#  Multivariate Beta Mixture Model: Probabilistic Clustering With Flexible
Cluster Shapes

Yung-Peng Hsu, Hung-Hsuan Chen
National Central Univer
Taoyuan, Taiwan
<EMAIL_ADDRESS><EMAIL_ADDRESS>

###### Abstract

This paper introduces the multivariate beta mixture model (MBMM), a new
probabilistic model for soft clustering. MBMM adapts to diverse cluster shapes
because of the flexible probability density function of the multivariate beta
distribution. We introduce the properties of MBMM, describe the parameter
learning procedure, and present the experimental results, showing that MBMM
fits diverse cluster shapes on synthetic and real datasets. The code is
released anonymously at https://github.com/hhchen1105/mbmm/.

_K_ eywords Mixture model $\cdot$ EM algorithm $\cdot$ Clustering.

##  1 Introduction

Data clustering groups data points into components so that similar points are
within the same component. Data clustering is commonly used for data
exploration and is sometimes used as a preprocessing step for later analysis
[1].

In this paper, the multivariate beta mixture model (MBMM), a new probabilistic
model for soft clustering, is proposed. As the MBMM is a mixture model, it
shares many properties with the Gaussian mixture model (GMM), including its
soft cluster assignment and parametric modeling. In addition, the MBMM allows
the generation of new (synthetic) instances based on a generative process.
Because the beta distribution is highly flexible (e.g., unimodal, bimodal,
straight line, or exponentially increasing or decreasing), MBMM can fit data
with versatile shapes. Figure 1 shows that various cluster shapes can be
obtained with a bivariate beta distribution. On the contrary, the shape of a
Gaussian distribution is symmetric and unimodal, which limits its fitting
capacity.

Figure 1: Examples of the versatile shape of the bivariate beta distribution.
The upper row shows three bivariate beta distributions with different
parameters. The bottom row shows the marginal distribution of $x_{1}$ (i.e.,
the variable on the horizontal axis in the top row). This distribution can be
symmetric unimodal (e.g., the left subfigure), skewed unimodal (e.g., the
middle subfigure), or bimodal (e.g., the right subfigure).

The multivariate beta distribution is defined in different ways. In some
studies, the Dirichlet distribution is considered a multivariate beta
distribution (e.g., [2]) because the beta distribution is a special case of
the Dirichlet distribution with two parameters. However, we apply the
definition provided in [3], which is even more general than the Dirichlet
distribution. The relationship between the Dirichlet distribution and our
multivariate beta distribution will be discussed in Section 2.1 when we
introduce the details of the multivariate beta distribution.

This paper presents several contributions. First, we propose a new
probabilistic model for soft clustering. Our model is similar to the Gaussian
Mixture Model (GMM), but the shape of each cluster is more versatile than
those generated by GMM. Second, we compare MBMM with well-known clustering
algorithms on synthetic and real datasets to demonstrate its effectiveness.
Finally, we release the code for reproducibility. Our implemented class offers
fit(), predict(), and predict_proba(), the common methods provided by scikit-
learn’s clustering algorithms, making it convenient to apply MBMM to new
domains.

The rest of the paper is organized as follows. Section 2 introduces the
multivariate beta distribution and the proposed MBMM. Section 3 describes
experiments on synthetic and real datasets. Section 4 reviews previous work on
data clustering. We conclude by discussing the limitations of the MBMM and the
ongoing and future work on the MBMM in Section 5.

##  2 Multivariate Beta Mixture Model

###  2.1 Multivariate beta distribution

The probability density function (PDF) of a multivariate beta distribution
(MB) has been defined in different ways [4, 3]. Here, we apply the definition
in [3]: given an instance $\boldsymbol{x}=[x_{1},\ldots,x_{M}]^{T}$ with $M$
variates (i.e., features) and the shape parameters
$a_{m}>0,b>0~{}(m=1,\ldots,M)$, its PDF is given by

$\mathit{MB}(\boldsymbol{x}|a_{1:M},b)=\frac{1}{Z}\times\frac{\prod_{m=1}^{M}\frac{x_{m}^{a_{m}-1}}{(1-x_{m})^{a_{m}+1}}}{(1+\sum_{k=1}^{M}\frac{x_{k}}{1-x_{k}})^{a_{1}+\ldots+a_{M}+b}},$
(1)

where $x_{m}\in(0,1),a_{m}>0,b>0$, and the normalizer $Z$ is defined by

$Z=\frac{\Gamma(b)\prod_{m=1}^{M}\Gamma(a_{m})}{\Gamma(b+\sum_{j=1}^{M}a_{j})},$
(2)

where $\Gamma$ is the gamma function.

Figure 2: A comparison of the support of the Dirichlet distribution (left) and
our multivariate beta distribution (right) with 3 variates. The Dirichlet
distribution is only defined on $x_{i}\in(0,1)$ such that
$x_{1}+x_{2}+x_{3}=1$ (the standard 2-simplex in $R^{3}$). On the contrary,
our multivariate beta distribution is defined on $(0,1)^{3}$ (the unit cube in
$R^{3}$), which is a superset of the Dirichlet distribution.

In some previous studies, the Dirichlet distribution was treated as a
multivariate generalization of the beta distribution (e.g., [2]) since the
Dirichlet distribution falls back to the beta distribution when the number of
parameters is 2. However, we describe a more general definition of the
multivariate beta distribution that regards the Dirichlet distribution as a
special case. The relationship between the Dirichlet distribution and the
proposed multivariate beta distribution is illustrated in Figure 2.
Specifically, the support of an $n$-variate Dirichlet distribution is
restricted to a standard $(n-1)$-simplex. However, the support of our
multivariate beta distribution is a hypercube in an $n$-dimensional space with
a length of 1 on each side. In other words, the Dirichlet distribution is the
multivariate beta distribution subject to
$\left\lVert\boldsymbol{x}\right\rVert_{1}=1$.

###  2.2 MBMM density function and generative process

In Table 1, we list the notations that will be used in this paper hereafter.

Table 1: Notation list Indices:
---
|  $M$ |  Dimensions of an observed instance ($m\in\\{1,\ldots,M\\}$)
|  $C$ |  Number of clusters ($c\in\\{1,\ldots,C\\}$)
|  $N$ |  Number of instances ($n\in\\{1,\ldots,N\\}$)
Parameters:
|  $a_{c,m}$ |  the $m$-th shape parameter for cluster $c$; $a_{c,m}>0$
|  $b_{c}$ |  the $(M+1)$-th shape parameter for cluster $c$; $b_{c}>0$
|  $\pi_{c}$ |  Mixture weight of cluster $c$, $0<\pi_{c}<1,\sum_{c=1}^{C}\pi_{c}=1$
|  $z_{n}$ |  Cluster that $\boldsymbol{x}_{n}$ belongs to; $z_{n}\in\\{1,\ldots,C\\}$
|  $\gamma_{n,c}$ |  Probability that $\boldsymbol{x}_{n}$ belongs to cluster $c$; $\sum_{c=1}^{C}\gamma_{n,c}=1$
|  $\boldsymbol{\theta}_{c}$ |  Set of parameters for cluster $c$; $\boldsymbol{\theta}_{c}=\left\\{a_{c,1},\ldots,a_{c,M},b_{c}\right\\}$
|  $\boldsymbol{\theta}$ |  Set of parameters for the MBMM; $\boldsymbol{\theta}=\left\\{a_{1:C,1:M},b_{1:C},\pi_{1:C}\right\\}$
Observed random variables:
|  $\boldsymbol{x}_{n}$ |  Observed instance; $\boldsymbol{x}_{n}=\left[x_{n,1},\ldots,x_{n,M}\right]^{T}\in R^{M}$

In MBMM, it is assumed that the data points are generated from a mixture of
multivariate beta distributions (whose PDF is defined in Equation 1).
Consequently, the probability of the MBMM given $C$ components is

$p(\boldsymbol{x}_{n}|\boldsymbol{\theta})=\sum_{c=1}^{C}\pi_{c}\mathit{MB}(\boldsymbol{x}_{n}|\boldsymbol{\theta}_{c}).$
(3)

The parameter $\pi_{c}$ determines the probability that a random instance
$\boldsymbol{x}_{n}$ belongs to cluster $c$ (before knowing the values of the
variates in $\boldsymbol{x}_{n}$), and
$\mathit{MB}(\boldsymbol{x}_{n}|\boldsymbol{\theta}_{c})$ gives the PDF if
$\boldsymbol{x}_{n}$ indeed belongs to cluster $c$.

Figure 3: Graphical representation of the multivariate beta mixture model

Figure 3 shows a graphical representation of the multivariate beta mixture
model. To generate a sample $\boldsymbol{x}_{n}$, we first sample a latent
variable $z_{n}$ (the cluster ID of the sample $\boldsymbol{x}_{n}$) from a
multinomial distribution with parameters $\pi_{1},\ldots,\pi_{C}$. Suppose
that $z_{n}=c$ after sampling, we further sample an instance
$\boldsymbol{x}_{n}$ from the multivariate beta distribution with parameters
$\boldsymbol{\theta}_{c}$:
$\mathit{MB}(\boldsymbol{x}_{n}|\boldsymbol{\theta}_{c})=\mathit{MB}(\boldsymbol{x}_{n}|a_{c,1},\ldots,a_{c,M},b_{c})$.

###  2.3 Parameter Learning for the MBMM

Data: Input data $\boldsymbol{x}_{1},\ldots,\boldsymbol{x}_{N}$, cluster
number $C$

Result: Parameters $\boldsymbol{\theta}=\\{\pi_{1:C},a_{1:C,1:M},b_{1:C}\\}$

Initialize $\boldsymbol{\theta}$ randomly;

while  _not converge_ do

// E-step

for  _$n\leftarrow 1$ to N_ do

for  _$c\leftarrow 1$ to C_ do

Update $\gamma_{n,c}$ by Equation 7;

end for

end for

// M-step

Update $a_{1:C,1:M}$ and $b_{1:C}$ with the SQP solver [5];

Update $\pi_{1:C}$ by Equation 8;

if  _iteration count reaches a pre-defined value_ then

Exit while loop;

end if

end while

Algorithm 1 Parameter learning algorithm for the MBMM

In reality, we do not know the values of the parameters
$\boldsymbol{\theta}=\left\\{a_{1:C,1:M},b_{1:C},\pi_{1:N}\right\\}$
(referring to Figure 3). We hope to recover these parameters based on the
observed $\boldsymbol{x}_{n}$-s to maximize the likelihood function:

$L(\boldsymbol{\theta})=p(\boldsymbol{x}_{1:N},z_{1:N}|\boldsymbol{\theta})=\prod_{n=1}^{N}\prod_{c=1}^{C}\left[\pi_{c}\mathit{MB}(\boldsymbol{x}_{n}|\boldsymbol{\theta}_{c})\right]^{I(z_{n}=c)},$
(4)

where $I$ is the indicator function.

As the likelihood function (Equation 4) involves the multiplication of
$N\times C$ terms, the result is numerically unstable. Instead, we compute the
log-likelihood function to convert multiplications to additions, as shown in
Equation 5. As a result, the computation is more numerically stable.

$\log
L(\boldsymbol{\theta})=\sum_{n=1}^{N}\sum_{c=1}^{C}I(z_{n}=c)\left(\log\pi_{c}+\log
MB(\boldsymbol{x}_{n}|\boldsymbol{\theta}_{c})\right).$ (5)

However, since we cannot observe the latent $z_{n}$ in practice, direct
optimization of Equation 5 is difficult. As an alternative, we compute the
expected value of the log-likelihood function with respect to the latent
variables $z_{1:N}$, which involves the expected (but not the true) values of
$z_{n}$:

$E_{z_{1:N}}\left[\log
L(\boldsymbol{\theta})\right]=\sum_{n=1}^{N}\sum_{c=1}^{C}\gamma_{n,c}\left(\log\pi_{c}+\log
MB(\boldsymbol{x}_{n}|\boldsymbol{\theta}_{c})\right).$ (6)

After the above reformulation, the parameters
($\boldsymbol{\theta}_{1:C},\pi_{1:C}$) that are used to maximize the expected
value of the log-likelihood function (Equation 6) can be learned via the EM
algorithm, as given by the pseudocode in Algorithm 1. In the E-step, we
compute $\gamma_{n,c}$ (the probability that instance $\boldsymbol{x}_{n}$
belongs to cluster $c$) that maximizes Equation 6 by assuming that the
randomly initialized or currently estimated $\pi_{1:C}$ and
$\boldsymbol{\theta}_{1:C}$ are correct. The assignment of $\gamma_{n,c}$ has
a simple closed-form solution, as shown below

$\gamma_{n,c}=\frac{\pi_{c}\mathit{MB}(\boldsymbol{x}_{n}|\boldsymbol{\theta}_{c})}{\sum_{k=1}^{C}\pi_{k}\mathit{MB}(\boldsymbol{x}_{n}|\boldsymbol{\theta}_{k})}.$
(7)

In the M-step, we search for the parameters $\pi_{1:C}$, $a_{1:C,1:M}$, and
$b_{1:C}$ by assuming that the estimated $\gamma_{n,c}$ values in the E-step
are correct. However, since the $a_{1:C,1:M},b_{1:C}$ parameters seem to lack
a closed-form solution, we resort to numerical optimization strategies,
specifically the sequential quadratic programming (SQP) iterative method, as
the minimization strategy [5] because SQP allows linear constraints on the
parameters (i.e., $a_{c,m}>0\textrm{ and }b_{c}>0~{}\forall c,m$). For
parameters $\pi_{1},\ldots,\pi_{C}$, we rely the efficient closed-form
solution:

$\pi_{c}=\frac{1}{N}\sum_{n=1}^{N}\gamma_{n,c}.$ (8)

We compute the difference between the log-likelihood estimation in successive
rounds for the convergence check. Additionally, if the number of iterations
reaches a predefined value, we terminate the loop.

###  2.4 The similarity score between data points

Most clustering algorithms define the distance between two samples by
converting them into a non-negative real value, i.e., given
$\boldsymbol{x}_{i},\boldsymbol{x}_{j}\in R^{M}$, the distance function is
represented by
$d_{i,j}:\boldsymbol{x}_{i}\times\boldsymbol{x}_{i}\xrightarrow{}\\{0,R^{+}\\}$.
However, if we define the distance of two samples based on their coordinates
and assign samples to the closest cluster centroid, the output shapes of the
clusters are inevitably convex.

In MBMM, we define the distance between two data points from a different
perspective. Since the PDF of a data point is an affine combination of $C$
multivariate beta distributions (Equation 3), we consider
$MB(\cdot|\boldsymbol{\theta}_{1}),\ldots,MB(\cdot|\boldsymbol{\theta}_{C})$
as the basis to form a function space. Consequently, the coordinate of the
data point $\boldsymbol{x}_{n}$ becomes
$\boldsymbol{\gamma}_{n}=\left[\gamma_{n,1},\gamma_{n,2},\ldots,\gamma_{n,C}\right]^{T}$
with respect to the basis functions. The vector $\boldsymbol{\gamma}_{n}$ is a
discrete probability distribution since $\sum_{c=1}^{C}\gamma_{n,c}=1$. Thus,
we can define the distance between the data points $\boldsymbol{x}_{i}$ and
$\boldsymbol{x}_{j}$ as the distance between the discrete probability
distributions $\boldsymbol{\gamma}_{i}$ and $\boldsymbol{\gamma}_{j}$. We use
the Kullback-Leibler divergence (KL divergence) to determine this distance:

$d_{i,j}^{\mathit{KL}}:=\sum_{c=1}^{C}\gamma_{i,c}\log\left(\frac{\gamma_{i,c}}{\gamma_{j,c}}\right).$
(9)

##  3 Experiments

###  3.1 Comparisons on the synthetic datasets

Figure 4: Clustering algorithms on synthetic datasets

####  3.1.1 Baseline models

Clustering algorithms can be classified into four types: centroid-based,
density-based, distribution-based, and hierarchical clustering algorithms. We
select representative models from each of these categories. For centroid-based
models, we choose the $k$-means algorithm, which is likely the most widely
used clustering algorithm. We select the DBSCAN algorithm for density-based
models, which received the Test of Time award at KDD 2014. We choose the GMM
as the distribution-based model and the agglomerative clustering (AC)
algorithm as the hierarchical clustering model.

####  3.1.2 Synthetic dataset generation

We generate synthetic datasets for the experiments. First, we create data
points from three isotropic 2D Gaussian distributions whose means are distant
and whose variances are small in each dimension. Consequently, a data point is
close to other data points of the same Gaussian distribution but far from
those in other Gaussian distributions (Figure 4, first row). This dataset
represents an ideal case for data clustering.

Second, we generate two distant 2D Gaussian distributions with small variances
in each dimension. However, the third distribution has a large variance.
Consequently, several data points sampled from the third distribution are
mixed with those from other distributions (Figure 4, second row).

Third, we generate three 2D Gaussian distributions with isolated means as
before. However, we introduce a high correlation between two covariates for
each Gaussian distribution. Consequently, data points are sometimes closer to
data points generated from other distributions (Figure 4, third row).

Fourth, we generate concentric circles (i.e., one circle within another). If a
point from the outer circle is selected, the most distant data point is
located on the other side of the same circle. Consequently, such a synthetic
dataset is highly challenging for centroid-based and distribution-based
methods to group the outer circle as one cluster (Figure 4, fourth row).

We scale the range of every feature to within 0 and 1 because the support of
each variate in the MB distribution is between 0 and 1, as explained in
Section 2.1. This constraint has little influence in practice because
clustering algorithms generally require the availability of all data sets
prior to training; thus, scaling each variate as a preprocessing step is
straightforward.

####  3.1.3 Visualizing clustering results

Figure 5: Upper row: data points clustered by the MBMM; bottom row: PDFs based
on the fitted parameters.

Figure 4 shows a visualized comparison of the clustering algorithms for the
four synthetic datasets.

All algorithms compared perform well on the first synthetic dataset. However,
some data points belonging to the middle cluster are incorrectly grouped as
the right cluster for the second dataset when using $k$-mean and AC. This is
because the middle cluster has a wider spread, making the centroid far away
from some points in the same cluster. With DBSCAN, many data points from the
middle cluster are regarded as outliers because density-based algorithms
usually have difficulty when the intracluster distance (the distance between
members of a cluster) differs greatly. For similar reasons, unsatisfied
performance is obtained using the $k$ means and DBSCAN on the third dataset.
Mediocre results are also obtained using the AC algorithm, likely because
Ward’s linkage function merges the wrong groups. On the concentric circles
dataset, poor performance is observed for the $k$-means and GMM algorithms
because they can only group geometrically neighboring nodes in one cluster.
Since the AC and DBSCAN algorithms can recursively group adjacent points into
the same cluster, it is possible to group two distant points into the same
cluster, so reasonable performance is achieved. Excellent results are obtained
using our proposed MBMM on all synthetic datasets because the shape of a
multivariate beta distribution is versatile. In particular, for the fourth
dataset, because a multivariate beta distribution can be bimodal, the MBMM can
group the data points in the outer circle as one cluster even though they are
geometrically distant.

We visualize the PDFs by fitting the MBMM to the four synthetic datasets
(Figure 5). The upper row shows the data points, and the bottom row shows the
PDFs estimated by MBMM, which indeed fits these datasets adequately.

###  3.2 Comparison on the real datasets

####  3.2.1 Real datasets

We used two open real datasets. The first dataset is MNIST, which includes
grayscale images of handwritten digits. The size of each image is $28\times
28$. Since image pixels should have spatial correlations, directly using the
pixel values as input features for clustering algorithms could be problematic.
Eventually, we reduce the dimension of each image to 2 dimensions using the
following procedure. First, we train a vanilla convolutional neural network
(ConvNet) using the Fashion MNIST dataset (not the MNIST dataset). Then, we
feed each MNIST image into the Fashion MNIST-trained ConvNet and take the
hidden layer before the output (a vector with 512 neurons) as the image
representation. Finally, we use a standard autoencoder to reduce the vector
into 2 dimensions, which are the inputs of the clustering algorithms.
Ultimately, we include only images of number 1 and number 9 in MNIST for
experiments.

The second dataset, the breast cancer Wisconsin (diagnostic) dataset, consists
of 569 instances. Each instance includes 32 attributes and a binary class
label indicating the status of the tumor (benign or malignant). We download
the dataset from the UCI Machine Learning Repository.

####  3.2.2 Results

We compare clustered IDs with ground truth labels to calculate the adjusted
Rand index (ARI) [6, 7] and adjusted mutual information (AMI) [8], two
standard metrics for clustering evaluation. If a clustering result perfectly
matches the referenced clusters (labels), both metrics return a score of 1.
However, ARI and AMI are biased toward different types of clustering results:
ARI prefers balanced partitions (clusters with similar sizes), and AMI prefers
unbalanced partitions [9]. For a fair comparison, we report both metrics.

Table 2: Comparison of the clustering algorithms on the MNIST dataset and the breast cancer dataset (mean $\pm$ standard deviation) | MNIST | breast cancer
---|---|---
| ARI | AMI | ARI | AMI
MBMM | $\boldsymbol{.937}\pm.000$ | $\boldsymbol{.884}\pm.000$ | $\underline{.664}\pm.000$ | $\underline{.558}\pm.000$
$k$-means | $.913\pm.000$ | $.850\pm.000$ | $.491\pm.000$ | $.464\pm.000$
AC | $\underline{.933}\pm.000$ | $\underline{.878}\pm.000$ | $\boldsymbol{.689}\pm.000$ | $\boldsymbol{.568}\pm.000$
DBSCAN | $.854\pm.009$ | $.745\pm.011$ | $.554\pm.018$ | $.447\pm.010$
GMM | $.909\pm.000$ | $.846\pm.000$ | $\underline{.664}\pm.000$ | $\underline{.558}\pm.000$

Table 2 shows the results. We repeat each experiment five times and report the
mean $\pm$ standard deviation. We highlight each metric’s first and second
highest values in bold and underlined. For MNIST, the top 3 methods are our
MBMM, followed by AC, and then $k$-means. For the cancer dataset, the best
performance is achieved using the AC algorithm, followed by our MBMM and GMM.
In general, our MBMM and AC perform best among all.

###  3.3 Distance between data points

Figure 6: Distance from red point to other points in concentric circles
dataset

As explained in Section 2.4, we define the distance between
$\boldsymbol{x}_{i}$ and $\boldsymbol{x}_{j}$ based on the KL divergence
between $[\gamma_{i,1},\ldots,\gamma_{i,C}]^{T}$ and
$[\gamma_{j,1},\ldots,\gamma_{j,C}]^{T}$. Consequently, even if
$\boldsymbol{x}_{i}$ and $\boldsymbol{x}_{j}$ are distant based on the
Euclidean distance, they could still have a small distance score if
$\gamma_{i,c}\approx\gamma_{j,c}$ for most $c$-s.

We illustrate the red data point’s distances to others using concentric
circles in Figure 6. Outer circle points are closer, showing that MBMM’s
distance function can assign small values even with large Euclidean distances.

##  4 Related Work

Table 3: Popular clustering algorithms and their properties | MBMM |  $k$-means | DBSCAN | AC | GMM
---|---|---|---|---|---
Type |  Distribution- based |  Centroid- based |  Density- based |  Hierarchical clustering |  Distribution- based
Assignment | Soft | Hard | Hard | Hard | Soft
Cluster shape | Versatile | Convex | Versatile | Versatile | Convex
Generative/Discriminative | Gen. | Discr. | Discr. | Discr. | Gen.

Clustering algorithms can be classified into four types based on how they
partition data points: hierarchical, centroid-based, density-based, and
distribution-based clustering.

Hierarchical clustering algorithms are top-down or bottom-up, corresponding to
the iterative division of each cluster into smaller clusters and the
aggregation of smaller clusters into larger clusters. Hierarchical clustering
algorithms allow dynamically adjusting the cluster numbers. However, users
must define the distance between not only data points but also between
clusters, which could sometimes be counterintuitive. Well-known hierarchical
clustering algorithms include agglomerative clustering (AC) and BIRCH [10].

Centroid-based algorithms represent each cluster using a centroid, assigning
each data point to the closest cluster. However, these algorithms generate
clusters with convex shapes, eliminating the possibility of fitting a bimodal
cluster. Well-known algorithms include $k$-means, $k$-medoids, $k$-medians,
and $k$-means$++$.

Density-based algorithms determine clusters by assuming that densely
distributed areas are clusters. Typical algorithms include DBSCAN [11] and
OPTICS [12]. Although they discover clusters of various shapes, hyperparameter
tuning can be time-consuming and heavily influence clustering results [13].
Additionally, density-based algorithms sometimes have difficulty clustering
data points when the distances between different data points vary widely.

Distribution-based models assume that each cluster follows a probability
distribution. One well-known is GMM, which assumes that each cluster follows a
Gaussian distribution. Distribution-based models naturally generate synthetic
data points by sampling a cluster ID and then one data point from the data
distribution of cluster $i$. One problem with GMM is that the shape of each
cluster must be convex since this is a fundamental property of a Gaussian
distribution. As a result, GMM cannot differentiate between inner and outer
circles.

Table 3 gives an overview of these clustering algorithms and their properties.
Both MBMM and GMM are distribution-based models, which thus allow for soft
clustering and synthetic generation of the data points. DBSCAN and AC allow
non-convex cluster shapes in which each data point within a cluster is close
to only a few data points from the same cluster. MBMM also supports non-convex
cluster shapes due to the versatility of the multivariate beta distribution.

The beta mixture model has been studied in the bioinformatics and biochemical
domains [14, 15]. However, they assumed that each cluster follows a standard
beta distribution, which limits the practical usage of these models because
each data point must be univariate. This constraint is probably due to the
fact that the definition of a multivariate beta distribution is still
ambiguous. Our study is more practical because we allow each data point to be
multivariate.

##  5 Discussion

This paper proposes a new probabilistic model, the multivariate beta mixture
model, for data clustering. We demonstrate MBMM’s effectiveness by thorough
experiments on synthetic and real datasets. Furthermore, MBMM is a generative
model that allows for the generation of new data points. Compared to another
famous generative clustering algorithm, the Gaussian mixture model, MBMM
allows for a more flexible cluster shape. To ensure reproducibility, we have
released our experimental code and encapsulated the MBMM module as a class
with typical class methods supported by the clustering algorithms in scikit-
learn, facilitating the utilization of the MBMM in various applications.

Although MBMM has these nice properties, we believe that different clustering
algorithms should be used in combination to jointly partition data points for
the following reasons. First, data clustering is ill-defined due to the lack
of ground truth labels during both training and testing, making the choice of
training objective and evaluation ad hoc [16]. Additionally, it has been shown
that, under reasonably general conditions, no single clustering algorithm can
satisfy the three fundamental properties introduced in [17].

The capacity of MBMM is limited by the need for a positive correlation among
all variates due to parameter $b$ [3]. We plan to explore multivariate beta
distributions allowing both positive and negative correlations based on [4].

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# Analysis, Modeling, and Representation of COVID-19 Spread: A Case Study on
India

Rahul Mishra, Hari Prabhat Gupta, and Tanima Dutta The authors are with the
Department of Computer Science and Engineering, Indian Institute of Technology
(BHU) Varanasi, India (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Coronavirus outbreak is one of the most challenging pandemics for the entire
human population of the planet Earth. Techniques such as the isolation of
infected persons and maintaining social distancing are the only preventive
measures against the epidemic COVID-19. The actual estimation of the number of
infected persons with limited data is an indeterminate problem faced by data
scientists. There are a large number of techniques in the existing literature,
including reproduction number, the case fatality rate, etc., for predicting
the duration of an epidemic and infectious population. This paper presents a
case study of different techniques for analysing, modelling, and
representation of data associated with an epidemic such as COVID-19. We
further propose an algorithm for estimating infection transmission states in a
particular area. This work also presents an algorithm for estimating end-time
of an epidemic from Susceptible Infectious and Recovered model. Finally, this
paper presents empirical and data analysis to study the impact of transmission
probability, rate of contact, infectious, and susceptible on the epidemic
spread.

###### Index Terms:

Coronavirus, epidemic, infectious, isolation, susceptible.

##  I Introduction

The disease is an abnormal condition that has stricken human civilisation
since their existence [1]. Every disease is the medical condition that has
some symptoms and signs for their identification. These symptoms and signs
provide the suitability to identify the disease in the early stage. However,
some diseases do not have identifiable symptoms, which makes them severe. Due
to significant growth in medical science, most of the existing diseases are
cured or halted from further damage to the human body. However, some diseases
spread due to some unknown virus or bacteria and lead to the most deadly
pandemic in humans. One of such pandemic is novel coronavirus named COVID-19
that has potential for maximal death [2, 3]. The controlling mechanism to
hamper the spread of an epidemic like COVID-19 requires an attentive and
strict application of quarantine and social distancing. The effect of the
epidemic (coronavirus) spread, quarantine, and social distancing can be
simulated using data analysis, modelling, and representation techniques of
data mining [4, 5, 6]. These techniques help in predicting disease behaviour
and its corresponding preventive measure.

One of the most widely used techniques for predicting the behaviour of an
epidemic in term of infection spread during an epidemic is the reproduction
number ($R_{o}$). It estimates the spread of secondary infections from the
primary infected persons. Next, the case fatality rate captures the number of
deaths due to an epidemic spread. The case fatality rate plays a vital role in
estimating the actual damage due to an epidemic. In other words, the number of
people that are infected (estimated through reproduction number) losses their
life during an epidemic. Further, there are four states of infection spread in
an epidemic, i.e., no contact, local, untraceable, and source missing
transmission. The first state is the least harmful and often occur at the
initial stage of the spread, whereas, the last state is the critical
transmission state indicating the spread of infection is severe. In the
literature for assessment of epidemic spread, different models are discussed
like Susceptible Infectious Recovered (SIR) [7], Susceptible Infectious (SI),
and Susceptible Infectious and Susceptible (SIS) model.

Figure 1: Illustration of data analysis, modelling and representation with the
prior step of data collection and processing.

The modern era of information encourages us to perform data analysis for
predicting the future course of actions against disease and its behaviour [8].
The prediction is performed by a model that takes the historical data to learn
a pattern (like the pattern of epidemic spread COVID-19). The performance of
these models depends heavily upon data representation and storage. Therefore,
in this paper, we have summarises different techniques presented in the
literature for the estimation of various factors that are responsible for an
epidemic (COVID-19). We further discuss the methods that are predicting the
end time of an epidemic and the total population that is going to be infected
and susceptible. This is the first work as a case study that covers data
analysis, modelling and representation of the coronavirus spread in India.
Fig. 1 illustrate the steps involved prior to the data analysis, modelling,
and representation. After collecting the epidemic data, the first and foremost
task is to perform data processing (like data cleaning, normalization, etc).
Further, analysis and modelling is performed on the processed data. The papers
answer the following question for predicting the spread of infection and
estimating the duration of an epidemic.

  * •

Question 1: What is the role of reproduction number in the estimation of the
population infected during an epidemic such as COVID-19? How to determine the
average of the reproduction number for an epidemic disease dataset? (Section
II-B)

  * •

Question 2: Is the reproduction number capable of estimating the number of
infected people in a particular area during an epidemic? What is the impact of
the case fatality rate on estimating the damage due to an epidemic like
COVID-19? (Section II-C)

  * •

Question 3: What are the infection transmission states in an epidemic? How to
identify the transmission state in a particular area? Is there any existing
algorithm for transmission state identification? (Section II-D)

  * •

Question 4: How to model an epidemic for accurate estimation of infection
spread and recovery rate? Which model is suitable for end-time estimation of
an epidemic? What is the Susceptible Infectious Recovered model? How
Susceptible Infectious Recovered (SIR), Susceptible Infectious (SI), and
Susceptible Infectious and Susceptible (SIS) models estimate the impact of an
epidemic? (Section III)

The rest of the paper is organised as follows. Next section illustrates
Covid-19 open access dataset, the parameters used for measuring the impact of
an epidemic, and the method to identify the state of the epidemic. Section III
illustrates the mathematical models of the epidemic. Section IV illustrates
the empirical results and paper is concluded in Section V.

##  II Dataset and Parameters for measuring the spread of an epidemic disease

In this section, we first illustrate an open access dataset of Covid-19 that
we use in the rest of the paper. We next define the parameters used for
measuring the impact of epidemic, as shown in Fig. 2.

Figure 2: Illustration of parameters for measuring the epidemic.

###  II-A Dataset of Covid-19

This work uses COVID-$19$ dataset available at [9] for predicting and
analyzing the outbreak of epidemic in India. The dataset illustrates the
detail description of the patients infected from coronavirus. Table I depicts
the first $10$ entries of the dataset having different columns including,
patient number, state patient number, date announcement, age bracket, gender,
detected city, detected district, detected state, state code, current status,
notes, contacted from which patient, nationality, transmission type, and
status change date. The column header patient number indicates the actual
patient count in India, and state patient number is a combination of state
code, detected district code and its patient count. For example, the infected
patient $1$ is from Thrissur district (district code: TS) of Kerala (state
code: KL). Therefore, state patient number is KL-TS-P1 for patient $1$. The
column current status indicates the current state of the patient, i.e.,
recovered, hospitalized, or deceased. The column with header notes holds the
details about travelling history of the patient. Next, the local transmission
from one patient to another is recorded in column  contacted from which
patient. Further, nationality followed by transmission type that depicts
whether the transmission of infection is imported from other countries or
local. Finally, the column status change date illustrates the status change of
infected patient to recovered or deceased from infected.

TABLE I: Dataset of Covid-19 [9] (total rows are 20000+ as on April 23, 2020).

| Patient
---
Number
| State
---
Patient
Number
| Date
---
Announced
| Age
---
Bracket
Gender |  | Detected
---
City
| Detected
---
District
| Detected
---
State
| State
---
code
| Current
---
Status
Notes |  | Contracted from
---
which Patient
(Suspected)
Nationality |  | Type of
---
transmission
| Status
---
Change
Date
1 | KL-TS-P1 | 30/01/2020 | 20 | F | Thrissur | Thrissur | Kerala | KL | Recovered | Travelled from Wuhan |  | India | Imported | 14/02/2020
2 | KL-AL-P1 | 02/02/2020 |  |  | Alappuzha | Alappuzha | Kerala | KL | Recovered | Travelled from Wuhan |  | India | Imported | 14/02/2020
3 | KL-KS-P1 | 03/02/2020 |  |  | Kasaragod | Kasaragod | Kerala | KL | Recovered | Travelled from Wuhan |  | India | Imported | 14/02/2020
4 | DL-P1 | 02/03/2020 | 45 | M | East Delhi | East Delhi | Delhi | DL | Recovered | Travelled from Austria |  | India | Imported | 15/03/2020
5 | TS-P1 | 02/03/2020 | 24 | M | Hyderabad | Hyderabad | Telangana | TG | Recovered | Travelled from Dubai |  | India | Imported | 02/03/2020
6 |  | 03/03/2020 | 69 | M | Jaipur | Jaipur | Rajasthan | RJ | Recovered | Travelled from Italy |  | Italy | Imported | 03/03/2020
7 |  | 04/03/2020 | 55 |  | Gurugram | Gurugram | Haryana | HR | Recovered | Travelled from Italy | P6 | Italy | Imported | 29/03/2020
8 |  | 04/03/2020 | 55 |  | Gurugram | Gurugram | Haryana | HR | Recovered | Travelled from Italy | P6 | Italy | Imported | 29/03/2020
9 |  | 04/03/2020 | 55 |  | Gurugram | Gurugram | Haryana | HR | Recovered | Travelled from Italy | P6 | Italy | Imported | 29/03/2020
10 |  | 04/03/2020 | 55 |  | Gurugram | Gurugram | Haryana | HR | Recovered | Travelled from Italy | P6 | Italy | Imported | 29/03/2020

###  II-B Epidemic first parameter: Reproduction number

The epidemic disease modelling incorporates the reproduction number, denoted
by $R_{o}$, to understand the disease outbreak by wrapping up all the
infection transmission in a specified area. The reproduction number specifies
the number of persons being infected by an infectious person (a person
suffering from the disease) [10]. It defines as

######  Definition 1.

Reproduction number of a given region is the ratio of the total number of
persons infected to the total number of persons who are infecting, i.e.,

$R_{o}=\frac{\text{Total persons infected}}{\text{Total who are infecting}}.$
(1)

Fig. 3 illustrates an example scenario where from left to right there is an
increment in the value of $R_{o}$. The number of persons infected by an
infectious person also increase in a similar pattern. The value of $R_{o}$
less then unity indicates that the epidemic elimination and greater than unity
leading to an epidemic outbreak.

Figure 3: An example scenario of basic reproduction number $R_{o}$.

$\bullet$ Importance of $R_{o}$: $R_{o}$ is an essential metric that indicates
how contiguous the disease is?. A disease is said to be highly contiguous when
an infected individual transmits the infection to an unexpectedly large number
of people. The mechanism of providing a proper treatment or appropriate
handling of any disease requires full information about how the infection
develops in the body, the mode of transmission, and the social structure where
it can spread. If all this information is available, then the outbreak can be
detected and monitored quickly and allows us to react accordingly. The
reproduction number $R_{o}$ captures all these necessary information by
incorporating three components: population density, rate of infection, and
rate of recovery or death [11].
$\bullet$ $R_{o}$ in given dataset: Fig. 4 illustrates the value of $R_{o}$
for the given day. It also shows the minimum and maximum $R_{o}$ in the states
of India on each day. We conclude from the result that the difference between
the minimum and maximum value of $R_{o}$ is very high, which indicates that
the epidemic is very high in some state while others are free. The result also
illustrates that minimum value of $R_{o}$ increase with time which indicates
that the epidemic has spread in India. Table II illustrates the value of
$R_{o}$ for the spread of COVID-19 in different states of India from the
dataset available at [9] up till April, $18$ $2020$. The table shows that few
states like Haryana and Punjab have person to person spread of coronavirus
reaches to $20$. The average $R_{o}$ value of some state is less than $1$,
which is a good indication of the lower spread of COVID-19 in India and sub-
continents. However, some states show a value of $R_{o}$ more than $3$ that
requires a significant effort in the spread control methodology in the
country. The overall $R_{o}$ for India is $1.79$ that indicates the COVID-19
is in its initial stage in the country.

Figure 4: Illustration of $R_{o}$ with time. TABLE II: State-wise reproductive
number ($R_{o}$) using Covid-19 dataset [9].

Andaman and Nicobar
---
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4
Andhra Pradesh
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 3 | 0 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.8
Arunachal Pradesh
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Assam
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Bihar
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5.33
Chandigarh
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 1 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2.5
Chattisgarh
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Dadra Nagar Haveli
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Delhi
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3.34
Goa
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Gujarat
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 4.5
Haryana
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 2 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 6
Himanchal Pradesh
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Jammu and Kashmir
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 3 | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2.83
Jharkhand
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1
Karnataka
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 35 | 14 | 10 | 7 | 2 | 2 | 0 | 2 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 1 | 2.73
Kerala
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 7 | 5 | 1 | 3 | 0 | 4 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2.8
Ladakh
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 2 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0.5
Madhya Pradesh
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 2 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3.6
Maharashtra
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 2 | 2 | 2 | 1 | 2 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 3.7
Manipur
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Mizoram
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Odisha
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1
Puducherry
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
Punjab
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 7 | 0 | 1 | 0 | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 3.91
Rajasthan
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 4 | 3 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.75
Tamil Nadu
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 6 | 0 | 2 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 3.30
Telangana
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 5 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1.16
Uttar Pradesh
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 7 | 0 | 1 | 1 | 2 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2.58
Uttrakhand
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0
West Bengal
Person contact | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | Avg.
Count | 0 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 2.5
India |  Avg. = 1.79

###  II-C Epidemic second parameter: Case fatality rate

The case fatality rate, denoted by $C_{r}$, provides an estimation of the
number of the person who dies among the total infected person from disease
during its spread in a particular area (state/country). It defines as,

######  Definition 2.

Case fatality rate of a given region is the ratio of the total number of
persons who died from the epidemic to the total number of the persons who are
infected, i.e.,

$C_{r}=\frac{\text{Total death from the epidemic}}{\text{Total number of
persons who are infected}}\times 100.$ (2)

$\bullet$ Importance of $C_{r}$: The reproduction number can only give the
idea of the rate of disease spread, but cannot estimate how many persons can
lose their life in the entire cycle of the spread. Therefore, the case
fatality rate is crucial for such estimation. An epidemic like measles with
$R_{o}$ around $18$ has a lower case fatality rate to that of Ebola with
$R_{o}$ around $1.5$ [12]. Thus, the case fatality rate plays a vital role in
estimating the actual effect of a disease on humanity. For example, Ebola with
lesser $R_{o}$ is more severe than the measles.
$\bullet$ $C_{r}$ in given dataset: Fig. 5 illustrates the value of $C_{r}$
for the given day. It also shows the minimum and maximum $C_{r}$ on the states
of India each day. We conclude from the result that the difference between the
minimum and maximum value of $C_{r}$ is very high, which indicates that an
epidemic is very high in some state while others are free. The result also
illustrates the minimum value of $C_{r}$ increases with time. It indicates the
epidemic is spreading in India at a slower rate. Table III illustrates the
case fatality rate of different states of India from COVID-19. The fatality
rate of Himanchal Pradesh is highest, i.e., $1.79$. High fatality rate
indicates that a large number of infected persons die in these states.
However, the fatality rate also increases if the number of infected patients
is low. It seems right in the case of Himanchal Pradesh too, as the reported
cases in the state are only $39$ out of which one is deceased. The fatality
rate of few states is zero as there is no casualty reported so far in them
like Andaman and Nicobar, Assam, Chandigarh, etc. The overall fatality rate of
India reported up till April, $18$ $2020$ is $0.34$, which indicates person
infection rate is low in India and they are gaining significant recovery from
the COVID-19.

Figure 5: Illustration of $C_{r}$ with time in India. TABLE III: Illustration of case fatality rate in states of India. State/UT |  | Fatality
---
rate (in%)
State/UT |  | Fatality
---
rate (in%)
Andaman and Nicobar | 0 | Andhra Pradesh | 0
Arunachal Pradesh | 0 | Assam | 0
Bihar | 0.71 | Chandigarh | 0
Chattisgarh | 0 | Dadra Nagar Haveli | 0
Delhi | 0.044 | Goa | 0
Gujrat | 0.17 | Haryana | 0
Himanchal Pradesh | 2.56 | Jammu and Kashmir | 0.246
Jharkhand | 0 | Karnataka | 2.48
Kerla | 0.46 | Ladakh | 0
Madhya Pradesh | 0.126 | Maharashtra | 0.195
Manipur | 0 | Mizoram | 0
Odisha | 1.21 | Puducherry | 0
Punjab | 0.72 | Rajasthan | 0.052
Tamil Nadu | 0.061 | Telangana | 0.11
Tripura | 0 | Uttar Pradesh | 0.069
Uttrakhand | 0 | West Bengal | 0.874
India | 0.34 (in%)

###  II-D Epidemic third parameter: State of given region

Measures such as the social and physical distancing reaccelerate the spread of
the disease, infections by stopping the chains of transmission from one person
to another. These preventive measures reduce the contamination from the
persons or surfaces while encouraging connections with family members. The
spread of the infection is broadly categorised in the following four infection
transmission states [13]:

  1. 1.

State 1 (No contact transmission): In this transmission state, the infection
from an infected person does not proceed further to any other person. The
infected person remains confined with the disease-causing virus without
further dissipation. For an epidemic disease like COVID-19, this state is the
most crucial state to control the spread of coronavirus through quarantine and
social isolation. However, several infected persons in this state are unaware
of their infection. Thus, passing the infection to the person in the contact
and accelerate the growth of the disease. For example, the topmost layer in
Fig. 6 demonstrates two persons (i.e., A and B) are infected with coronavirus
and there is no any further transmission. This may result due to the complete
isolation of both A and B.

  2. 2.

State 2 (Local transmission): The isolation compromise and delay in the
detection of infection in a person leads this transmission state. The infected
person transfers the infection to the person in its close contact. This type
of transmission is usually observed in the case when the infected person comes
in close contact with a person like a friend or a family member. At this
stage, it is easy to trace spread and quarantine people. The quarantine of the
infected person must be strictly followed to avoid this state to enter into
the next state. The second layer in Fig. 6 indicates the transmission is from
infected person A to C and B to D.

  3. 3.

State 3 (Untraceable transmission): The unaltered social contact between the
persons results in transmission state third. In this state, the concept of
transmitting and being transmitted is followed. Here, an infected person
transfers the infection to another person who in turn transfers to the third
person. The third stage is also when the source of the infection is
untraceable. The infected people in this stage are identified who haven’t had
travel history. This state, in case of COVID-19, is controlled by adopting a
hard mechanism like lockdowns over the entire country. The third stage in Fig.
6 illustrates State 3. In this state, A transfers infection to C that further
propagates to E. Similar transmission is observed among B, D, and F.

  4. 4.

State 4 (Source missing transmission): This is the final infection
transmission state where the source of infection is unidentified. It can be
understood by an example where a person is infected in spite of not coming in
close contact with an infected person and had no travel history from an
infected country. This transmission state is most severe and leads to
infection spread to all the persons in a particular area where the infection
persists. The last layer in Fig. 6 illustrates the fourth state of
transmission, where, the source is unidentified.

Figure 6: An example scenario representing four transmission state of an
epidemic.

$\bullet$ Algorithm for identify state of epidemic: Algorithm $1$ illustrates
the steps involved in the identification of the infected population in the
different transmission state. In the algorithm, we introduce a variable
${infection\\_tramission}$ that predicts the chance of a person to lie in a
particular transmission state depending upon its values (i.e., $0$, $1$, and
more). ${infection}_{transmission}=0$ indicates that there is no person-to-
person infection transmission and ${infection}_{transmission}=1$ indicate that
infection transmission is in State $2$. Further, if the infection is
transmitted from the first person to second person and then to the third or
the source is untraceable indicates the transmission State $3$. If the
transmission state does not belong to the above three, then the infection
transmission State $4$ is reached. At State $4$, it is assumed that the entire
population is equally likely to be infected. The estimation of infection
transmission State $4$ is out of scope from this algorithm. To hamper the
infection transmission, the measures for the peoples includes the closure of
non-essential services, work from home, avoid social gathering, isolating
vulnerable groups, etc. These measures are successful if peoples follow
conjunctive measures such as hand washing frequently.

Input:  Population $M$ of a particular area (state/country);

Output: States in which all the infected population lie;

/*Variable initialization*/

1 ${Set}_{1}$={}, ${Set}_{2}$={}, and ${Set}_{3}$={}.

2 ${infection\\_tramission}\leftarrow 0$.

3 Identify the infectious population $I$.

4 for _ $i\leftarrow$ 1 to $\|I\|$_ do

5 if _${infection\\_tramission}_{i}==0$_ then

6 ${Set}_{1}\leftarrow{Set}_{1}.\text{append}(\text{person}_{i})$.

7 else if _${infection\\_tramission}_{i}==1$_ then

8 ${Set}_{2}\leftarrow{Set}_{2}.\text{append}(\text{person}_{i})$.

9 else

10 ${Set}_{3}\leftarrow{Set}_{3}.\text{append}(\text{person}_{i})$.

11 The resultant sets ${Set}_{1}$, ${Set}_{2}$, and ${Set}_{3}$ contain the
infected person in different transmission state, i.e., State $1$, State $2$,
and State $3$, respectively.

Algorithm 1 Epidemic state identification

$\bullet$ State in given dataset: Fig 7 illustrates the number of persons each
state with time. As we expect, the number of cases are registered more with
time. An interesting observation from this result is that the number of
persons in state 2 increases sharply then state 1. This is because of the
state 1 person infected to state 2 with a high value of $R_{o}$. Table IV
illustrates the different state of India, having different person indulge in
all four transmission states. Due to a large number of cases in the state of
Maharashtra, the number of persons in transmission state four is high. The
table is created using the data available at [9] (April $18$ $2020$). An
important conclusion from the table is that India is in Stage 2 ($90.11\%$)
and shifting in Stage 3 ($5.13\%$).

Figure 7: Illustration of state 1, state 2, and state 3 persons. TABLE IV: Identify the state of transmission. State/UT | State1 | State2 | State3
---|---|---|---
Andaman and Nicobar | 5 | 1 | 2
Andhra Pradesh | 18 | 731 | 64
Arunachal Pradesh | - | - | 1
Assam | 1 | 10 | 25
Bihar | 24 | 111 | 8
Chandigarh | 12 | 14 | 1
Chattisgarh | 6 | 29 | 1
Dadra Nagar Haveli | - | - | -
Delhi | 22 | 2192 | 34
Goa | 7 | - | -
Gujrat | 121 | 2280 | 6
Haryana | 29 | 219 | 16
Himanchal Pradesh | 3 | 21 | 15
Jammu and Kashmir | 21 | 380 | 6
Jharkhand | 2 | 47 | -
Karnataka | 89 | 224 | 114
Kerla | 259 | 156 | 22
Ladakh | 7 | - | -
Madhya Pradesh | 17 | 1568 | 2
Maharasthra | 82 | 5547 | 20
Manipur | 1 | - | 1
Odisha | 19 | 64 | 1
Puducherry | 1 | 2 | 4
Punjab | 30 | 223 | 25
Rajasthan | 96 | 1811 | 28
Tamil Nadu | 70 | 1046 | 513
Telangana | 29 | 888 | 26
Tripura | - | 2 | -
Uttar Pradesh | 25 | 1264 | 160
Uttrakhand | 5 | 37 | 4
West Bengal | 16 | 438 | 2
INDIA (Total) | 1017 | 19305 | 1101
INDIA (in %) | 4.74 | 90.11 | 5.13

##  III Epidemic Models and Contact tracking

Data models help in quantifying the damage caused by a disease like epidemic
COVID-19. The model simulates the spread of infection from one person to
another and its geographical spread on the entire population. The models also
help in identifying the time after which the epidemic end or its effect can be
reduced. Data models predict the number of population that is infectious,
susceptible and recovered in the near future. In this section, we discuss the
exiting epidemic models and how to use on the dataset.

###  III-A Susceptible, Infectious, and Recovered (SIR) model

The most generalised framework for such simulation is based on Susceptible,
Infectious, and Recovered (SIR) model [14]. The SIR model is a three-phase
model where a person partially stays in a phase and switches to another with a
specific course of action. Let an epidemic spread of disease covers the entire
population of $M$ persons having Susceptible $S$ and Infectious $I$ with
Recovered $R$ persons, then $M=S+I+R$. The phases of SIR models are
illustrated in Fig. 8. The description of each phase is as follows.

Figure 8: Illustration of susceptible, infectious, and recovered persons of a
particular area.

Susceptible: All the person in a particular area who are not infected with the
disease and have no reported immunity against the infection are categorised as
Susceptible (S). The absence of immunity in a person is the result of no prior
exposure to the infection, non-vaccinated against such infection, and no
confirm disease resistance. The susceptible person moves to the infectious
phase by social contact with an infected person.

Infectious: During the disease outbreak like COVID-19, some person come in
close contact with the infected person and got infected afterwards, forming a
group of persons termed as Infectious (I). These persons are solely
responsible for the spread of the disease in the whole community. The
quarantine of such infected persons preserves the community from being
infectious. Social isolation from such a person is the only remedy against any
epidemic outbreak like COVID-19.

Recovered: Every disease like epidemic COVID-19 has a specific cycle for
infection persistence. After this cycle, the infected person recovers from the
infection with proper treatment. However, due to the severe spread of
infection in the human body and inadequate cure, the infection can result in
death. The summation of total recovered and total dead is termed as Recovered
(R) or sometimes removed. The persons that are in this category are free from
further infection.

There is always a phase of transmission among the persons, i.e., from
susceptible to infectious and infectious to susceptible. The transmission of
the disease changes the number of persons in a phase over time. The SIR model
can be represented using [7] as follows

$\displaystyle R=$ $\displaystyle\alpha_{2}I-\gamma I,$ (3) $\displaystyle I=$
$\displaystyle\alpha_{1}S,$ (4) $\displaystyle M=$ $\displaystyle S\cup I\cup
R,$ (5)

where, $\alpha_{1}$ and $\alpha_{2}$ are the force of infection and recovery
rate, respectively. $\gamma$ denotes the death rate and $\cup$ is the union
operator. Eq. 5 holds when the entire population of a specified area is
susceptible to the epidemic outbreak.

Force of infection $\alpha_{1}$ is the rate at which the susceptible person
becomes infectious on the close contact with the infected person. Recovery
rate $\alpha_{2}$ is the rate at which an infected person recovers from the
infection using the proper treatment. The force of infection is not a constant
quantity and is proportional to the transmission rate $\tau$. The transmission
rate $\tau$ is the product of the rate of contact $r_{c}$ and the probability
of transmission $p_{t}$ from susceptible to infectious phase. $\tau$ is
defined as

$\tau=r_{c}\times p_{t}.$ (6)

Using Eq. 6 we can define force of infection $\alpha_{1}$ as

$\alpha_{1}(I)=\tau\Big{\\{}\frac{I}{M}\Big{\\}}\times 100,$ (7)

where, $\\{\frac{I}{M}\\}\times 100$, represents percentage of population,
which is infectious among the total population in a particular area.

To capture the population change in a particular phase, SIR model adopts
differential equations. A differential equation captures the progression of
the disease. The rate of change in the susceptible population (S), infectious
population (I), and recovered population (R) as defined as follows

$\displaystyle\frac{\mathrm{d}S}{\mathrm{d}t}$
$\displaystyle=-\alpha_{1}(I)S.$ (8)
$\displaystyle\frac{\mathrm{d}I}{\mathrm{d}t}$
$\displaystyle=\alpha_{1}(I)S-\alpha_{2}I.$ (9)
$\displaystyle\frac{\mathrm{d}R}{\mathrm{d}t}$ $\displaystyle=\alpha_{2}I.$
(10)

An essential step after estimating the rate of change in $S$, $I$, and $R$
using the differential equation, is to determine the equilibrium state of the
SIR model. To describe the epidemic methodology, two equilibria exist side-by-
side, i.e., Disease Free Equilibrium and Endemic Equilibrium [15]. The Disease
Free Equilibrium comes into the picture when there is no person left with the
infection in a particular area. However, in the case of Endemic Equilibrium
there always persists an infectious individual. It requires a constant supply
of susceptible person to maintain the equilibrium. Both the equilibrium
persists when the value of $R_{o}$ is less than $1$. The SIR model reaches an
equilibrium state if the rate of transmission form susceptible to infectious
is equal to infectious to recover. Algorithm $2$ illustrates the process of
estimating the time $\mathbf{t}$ for which an epidemic persists using
Susceptible, Infectious, and Recovered model.

Input: Population $M$ of a particular area facing epidemic;

Output: Time $T$ for which the epidemic persist;

/*Variable initialization*/

1 $\alpha_{1}\leftarrow 0,\alpha_{2}\leftarrow 0,\tau\leftarrow 0,\text{ and
}r_{c}\leftarrow 0$.

2 Time $\mathbf{t}\leftarrow 0$ and change in time $\Delta t$.

3 Calculate reproduction number $R_{o}$.

4

5while _$R_{o} <1$ or
$\frac{\mathrm{d}S}{\mathrm{d}t}=\frac{\mathrm{d}I}{\mathrm{d}t}=\frac{\mathrm{d}R}{\mathrm{d}t}=0$_
do

6

7 Estimating transmission rate $\tau$

8 Calculate rate of contact ($r_{c}$) among the persons.

9 Calculate probability of transmission ($p_{t}$).

10 $\tau\leftarrow r_{c}\times p_{t}$.

11 Estimate force of infection $\alpha_{1}$ using Eq. 7.

12 Calculate $R$ and $I$ for given $S$.

13 $I=\alpha_{1}S$ and $R=\alpha_{2}I-\gamma I$ using Eqs. 4 and 3.

14 Calculate the derivatives $\frac{\mathrm{d}S}{\mathrm{d}t}$,
$\frac{\mathrm{d}I}{\mathrm{d}t}$, and $\frac{\mathrm{d}R}{\mathrm{d}t}$.

15 $\mathbf{t}\leftarrow\mathbf{t}+\Delta t$.

16 Either Disease Free or Endemic Equilibrium is reached.

The epidemic will be ended after $\mathbf{t}$ time.

Algorithm 2 Susceptible, Infectious, and Recovered

$\bullet$ SIR model on the given dataset: Fig 9 illustrates the relationship
between susceptible, infectious, and recovered cases in India up till April
$18$ $2020$. The figure demonstrates that infectious persons grow up to a
limiting value, where recovery starts at a higher pace. It compels most of the
people to come out of the susceptible, and the outbreak is about to die.
During the construction of the result, we have taken the value of reproduction
number $R_{o}=1.79$, i.e., current status for India with a total $12,079$
active cases. We can observe that if the rate of new case generation in India
is decreasing or not exceeding the current rate, then the COVID-19 breakout
will be ended in the next $70$ days.

Figure 9: Illustration of SIR model on coronavirus dataset of India up till
April $18$ $2020$.

###  III-B Susceptible Infectious (SI) model

Chermack and McKinsey [16] proposed the Susceptible Infectious (SI) model. The
authors assumed that the total population in a particular area during epidemic
breakout belongs either to susceptible or infectious. Apart from the SIR
model, they only consider the susceptible and infectious population without
considering the recovered cases in a particular location. Let at time $t=0$
(time at which the estimation begins), the number of infectious and
susceptible peoples are represented as $I_{o}$ and $S_{o}$, respectively. Let
$\tau$ represents the transmission rate given in Eq. 6. Then, the rate of
change in susceptible and infectious is given as

$\displaystyle\frac{\mathrm{d}S}{\mathrm{d}t}=-\tau SI.$ (11)
$\displaystyle\frac{\mathrm{d}I}{\mathrm{d}t}=\tau I(1-I).$ (12)

Upon further calculation as given in [16], the number of infectious during an
epidemic breakdown can be obtained as follows

$I=\frac{I_{o}e^{\tau t}}{1-I_{o}+I_{o}e^{\tau t}}.$ (13)

$\bullet$ SI model on the given dataset: Fig. 10 illustrates the fraction of
the infectious population for past $30$ days of the epidemic (COVID-19)
breakout in India. The result shows that the curved we obtained is an S-shaped
curve for infectious. The curve grows exponentially from the early stage of
infection in the country the infection starts spreading. During the result
estimation, we have taken the value of reproduction number $R_{o}=1.79$, i.e.,
current status for India with a total $12,079$ active cases as on April $18$
$2020$. We can observe that if the rate of new case generation in India is
increasing, which in turn increases the infectious population in the country.
This result seems to be an alarming situation for taking more preventive
measures to hamper the growing cases in the country.

Figure 10: Illustration of infectious population using SI model on coronavirus
dataset of India up till April $18$ $2020$.

###  III-C Susceptible Infectious and Susceptible (SIS) model

Susceptible Infectious and Susceptible (SIS) model is an extension of the SI
model [17]. The authors assumed that the people who recovered from the
infection are still susceptible. It means the those who recovered from
infection during an epidemic has not developed immunity against it. The rate
of change in susceptible is given as

$\frac{\mathrm{d}S}{\mathrm{d}t}=-\tau SI+\alpha I,$ (14)

where, $\tau$ and $\alpha$ represents the transmission rate and fraction of
the infected population, respectively. The term $\tau SI$ can be understood as
an average infection transmitted by an individual through sufficient contacts
for transmitting the infection to $\tau M$ susceptible population, for $M$ as
the population size of a particular area. The probability that a given person
can infect other is $S/M$ in this case. Therefore, the infection transmission
per unit time is $\tau S$ by a single individual. The rate of change in
Infectious population is as follows

$\frac{\mathrm{d}I}{\mathrm{d}t}=\tau SI-\alpha I.$ (15)

Using the assumption that the epidemic could infect the entire population such
that $M=S+I$. Therefore, we can obtain

$\displaystyle\frac{\mathrm{d}I}{\mathrm{d}t}=$ $\displaystyle\tau
I(M-I)-\alpha I,$ $\displaystyle=$ $\displaystyle(\tau M-\alpha)I-\tau I^{2}.$
(16)

Solving for $\frac{\mathrm{d}I}{\mathrm{d}t}=0$, we can obtain two possible
equilibria, i.e, at $I=0$ and $I=M-\frac{\alpha}{\tau}$. When we achieve the
equilibrium then the epidemic is about to end.

$\bullet$ SIS model on the given dataset: The result illustrated in Fig 11
depicts the relationship between susceptible and the infectious fraction of
cases in India up till April $18$ $2020$. The result is estimated by taking
the value of the reproduction number as $R_{o}=1.79$, i.e., current status for
India with a total $12,079$ active cases up to April $18$ $2020$. Similarly,
to previous observations in the SIR model in Fig. 9, here also, we can observe
that if the rate of new case generation in India is decreasing or not
exceeding the current rate, then the COVID-19 breakout will be ended in the
next $70$ days.

Figure 11: Illustration of infectious and susceptible population curve of SIS
model on dataset of India upto April $18$ $2020$.

##  IV Results using empirical analysis

This work carried experiment using empirical analysis of population size
$1000$ persons. We consider the value of the reproduction number ($R_{o}$) as
$1.79$. This value is taken as the average reproduction number for COVID-19 in
India is $1.79$. The empirical analysis in this paper is based on the
following formula derived from SIR model [18].

$I_{total}=I_{o}+r_{c}\Big{(}\frac{I}{M}\Big{)}\times 100\times S\times
p_{t},$ (17)

where, $I_{o}$ is the initial infected population, $r_{c}$ is the rate of
contact, and $p_{t}$ is the transmission probability.

####  IV-1 Impact of Infectious and Susceptible

This work estimates the infectious population when the susceptible increase
from $25\%$ to $100\%$, with the rate of contact $r_{c}=1$ and the probability
of transmission $p_{t}=0.3$. Part (a) of Fig. 12 illustrates that the number
of infected persons increases with the increase in the % susceptible. If the
susceptible reaches to $100\%$ on population size $1000$, then the infection
can spread up to $487$ persons. Similarly, part (b) of Fig. 12 illustrates
that the increment in the infectious accelerates the increment in the
susceptible population and reaches $564$ when population size is $1000$ and
infectious is $40\%$. These results illustrate that infectious and susceptible
are directly related to each other and increases in a similar manner.
Therefore, the effect of an epidemic such as COVID-19 can be reduced by
decreasing either the susceptible or infectious population.

Figure 12: Impact of number of population on infectious and susceptible,
respectively. Figure 13: Impact of rate of contact ($r_{c}$) and probability
of transmission ($p_{t}$) on number of infectious and susceptibles. Figure 14:
Impact of case fatality rate and recovery rate on infectious and susceptible.
(a) having both case fatality and recovery increasing, (b) increasing case
fatality and decreasing recovery, (c) increasing recovery and decreasing case
fatality, and (d) both decreasing.

####  IV-2 Impact of rate of contact and probability of transmission

The rate of contact and the probability of transmission play a vital role in
determining the infection spread of an epidemic like COVID-19 in a particular
area. Therefore, this paper calculates the number of infected and susceptible
at a varying rate of contact and probability of transmission separately.
Firstly, we put the probability of transmission as a constant $(p_{t}=0.3)$
and the rate of contact as a variable. Part (a) of Fig. 13 illustrates that
the increment in the infectious is nearly linear with the rate of contact and
population size. At higher population size beyond $800$, the curve starts
flattening as the infection has covered roughly half of the population.
Similarly, a linear increment is observed for susceptible to the increase in
the rate of contact.

Further, parts (c) and (d) of Fig. 13 show that the transmission probability
is highly sensitive in determining the rate of infection spread and increment
in the susceptible. We have considered the rate of contact ($r_{c}=1$) for
calculating the value of infectious and susceptible. Part (c) of Fig. 13
demonstrates that the infectious count is directly proportional to the
transmission probability and shows an exact linear increment with the increase
in the population size. In a similar pattern, part (d) of Fig. 13 illustrates
that the number of susceptible increases with the increase in the transmission
probability as the chance to get infected is higher when the population
increases.

####  IV-3 Impact of case fatality rate and recovery rate

Next, we evaluate the impact of the case fatality rate and recovery rate on
the rate of change in infectious and susceptible. The case fatality rate and
recovery rate forms following four cases

  1. 1.

Both case fatality rate and recovery rate increase: When both case fatality
rate and recovery rate increase, then the rate of infectious and susceptible
people are increased simultaneously. The susceptible population changes more
rapidly to that of infectious. The increment in both infectious and
susceptible continues up till a thresholding limit, where the influence of the
case fatality rate start degrading to that of recovery rate. Part (a) of Fig.
14 shows the variation in infectious and susceptible when both case fatality
and recovery rates increases.

  2. 2.

Case fatality rate increase and recovery rate decrease: It is the most adverse
situation for an epidemic breakout, where the case fatality is increasing, and
recovery is decreasing. This clearly indicates that the epidemic is in the
worst phase, and it can infect the entire population in a particular area.
Part (b) of Fig. 14 illustrates the adverse situation where both infectious
and susceptible are increasing.

  3. 3.

Both case fatality rate and recovery rate decrease: This is similar to that of
the previously discussed case, but one positive thing that mitigates form
severe damage is the decrement in the case fatality rate. Part (c) of Fig 14
indicates that the simultaneous decrement in both case fatality rate and
recovery rate increases the infectious and susceptible.

  4. 4.

Case fatality rate decreases and recovery rate increases: The last combination
for case fatality and recovery rate is the case when the case fatality rate
decreases and recovery rate increases. It is the most favourable situation
where there is a clear indication that the epidemic is reducing. Part (d) of
Fig. 14 illustrates that both infectious and susceptibles decrease under such
circumstances.

##  V Conclusion

The pandemic COVID-19 is spreading at a higher pace around the globe. The
empirical and data analysis in this paper tries to figure out the impact of
various parameters such as the probability of transmission, rate of contacts,
infectious and susceptible. The results indicate that the spread of COVID-19
can be analysed, modelled, and represented in the tabular and graphical
format. The data analysis provides the complete in-site of the coronavirus
spread and also highlights the reproduction number of COVID-19 is between
$1.5$ to $4$, and its fatality rate is low. The paper covers those models that
are widely in the various estimations in an epidemic and identifies the
advantage and disadvantage of one model over others. The representation
provides a more natural way to visualise the actual effect of the epidemic
spread.

## Acknowledgements

The authors thank to Mr. Vinit Kumar Patra for doing the implementation and
data preprocessing.

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|
Rahul Mishra received M.Tech. degree in Computer Science and Engineering from
Madan Mohan Malaviya Univesity of Technology, U.P., India, in 2017. Presently,
he is pursuing Ph.D. in Computer Science and Engineering, IIT (BHU) Varanasi,
India. His research interests include wireless sensor networks, deep learning,
and fog computing.
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Hari Prabhat Gupta received Ph.D degree in Computer Science and Engineering
from IITG, Guwahati, India. He is currently working as Assistant Professor in
Department of Computer Science and Engineering, IIT (BHU), Varanasi, India.
His research interests include wireless sensor networks, machine learning, and
distributed algorithms.
---|---
|
Tanima Dutta received Ph.D degree in Computer Science and Engineering from
IITG, Guwahati, India. She is currently working as Assistant Professor in
Department of Computer Science and Engineering, IIT (BHU), Varanasi, India.
Her research interests include multimedia, digital forensics, wireless sensor
networks, and algorithms.
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