows the fitting of data with the
Richard growth function. The main difference between this model of fitting
from the others is that the data has to be fitted from the 186th day after the
start of the pandemic. The fit could not be performed piecewise as the others.
Though this lacuna was present, the fit parameters were accurate(within a
certain error limit) in obtaining the information about when the peak of the
daily number of new cases occurred.

The growth curves of Gompertz and RG functions were fitted for the data of
states of Delhi, Maharashtra, and Karnataka and have been shown in the
supplementary information[Figures 5-7]. The fit statistics show similar
behavior as seen for the country-wide data. This feature of behaviors of fit
statistics supplants the claim that the dynamics of the disease on a country-
wide scale emerges from local state-wide dynamics. Comparatively, one can also
see which state is far better in the case of control strategies and response
designs. These can serve as crucial precedents in states with the pandemic in
an early stage of growth.

####  III.0.2 Spatiotemporal Mapping of growth of Covid-19 in individual
states

The systematic analysis of the data with the visualization procedure can be
interpreted to provide insights of the nature of the spread of the pandemic
and correlational information about the effects of the interventions and
response strategies. The results are presented here from the analyses
mentioned in the previous section i.e. firstly the fit statistics from the
growth function fitting and then the harmonic analyses of the spatiotemporal
data. Some of the inferences are:

Figure 4:  Sequential analysis of Spatio-temporal plots showing the evolving
wave like features in the dynamics of the pandemic. (a)Plot showing the
normalized data of states 381 days after the first infection incident. (b)Plot
showing the normalized data of states 450 days from the first infection
incident.

1\. The wave speed in essence depends on the various interactions and
demography effects and thus changes the speed (accelerates or deaccelarates)
of the advance of the disease in the population habitatBirzu2018  . Since the
above model is also analogous to the modelling of invasion of infection into a
susceptible population the rate of invasion can be seen depending on the
perturbation term ie.e the interaction terms arising due to various effects
like inherent variation of the degree of susceptibility of the population to
the viral infection.

2\. Diffusion plays an important role by creating instabilities hence it
raises the need to monitor the mobility of infected popualtions in real time
as these will be sequentially used as initial conditions for model simulations
to infer parameters.

3\. We reiterate that the model presented is an highly abstracted and
simplified form of a large numer of complex factors, trade-offs such as
demographic variation of susceptitbility to infection, time-delays caused due
to discrepancies between onset of infection, testing of the individual and
finally reporting, population heterogenity and density etc. (all these factors
can play a confounding role while inferring parameters).

4 Inspite of the simplifications as seen from the wave picture, an essential
inference can be drawn is that to stop or slow the advance of the disease we
need to create as given by Fisher’s FISHER1937  terms ”centres of extinctions”
and identify then isolate the ”centres of multiplications” by building buffer
zones around them(to reduce the motility ).

Latent features in the time-series data can be extracted using harmonic
analysis. Without taking into account the spatial regions this temporal
information of the dynamics of the pandemic is only half the picture. The
analysis and subsequent visualization of the geo-encoded data are analogous to
the contextualization of the information provided by the harmonic analysis to
the concepts of demography, population scales, etc.

####  III.0.3 Geo-encoded harmonic analysis shows synchronous incidence
patterns

The time-series data as shown in the Figure 4(a) have been geo-coded and
plotted as a Hovmoller type plot in the Figure 4 (b) . As can be seen in
Figure 4 (b) there is a signature of a traveling wave-like feature present
across the country. To further cross-analyze whether this spatio-temporal
dynamics is an artifact of visualizations, cross-correlation (CCF) analysis is
performed using R softwareR . In the Figure 4 (b), the states are arranged
spanning from north to south. As can be seen in each panel the peaks of the
incidence occur in the same months. This feature can be labeled as countrywide
incidence synchrony. One of the causes of this synchrony can be the
simultaneous uplifting of lockdown across the country and a correlated
increase in the diffusive-mobility of individuals. Further, as the pandemic
dynamics in states are visualized by clubbing them together local next-
neighbor state-wide synchrony is also prominent.

####  III.0.4 EMD and CCF analysis

Figure 5: Plots of the time series data of various states after EMD analysis.

To verify whether these features are due to the architecture of visualizations
only, EMD analysis was performed on the time-series data which yielded
Intrinsic Mode Frequencies (IMFs) present in the data. These IMFs are then
summed up and plotted in Figure 7. A prominent and similar feature in the
plots is the existence of multiple maxima or an initial decrease in the
function .The behavior depicted in these plots again can be correlated to the
increase in the mobility amongst the population after simultaneous uplifting
of lockdown measures (on the 80th day) and the variant evolution.

Figure 6: (a-b)CCf of time series data of various states with Delhi time
series data.(a)Plot showing the correlation values for 381 days after the
first infection incident. (b) Plot showing the correlation values 450 days
from the first infection incident. (c-d) CCf of time series data of various
states with Maharashtra time series data.(c)Plot showing the correlation
values for 381 days after the first infection incident. (d) Plot showing the
correlation values 450 days from the first infection incident.  Figure 7:
Plots of the residuals after EMD analysis of the time series data of various
states. These represent latent trends present in the data which are not
observed when simply plotted.

At this junction, one hypothesis is introduced to simplify the complexity
associated with the causes of this wave-like feature. Since all states have
open borders, a high variation in population density, and a diverse set of
demographic variables, they can be described as coupled components in a
complex system. In this context, one can hypothesize the presence of varied
sources of traveling waves. The patterns observed in the cross-correlation
between the data of Delhi, Maharashtra, and the other states as shown in the
Figure 6 (a-d) .The spatial extent of this wave is very large and can be said
to cover almost the whole country as seen in the plots Figure 6 (a-b) and the
Figure 6 (c-d) by the presence of a strong correlation across time between the
time series data of Delhi, Maharashtra with rest of the states.

Table 1: Summary table of Inferences. Mode of analysis |  |  Inferences
---|---|---
Traveling wave |  |  Identification of patches of high infection rates based on speed of advance of the disease wave and then creation of buffering zones to reduce the mobility or vaccination drives in or near by areas where past data shows high degree of infections.
Growth functions fitting |  |  1\. If the data is fitted and the fit statistics are tracked piecewise (on a weekly or daily basis), one could infer when the peak of the daily number of new cases has occurred. This is critical for the agencies involved as it may allow for adaptive changes in policies and interventions taken e.g. to design and decide for strict or moderate lockdowns, the degree upto which the bus/train/airport terminals etc. are opened or closed.
|  |  2\. Each growth function has it’s own advantage. The information provided could be compared amongst models or separately stored as a repertoire in a data-base. This pluralistic mode of analysis may be used to make components of a Decision Support System(DSS) during designing of sequential response strategies.
|  |  3\. The behavior of the fit statistics are similar when done on state wide scales and on country wide scale.
Harmonic analysis |  |  1\. Spatio-temporal synchrony is an important driver of infection incidence across country. To reduce the speed of the invasion of the disease outbreak measures could be taken to break down this synchrony
|  |  2\. Dense transportation hubs such as port cities or densely populated cities should be identified and monitored as they could act as centres for emanating travelling waves.

##  IV DISCUSSION

The effects of control strategies of Covid-19 pandemic are reflected in the
observed data and the analysis of these data could reveal the magnitude of the
importance of the strategies and other properties. Implementable response
strategies take their inputs from precedents (if present) and inferences from
iteratively validated models. Models, both qualitative and quantitative are
built on tradeoffs. Thus inferences from models are prone to biases that
become significant at later stages of a sequential response program.
Similarly, precedents contain entangled information about context-based
solutions Wauchope2020  that can create long term problems if the information
is used without proper analyses of cause and effects. We have tried to factor
these in by correctly stating whether the inference is a causative one or a
correlative one. The dynamic character of the parameters as elucidated from
the above analyses may provide probabilistic information about the effects of
the interventions taken Lilleri2020 . Time to time analysis of the growth of
the disease outbreak through the fitting of curves and tracking the fit
parameters must be undertaken for a continuous adaptive mitigation program.

The wave-like features illustrate the nature of the invasion of the disease
into suspected population pockets driven by an increase in diffusion mobility
of individuals. Here a two-pronged methodology can be adopted. Non-linear
dynamics of synchrony in population growth can be helpful to model the
epidemic development in terms of number. There is a well-established
literature on this subject which can help to model the effects of inherent
demographic stochasticity in the growth of epidemic inside the country. These
analyses can be performed on national or statewide scales. One future work is
mapping the edge of these waves Panja2003  where the number of infected
individuals is small and the character of the wave is dominated by a pulled
front wave Birzu2018 . Modeling these aspects together with the population
heterogeneity and influx can help understand the dynamics of the epidemic
development.

Another factor that adds to the stochasticity in the dynamics is the evolution
of the virus. Thus, genetic changes in the virus and its correlation with the
infected population heterogeneity must be tracked in a Spatio-temporal manner
Anderson1984 ; Langwig2017 . The inferences from the growth curve fitting,
harmonic analyses together with inferences from genetic analyses could provide
a holistic picture of the growth and evolution of the pandemic.

## References

  * (1) Wang C., Horby P.W.,Hayden F. G. ,Gao G. F. A novel coronavirus outbreak of global health concern. Lancet, 395, 470-473 (2020).
  * (2) Marciano, H., Eshel, Y., Kimhi, S. and Adini, B. (2022), “Hope and fear of threats as predictors of coping with two major adversities, the COVID-19 pandemic and an armed conflict”, International Journal of Environmental Research and Public Health, Vol. 19 No. 3, p. 1123.
  * (3) Bano, I., Sharif, M. and Alam, S. Genetic drift in the genome of SARS COV-2 and its global health concern. J. Med. Virol. 94, 88–98 (2022).
  * (4) Mohanan P, Islam Z, Hasan MM, Adedeji OJ, Dos Santos Costa AC, Aborode AT, et al. Malaria and COVID-19: a double battle for Burundi. Afr J Emerg Med. 2022;12(1):27–9.
  * (5) Maia CMF, Martelli DRB, Silveira DMML, Oliveira EA, Martelli Jr H. Tuberculosis in Brazil: the impact of the COVID-19 pandemic. J Bras Pneumol.2022;48(2):e20220082.
  * (6) León-Figueroa DA, Bonilla-Aldana DK, Pachar M, Romaní L, Saldaña-Cumpa HM, Anchay-Zuloeta C, Diaz-Torres M, Franco-Paredes C, Suárez JA, Ramirez JD, Paniz-Mondolfi A, Rodriguez-Morales AJ. The Never-Ending Global Emergence of Viral Zoonoses After COVID-19? The rising concern of Monkeypox in Europe, North America and Beyond. Travel Med Infect Dis. 2022;49:102362. https://doi.org/10.1016/j.tmaid.2022.102362.
  * (7) Yang L, Chai P, Yu J, Fan X. Effects of cancer on patients with COVID-19: a systematic review and meta-analysis of 63,019 participants. Cancer Biol Med. 2021;18: 298–307. pmid:33628602
  * (8) Steenblock, C.; Schwarz, P.E.H.; Perakakis, N.; Brajshori, N.; Beqiri, P.; Bornstein, S.R. The interface of COVID-19, diabetes, andn depression. Discov. Ment. Health 2022, 2, 5.
  * (9) Adu-Amankwaah J, Mprah R, Adekunle AO, Noah MLN, Adzika GK, Machuki JO et al (2021) The cardiovascular aspect of COVID-19. Ann Med 53(1):227–236.
  * (10) Wilder-Smith A. COVID-19 in comparison with other emerging viral diseases: risks of geographic spread via travel. Tropical Diseases, Travel Medicine and Vaccines. 2021;7:Article 3 https://tdtmvjournal.biomedcentral.com/track/pdf/10.1186/s40794-020-00129-9.pdf.
  * (11) Morgan AK, Awafo BA, Quartey T (2021) The effects of COVID-19 on global economic output and sustainability: evidence from around the world and lessons for redress. Sustainability: Science, Practice and Policy 17(1):77–81
  * (12) Christopher I., Rocha N., Mehedi Hasan M., Goyal S., Patel T., Jain S., et al. COVID-19 and mucormycosis syndemic: double health threat to a collapsing healthcare system in India. Trop Med Int Health. 2021 doi: 10.1111/tmi.13641.
  * (13) R. Barouki, M. Kogevinas, K. Audouze, K. Belesova, A. Bergman, L. Birnbaum, S. Boekhold, S. Denys, C. Desseille, E. Drakvik, H. Frumkin, J. Garric, D. Destoumieux-Garzon, A. Haines, A. Huss, G. Jensen, S. Karakitsios, J. Klanova, I.M. Koskela, F. Laden, F. Marano, E. Franziska Matthies-Wiesler, G. Morris, J. Nowacki, R. Paloniemi, N. Pearce, A. Peters, A. Rekola, D. Sarigiannis, K. Šebková, R. Slama, B. Staatsen, C. Tonne, R. Vermeulen, P. Vineis The COVID-19 pandemic and global environmental change: Emerging research needs Environ. Int., 146 (2021), p. 106272.
  * (14) Li Z., Dake W., Jaffar A., Saad H., Riaqa M. (2022). Tourists’ health risk threats amid COVID-19 era: role of technology innovation, Transformation, and recovery implications for sustainable tourism. Front. Psychol. 12:769175. doi: 10.3389/fpsyg.2021.769175
  * (15) E.D. Lioutas, C. Charatsari. Enhancing the ability of agriculture to cope with major crises or disasters: What the experience of COVID-19 teaches us. Agric. Syst., 187 (2021), 103023
  * (16) Jaspal, R., and Breakwell, G. M. (2020). Socio-economic inequalities in social network, loneliness and mental health during the COVID-19 pandemic. International Journal of Social Psychiatry, 002076402097669. Advance online publication. https://doi.org/10.1177/0020764020976694.
  * (17) Ndwandwe D, Wiysonge CS. COVID-19 vaccines. Curr Opin Immunol. 2021;71:111–6.
  * (18) S.P. Otto, T. Day, J. Arino, C. Colijn, J. Dushoff, M. Li, S. Mechai, G. Van Domselaar, J. Wu, D.J.D. Earn, N.H. Ogden. The origins and potential future of SARS-CoV-2 variants of concern in the evolving COVID-19 pandemic. Curr. Biol., 31 (2021), pp. R918-R929
  * (19) Perra, N. Non-pharmaceutical interventions during the COVID-19 pandemic: a review. Phys. Rep. 913, 1–52 (2021).
  * (20) Dinleyici Ener Cagri, Borrow Ray, Safadi Marco Aurélio Palazzi, van Damme Pierre, Munoz Flor M. Vaccines and routine immunization strategies during the COVID-19 pandemic. Hum. Vaccin. Immunother. 2020:1–8.
  * (21) Rohith, G. & Devika, K. B. Dynamics and control of COVID-19 pandemic with nonlinear incidence rates Nonlinear Dynamics, Springer Science and Business Media LLC, 2020, 101, 2013-2026
  * (22) Biswas, S. K.; Ghosh, J. K.; Sarkar, S. $\&$ Ghosh, U. COVID-19 pandemic in India: a mathematical model study Nonlinear Dynamics, Springer Science and Business Media LLC, 2020, 102, 537-553
  * (23) Tiwari, I.; Sarin, P. & Parmananda, P. Predictive modeling of disease propagation in a mobile, connected community using cellular automata Chaos: An Interdisciplinary Journal of Nonlinear Science, AIP Publishing, 2020, 30, 081103
  * (24) Zadeh, L. What is optimal? IRE Transactions on Information Theory, Institute of Electrical and Electronics Engineers (IEEE), 1958, 4, 3-3
  * (25) Shea, K.; Bjørnstad, O. N.; Krzywinski, M. & Altman, N. Uncertainty and the management of epidemics Nature Methods, 2020, 17, 867-868
  * (26) Bjørnstad, O. N.; Shea, K.; Krzywinski, M.$\&$ Altman, N. Modeling infectious epidemics Nature Methods, Springer Science and Business Media LLC, 2020, 17, 455-45
  * (27) Drake, J. M.; Brett, T. S.; Chen, S.; Epureanu, B. I.; Ferrari, M. J.; Marty, É.; Miller, P. B.; O’Dea, E. B.; O’Regan, S. M.; Park, A. W.$\&$ Rohani, P. Regoes, R. R. (Ed.) The statistics of epidemic transitions PLOS Computational Biology, Public Library of Science (PLoS), 2019, 15, e1006917
  * (28) Iwasaki, Y. & Simon, H. A. Causality and model abstraction Artificial Intelligence, Elsevier BV, 1994, 67, 143-194
  * (29) Whitman, J. & Jayaprakash, C. d’Onofrio , A. (Ed.) Stochastic modeling of influenza spread dynamics with recurrences PLOS ONE, Public Library of Science (PLoS), 2020, 15, e0231521
  * (30) Cacciapaglia, G.; Cot, C. $\&$ Sannino, F. Second wave COVID-19 pandemics in Europe: a temporal playbook Scientific Reports, Springer Science and Business Media LLC, 2020, 10
  * (31) Hitz, D. A.; Tewald, F. & Eggers, M. Seasonal Bordetella pertussis pattern in the period from 2008 to 2018 in Germany BMC Infectious Diseases, Springer Science and Business Media LLC, 2020, 20
  * (32) Mollison, D. Modelling biological invasions: chance, explanation, prediction Philosophical Transactions of the Royal Society of London. B, Biological Sciences, The Royal Society, 1986, 314, 675-693
  * (33) Bartlett, M. S. Deterministic and Stochastic Models for Recurrent Epidemics Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, Volume 4: Contributions to Biology and Problems of Health, University of California Press, 1956, 81-109
  * (34) Anderson, R. M.; Grenfell, B. T. $\&$ May, R. M. Oscillatory fluctuations in the incidence of infectious disease and the impact of vaccination: time series analysis Journal of Hygiene, Cambridge University Press (CUP), 1984, 93, 587-608
  * (35) Schwartz, I. B. & Smith, H. L. Infinite subharmonic bifurcation in an SEIR epidemic model Journal of Mathematical Biology, Springer Science and Business Media LLC, 1983, 18, 233-253
  * (36) Gomes, M. G. M.; Lipsitch, M.; Wargo, A. R.; Kurath, G.; Rebelo, C.; Medley, G. F. & Coutinho, A. Rall, G. F. (Ed.) A Missing Dimension in Measures of Vaccination Impacts PLoS Pathogens, Public Library of Science (PLoS), 2014, 10, e1003849
  * (37) Krylova, O. & Earn, D. J. D. Effects of the infectious period distribution on predicted transitions in childhood disease dynamics Journal of The Royal Society Interface, The Royal Society, 2013, 10, 20130098
  * (38) Korevaar, H.; Metcalf, C. J. & Grenfell, B. T. Structure, space and size: competing drivers of variation in urban and rural measles transmission Journal of The Royal Society Interface, The Royal Society, 2020, 17, 20200010
  * (39) Cui, M.; Ma, T.-H.$\&$ Li, X.-E. Spatial behavior of an epidemic model with migration Nonlinear Dynamics, Springer Science and Business Media LLC, 2010, 64, 331-338
  * (40) Cummings, D. A.; Irizarry, R. A.; Huang, N. E.; Endy, T. P.; Nisalak, A.; Ungchusak, K.$\&$ Burke, D. S. Travelling waves in the occurrence of dengue haemorrhagic fever in Thailand Nature, Springer Science and Business Media LLC, 2004, 427, 344-34
  * (41) Grenfell, B. T.; Bjørnstad, O. N. & Kappey, J. Travelling waves and spatial hierarchies in measles epidemics Nature, Springer Science and Business Media LLC, 2001, 414, 716-723
  * (42) Bjørnstad, O. N.; Ims, R. A. $\&$ Lambin, X. Spatial population dynamics: analyzing patterns and processes of population synchrony Trends in Ecology $\&$ Evolution, Elsevier BV, 1999, 14, 427-432
  * (43) He, D. & Stone, L. Spatio-temporal synchronization of recurrent epidemics Proceedings of the Royal Society of London. Series B: Biological Sciences, The Royal Society, 2003, 270, 1519-1526
  * (44) Li, R.; Pei, S.; Chen, B.; Song, Y.; Zhang, T.; Yang, W. & Shaman, J. Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2) Science, American Association for the Advancement of Science (AAAS), 2020, 368, 489-493
  * (45) Wang, S.; Pan, Y.; Wang, Q.; Miao, H.; Brown, A. N. & Rong, L. Modeling the viral dynamics of SARS-CoV-2 infection Mathematical Biosciences, Elsevier BV, 2020, 328, 10843
  * (46) T. Hale, N. Angrist, A.J. Hale, B. Kira, S. Majumdar, A. Petherick, T. Phillips, D. Sridhar, R.N. Thompson, S. Webster Government responses and COVID-19 deaths: Global evidence across multiple pandemic waves. PLoS ONE, 16 (2021), Article e0253116.
  * (47) Bittihn P, Golestanian R. Stochastic effects on the dynamics of an epidemic due to population subdivision. Chaos. 2020;30(10):12.
  * (48) Gardner Billy J and Kilpatrick A Marm. Contact tracing efficiency, transmission heterogeneity, and accelerating COVID-19 epidemics. PLOS Computational Biology, 17(6):e1009122, 2021.
  * (49) Chanu, A.L., Bhadana, J., Singh, R.B.: Stochastic fluctuations as a driving force to dissipative non-equilibrium states. J. Phys. A Math. Theor. 53(42), 425002 (2020).
  * (50) Lega, J. and Brown, H. E. Data-driven outbreak forecasting with a simple nonlinear growth model. Epidemics 17, 19–26 (2016).
  * (51) PRS Legislative research database: https://prsindia.org/covid-19/cases.
  * (52) Richards, F. J. A Flexible Growth Function for Empirical Use Journal of Experimental Botany, Oxford University Press (OUP), 1959, 10, 290-30
  * (53) Winsor, C. P. The Gompertz Curve as a Growth Curve Proceedings of the National Academy of Sciences, Proceedings of the National Academy of Sciences, 1932, 18, 1-8
  * (54) von Bertalanffy, L. Quantitative Laws in Metabolism and Growth The Quarterly Review of Biology, University of Chicago Press, 1957, 32, 217-231
  * (55) Tjørve, K. M. C. & Tjørve, E. Merks, R. M. (Ed.) The use of Gompertz models in growth analyses, and new Gompertz-model approach: An addition to the Unified-Richards family PLOS ONE, Public Library of Science (PLoS), 2017, 12, e017869
  * (56) Somers, I. On a seasonally oscillating growth function Fishbyte, 1988, 6, 8-11
  * (57) Consolini, G. & Materassi, M. A stretched logistic equation for pandemic spreading Chaos, Solitons $\&$ Fractals, Elsevier BV, 2020, 140, 110113
  * (58) M. E. Fisher, Renormalization group theory: Its basis and formulation in statistical physics. Rev. Mod. Phys. 70, 653 (1998).
  * (59) G. Jona-Lasinio, Renormalization group and probability theory, Phys. Rep. 352, 439 (2001).
  * (60) Cacciapaglia, G. $\&$ Sannino, F. Interplay of social distancing and border restrictions for pandemics via the epidemic renormalisation group framework Scientific Reports, 2020, 10
  * (61) Morte, M. D.; Orlando, D. & Sannino, F. Renormalization Group Approach to Pandemics: The COVID-19 Case Frontiers in Physics, Frontiers Media SA, 2020, 8
  * (62) Huang, N. E.; Shen, Z.; Long, S. R.; Wu, M. C.; Shih, H. H.; Zheng, Q.; Yen, N.-C.; Tung, C. C. & Liu, H. H. The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, The Royal Society, 1998, 454, 903-995
  * (63) Bjornstad, O. N. $\&$ Falck, W. Nonparametric spatial covariance functions: Estimation and testing Environmental and Ecological Statistics, Springer Science and Business Media LLC, 2001, 8, 53-70
  * (64) Stallone, A.; Cicone, A. & Materassi, M. New insights and best practices for the successful use of Empirical Mode Decomposition, Iterative Filtering and derived algorithms Scientific Reports, Springer Science and Business Media LLC, 2020, 10
  * (65) Python Software Foundation. Python Language Reference, version 3. Available at http://www.python.org
  * (66) Birzu G, Hallatschek O, and Korolev K. S., Fluctuations uncover a distinct class of traveling waves 2018 Proceedings of the National Academy of Sciences 115 (16) E3645-E3654
  * (67) Fisher R. A. The wave of advance of advantageous genes Annals of Eugenics, Wiley, 1937, 7, 355-369
  * (68) R Core Team (2021). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/
  * (69) Wauchope, H. S.; Amano, T.; Geldmann, J.; Johnston, A.; Simmons, B. I.; Sutherland, W. J. & Jones, J. P. Evaluating Impact Using Time-Series Data Trends in Ecology & Evolution, Elsevier BV, 2020
  * (70) Lilleri, D.; Zavaglio, F.; Gabanti, E.; Gerna, G. & Arbustini, E. Tang, P. (Ed.) Analysis of the SARS-CoV-2 epidemic in Italy: The role of local and interventional factors in the control of the epidemic PLOS ONE, Public Library of Science (PLoS), 2020, 15, e0242305
  * (71) Panja, D.; Tripathy, G. & van Saarloos, W. Front propagation and diffusion in theAleftrightarrowsA+Ahard-core reaction on a chain Physical Review E, American Physical Society (APS), 2003, 67
  * (72) Langwig, K. E.; Wargo, A. R.; Jones, D. R.; Viss, J. R.; Rutan, B. J.; Egan, N. A.; Sá-Guimarães, P.; Kim, M. S.; Kurath, G.; Gomes, M. G. M. & Lipsitch, M. Bansal, S. & Pettigrew, M. M. (Eds.) Vaccine Effects on Heterogeneity in Susceptibility and Implications for Population Health Management mBio, American Society for Microbiology, 2017, 8