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Journal of Applied Nonlinear Dynamics 12(2) (2023) 405--425 | DOI:10.5890/JAND.2023.06.015

Kalyan Das$^1$, M. N. Srinivas${}^{2} $, Pabel Shahrear${}^{3} $, S. M. Saydur Rahman${}^{3 }$, Md M.H. Nahid${}^{4 }$, \\ B. S. N. Murthy$ {}^{5} $

${}^{1 }$ Department of Basic and Applied Sciences, National Institute of Food Technology Entrepreneurship and Management, HSIIDC Industrial Estate, Kundli-131028, Haryana, India

${}^{2 }$ Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamilnadu, India

${}^{ 3 }$ Department of Mathematics, Shahjalal University of Science and Technology, Sylhet, Bangladesh

${}^{4}$ Department of Computer Science and Engineering, Shahjalal University of Science and Technology, Sylhet, $ Department of Mathematics, Aditya College of Engineering and Technology, JNTUK, Andhrapradesh, India

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Abstract

We look at the SQIRP mathematical model for new coronavirus transmission in Bangladesh and India in this study. The basic reproduction number of the SQIRP system is designed using the next cohort matrix process. The SQIRP system has asymptotically stable locally at an infection-free equilibrium point when the basic reproduction number is not more than unity and unsteady when the value is greater than unity. The SQIRP system is found to go through a backward bifurcation, which is a novel perspective for Coronavirus infection transmission. The infection-free equilibrium and endemic equilibrium are shown to be asymptotically stable globally using the Lyapunov function hypothesis and the invariance principle of Lasalle. A SQIRP system with backward bifurcation is explored using stochastic analysis. The ecological stochasticity in the appearance of white noise best describes the system's value. To verify the results, more numerical simulations are run. } [\hfill Covid-19\par \hfill Bangladesh\par \hfill India\par \hfill Stability\par \hfill Bifurcation\par \hfill Stochasticity][Jorge Duarte][17 December 2021][23 August 2022][1 April 2023][2023 L\&H Scientific Publishing, LLC. All rights reserved.] \maketitle %\thispagestyle{fancy} \thispagestyle{firstpage} \renewcommand{\baselinestretch}{1} \normalsize \section{Introduction} The world health organisation has declared a pandemic and a worldwide general health crisis due to a continuous epidemic of the new coronavirus, which the entire human race is seeking to handle. It is a rapidly changing and rising situation \cite{1}. Since the virus's first appearance in December 2019, around 2,000,000 persons in 188 countries in the region of the earth have been confirmed as new coronavirus cases. Because no vaccinations or antiviral drugs have been approved for the infection, the best general health measures to prevent the Coronavirus are on-remedial mediations to control the spread of the infection \cite{2, 3}. The main goals of such general well-being interventions are to prevent disease spread by separating individuals and interfering with the transmission. Throughout the world, billions of individuals are staying in residence to prevent the spread of the disease. Isolate, quarantine, social distance, and network restriction are among the preventive measures being used in several countries \cite{4, 5}. Analysts from all around the world are working to improve the science of emerging coronaviruses and the research on disease transmission in the Coronavirus \cite{6}. Bangladesh one of the lowest middle-income countries and the most population density in the globe is fighting the infection extend. On March 7, the country confirmed the first Coronavirus case in a long time, but several experts speculated that Covid-19 may have infiltrated the nation prior and gone unnoticed due to insufficient surveillance. Beginning on March 16, the nation forced a 14-day compulsory isolate on all voyagers who entered the nation. On March 19, the nation sent the military to manage two isolated offices in Dhaka. From the principal seven-day stretch of April, Bangladesh began to defer every single mass get-together. Subsequently, the legislature restricted all political, social, social, and strict meetings and get-togethers in the nation \cite{7}. On March 25, Bangladesh pronounced the requirement of lockdown for 10 days powerful from March 26. With the requirement of this lockdown, travel on water, rail, and air courses is prohibited and street transmission is incomplete. All superfluous associations, organizations, and instructive foundations are shut, aside from drug stores and food supplies. The nation's endeavors to lessen the spread of the infection in Bangladesh endured in their execution because of the absence of coordination between various specialists and groups \cite{8, 9, 10, 11}. Later, in two occurrences, the nation proclaimed expansions of the cross-country lockdown, keeping it set up through April 25. Five weeks after the identification of the first Coronavirus case in Bangladesh, the IEDCR had just tried 11, 223 individuals, establishing around 68 tests for every million populaces. It is maybe among the most exceedingly awful positioned nations for novel coronatesting rate, however, the death rate is nearly higher. The Government should approach to ensure that its minor populace approaches legitimate cleanliness, which might be by providing free sanitizer and versatile washrooms. Extra estimates must be taken immediately, foreseeing the potential test that would be looked at by the emergency clinics on account of an upsurge of Coronavirus cases \cite{12}. Mathematical modelling based on differential equations has the potential to give a comprehensive mechanism for disease dynamics as well as to examine the efficacy of control strategies to reduce disease burden \cite{13, 14, 15}. There have also been some mathematical studies of Covid-19 transmission dynamics in Bangladesh and India \cite{10}. Kucharski \cite{16} recently suggested an epidemic model based on a mathematical model that accommodates the impacts of infection transmission and control remedies. However, due to the tiny size of Bangladesh, their prediction model may not produce credible findings. There are a variety of models for modelling infectious diseases in the literature, but only a few have been used primarily for nations with a large number of cases, such as China, Italy, and Spain. Various mathematical models have been employed by Phuaet et al \cite{17} to determine the transmission of the disease, predict the number of cases, and healthcare facilities in combating COVID-19 spread. However, according to WHO recommendations \cite{18}, Bangladesh's existing healthcare infrastructure is not very strong, especially in the event of community spread. Using time scale data, to forecast novel corona outbreaks in China underneath particular community actions was investigated in SEIR and the non-natural intellect approach by Yang et al \cite{19}, Engbert et al \cite{20} used a stochastic SEIR system to anticipate the behaviour of novel corona cases in Germany using various scenarios. Many studies \cite{21,22,23,24,25,26,27,28,29, 30, 31,32} have looked into the current situation of the new coronavirus in various epidemiological areas. To solve the epidemic model SAEIQRS, Chakraborty \cite{33} employed a compartmental model to develop a series of solutions based on the differential transformation method (DTM) (Susceptible-Antidotal-Exposed-Infected-Quarantined-Recovered-Susceptible). Godia et al \cite{34} Used the SEIR model to assess the novel corona cases in Italy, taking into account strong social distancing attempts to restrict disease spread. Allen and Cruz \cite{35, 36} is a decent treatment of the basic reproduction number and its consequences, which is crucial in treating novel corona. Cruz \cite{36} investigated the intervention effect on COVID-19 in Brazil using the SEIR-A model. Chatterjee \cite{37} used a mathematical approach to apply a stochastic SEIQRD model on novel corona data from India and its different regions through May 12, 2020. Mandal et al \cite{38} used five-time dependent classes to analyse novel corona cases in distinct states in India: susceptible, exposed, hospitalised infected, quarantined, and recovered. Influence of covid to sacrifice lives due to xenophobia proposed by Manun and Griffiths \cite{39}. Using a conventional SIRDC system of the novel corona, Fernandez \cite{40} evaluated the parameters of many nations and states in the universe. \section{Formulation of mathematical model} Using epidemiological models, one can obtain a piece of dependable and precious order regarding infection control and extension. As a result, our purpose is to look into the novel corona situation in Bangladesh. In this state, we designed the SQIRP model from the SIR model's creator. In this process, the system SQIRP divides into five compartments, each representing a different instance $t$. Here $S (t)$ denotes the uninfected susceptible category, $Q(t)$ denotes the quarantine, which may include non-contaminated, contaminated, and decease, $I(t)$ denote the number of contaminated persons who must be treated or die, and $R(t)$ denotes the recovered persons who have been cured of the infection. At last, $P (t)$ symbolizes the infection that kills people, and $N (t)$ represents the whole in-habitants at instance t. i.e., $N (t )=S (t )+Q (t )+I (t )+R (t )+P (t )$. Fig.1 shows the schematic representation of the SQIRP Covid-19 transmission model. The following properties are maintained by the model. \begin{enumerate} \item At a rate of $\beta _{1} $ and $\beta _{2} $ susceptible people are transferred to quarantine and infected classes. \item People under quarantine are moved into the susceptible category at a specific time $\varepsilon $ and they are also transferred into the infected and death classes at a certain rate $\varphi $ and $\sigma .$ contaminated individuals are being separated from the infection at a rapid time $\gamma $. \item The infection kills affected persons at an alarming pace $\delta $. \item People who have recovered become susceptible again at a rapid rate $\alpha $. \end{enumerate} The compartmental arrangement and flow instructions of the SQIRP system can be expressed using a focussed schematic diagram based on these assumptions, as shown below, the following system of differential equations has a modified model construction. \begin{equation} \label{Eq__1_} \left. \begin{split} &{\frac{dS(t)}{dt} =-\frac{\left(\beta _{1} +\beta _{2} \right)}{N} S\left(t\right)I\left(t\right)+\varepsilon Q\left(t\right)+\alpha R\left(t\right)} \\ &{\frac{dQ(t)}{dt} =\frac{\beta _{1} S\left(t\right)I\left(t\right)}{N} -\varepsilon Q\left(t\right)-\varphi Q\left(t\right)-\sigma Q\left(t\right)} \\ & {\frac{dI\left(t\right)}{dt} =\frac{\beta _{2} S\left(t\right)I\left(t\right)}{N} -\gamma I\left(t\right)-\delta I\left(t\right)+\varphi Q\left(t\right)} \\ &{\frac{dR\left(t\right)}{dt} =\gamma I\left(t\right)-\alpha R\left(t\right)} \\ & {\frac{dP\left(t\right)}{dt} =\sigma Q\left(t\right)+\delta I\left(t\right)} \end{split} \right\} \end{equation} We assume that the natural birth and death rates are the same. The parameters utilised in system \eqref{Eq__1_} are listed in the table below. \newpage \clearpage \begin{figure

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