Dynamical analysis of novel COVID‐19 epidemic model with non‐monotonic incidence function - Wiley
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 commitment 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 Management 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 physical 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 rising 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 mechanisms 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 infectives is getting larger. Xu and Ma (2009) investigated a SIR epidemic 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 recovered (SEIR) model with varying population size and vaccination strategy. Zhou and Cui (2011) studied an SEIR epidemic model with a saturated recovery rate. Bai and Zhou (2012) proposed an SEIRS epidemic model with a general periodic vaccination strategy and seasonally varying contact rates. Khan et al. (2015) considered an SEIR model with nonlinear saturated incidence rate and temporary immunity. Elkhaiar and Kaddar (2017) studied the dynamics of an SEIR epidemic 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 coronavirus 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 epidemic 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) proposed a COVID-19 epidemic model where the population is partitioned into classes corresponding to ages. Roda et al. (2020) demonstrated the reasons for wide variations in numerous model predictions of the COVID-19 epidemic in China. Liu et al. (2020a) developed two differential equations models to account for the latency period of COVID-19 infection. Basnarkov (2021) studied a SEAIR epidemic spreading model of COVID-19. Yang and Wang (2020) proposed a mathematical model for the novel coronavirus epidemic in Wuhan, China. Wang, Lu, et al. (2020) performed the dynamical analysis 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 mathematical 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 epidemic 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) investigated 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 strategy 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 fractional order models (Askar et al., 2021; Awais et al., 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) explained the linear behavior in COVID-19 epidemic as an effect of lockdown. Alberti and Faranda (2020) presented statistical predictions of COVID-19 infections 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) presented a COVID-19 model with optimal treatment of infected individuals and the cost of necessary treatment. Khan et al. (2021) focused on the 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 individuals 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) 
is 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
. Here 
describes the infection force of the disease and 
measures the inhibition effect from the behavioral change of the susceptible individuals when the number of infectious individuals increases. Some susceptible class individuals move to the reserved area, which is considered a safe zone during the pandemic. The known infected individuals entered the recovered class after recovered from the COVID-19. Here in this model, we consider the recovered class and the reserved class as the same and denote the density at time 
by 
. The model of the study has been taken in the following form 
(1)
subject to initial conditions 
(2)
and 
are the densities at the time 
of susceptible population, unknown infected population (incubate the illness but do not have any symptoms and not identified), known infected population (in the isolated ward), and recovered or reserved population, respectively, and the parameters 
, 
, 
, 
, and 
are all positive. Here, 
is defined as the total number of population under risk at the time 
(Figure 1). 
Transfer diagram of the COVID-19 model
The biological meanings of the model parameters are listed below:
The recruitment rate at which new individuals enter the Indian population.
The homogeneous transmission coefficient from the susceptible population (
) to the unknown infected population (
). The rate of transmission of infection is given by 
.
The parameter measures the psychological or inhibitory effect.
The transmission coefficient from the unknown infected population (
) to the known infected population (treatment population) (
).
The transmission coefficient from the susceptible population (
) to the recovered or reserved population (
).
The transmission coefficient from the known infected population (treatment population) (
) to the recovered population or reserved population (
).
The natural death rate.
The death rate of the known infected class (
).
3 ANALYSIS OF THE MODEL AND BASIC PROPERTIES
3.1 Non-negativity of solutions
Theorem 1.Every solution of the system (1) with initial conditions (2) are non-negative for every t 
0.
3.2 Boundedness of the system and invariant region
3.3 Equilibrium of system
4 DFE AND STABILITY ANALYSIS
To eradicate the disease from a varying size population, the more stringent way requires that the total number of the virus-infected population 
, while a weaker requirement is that proportion sum of the same tends to zero (Busenberg et al., 1991). Thus we need to find the conditions for the existence and stability of the DFE 
and the endemic equilibrium 
. Therefore, 
is the DFE of (4), which exists for all positive parameters.
4.1 The basic reproduction number
The BRN is the average number of secondary infections generated by a single infection and is one of the most vital threshold quantities which mathematically represent the spreading of the virus infection.
becomes 
(5)The stability of 
is equivalent to all the eigenvalues of the characteristic equation of 
at 
being with negative real parts, which can be assured by the BRN (
) obtained by the next-generation matrix method (Van den Driessche & Watmough, 2002), where 
is the epidemiological threshold parameter.
. Then the system (1) can be written as 
The next generation matrix for the system (1) is 
.
The spectral radius of the matrix 
is 
which is the BRN 
. Now, it has been observed that 
. From this observation, it is obvious that if the transmission coefficient 
from the susceptible population (
) to unknown infected population (
) decreases, then the BRN 
also decreases and therefore reduces the burden on the infection. Otherwise, if 
increase, then 
would also increase, and thus, the transmission of virus infection will also rise; therefore, the scenario will be very harmful to society.
Now, the BRN 
has been presented graphically in Figure 2 with respect to related estimated or hypothetical parameter values given in Table 1. From Figure 2, it is observed that as the value of 
increases, 
also increases simultaneously and become greater than unity after a certain value of 
. Therefore, it is said that up to a certain value of 
, the DFE point is stable (Theorem 3) and beyond that value of 
, the endemic equilibrium point is stable (Theorem 6).
with respect to 
4.2 Local stability of the DFE
This section will discuss the parameter restrictions of the local stability of DFE.
Theorem 3.The disease free equilibrium 
of the system (4) is locally asymptotically stable if 
. Whereas, it is unstable if 
.
4.3 Global stability of the DFE
In this section, we will discuss the parameter restrictions of the global stability of DFE.
Theorem 4.When 
, the disease free equilibrium 
is globally asymptotically stable in 
.
5 DISEASE PERSISTENCE
5.1 Uniformly persistence
In this subsection, an effort is made to understand the uniform persistence of the dynamical system (4) for the threshold parameter by applying the acyclicity theorem (Sun & Hsieh, 2010).
Definition 1.The system (4) is said to be uniformly persistent (Butler et al., 1986) if there exists a constant 
such that all solutions 
with positive initial 
satisfy the following inequality
(7)Let 
be a locally compact metric space with metric 
, and let 
is a closed non-empty subset of 
with the boundary 
and interior 
. Clearly, 
is a closed subset of 
and let 
be a dynamical system on 
. Then set 
in 
is said to be invariant if 
.
Theorem 5.Suppose the conditions 
and 
holds true for the dynamical system 
H1:.The system 
has a global attractor.
6 ENDEMIC EQUILIBRIUM AND STABILITY ANALYSIS
6.1 Local stability analysis of the endemic equilibrium
Theorem 6.The endemic equilibrium 
of the system (4) is locally asymptotically stable in 
if 
.
6.2 Global stability analysis of the endemic equilibrium
Let 
be an 
matrix valued function which is 
in 
and 
, where 
is the matrix obtained by replacing each entry 
in 
by its directional derivative in the direction of
