Modelling the effect of self-immunity and the impacts of asymptomatic and symptomatic individuals on COVID-19 outbreak

COVID-19 is one of the most highly infectious diseases ever emerged and caused by newly discovered severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) It has already led the entire world to health and economic crisis It has invaded the whole universe all most every way The present study demonstrates with a nine mutually exclusive compartmental model on transmission dynamics of this pandemic disease (COVID-19), with special focus on the transmissibility of symptomatic and asymptomatic infection from susceptible individuals Herein, the compartmental model has been investigated with mathematical analysis and computer simulations in order to understand the dynamics of COVID-19 transmission Initially, mathematical analysis of the model has been carried out in broadly by illustrating some well-known methods including exactness, equilibrium and stability analysis in terms of basic reproduction number We investigate the sensitivity of the model with respect to the variation of the parameters' values Furthermore, computer simulations are performed to illustrate the results Our analysis reveals that the death rate from coronavirus disease increases as the infection rate increases, whereas infection rate extensively decreases with the increase of quarantined individuals The quarantined individuals also lead to increase the concentration of recovered individuals However, the infection rate of COVID-19 increases more surprisingly as the rate of asymptomatic individuals increases than that of the symptomatic individuals Moreover, the infection rate decreases significantly due to increase of self-immunity rate © 2020 Tech Science Press All rights reserved


Introduction
In the earth, viruses are the old detected human killer and in different times the world had to face a big challenge to fight against world pandemic diseases causing a huge loss of life and wealth. Different pandemic situations occurred in several parts of the world over the years [1]. COVID-19 is the most recent emerged devastating fatal disease, caused by coronavirus to make the jump to human infection [2]. However, there are several coronaviruses known to be circulating in different animal population that have not yet been infected human. Middle East Respiratory Syndrome (MERS-CoV) and Severe Acute Respiratory Syndrome (SERS-CoV-2) are the two zoonotic (transmits from animals to humans) corona viruses' outbreaks which have recently been experienced in the world. Coronaviruses are such type of deadly viruses that cause illness ranging from the common cold to severe respiratory disease. The genome of coronavirus is fully sequenced. The genomic sequence of SARS CoV-2 showed identical, but distinct genome composition of SARS-CoV and MERS-CoV. Since its first case reported in late 2019, the infection has spread to other regions in China and other countries, and the transmission rate, the mortality rate and the clinical manifestation slowly emerged [3].
Mathematical modelling is playing a significant role to describe the epidemiology of infectious diseases [4]. Mathematical modelling aims at the mathematical representation of various biological processes such as wound healing, morphogenesis, blood-cell production and dynamics of infectious diseases, using techniques and tools of applied mathematics [5,6]. There exist a number of models for infectious diseases as well as chronic diseases; as for compartmental models and optimal control problem, starting from the very classical SIR model to more complex proposals [7][8][9]. The classical SIR type compartmental model was first introduced by Kermack and McKendrick in 1927 [10]. Since coronavirus is a zoonotic virus, it first transmits from animals to humans. Once people become infected, then it spreads from human to human by the physical contact with infected human owing to its tremendous infectiousness. So the dynamics of COVID-19 can be described by SIR type epidemic model. It is sometimes more realistic to study such epidemic disease in terms of SEIR model when there is a certain incubation period before showing the symptom. This global outbreak has attracted the interest of researchers of different areas. Several researches of COVID-19 have been carried out focusing on mathematical modelling of the mysterious mechanisms of this disease [11,12]. An estimation of the reproductive number of coronavirus by using simulation has been introduced in [13]. Mathematical modelling of COVID-19 transmission dynamics in Wuhan has been described with stability analysis in [14]. Evolution of the novel coronavirus from the ongoing Wuhan outbreak is discussed in [15]. Aguilar et al. [16] investigate the impact of asymptomatic carriers on COVID-19 transmission. However, some articles [17][18][19][20][21][22][23][24][25][26][27] are referred for more details on SIR and SEIR type analysis and development of mathematical model on the dynamics of COVID-19.
COVID-19 transmission now becomes a worldwide pandemic. Our main contribution in this study is to consider the symptomatic and asymptomatic individuals who have remarkable influenced on the spread of COVID-19. In this paper, we have developed a mathematical model to study the dynamics of the deadly coronavirus disease in terms of nine ordinary nonlinear coupled equations. We have considered rates of change for susceptible, exposed, quarantined, symptomatic, asymptomatic, infected, hospitalized, death as well as recovered individuals. In our study, we have determined the basic reproduction number and studied the existence of the solution of the model with stability or instability criteria at disease-free and endemic equilibrium points. We have also performed sensitivity analysis of the model. Finally, numerical simulations have been performed to show the dynamic behavior of COVID-19.
After Europe, the COVID-19 spread out in American continent being USA as the epicenter in North America followed by Brazil in Latin America. In North America the infected, recovered and death are more than 3.49 million, 1.57 million and 1.75 lakhs respectively and in South America the infected, recovered and death are more than 2.72 million, 1.79 million and 1.049 lakhs as on 10 July, 2020 [32]. India and Bangladesh are in the top most position in COVID-19 infections in Asian continent. According to World Health Organization, Asia is within the foremost risk position within the world of COVID-19 transmission including more than 1.37 million infected, 84.36 lakhs recovered and 35,745 deaths till July 10, 2020 [33]. The second leading infected continent is Africa, having more than 5.59 lakhs infected, 1,93,481 recovered, and 12,769 deaths [34]. The pandemic situation in Australia and Oceania is lowest under control including infected more than 9,553 and 11,834, deaths 107 and 118 respectively [35]. The statistics of global pandemic situation of COVID-19 outbreaks is studied graphically by drawing bar diagram which is presented in Fig. 1. The age and gender have great influence on COVID-19 infection and deaths. Research shows that the older people are most likely vulnerable to get infected by coronavirus and deaths [30]. Mortality rates are significantly higher at the age of 80 years above and about 14.8%. The worldwide death report in respect to age distribution is given in Fig. 2

Basic Assumption and Formulation of the Model
Coronavirus is primarily spread during close contact via small droplets produced by coughing, sneezing, or talking. The transmission dynamics of COVID-19 occurs when a person is in close contact (within 1 m) with someone who has already infected [27]. Any individual may also become infected by touching a contaminated surface and then touching their face. It is most contagious during the first three days after the onset of symptoms, although spread hours or days may be possible before symptoms appear and in later stages of the disease. However, evidence to date suggests that older people (over 60 years old) and those are already affected by diabetes, chronic respiratory disease and cancer are at a higher risk to be infected by coronavirus. World Health Organization (WHO) has issued advice for these two groups and for community support to ensure that they are protected from COVID-19 without being isolated, stigmatized, left in a position of increased vulnerability or unable to access basic provisions and social care.
So, the whole dynamics of MERS and SARS-CoV-2 (COVID-19) can be described by a SEIR type infectious disease model in terms of a set of nonlinear ordinary differential equations (ODEs). In this paper, we extend the basic SEIR model to nine compartments to show the individual significance of each compartment. We consider quarantine, symptomatic, asymptomatic, hospitalized (or Isolation) and death compartments. The susceptible S (t), who can acquire the infection; exposed E (t), when the virus exposed itself into human bodies; the separation of a person or group of people reasonably believed to have been exposed to a communicable disease, such type of populations are considered as a quarantined populations in this model and these populations are represented by Q (t). The populations who are pertaining to a symptom or symptoms of the COVID-19 disease are considered as symptomatic populations (M (t)). The populations presenting no symptoms of the disease, we consider as asymptomatic populations and denoted by A (t). The novel coronavirus can transmit through direct contact with infected people or with objects used on the infected person. Thus the population, who can transmit infection to susceptible, is defined as infected I (t) populations. Since, there is no vaccine and no specific antiviral medicines for COVID-19, those with serious illness, may need to be hospitalized so that they can receive life-saving treatment for complications. In our model, we consider such kind of people as the hospitalized populations presented by H (t). The recovered class R (t) are those who are immunized from infection. Finally, the populations who have died of the COVID-19 are considered as death compartment and denoted by D (t). Let, N (t) be the total population at time t where, The transmission mechanisms of the novel coronavirus disease COVID-19 are shown in Fig. 3. Taking the above diagram presented in Fig. 3 into consideration, we formulate a nine compartmental model in terms of a set of nonlinear ordinary differential equations (ODEs) of the following form: with initial conditions, In the model (1), we have considered the parameter as the recruitment rate of susceptible individuals and α as the exposed rate of the individuals. β 1 and β 2 are the rate at which the symptomatic and asymptomatic individuals becoming infected. The probabilities of transmission of infections from symptomatic, asymptomatic and susceptible individuals are represented by the parameters γ 1 , γ 2 and ϕ respectively. In order to maintain physical distancing, some exposed individuals go to self-quarantined and this rate is represented by the parameter γ 3 . Quarantined individuals are divided into parts when they get tested. If these individuals show negative result of infection, they become infection free and hence they again enter into the susceptible individual compartment at a rate ρ. On the other hand, if they express positive result of infection, they enter into the symptomatic individual compartment at rate γ 4 . The infected individuals get hospitalized at rate δ. A portion of hospitalized individuals become dead because of the severity of the infection and this phenomenon is denoted by rate λ 1 . Another portion of hospitalized individuals get recovered at rate λ 2 . Besides, individuals also get recovered due to their strong immunity system and this rate is represented by the parameter ψ 1 . Before getting into hospitalization, infected individuals are died at rate ψ 2 . Finally, μ denotes the natural death rate of each individual.

Mathematical Analysis of COVID-19 Model
In this section, we discuss the boundedness, Positivity, equilibrium analysis, dynamical behavior of those points, local and global stability of the model (1).

Boundedness of the Model
+ is a positively invariant set of the proposed model.

Proof: Let the total population size is
Then, the growth rate of the total population is After solving the above equation, we have On the other hand, if N 0 > μ , then N (t) will decrease to μ as t → ∞ i.e., the solutions approaches the region asymptotically. Therefore, the model is mathematically and epidemiologically well-posed in the region . Hence, the Theorem 1 is proved.

Positivity of the Model
Here, we will show that all the variables in the model (1) are positive.  (1), In order to find the positivity of the Eq. (2), we can write Integrating Eq. (3), we have We apply the initial condition at t = 0, then S (0) − μ ≥ c. Then putting the value of c, Hence, S (t) ≥ 0 at t = 0 and t → ∞. Therefore, Similarly, with the help of [5,9], we obtain Hence the Theorem 2 is proved.

Disease Free Equilibrium (DFE) Point
For the disease-free equilibrium point of the model (1), we have to solve In case of disease free, all the state variables are zero except the susceptible individuals. By solving the system (5), we obtain Hence, the disease-free equilibrium point (DFE) of COVID-19 model is μ , 0, 0, 0, 0, 0, 0, 0, 0 .

Basic Reproductive Ratio
An important measure of transmissibility of the disease is the epidemiological concept of basic reproductive ratio. It provides an invasion criterion for the initial spread of the virus in a susceptible population.

Definition 4.1: (See [4])
The basic reproduction number, denoted by R 0 is defined as the average number of secondary infections that occurs when one infective is introduced into a completely susceptible population.

Basic Reproduction Number at DFE
Using the next generation matrix approach outlined in [36,37] to our model (1), the basic reproduction number can be computed by considering the below generation matrices F and V , that is, the Jacobian matrices associated to the rate of appearance of new infections and the net rate out of the corresponding compartments.
Matrix for the gain term, Now we have to evaluate next generation matrix G, Hence, the largest eigen value of the matrix G is . Thus, the basic reproduction number of the model (1) is .
For the parameters used in our simulations (see Tab. 1), we compute this basic reproduction number to obtain R 0 = 0.8724.

Local Stability at Disease Free Equilibrium Point
Firstly, we investigate the local stability at disease free equilibrium point W 0 . Before further proceeding, we need the following Theorem 3.

Theorem 3:
The disease-free equilibrium of model (1) is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

Proof:
To prove the Theorem 3, the following variation matrix [38] is computed corresponding to equilibrium point W 0 . From the model (1), the Jacobean matrix of the model is where, a 11 = − (αE + ϕI + μ) + ρQ and At disease free equilibrium, we get In order to determine the stability of disease-free equilibrium point, we utilize |J (W 0 ) − λI| = 0, where λ be the eigen value and I be the identity matrix.
By factoring out from the above matrix, we have The remaining two eigen values can be obtained from this characteristic equation, After simplifying, the eigen values will be The eigenvalues of the equation are negative when 1 − R 0 > 0 i.e., R 0 < 1. Since all the eigen values are negative, the diseases free equilibrium point is locally stable when R 0 < 1.
This holds the Theorem 3.

Global Stability of the Disease-Free Equilibrium Point
In this section, we use the Lyapunov direct method [39,40] to show the conditions for the global asymptotic stability of the disease-free equilibrium point in int R + 9 . Theorem 4: The disease-free equilibrium point of the system (1) is globally asymptotically stable if 0 < δ + ψ 1 ϕ < R 0 < 1 in the interior of the feasible region, otherwise it is unstable.
Proof: Theorem 4 can be proved based on the Lyapunov stability theorem [35,39]. For this purpose, we consider the following nonlinear Lyapunov function, Then V is C 1 on the interior of R + 9 , W 0 S, E, Q, M, A, I, H, R, D is the disease-free equilibrium point.
The derivative of V along the solution curves of (1) is given by the expression:

M,A,H)
At the disease-free equilibrium point, putting = μS and simplifying the above equation, we geṫ M, A, H) .
Since all the parameter values and state variables are nonnegative, it follows thatV ≤ 0 for Thus, we can say that the disease-free equilibrium point is globally asymptotically stable when This holds the Theorem 4.

Endemic Equilibrium Point (EEP)
Endemic equilibrium point (EEP) of the model (1) can be obtained by setting The differential equation for the death compartment (D (t)) is not present here. This is due to the fact that the state variable D (t) only appears in the corresponding differential equation and so it has no significance in the overall system. Also, the number of death individuals at each instant t can be obtained from Let W * (S * , E * , Q * , M * , A * , I * , H * , R * ) be the endemic equilibrium point and by solving the system (8), we get

Basic Reproduction Number at EEP
By applying the similar approach as given in Sub-section 4.4.1, we have basic reproduction number at the EEP.
For the parameters used in our simulations (Tab. 1), it is easy to compute this basic reproduction number as R * 0 = 4.0190. This means that the pandemic outbreak has not been controlled in the world.

Local Stability at the Endemic Equilibrium Point
Theorem 5: The endemic equilibrium point of the model (1) is locally asymptotically stable if R 0 > 1 otherwise, it is unstable.
Proof: To determine the local stability at endemic equilibrium point, the characteristic equation of the model (1) is Therefore, the eigenvalues are In this case, the basic reproduction number is more than one, i.e., R 0 > 1 according to the given data. Therefore, the endemic equilibrium point of the model (1) is locally asymptotically stable, which proves the Theorem 5.

Global Stability of the Endemic Equilibrium Point
In this section, we use the Lyapunov direct method to establish sufficient conditions for the global asymptotic stability of the endemic equilibrium point W * in int R + 9 when R 0 > 1. Theorem 6: The endemic equilibrium point W * of the system (1) is globally asymptotically stable if R 0 > 1 in the interior of the feasible region, otherwise it is unstable.
Proof: Theorem 6 can be proved based on the Lyapunov stability theorem [21,39]. For that purpose, we consider the following nonlinear Lyapunov function, Then L is C 1 on the interior of R + 9 , W * is the endemic equilibrium point. The derivative of (9) along the solution curves of (1) is given by the expression: At the endemic equilibrium point W * , we have Using the Eqs. (11)- (13) in (10), we obtaiṅ

M,A,H).
Since the arithmetic mean is greater than or equal to the geometric mean, it follows that : dL/dt = 0 is the singleton {W * }, where W * is the endemic equilibrium point. By LaSalle's invariance principle [41,42], it implies that W * is globally asymptotically stable in the interior of R + 9 . Hence, the Theorem 6 is proved.

Sensitivity Analysis
In determining the best strategy to reduce the disease transmission and human mortality due to pandemic outbreak of COVID-19, it is necessary to know the relative importance of the different factors responsible for its transmission and prevalence. Initial disease transmission is directly related to R * 0 , and disease prevalence is directly related to the endemic equilibrium point. Sensitivity analysis is the study of how the uncertainty in the output of a mathematical model or system can be divided and allocated to different sources of uncertainty in its inputs. It tells us how important each parameter is to disease transmission. Such information is crucial not only for experimental design, but also to data assimilation and reduction of complex nonlinear models [43]. Sensitivity analysis is commonly used to determine the robustness of model predictions to parameter values, since there are usually errors in data collection and presumed parameter values. It is used to discover parameters that have a high impact on R * 0 and should be targeted by intervention strategies [44].
More accurately, sensitivity indices allow us to measure the relative change in a variable when a parameter changes. The normalized forward sensitivity index of a variable with respect to a parameter is the ratio of the relative change in the variable to the relative change in the parameter. When the variable is a differentiable function of the parameter, the sensitivity index may be alternatively defined using partial derivatives.

Definition 4.2:
( [14]) The normalized forward sensitivity index of R * 0 , which is differentiable with respect to a given parameter P, is defined by As we have an explicit formula for R * 0 , we derive an analytic expression for the sensitivity R * 0 to each of the different parameters described in Tab. 1. For example, the sensitivity index of R * 0 with respect to ϕ is These values have been calculated analytically from the real data collected from [30] (see also [13,15]   Note that, the sensitivity index may depend on several parameters of the system, but also can be constant, independent of any parameter. For example, γ R * 0 P = +1 means that increasing (decreasing) P by a given percentage increases (decreases) always R * 0 by that same percentage. The estimation of a sensitive parameter should be carefully done, since a small perturbation in such parameter leads to relevant quantitative changes. On the other hand, the estimation of a parameter with a rather small value for the sensitivity index does not require as much attention to estimate, because a small perturbation in that parameter leads to small changes. From Tab. 2, we conclude that the most sensitive parameters to the basic reproduction number R * 0 of the COVID-19 model (1) are ϕ, α and ψ 1 . In concrete, an increase of the value of ϕ will increase the basic reproduction number by 99.73%. In contrast, an increase of the value of α will decrease R * 0 by 77.11%.

Numerical Simulations
We perform numerical simulations to compare the results of our model with the real data published by Worldometer [30] till July 10, 2020. We show that our COVID-19 model describes well the real data of daily confirmed cases during the 2 months outbreak. The computational study for graphical representation of the model (1) was performed by ode45 solver using MAT-LAB programming language. We use a set of suitable parameter values as presented in Tab. 1 for the simulations. We have considered the initial condition S 0 = 100 × 10 5 , E 0 = 70 × 10 5 , Q 0 = 60 × 10 5 , M 0 = 40 × 10 5 , A 0 = 30 × 10 5 , I 0 = 10 × 10 5 , H 0 = 7 × 10 5 , R 0 = 8 × 10 5 and D 0 = 2 × 10 4 . Firstly, we solve the model (1) considering the initial values and all other parameters that are shown in Tab. 1. Also, we have performed the numerical simulations for time interval t ∈ [0, 60] for 60 days.
Our object is to study the effects of infection rate (ϕ) from susceptible individuals, probability of transmission of infection from symptomatic individuals (γ 1 ), probability of transmission of infection from asymptomatic individuals (γ 2 ), quarantined rate (γ 3 ) and effective rate of recovery using self-immunity system (ψ 1 ) in case of disease transmission. We have selected these parameters because they have a large impact in determining the best strategy to reduce the disease transmission and human mortality due to pandemic outbreak of COVID-19. So, if it is possible to minimize the infection rate from susceptible individuals, rate of transmission of infection from symptomatic individuals, rate of transmission of infection from asymptomatic individuals and maximize the quarantined rate and effective rate of recovery using self-immunity system then the spreading of novel coronavirus will be controlled.
Considering these parameters into account, we have run the program for the state variables to show all state trajectories simultaneously. The result of simulation of the combined class is presented in Fig. 4.
From Fig. 4, we observe that the susceptible and quarantined individuals are decreased monotonically. At the same time, the exposed individuals initially increase but after some days it gradually decreases. From the very beginning of this pandemic, the symptomatic curve is increased rapidly than asymptomatic individual but after some days asymptomatic curve is also increased surprisingly. However, a massive number of individuals having no symptoms of COVID-19 transmit the virus to others. Due to increase these symptomatic and asymptomatic individuals the infected individual increases extensively. As a result, hospitalized and death individuals are also increased than that of recovered individuals. From the dynamical behavior of the graph, it is anticipated that the infected individuals will continue to grow up with the passes of time until the initiation of vaccine or proper medicine. Though, it can be controlled by maintaining physical distances, increasing quarantined rate and developing self-immunity system which reflect our study.
Again, we run the program keeping all other values of the parameters same as before for the susceptible, infected and death individuals. The result of simulation in this case is presented in Fig. 5.  5 shows the state trajectories of three compartments such as susceptible, infected and death individuals in the absence of any control measures. We have observed that the infected individuals increase sharply whereas the death individuals increase steadily from the initial state. Thus, these two individuals lead the susceptible individuals to be decreased dramatically. Now, we run the program for the quarantined, infected and recovered individuals to show the effect of quarantine rate keeping the parameters value same as before. The result of simulation in this case is shown in Fig. 6. In Fig. 6, we see the variation of three state trajectories of quarantined, infected and recovered individuals with time. It has been observed that the infected individuals decrease significantly as the quarantined rate increases. As a result, the decreasing rate of infected individuals bolsters the recovered individuals to be increased extensively.
We also observe from Fig. 7 that the infected populations are significantly decreased due to maintain the quarantine system strictly. The figure shows that if the quarantined rate is increased from γ 3 = 0.25 to γ 3 = 0.50, the probability to become COVID-19 positive is very little. In this case, the recovered individuals are also enhanced for the high rate of quarantine rate.  We observe that the infected population is extremely increased due to increase of asymptomatic individuals. Because an asymptomatic individual does not exhibit the symptoms of COVID-19 outbreak but can transmit the virus to others susceptible individuals and due to scarcity of symptoms the family members and others live together with him and they become COVID-positive in absence of mind and still transmit the virus to other members. Therefore, the population presenting no symptoms of the disease can transmit the coronavirus rapidly.
In Fig. 9, we notice that infected populations of COVID-19 are increased with the increase rate of symptomatic individuals. The important thing is that, the infected individuals are increased due to symptomatic individuals but not as like in asymptomatic case. Figure 9: Dynamics of symptomatic and infected individuals where the infected population is increased but not tremendously due to increase of symptomatic population Next, we solve the model for the class of infected individuals to show how the change in the infected individuals for different values of self-immunity rate. The result in this case is presented in Fig. 10.
From Fig. 10, it has been observed that the infected populations are decreased tremendously due to increase of self-immunity system (i.e., by increasing the self-immunity rate from ψ 1 = 0.27 to = 0.37). Hence, to reduce the infected individuals from this outbreak, self-immunity system must be developed for all the populations of a community. For that reason, it is mandatory for each of the individuals to develop a strong immune system through indoor and outdoor activities, by trying muscle strength training, by eating a diet high in fruits, vegetables, and whole grains and to restrict saturated fats and sugars to 10% of total calories while minimizing the consumption of red and processed meats. In the epidemiology, disease transmission and disease prevalence are directly related to value of basic reproduction number and endemic equilibrium point. It provides an incursion formula for the initial spread of the disease in a susceptible population and it's defined as the average number of secondary infectious population that occurs when one infective people is introduced it into others susceptible population. It is cleared to define that the outbreak of corona virus infection will be eliminated if R 0 < 1 and remained in the community if R 0 > 1. Now, we run the program to know which parameters are responsible for the corona virus transmission by considering the initial values and parameters value that are shown in Tab. 1. The result obtained in this case is given in Figs. 11-14. Figure 11: Numerical simulation of basic reproduction number with respect to the infectious rate ϕ, where the value of R 0 is increased linearly as the increase of ϕ, i.e., the disease persist in the community with the increase of infectious rate Figure 12: The value of R 0 is decreased with respect to the rate of self-immunity system ψ 1 , and R 0 < 1 when 0.04 < ψ 1 Figure 13: The value of R 0 is decreased significantly with respect to the increase of isolation rate, δ, and R 0 < 1, i.e., asymptotically stable in the region 0.1 < ψ 1 Figure 14: The overall stability of pandemic corona virus infection using the parameter values in Tab. 1, that may be wiped out when R 0 < 1 and persist in the community when R 0 > 1 Figs. 11-14 show the dynamics of the basic reproduction number with respect to the important parameters that are responsible to continue this pandemic situation. It is observed from Fig. 11 that the value of the basic reproduction number is increased linearly as the increase of the value of infectious rate. In this case, the disease persists in the community. It is also clear from Figs. 12 and 13 that the value of basic reproduction number is extensively decreased with respect to the self-immunity system as well as the increase of isolation rate. Fig. 14 illustrates the overall scenario of the COVID-19 situation with respect to the basic reproduction number. The pandemic corona virus infection may be wiped out when R 0 < 1 and persist in the community when R 0 > 1. Thus, to eradicate the outbreak of coronavirus, it is important to develop the individuals' self-immunity system and maintain the physical distances strictly.

Conclusions
COVID-19 is a highly infectious pandemic disease which is impendence for the whole world. There is no specific treatment for this novel coronavirus disease which leads it more deadlier. This paper deals with a nine mutually exclusive compartmental model on transmission dynamics of COVID-19. The compartmental model has been investigated with mathematical analysis and computer simulations in order to understand the dynamics of this disease transmission. In our study, we have observed that the spread of novel coronavirus largely depends on the rate of close contact between susceptible and infected individuals. From Figs. 6 and 7, it has been observed that the death rate from coronavirus disease increases as the infection rate increases whereas infection rate extensively decreases with the increase of quarantined individuals. The quarantined individuals also lead to increase of recovered individuals. In Figs. 8 and 9, we have noticed that the infection rate of COVID-19 increases more surprisingly as the rate of asymptomatic individuals increases than that of the symptomatic individuals. Figs. 7 and 10 show that the infection rate significantly reduces due to increase of quarantined rate as well as the self-immunity rate. From the sensitivity analysis, we have obtained that the most sensitive parameters to the basic reproductive ratio of the COVID-19 model (1) are infection rate (ϕ), exposed rate (α) and the self-immunity rate (ψ 1 ). Our findings suggest that to combat or eradicate this pandemic outbreak, the physical distances must be maintained rigorously. It is also highly recommended for eating immune-boosting foods to increase immunity and taking immune-boosting drugs to defend the virus.

Authors' Contributions:
This research is a group work carried out in collaboration among all authors. Authors MHAB and MAI designed the study, performed the conceptualization and methodological analysis and model formulation of the first draft of the manuscript. Authors SA and SM analyzed the model analytically and wrote some literature of the study. Author AKP wrote the programming codes and performed some part of the computational analysis. Author SAS contributed to literature searches and calculated the real data to estimate the parameters, MSK and MRK verified the parameters and checked the literature. All authors have read and agreed to publish the final version of the manuscript.

Data Availability:
The data used to support the findings of this study are included within the article.

Conflicts of Interest:
The authors declare that they have no conflicts of interest to report regarding the present study.