Dynamical analysis of novel COVID‐19 epidemic model with non‐monotonic incidence function

• Published date: 01 December 2022
• Author: R. Prem Kumar,Sanjoy Basu,Dipankar Ghosh,Prasun Kumar Santra,G. S. Mahapatra
• Date: 01 December 2022
• DOI: http://doi.org/10.1002/pa.2754

RESEARCH ARTICLE

Dynamical analysis of novel COVID-19 epidemic model with

non-monotonic incidence function

R. Prem Kumar

1,2

| Sanjoy Basu

3

| Dipankar Ghosh

1

| Prasun Kumar Santra

4

|

G. S. Mahapatra

1

1

Department of Mathematics, National

Institute of Technology Puducherry, Karaikal,

Puducherry, India

2

Avvaiyar Government College for Women,

Karaikal, Puducherry, India

3

Arignar Anna Government Arts and Science

College, Karaikal, Puducherry, India

4

Moulana Abul Kalam Azad University of

Technology, Kolkata, West Bengal, India

Correspondence

Sanjoy Basu, Department of Mathematics,

Arignar Anna Government Arts and Science

College, Karaikal 609605, Puducherry, India.

Email: basu1982@gmail.com

In this study, we developed and analyzed a mathematical model for explaining the

transmission dynamics of COVID-19 in India. The proposed SIuIkRmodel is a modi-

fied version of the existing SIR model. Our model divides the infected class Iof SIR

model into two classes: Iu(unknown infected class) and Ik(known infected class). In

addition, we consider Ra recovered and reserved class, where susceptible people can

hide them due to fear of the COVID-19 infection. Furthermore, a non-monotonic

incidence function is deemed to incorporate the psychological effect of the novel

coronavirus diseases on India's community. The epidemiological threshold parameter,

namely the basic reproduction number, has been formulated and presented graphi-

cally. With this threshold parameter, the local and global stability analysis of the

disease-free equilibrium and the endemic proportion equilibrium based on disease

persistence have been analyzed. Lastly, numerical results of long-run prediction using

MATLAB show that the fate of this situation is very harmful if people are not follow-

ing the guidelines issued by the authority.

KEYWORDS

basic reproduction number, epidemiological model, local and global stability, novel

coronavirus, persistence

1|INTRODUCTION

The first case of the coronavirus pandemic was reported on January

30, 2020 in India (Zhu et al., 2020). India's response to COVID-19 has

been graded, pro-active, and pre-emptive with high-level political commit-

ment and a “whole government”approach to respond to the COVID-19

pandemic. Educational institutions, various Governments, and non-

Government offices, and many commercial establishments have been

shut down immediately. Government of India, exercise the Disaster Man-

agement Act, 2005, issued an order for State/UT's prescribing lockdown

for containment of COVID-19 pandemic in the country for 21 days with

effect from March 25, 2020, and announced to maintain mandatory phys-

ical distancing in India. Later Government extent the lockdown period up

to May 3, 2020 as the number of active cases increases daily. The most

common symptoms are dry cough, fever, and tiredness, but some infected

people may have aches and pains, running nose, nasal congestion, sore

throat, vomiting, or diarrhea. These symptoms usually are severe and

begin gradually. Some people become infected, but neither has symptoms

nor feels ill. Disease recovery is high without requiring special treatment.

The guidelines for preventing the spreading of COVID-19 are wearing a

face mask, staying more than 3 ft away from a sick person, washing hands

with soap, or using alcohol-based hand rub, and so forth.

During an epidemic, reported cases of coronavirus disease are ris-

ing worldwide day by day due to human-to-human transmission; the

study for prevention and control of infectious COVID-19 disease is

essential. The modernized mathematical model is necessary to give a

deeper understanding and perception of disease transmission mecha-

nisms and find how to control the spread of the COVID-19 disease.

Xiao and Ruan (2007) studied an epidemic model with a non-

monotonic incidence rate, which describes the psychological effect of

certain serious diseases on the community when the number of infec-

tives is getting larger. Xu and Ma (2009) investigated a SIR epidemic

Received: 11 March 2021 Revised: 26 June 2021 Accepted: 14 August 2021

DOI: 10.1002/pa.2754

J Public Affairs. 2022;22(Suppl. 1):e2754. wileyonlinelibrary.com/journal/pa © 2021 John Wiley & Sons Ltd. 1of18

https://doi.org/10.1002/pa.2754

model with nonlinear incidence rate and time delay. Yang et al. (2010)

formulated a SIR model with vaccination and varying population. Sun

and Hsieh (2010) investigated an susceptible exposed infected recov-

ered (SEIR) model with varying population size and vaccination strat-

egy. Zhou and Cui (2011) studied an SEIR epidemic model with a

saturated recovery rate. Bai and Zhou (2012) proposed an SEIRS epi-

demic model with a general periodic vaccination strategy and season-

ally varying contact rates. Khan et al. (2015) considered an SEIR

model with nonlinear saturated incidence rate and temporary immu-

nity. Elkhaiar and Kaddar (2017) studied the dynamics of an SEIR epi-

demic model with nonlinear treatment function that takes into

account the limited availability of resources in the community. Wang

et al. (2018) extended the incidence rate of an SEIR epidemic model

with relapse and varying total population size to a general nonlinear

form. Tiwari et al. (2017) investigated an SEIRS epidemic model with

nonlinear saturated incidence rate. Lahrouz et al. (2012) studied the

global dynamics of a SIRS epidemic model for infections with non-

permanent acquired immunity. Tian and Wang (2011) discussed the

global stability analysis for several deterministic cholera epidemic

models. Samanta (2011) discussed the permanence and extinction of

a non-autonomous HIV/AIDS epidemic model with distributed time

delay. Cai et al. (2014) investigated an HIV/AIDS treatment model.

Gralinski and Menachery (2020) studied the return of novel coro-

navirus in 2019. Chen et al. (2020) developed a mathematical model

for calculating the transmissibility of the novel coronavirus. Saldana

et al. (2020) developed a compartmental epidemic model to study the

transmission dynamics of the COVID-19 epidemic outbreak, with

Mexico as a practical example. Silva et al. (2020) proposed a new SEIR

agent-based COVID-19 model to simulate the pandemic dynamics

using a society of agents emulating people, business, and government.

Pal et al. (2020) proposed a COVID-19 model for stability analysis

with five compartments. Lee et al. (2020) proposed a COVID-19 epi-

demic model for estimating the unidentified infected population in

China. Maheshwari et al. (2020) forecasted the epidemic spread of

COVID-19 in India using the ARIMA model. Zakharov et al. (2020)

predicted the dynamics of the COVID-19 epidemic in real-time using

the case-based rate reasoning model. Bonnas and Gianatti (2020) pro-

posed a COVID-19 epidemic model where the population is par-

titioned into classes corresponding to ages. Roda et al. (2020)

demonstrated the reasons for wide variations in numerous model pre-

dictions of the COVID-19 epidemic in China. Liu et al. (2020a) devel-

oped two differential equations models to account for the latency

period of COVID-19 infection. Basnarkov (2021) studied a SEAIR epi-

demic spreading model of COVID-19. Yang and Wang (2020) pro-

posed a mathematical model for the novel coronavirus epidemic in

Wuhan, China. Wang, Lu, et al. (2020) performed the dynamical analy-

sis of a COVID-19 epidemic model. Zlatic et al. (2020) developed a

COVID-19 epidemics model spreading on the availability of tests for

the disease. Xue et al. (2020) proposed a data-driven network model

for the COVID-19 epidemics in Wuhan, Toronto, and Italy. Neves and

Guerrero (2020) presented the A-SIR model to predict the evolution

of the COVID-19 epidemic. Ndairou et al. (2020) proposed a mathe-

matical model for COVID-19 epidemic with a case study of Wuhan.

Jiao and Huang (2020) proposed a SIHR COVID-19 epidemic model

with effective control strategies. Zhao and Chen (2020) modeled the epi-

demic dynamics and control of the COVID-19 outbreak in China. Li

et al. (2020) modeled the impact of mass influenza vaccination and public

health interventions on COVID-19 epidemics. Pizzuti et al. (2020)investi-

gated the prediction accuracy of the SIR model on networks for Italy.

Wang, Zheng, et al. (2020) used the logistic model and machine learning

technics to predict the COVID-19 epidemics. Pongkitivanichkul

et al. (2020) estimated the size of the COVID-19 epidemic outbreak. Liu

et al. (2020b) predicted the cumulative number of cases for the COVID-

19 epidemic in China. Zhu and Zhu (2020) devised a method to analyze

the COVID-19 epidemic. Kantner and Koprucki (2020) computed a strat-

egy for the case that a vaccine is never found and complete containment

is impossible. Engbert et al. (2021) presented a Stochastic SEIR epidemic

model for regional COVID-19 dynamics by sequential data assimilation.

Several researcher investigated the dynamics of COVID-19 using frac-

tional order models (Askar et al., 2021;Awaisetal.,2020;Rezapour

et al., 2020). Rihan et al. (2020) analyzed a stochastic SIRC epidemic model

with time-delay for COVID-19. Bambusi and Ponno (2020)explainedthe

linear behavior in COVID-19 epidemic as an effect of lockdown. Alberti

and Faranda (2020) presented statistical predictions of COVID-19 infec-

tions by fitting asymptotic distributions to actual data. Abbasi et al. (2020)

discussed the Optimal control for Impulsive SQEIAR Epidemic model on

COVID-19 epidemic. Lobato et al. (2020) identified an epidemiological

model to simulate the COVID-19 epidemic. Khan and Atangana (2020)

modeled the dynamics of novel coronavirus with fractional derivative.

Alshammari and Khan (2021) analyzed the dynamics of modified SIR

model with nonlinear incidence and recovery rates. Pal et al. (2021)pres-

ented a COVID-19 model with optimal treatment of infected individuals

and the cost of necessary treatment. Khan et al. (2021)focusedonthe

novel coronal virus model to understand its dynamics and possible control.

Khajanchi and Sarkar (2020) developed a new compartment model that

explains the transmission dynamics of COVID-19. Rai et al. (2021) studied

the social media advertisements in combating the coronavirus pandemic

in India. Tuncer (2020) explored globalization's effect on the spread of fear

across the world by focusing on the case of COVID-19. Adekola

et al. (2020) examined various forms of mathematical models relevant to

the containment, risk analysis, and features of COVID-19.

This paper determines the fate of coronavirus infective individ-

uals introduced into the population in India. The dynamics of the

nonlinear system have been considered in the study with reinfection

turned off. The basic reproduction number (BRN) ℜ0is estimated and

analyzed as a threshold parameter for the stability analysis of the

disease-free equilibrium (DFE) and endemic equilibrium. The uniform

persistence of the disease near the threshold parameter is also

determined.

2|FORMULATION OF COVID-19

MATHEMATICAL MODEL

The proposed COVID-19 model involves a specific postulate consid-

ered for developing mathematical modeling in the Indian perspective.

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