# Mathematical Model and Analysis on the Impact of Awareness Campaign and Asymptomatic Human Immigrants in the Transmission of COVID-19

Authors: Alemzewde Ayalew Anteneh; Yezbalem Molla Bazezew; Shanmugasundaram Palanisamy
Journal: BioMed Research International
 (2022)

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## Abstract

In this study, an autonomous type deterministic nonlinear mathematical model that explains the transmission dynamics of COVID-19 is proposed and analyzed by considering awareness campaign between humans and infectives of COVID-19 asymptomatic human immigrants. Unlike some of other previous model studies about this disease, we have taken into account the impact of awareness c between humans and infectives of COVID-19 asymptomatic human immigrants on COVID-19 transmission. The existence and uniqueness of model solutions are proved using the fundamental existence and uniqueness theorem. We also showed positivity and the invariant region of the model system with initial conditions in a certain meaningful set. The model exhibits two equilibria: disease (COVID-19) free and COVID-19 persistent equilibrium points and also the basic reproduction number,R0 which is derived via the help of next generation approach. Our analytical analysis showed that disease-free equilibrium point is obtained only in the absence of asymptomatic COVID-19 human immigrants and disease (COVID-19) in the population. Moreover, local stability of disease-free equilibrium point is verified via the help of Jacobian and Hurwitz criteria, and the global stability is verified using Castillo-Chavez and Song approach. The disease-free equilibrium point is both locally and globally asymptotically stable whenever R0<1, so that disease dies out in the population. If R0>1, then disease-free equilibrium point is unstable while the endemic equilibrium point exists and stable, which implies the disease persist and reinvasion will occur within a population. Furthermore, sensitivity analysis of the basic reproduction number, R0 with respect to model parameters, is computed to identify the most influential parameters in transmission as well as in the control of COVID-19. Finally, some numerical simulations are illustrated to verify the theoretical results of the model.


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## Body

## 1. Introduction

Coronaviruses are a large family of viruses that may cause illness in animals and humans. In humans, several coronaviruses are known to cause respiratory infections ranging from the common cold to more severe diseases such as Middle East respiratory syndrome (MERS) and severe acute respiratory syndrome (SARS). The most recently discovered human coronavirus disease which is caused by a novel coronavirus severe acute respiratory syndrome 2 (SARS-CoV-2) is coronavirus disease 2019 (COVID-19) [1, 2]. This disease was first identified in December 2019 in Wuhan, China, with common manifestation appears to be pneumonia [3–5]. One of the things which makes this new virus so dangerous is that it spreads very quickly between peoples. COVID-19 is transmitted from human to human via direct contact with contaminated surfaces and through respiratory droplets’ inhalation from infected individuals [4, 6, 7]. The most common symptoms of COVID-19 include the following: fever, dry cough, shortness of breath, and fatigue [3, 8]. Other fewer symptoms may include the following: sore throat, headache, chills, congestion, nausea, and diarrhea. The incubation period (time from exposure to the development of symptoms) of COVID-19 is somewhere between 2 and 14 days, on average around five days after exposure to the virus [9, 10].The World Health Organization (WHO) declared the outbreak as a Public Health Emergency of International Concern on January 2020 and a pandemic on 11 March 2020 [11]. In China mainland until March 8, 2020, a total of 80,868 confirmed cases and 3,101 deaths were recorded due to COVID-19 [3]. The outbreak of COVID-19 has spread to 222 countries and territories, inflicted more than 116.17 million confirmed cases, and claimed more than 2.58 million lives, as reported on March 7th, 2021 [12]. Globally, over 3.1 million new cases and just over 54,000 new deaths were reported during the week of 27 September to 3 October 2021 and more than 197 million new cases, and among these, more than 4 million individuals have died up to the end of July 2021 [13, 14].Mathematical model is a mathematical equation that describes changes in the system with time, and it is useful to accurately predict the evolution of infectious disease (COVID-19), and this in turn helps to give an insight for health workers and government to deal on the most influential parameters in COVID-19 transmission [15]. Since COVID-19 outbreak, different scholars studied the transmission dynamics of the disease (COVID-19) by considering different scenarios [1, 2, 5, 9–11, 14–21] to curb its spread with the help of epidemiological mathematical model. However, all the above studies failed to consider the impact of awareness campaign and influx of COVID-19 asymptomatic human immigrants in the control of COVID-19 transmission. In this study, we inspire to modify the model [16] to fulfill the entire gap. Our proposed model is different from others, and it is that the class of susceptible human population is subdivided into two: individuals who have awareness about COVID-19 and apply all the recommended mitigation of COVID-19 to save themselves and others are classified as aware susceptible human population class, denoted by Ah and individuals who have no awareness about COVID-19 or even they have awareness but they do not give an emphasis for the severity of COVID-19 are classified as unaware susceptible human population class, denoted by Uh. The rate of COVID-19 asymptomatic human immigrants is also included. The remaining part of this study is structured as follows: in Section 2, we developed our mathematical model for the transmission of COVID-19 dynamics. In Section 3, the analytical analysis of the model established. In Section 4, numerical simulations of the model are illustrated. The conclusion and concluding remark with suggestions are provided in Sections 5 and 6.


## 2. Model Formulation

In this section, the total number of human population at a given timet is denoted by Nht and it is catagorized into six subclasses named as susceptible human population class (Sh), aware susceptible human population class (Ah), unaware susceptible human population class (Uh), exposed human population class (Eh), infectious human population class (Ih), and recovered human population class (Rh). The class of aware susceptible human will increase depending on the movement of susceptible human population and recovered human population class due to awareness created. Unaware susceptible human population class increases due to the movement of susceptible human population class and decreases due to contact with infectives of COVID-19 humans and progress into exposed and infectious human population class. In the formulation of our model, the following additional assumptions are important:
(1)
It is assumed that all recruited humans either by birth or immigration into susceptible human class are not carriers of COVID-19(2)
We consider only immigrants of COVID-19 asymptomatic human in our model with a rate ofηh(3)
It is assumed that the influx rateηh of COVID-19 asmptomatic human immigrants is not constant(4)
It is assumed that the class of unaware susceptible human populations becomes infective if they contact with exposed or infectious individuals at the rateβ with probability of α2and α1, respectively(5)
It is assumed that the rate of asymptomatic human immigrantsηh is less than the progression rate of exposed human into infectious or recovered human classδh(6)
It is assumed that a proportion ofθ of susceptible population properly apply COVID-19 mitigation measures and hence progress into aware susceptible human class and the remaining proportion (1−θ) move to unaware susceptible human class by the rate of ϕh(7)
It is assumed that recovered human populations develop permanent immunity due to their improved immunity or successful hospitalized treatment, and they join the aware susceptible human population class at the rate ofωh(8)
It is assumed that all parameters involved in the model are nonnegative. With regard to the above considerations, the compartmental flow diagram is shown below in Figure1Figure 1
Flow diagram of COVID-19 model.Based on assumptions and the flow diagram of COVID-19 dynamics (Figure1), Table 1 the model is governed by the following system of autonomous type nonlinear ordinary differential equations:
(1)dShdt=Λh−αh+ϕhSh,dAhdt=1−θϕhSh−αhAh+ωhRh,dUhdt=θϕhSh−βα2Eh+α1IhUh−αhUh,dEhdt=βα2Eh+α1IhUh−αh+δh−ηhEh,dIhdt=τδhEh−αh+ρh+γhIh,dRhdt=1−τδhEh+γhIh−αh+ωhRh,with initial conditions: Sh0>0, Ah0≥0, Uh0, Eh0≥0, Ih0≥0, Rh0≥0, and 0≤θ≤1,0≤τ≤1.Table 1
Model parameters with their description.
ParameterDescription of parameterΛhThe recruitment rate of susceptible human populationβThe contact rate of unaware susceptible humans with exposed and infectious humansθThe proportion rate that susceptible humans join unaware susceptible humansϕhThe progression rate of susceptible into aware and unaware susceptible humansα2The probability that unaware susceptible humans will contact with infectious humansα1The probability that unaware susceptible humans will contact with exposed humansδhThe progression rate of exposed humans into infectious and recovered human classγhThe recovery rate of infectious humans into recovered human classωhThe progression rate of recovered humans into aware susceptible human classαhThe natural death rate of all human population classesρhThe disease induced death rate of infectious humansτThe proportion rate of exposed humans to be infectiousηhThe rate of asymptomatic human immigrants


## 3. Model Analysis

The validity and the authenticity of any mathematical model depend on the existence and uniqueness of its solutions. In this section, the basic properties of the model system of equations (1) include the following: the existence and uniqueness of solutions, positivity of solutions, invariant region, and basic reproduction number, and equilibria with their stability analysis are focused.Lemma 1 (Existence and Uniqueness of Solutions).
If the initial data beSh0>0,Ah0≥0,Uh0≥0,Eh0≥0,Ih0≥0,Rh0≥0to the system equation (1), then there exists a unique solution in Cℝ+,ℝ+6, ∀t≥0.Proof.
The model system of equation (1) can be rewritten in the form of ẋ=fx, where
(2)ẋ=ṠhȦhU̇hĖhİhṘhfx=Λh−αh+ϕhSh1−θϕhSh−αhAh+ωhRhθϕhSh−βα2Eh+α1IhUh−αhUhβα2Eh+α1IhUh−αh+δh−ηhEhτδhEh−αh+ρh+γhIh1−τδhEh+γhIh−αh+ωhRh.
Each of the right hand side components offx in system [2] is continuously differentiable almost everywhere in Cℝ+,ℝ+6, which implies f is locally Lipchitz (f∈c1). Hence, by the fundamental existence and uniqueness theorem [22], the model system of equation (1) exhibits a unique solution locally inℝ+6 for all time t≥0. Since the model system of equations [1] monitors human population, it is necessary to show that all its solutions must be nonnegative for future time t. This will be established by the following theorem.Theorem 2 (Positivity of Model Solutions).
If the initial condition beΣ=Sh0>0,Ah0≥0,Uh0≥0,Eh0≥0,Ih0≥0,Rh0≥0, then the solution set Sht,Aht,Uht,Eht,Iht,Rht of model (1) is positively invariant for all time t≥0.Proof.
From the model system of equation (1), consider the first equation:
dSh/dt=Λh−αh+ϕhSh; then, by integrating factor method, the solution becomes the following:
(3)Sht=Λh/αh+ϕh+Sh0−Λh/αh+ϕhe−αh+ϕht>0.
Hence,Sht>0 for all time t≥0. From model system of equation (1), consider the third equation:
dUh/dt=θϕhSh−βα2Eh+α1IhUh−αhUh. Assume that for the first time t1, Uht1=0, dUh/dt≤0, and it is true that Sht,Aht,Eht,Iht,Rht>0 for t∈0,t1. Based on our assumption, dUh/dt=θϕhSh, and then, integrate both sides and substitute the solution of Sht obtained from the above, and we get that Uht1=θϕhΛh/αh+ϕh+Sh0−Λh/αh+ϕhe−αh+ϕht1≥0 which contradicts with our assumption. Hence, Uht≥0, ∀t≥0.
Similarly, it can be shown analogously thatAht,Eht,Iht,Rht≥0, ∀t≥0.
Hence, all solutions of model system [1] are positive for all future time t≥0.Theorem 3 (Invariant Region).
There exists a domainΣ in which the solution set Sht,Aht,Uht,Eht,Iht,Rht of model equation (1) is positively invariant.Proof.
The total human population size can be determined byNht=Sht+Aht+Uht+Eht+Iht+Rht. Then, the time derivative of Nht along the solutions of model system (1) gives the following:
(4)dNhdt=Λh−αhNht−ρhIht+ηhEht.
In the absence of disease (COVID-19) in the population and COVID-19 asymptomatic human immigrants,(5)dNhdt≤Λh−αhNht⇒Nht=Λhαh+Nh0−Λhαhe−αht,
where(6)Nh0=Sh0+Ah0+Uh0+Eh0+Ih0+Rh0.
Thus, ifNh0≤Λh/αh, then Nht≤Λh/αh as t⟶∞.
Therefore,Σ=Sht,Aht,Uht,Eht,Iht,Rht∈ℝ+6:Nh0≤Nht≤Λh/αh is the feasible solution of model equation (1) which implies the total number of human population is positively invariant; hence, each solution in the system of model equation (1) is positively invariant. Therefore, the model is biologically meaningful and mathematically well-posed in the region Σ.


### 3.1. COVID-19-Free Equilibrium Point of the Model

The disease (COVID-19)-free equilibrium pointE0 of model system (1) is calculated by equating all the right hand side equations to zero and puttingEht=Iht=0; then, we obtained
(7)E0=Sh0,Ah0,Uh0,Eh0,Ih0,Rh0=Λhαh+ϕh,1−θϕhΛhαhαh+ϕh,θϕhΛhαhαh+ϕh,0,0,0.In epidemiological point of view, the implication ofE0 in the absence of infective populations and immigrants of COVID-19 asymptomatic humans only, susceptible, aware susceptible, and unaware susceptible class of human populations will live in the community.


### 3.2. Basic Reproduction Number of the Model

In this subsection, we will find the basic reproduction number, denoted byR0 for the considered model (1) using next generation approach [23]. From model system (1), matrix that consists the rate of new infections Fix and the rate of transfer Vix is
(8)Fix=βα2Eh+α1IhUh0,Vix=αh+δh−ηhEh−τδhEh+αh+ρh+γhIh,respectively.The Jacobian matrix ofFix and Vix at the disease-free equilibrium point E0 is given by F and V, respectively, as follows:
(9)F=βα2Uh0βα1Uh000,V=αh+δh−ηh0−τδhαh+ρh+γh,and then, the inverse ofV is given by
(10)V−1=1αh+δh−ηh0τδhαh+δh−ηhαh+ρh+γh1αh+ρh+γh.Finally,FV−1=Λhβθϕhα2αh+ϕh+α1τδh/αhαh+ϕhαh+δh−ηhαh+ρh+γhΛhβθϕhα1/αhαh+ϕhαh+ρh+γh00.The two eigenvalues ofFV−1 are as follows:
(11)λ1=0,λ2=Λhβθϕhα2αh+ϕh+α1τδhαhαh+ϕhαh+δh−ηhαh+ρh+γh.It follows that(12)R0=maxλ1,λ2=Λhβθϕhα2αh+ϕh+α1τδhαhαh+ϕhαh+δh−ηhαh+ρh+γh.Here,1/αh+δh−ηh refers the average duration of human population in exposed state to become infectious or recovered, and 1/αh+ρh+γh is the average duration of the infectious period of human populations until they die or recover.


### 3.3. Stability Behavior of the COVID-19-Free Equilibrium Point

The following theorems discuss the local and global stability analysis of disease-free equilibrium point of model system (1).Theorem 4.
The disease-free equilibrium pointE0 of model system (1) is locally asymptotically stable if R0<1 and unstable if R0>1.Proof.
The Jacobian matrix of system (1) at the disease-free equilibrium E0 is given by
(13)JE0=−αh+ϕh000001−θϕh−αh000ωhθϕh0−αh−βα2θϕhΛhαhαh+ϕh−θβα1ϕhΛhαhαh+ϕh0000βα2θϕhΛhαhαh+ϕh−αh+δh−ηhβα1θϕhΛhαhαh+ϕh0000τδh−αh+ρh+γh00001−τδhγh−αh+ωh,
and the characteristic polynomial ofJE0 is expressed by the following:
(14)Pλ=λ+αh+ϕhλ+αh2λ+αh+ωhA0λ2+A1λ+A2=0.
The four eigenvalues ofJE0are as follows:
λ1=−αh+ϕh<0, λ2=λ3=−αh<0, and λ4=−αh+ωh<0, and the remaining eigenvalues are determined from kλ=A0λ2+A1λ+A2=0, where
(15)A0=1>0,A1=αh+ρh+γh1−βα2θϕhΛhαhαh+ϕhαh+ρh+γh+αh+δh−ηh,A2=αh+ρh+γhαh+δh−ηh1−Λhβθϕhα2αh+ϕh+α1τδhαhαh+ϕhαh+δh−ηhαh+ρh+γh=αh+ρh+γhαh+δh−ηh1−R0.
FromA0λ2+A1λ+A2=0, we recall Routh-Hurwitz criteria [24, 25] and characteristic equation A0λ2+A1λ+A2=0 has strictly negative real root if and only if A0>0, A1>0, A2>0, and A1A2>0. It is obvious that A0 and A1 are positive, and A2 is positive provided that 1−R0>0, which leads to R0<1. Therefore, the disease-free equilibrium point E0 is locally asymptotically stable if R0<11, and COVID-19 cannot invade the population. For R0>1, we see that A2<0. This shows as there is one eigenvalue with positive real part, and hence, the disease-free equilibrium is unstable and the invasion of COVID-19 is always possible.Theorem 5.
The COVID-19-free equilibrium pointE0 of model system of equation (1) is globally asymptotically stable if R0<1 and vice versa.Proof.
To prove this, we follow Castillo-Chavez et al. theorem [26].
Let us rewrite model system (1)
(16)dXdt=FX,Z,dZ/dt=GX,Z,
withGX,0=0, where X=Sht,Aht,Uht,Rht∈ℝ4 represents nondisease state variables and X=Eht,Iht∈ℝ2 represents disease state variables in model (1).
To beE0 globally asymptotically stable for model system (1), the following two scenarios must satisfied:
H1 For dX/dt=FX,0, X∗ is globally asymptotically stable, where FX∗,0=0.
Clearly one can see that at disease-free equilibrium pointE0 of model system (1),
(17)Limt⟶∞Sht,Aht,Uht,Rht=Λhαh+ϕh,1−θϕhΛhαhαh+ϕh,θϕhΛhαhαh+ϕh,0.
Hence,Sht,Aht,Uht,Rht⟶Λh/αh+ϕh,1−θϕhΛh/αhαh+ϕh,θϕhΛh/αhαh+ϕh,0 which implies the global convergence of model system (1) in Σ.
H2GX,Z=AZ−G∧X,Z, G∧X,Z≥0 for X,Z∈Σ, where A=∂ZE0/∂Eh,Ih, Z=Eh,IhT, and GX,Z=Eḣ,İhT. Thus,
(18)A=βα2θϕhΛhαhαh+ϕh−αh+δh−ηhβα1θϕhΛhαhαh+ϕhτδh−αh+ρh+γh.
FromGX,Z=AZ−G∧X,Z, we have that G∧X,Z=AZ−GX,Z, and after simplification, we get that
(19)G∧X,Z=βθϕhΛhα2Eh+α1Ihαhαh+ϕh−βα2Eh+α1IhUh0.
ClearlyA is an M-matrix (off diagonal elements of A are nonnegative inside Σ) and G∧X,Z≥0 because Uht=θϕhΛh/αh+ϕh+Sh0−Λh/αh+ϕhe−αh+ϕht≤βθϕhΛh/αhαh+ϕh as t⟶∞.
Therefore, the disease-free equilibrium pointE0is globally asymptotically stable for model equation (1) when R0<1, and the epidemiological implication of this result is that in the long run, the disease (COVID-19) will die out in the population if the awareness is highly created in the population and in the absence of COVID-19 asymptomatic human immigrants regardless of infective populations within the community.


### 3.4. Existence and Local Stability of COVID-19 Persistent Steady State

LetE∗=Sh,∗Ah∗,Uh∗,Eh∗,Ih∗,Rh∗ be COVID-19 persistent equilibrium point of model equation (1) in which all state variables are to be positive, and hence, the disease (COVID-19) persists in the population. The components of E∗ are obtained by making the right hand sides of model equation (1) to be zeros, and after some mathematical manipulation we get the following:
(20)Sh∗=Λhαh+ϕh,Uh∗=αh+δh−ηhEh∗αhR0−1,Eh∗=αhαh+ρh+γhR0−1βα1τδh+βα2αh+ρh+γh,Ih∗=τδhEh∗αh+ρh+γh,Rh∗=γhτδh+αh+ρh+γh1−τδhEh∗αh+ωhαh+ρh+γh,Ah∗=γhτδh+αh+ρh+γhαh+ωh1−θϕhΛh+γhτδh+1−τωhδhMEh∗αhαh+ωhαh+ρh+γhαh+ϕh,where(21)M=αh+ρh+γhαh+xϕh.As we observe,Sh∗is positive and from the value of Eh∗, it is obvious that all the values of Ah∗,Uh∗,Ih∗,Rh∗ are positive for R0>1.Corollary 6.
The COVID-19 persistent equilibrium pointE∗of model system (1) exists only when R0>1.Theorem 7.
The COVID-19 persistent steady stateE∗=(Sh∗,Ah∗,Ut∗,Eh∗,Ih∗,Rh∗) of model (1) is locally asymptotically stable if and only if R0>1.Proof.
The linearized matrix of the model system (1) at the endemic steady state E∗ is given by
(22)JE∗=−k100000k2−αh000ωhk30−k6Eh∗+k4Ih∗+αh−k6Uh∗−k4Uh∗000k6Eh∗+k4Ih∗k6Uh∗−αh+δh−ηhk4Uh∗0000k7−k80000k9γh−k10,
where(23)k1=αh+ϕh,k2=1−θϕh,k3=θϕh,k4=α1β,k5=ϕh+δh,k6=βα2,k7=τδh,k8=αh+ρh+γh,k9=1−τδh,k10=αh+ωh,A=k4k7+k6k8.
The three eigenvalues ofJE∗ are λ1=−k1=−αh+ϕh<0,λ2=−αh<0,λ3=−k10=−αh+ϕh<0, and the remaining eigenvalues are determined from the submatrix given by
(24)J4E∗=−k6k8Eh∗+αhk8+k4k7Eh∗k8−k6k5−ηhEh∗αhR0−1−k4k5−ηhEh∗αhR0−1k6k8Eh∗+k4k7Eh∗k8k6k5−ηhEh∗αhR0−1k4k5−ηhEh∗αhR0−10k7−k8.
From COVID-19 persistent equilibrium pointE∗, further substitution of the value of k and A, we get that
(25)Eh∗=αhk8R0−1k4k7+k6k8=αhk8R0−1A,Uh∗=k8k5−ηhEh∗k4k7+k6k8=k8k5−ηhEh∗AandIh∗=k7Eh∗k8..
Now without calculating the eigenvalues ofJ4E∗, we can simply look the signs of its eigenvalues by using trace determinant rule as follows: traceJ4E∗=−αhR0A+k4k7k5−ηh+k8A/A<0. Therefore, traceJ4E∗<0. Furthermore, detJ4E∗=k6k82αhk5−ηhR0−R0−1/A+αhk5−ηhk4R0+k6k8R0+k4k7/A>0, which can be verified by substitute traceJ4E∗ inequality into detJ4E∗. Thus, the above scenarios, i.e., λ1=−k1=−αh+ϕh<0, λ2=−αh<0, λ3=−k10=−αh+ϕh<0, and traceJ4E∗<0 and detJ4E∗>0, lead to the COVID-19 persistent steady state E∗ of model system (1) which is locally asymptotically stable whenever R0>1. Hence, it is the required result.


### 3.5. Sensitivity Analysis of the Basic Reproduction Number

Sensitivity analysis helps to identify the most influential parameters on the basic reproduction number so efforts to control the problem are directed to these parameters. Mathematically, we compute the sensitivity analysis of our model system of equations based on the classical definition [14, 15] defined as the normalized forward sensitivity index of a variable R0, which depends differentiably on a parameter p, given by △pR0=p/R0×∂R0/∂p.For example,△βR0≡1, means increasing (or decreasing) the contact rate βof unaware susceptible with exposed and infectious individuals by 10%, will result to increase (or decrease) the value of R0 by 10%, whereas△δhR0≡−5.11, means increasing (or decreasing) the recovery rate of infectious humans, will result to decrease (or increase) the value of R0 by 51.1%. The remaining sensitivity analysis of our model can be obtained as follows:
(26)△βR0=1>0,△α1R0=α1τδhα1τδh+αhαh+ρh+γh>0,△τR0=α1τδhα1τδh+α2αh+ρh+γh>0,△ϕhR0=αhαh+ϕh>0,△θR0=1>0,△α2R0=α2αh+ρh+γhα1τδh+α2αh+ρh+γh>0,△ηhR0=ηhα1+δh−ηh>0,△γhR0=α2γhα1τδh+α2αh+ρh+γh−1αh+ρh+γh<0,△δhR0=δhτα1α1τδh+α2αh+ρh+γh−1α1+δh−ηh<0.The sensitivity analysis indices evaluated at the baseline model parameters values are resembled from Table2. The sensitivity indices are arranged in descending order as follows.Table 2
Parameter values and its sensitivity indices.
ParameterValuesSensitivity indicesSourceβ0.0143+1[16]Λh13.5+1[16]θ0.5+1Assumedα20.02+0.998[16]ϕh0.01+0.615Assumedα10.0001+0.0018[16]ηh0.0001+0.0016Assumedτ0.7+0.0015[16]γh0.15-5.11[16]δh0.07-11.64[16]


#### 3.5.1. Interpretation of Sensitivity Indices

From Table2, parameters that have positive indices have a negative impact in the control of COVID-19 transmission if their values are increasing. On the other hand, parameters which have negative indices have a positive impact to minimize the burden of the disease (COVID-19) transmission in the society. Thus, the most sensitive parameters for the transmission of disease (COVID-19) are βand θ, and the most sensitive parameter for the control of the disease (COVID-19) is δh followed byγh.


## 3.1. COVID-19-Free Equilibrium Point of the Model

The disease (COVID-19)-free equilibrium pointE0 of model system (1) is calculated by equating all the right hand side equations to zero and puttingEht=Iht=0; then, we obtained
(7)E0=Sh0,Ah0,Uh0,Eh0,Ih0,Rh0=Λhαh+ϕh,1−θϕhΛhαhαh+ϕh,θϕhΛhαhαh+ϕh,0,0,0.In epidemiological point of view, the implication ofE0 in the absence of infective populations and immigrants of COVID-19 asymptomatic humans only, susceptible, aware susceptible, and unaware susceptible class of human populations will live in the community.


## 3.2. Basic Reproduction Number of the Model

In this subsection, we will find the basic reproduction number, denoted byR0 for the considered model (1) using next generation approach [23]. From model system (1), matrix that consists the rate of new infections Fix and the rate of transfer Vix is
(8)Fix=βα2Eh+α1IhUh0,Vix=αh+δh−ηhEh−τδhEh+αh+ρh+γhIh,respectively.The Jacobian matrix ofFix and Vix at the disease-free equilibrium point E0 is given by F and V, respectively, as follows:
(9)F=βα2Uh0βα1Uh000,V=αh+δh−ηh0−τδhαh+ρh+γh,and then, the inverse ofV is given by
(10)V−1=1αh+δh−ηh0τδhαh+δh−ηhαh+ρh+γh1αh+ρh+γh.Finally,FV−1=Λhβθϕhα2αh+ϕh+α1τδh/αhαh+ϕhαh+δh−ηhαh+ρh+γhΛhβθϕhα1/αhαh+ϕhαh+ρh+γh00.The two eigenvalues ofFV−1 are as follows:
(11)λ1=0,λ2=Λhβθϕhα2αh+ϕh+α1τδhαhαh+ϕhαh+δh−ηhαh+ρh+γh.It follows that(12)R0=maxλ1,λ2=Λhβθϕhα2αh+ϕh+α1τδhαhαh+ϕhαh+δh−ηhαh+ρh+γh.Here,1/αh+δh−ηh refers the average duration of human population in exposed state to become infectious or recovered, and 1/αh+ρh+γh is the average duration of the infectious period of human populations until they die or recover.


## 3.3. Stability Behavior of the COVID-19-Free Equilibrium Point

The following theorems discuss the local and global stability analysis of disease-free equilibrium point of model system (1).Theorem 4.
The disease-free equilibrium pointE0 of model system (1) is locally asymptotically stable if R0<1 and unstable if R0>1.Proof.
The Jacobian matrix of system (1) at the disease-free equilibrium E0 is given by
(13)JE0=−αh+ϕh000001−θϕh−αh000ωhθϕh0−αh−βα2θϕhΛhαhαh+ϕh−θβα1ϕhΛhαhαh+ϕh0000βα2θϕhΛhαhαh+ϕh−αh+δh−ηhβα1θϕhΛhαhαh+ϕh0000τδh−αh+ρh+γh00001−τδhγh−αh+ωh,
and the characteristic polynomial ofJE0 is expressed by the following:
(14)Pλ=λ+αh+ϕhλ+αh2λ+αh+ωhA0λ2+A1λ+A2=0.
The four eigenvalues ofJE0are as follows:
λ1=−αh+ϕh<0, λ2=λ3=−αh<0, and λ4=−αh+ωh<0, and the remaining eigenvalues are determined from kλ=A0λ2+A1λ+A2=0, where
(15)A0=1>0,A1=αh+ρh+γh1−βα2θϕhΛhαhαh+ϕhαh+ρh+γh+αh+δh−ηh,A2=αh+ρh+γhαh+δh−ηh1−Λhβθϕhα2αh+ϕh+α1τδhαhαh+ϕhαh+δh−ηhαh+ρh+γh=αh+ρh+γhαh+δh−ηh1−R0.
FromA0λ2+A1λ+A2=0, we recall Routh-Hurwitz criteria [24, 25] and characteristic equation A0λ2+A1λ+A2=0 has strictly negative real root if and only if A0>0, A1>0, A2>0, and A1A2>0. It is obvious that A0 and A1 are positive, and A2 is positive provided that 1−R0>0, which leads to R0<1. Therefore, the disease-free equilibrium point E0 is locally asymptotically stable if R0<11, and COVID-19 cannot invade the population. For R0>1, we see that A2<0. This shows as there is one eigenvalue with positive real part, and hence, the disease-free equilibrium is unstable and the invasion of COVID-19 is always possible.Theorem 5.
The COVID-19-free equilibrium pointE0 of model system of equation (1) is globally asymptotically stable if R0<1 and vice versa.Proof.
To prove this, we follow Castillo-Chavez et al. theorem [26].
Let us rewrite model system (1)
(16)dXdt=FX,Z,dZ/dt=GX,Z,
withGX,0=0, where X=Sht,Aht,Uht,Rht∈ℝ4 represents nondisease state variables and X=Eht,Iht∈ℝ2 represents disease state variables in model (1).
To beE0 globally asymptotically stable for model system (1), the following two scenarios must satisfied:
H1 For dX/dt=FX,0, X∗ is globally asymptotically stable, where FX∗,0=0.
Clearly one can see that at disease-free equilibrium pointE0 of model system (1),
(17)Limt⟶∞Sht,Aht,Uht,Rht=Λhαh+ϕh,1−θϕhΛhαhαh+ϕh,θϕhΛhαhαh+ϕh,0.
Hence,Sht,Aht,Uht,Rht⟶Λh/αh+ϕh,1−θϕhΛh/αhαh+ϕh,θϕhΛh/αhαh+ϕh,0 which implies the global convergence of model system (1) in Σ.
H2GX,Z=AZ−G∧X,Z, G∧X,Z≥0 for X,Z∈Σ, where A=∂ZE0/∂Eh,Ih, Z=Eh,IhT, and GX,Z=Eḣ,İhT. Thus,
(18)A=βα2θϕhΛhαhαh+ϕh−αh+δh−ηhβα1θϕhΛhαhαh+ϕhτδh−αh+ρh+γh.
FromGX,Z=AZ−G∧X,Z, we have that G∧X,Z=AZ−GX,Z, and after simplification, we get that
(19)G∧X,Z=βθϕhΛhα2Eh+α1Ihαhαh+ϕh−βα2Eh+α1IhUh0.
ClearlyA is an M-matrix (off diagonal elements of A are nonnegative inside Σ) and G∧X,Z≥0 because Uht=θϕhΛh/αh+ϕh+Sh0−Λh/αh+ϕhe−αh+ϕht≤βθϕhΛh/αhαh+ϕh as t⟶∞.
Therefore, the disease-free equilibrium pointE0is globally asymptotically stable for model equation (1) when R0<1, and the epidemiological implication of this result is that in the long run, the disease (COVID-19) will die out in the population if the awareness is highly created in the population and in the absence of COVID-19 asymptomatic human immigrants regardless of infective populations within the community.


## 3.4. Existence and Local Stability of COVID-19 Persistent Steady State

LetE∗=Sh,∗Ah∗,Uh∗,Eh∗,Ih∗,Rh∗ be COVID-19 persistent equilibrium point of model equation (1) in which all state variables are to be positive, and hence, the disease (COVID-19) persists in the population. The components of E∗ are obtained by making the right hand sides of model equation (1) to be zeros, and after some mathematical manipulation we get the following:
(20)Sh∗=Λhαh+ϕh,Uh∗=αh+δh−ηhEh∗αhR0−1,Eh∗=αhαh+ρh+γhR0−1βα1τδh+βα2αh+ρh+γh,Ih∗=τδhEh∗αh+ρh+γh,Rh∗=γhτδh+αh+ρh+γh1−τδhEh∗αh+ωhαh+ρh+γh,Ah∗=γhτδh+αh+ρh+γhαh+ωh1−θϕhΛh+γhτδh+1−τωhδhMEh∗αhαh+ωhαh+ρh+γhαh+ϕh,where(21)M=αh+ρh+γhαh+xϕh.As we observe,Sh∗is positive and from the value of Eh∗, it is obvious that all the values of Ah∗,Uh∗,Ih∗,Rh∗ are positive for R0>1.Corollary 6.
The COVID-19 persistent equilibrium pointE∗of model system (1) exists only when R0>1.Theorem 7.
The COVID-19 persistent steady stateE∗=(Sh∗,Ah∗,Ut∗,Eh∗,Ih∗,Rh∗) of model (1) is locally asymptotically stable if and only if R0>1.Proof.
The linearized matrix of the model system (1) at the endemic steady state E∗ is given by
(22)JE∗=−k100000k2−αh000ωhk30−k6Eh∗+k4Ih∗+αh−k6Uh∗−k4Uh∗000k6Eh∗+k4Ih∗k6Uh∗−αh+δh−ηhk4Uh∗0000k7−k80000k9γh−k10,
where(23)k1=αh+ϕh,k2=1−θϕh,k3=θϕh,k4=α1β,k5=ϕh+δh,k6=βα2,k7=τδh,k8=αh+ρh+γh,k9=1−τδh,k10=αh+ωh,A=k4k7+k6k8.
The three eigenvalues ofJE∗ are λ1=−k1=−αh+ϕh<0,λ2=−αh<0,λ3=−k10=−αh+ϕh<0, and the remaining eigenvalues are determined from the submatrix given by
(24)J4E∗=−k6k8Eh∗+αhk8+k4k7Eh∗k8−k6k5−ηhEh∗αhR0−1−k4k5−ηhEh∗αhR0−1k6k8Eh∗+k4k7Eh∗k8k6k5−ηhEh∗αhR0−1k4k5−ηhEh∗αhR0−10k7−k8.
From COVID-19 persistent equilibrium pointE∗, further substitution of the value of k and A, we get that
(25)Eh∗=αhk8R0−1k4k7+k6k8=αhk8R0−1A,Uh∗=k8k5−ηhEh∗k4k7+k6k8=k8k5−ηhEh∗AandIh∗=k7Eh∗k8..
Now without calculating the eigenvalues ofJ4E∗, we can simply look the signs of its eigenvalues by using trace determinant rule as follows: traceJ4E∗=−αhR0A+k4k7k5−ηh+k8A/A<0. Therefore, traceJ4E∗<0. Furthermore, detJ4E∗=k6k82αhk5−ηhR0−R0−1/A+αhk5−ηhk4R0+k6k8R0+k4k7/A>0, which can be verified by substitute traceJ4E∗ inequality into detJ4E∗. Thus, the above scenarios, i.e., λ1=−k1=−αh+ϕh<0, λ2=−αh<0, λ3=−k10=−αh+ϕh<0, and traceJ4E∗<0 and detJ4E∗>0, lead to the COVID-19 persistent steady state E∗ of model system (1) which is locally asymptotically stable whenever R0>1. Hence, it is the required result.


## 3.5. Sensitivity Analysis of the Basic Reproduction Number

Sensitivity analysis helps to identify the most influential parameters on the basic reproduction number so efforts to control the problem are directed to these parameters. Mathematically, we compute the sensitivity analysis of our model system of equations based on the classical definition [14, 15] defined as the normalized forward sensitivity index of a variable R0, which depends differentiably on a parameter p, given by △pR0=p/R0×∂R0/∂p.For example,△βR0≡1, means increasing (or decreasing) the contact rate βof unaware susceptible with exposed and infectious individuals by 10%, will result to increase (or decrease) the value of R0 by 10%, whereas△δhR0≡−5.11, means increasing (or decreasing) the recovery rate of infectious humans, will result to decrease (or increase) the value of R0 by 51.1%. The remaining sensitivity analysis of our model can be obtained as follows:
(26)△βR0=1>0,△α1R0=α1τδhα1τδh+αhαh+ρh+γh>0,△τR0=α1τδhα1τδh+α2αh+ρh+γh>0,△ϕhR0=αhαh+ϕh>0,△θR0=1>0,△α2R0=α2αh+ρh+γhα1τδh+α2αh+ρh+γh>0,△ηhR0=ηhα1+δh−ηh>0,△γhR0=α2γhα1τδh+α2αh+ρh+γh−1αh+ρh+γh<0,△δhR0=δhτα1α1τδh+α2αh+ρh+γh−1α1+δh−ηh<0.The sensitivity analysis indices evaluated at the baseline model parameters values are resembled from Table2. The sensitivity indices are arranged in descending order as follows.Table 2
Parameter values and its sensitivity indices.
ParameterValuesSensitivity indicesSourceβ0.0143+1[16]Λh13.5+1[16]θ0.5+1Assumedα20.02+0.998[16]ϕh0.01+0.615Assumedα10.0001+0.0018[16]ηh0.0001+0.0016Assumedτ0.7+0.0015[16]γh0.15-5.11[16]δh0.07-11.64[16]


### 3.5.1. Interpretation of Sensitivity Indices

From Table2, parameters that have positive indices have a negative impact in the control of COVID-19 transmission if their values are increasing. On the other hand, parameters which have negative indices have a positive impact to minimize the burden of the disease (COVID-19) transmission in the society. Thus, the most sensitive parameters for the transmission of disease (COVID-19) are βand θ, and the most sensitive parameter for the control of the disease (COVID-19) is δh followed byγh.


## 3.5.1. Interpretation of Sensitivity Indices

From Table2, parameters that have positive indices have a negative impact in the control of COVID-19 transmission if their values are increasing. On the other hand, parameters which have negative indices have a positive impact to minimize the burden of the disease (COVID-19) transmission in the society. Thus, the most sensitive parameters for the transmission of disease (COVID-19) are βand θ, and the most sensitive parameter for the control of the disease (COVID-19) is δh followed byγh.


## 4. Numerical Results and Discussion

### 4.1. Graphs for General Population Dynamics

In this section, to verify the theoretical results of the model, numerical simulations are carried out by using MATLAB ode45 solver with the following initial conditions:Sh0=120,000,Ah0=30,000, Uh0=50,000,Eh0=25,000,Ih0=35,000,Rh0=20,000. It is important to note that parameter values and initial number of populations are taken for illustrative purpose.Figure2(a)reflects that when the value of Λh=7.99, θ=0.47, α1=0.19, α2=0.1399, αh=0.22, ϕh=0.59, ωh=0.1, τ=0.09, β=0.389, δh=0.6, ρh=0.99, γh=0.1, and ηh=0.0898, then R0=0.9785<1. It is shown that all trajectories of the solutions of the model system (1) converge towards disease-free equilibrium point components, or noninfective class of human population tends to nonzero components, and the infective class of human population tends to zero component. In this case, the basic reproduction number is less than unity (R0=0.9785<1) which confirms with our local stability analysis of disease (COVID-19)-free equilibrium point for model system (1) whenever R0<1 stated from Theorem 4.Figure 2
Change of human populations at disease free and endemic equilibrium points.
(a)
R0=0.9785<1(b)
R0=21.2443>1Figure2(b)reflects that when parameter values changed as Λh=12.4, θ=0.99, α1=0.2, α2=0.14, αh=0.0701, ϕh=0.79, ωh=0.1, τ=0.79, β=0.39, δh=0.5, ρh=0.75, γh=0.0391, and ηh=0.0899, then R0=21.2443>1. It is shown that all trajectories of solutions of model system (1) converge towards the endemic (COVID-19 persistent) equilibrium point components of model (1), or all distinct classes of human populations coexist. In this case, the basic reproduction number is greater than unity (R0=21.2443>1) which supports our analytical result about local stability of endemic equilibrium point E∗ for model system (1) whenever R0>1 stated from Theorem 7.


### 4.2. Graphs on Newly Included Parameters in the Model

As we observe from Figure3(a), when the values of ωh and γh increase and the values of θ decreases while other parameter values remain constant, then the value of the secondary infection, R0, decreases and this leads to increase the number of aware susceptible human population. If we implement effective awareness creation mechanisms between individuals, then the transmission rate and the spread of (COVID-19) will be eliminated. As we observe from Figure 3(b), when the values of ηh, θ, and β increase, then the values of secondary infection, R0, increases. If we implement effective mechanisms to create awareness between humans to avoid contacts and reduce the immigration rate of COVID-19 asymptomatic humans, then the number of exposed human population will be reduced and also the transmission of COVID-19 pandemic will be reduced.Figure 3
The impacts of some parameters on aware susceptible and exposed human populations.
(a)
The impact ofωh, γh, and θ on aware susceptible humans(b)
The impact ofηh, θ, and β on exposed humans


### 4.3. Graphs for Sensitive Analysis of the Model

The authors can find some significant results which have shown in Figures4(a) and 4(b), and one can observe that the large value of θ or ηhcan lead to the large value of secondary infection R0. This implies that high proportion rate from susceptible human into unaware susceptible human or high rate of COVID-19 asymptomatic human immigrants can increase the opportunity of COVID-19 outbreak. Generally, from Figure 4, we found that R0 is more sensitive to the proportion rate (θ) of susceptible humans into unaware susceptible human than the rate of COVID-19 asymptomatic human immigrant (ηh). This supports the idea that θ is the most sensitive parameter in the transmission of COVID-19. Therefore, decreasing the proportion rate of humans into unaware human class by creating unlimited awareness between individuals helps to reduce COVID-19 outbreak. From Figures 5(a) and 5(b), we can see that the large value of δh or γhcan lead to the small value of secondary infection, R0. This is to say that the high progression rate from exposed into infectious or recovered class and the recovery rate from infectious human class into recovered human class can decrease the opportunity of COVID-19 outbreak. Generally, from Figure 5, we found that R0 is less sensitive to the progression rate (δh) of exposed humans into infectious and recovered human class than the recovery rate γh of individuals from their infection. This supports the idea of sensitivity analysis result that δh is the most sensitive parameter in the control of COVID-19 transmission than γh.Figure 4
The impact ofθ and ηh on basic reproduction number R0.
(a)
Basic reproduction numberR0 vs. θ(b)
Basic reproduction numberR0 vs. ηhFigure 5
The impact ofδh and γh on basic reproduction number R0.
(a)
Basic reproduction numberR0 vs. δh(b)
Basic reproduction numberR0 vs. γhIn Figure6(a), it can be seen that large value of δh or γh and small value of ηh in the presence of β can lead to small value of R0. That is to say, if we increase the progression rate of exposed into infectious by diagnosis to be quarantined or recovered human class by treatment and the rate of COVID-19 asymptomatic humans into exposed human class by prediagnosis in the presence of human to human contact, the transmission of COVID-19 pandemic will decrease in the population. From Figure 6(b), it can be seen that large value of θ or ηh or τ can lead to the large value of R0. If we reduce the proportion rate from susceptible human class into unaware susceptible human class by creating awareness and the rate of infective COVID-19 asymptomatic human immigrants by effective prediagnosis, then the disease (COVID-19) outbreak will end.Figure 6
The impact of onηh, γh, δh, θ, and τ on R0in the presence of contact rate β.
(a)
The impact ofηh, γh, and δh on R0 in the presence of contact rate β(b)
The impact ofηh, θ, and τ on R0 in the presence of contact rate β


## 4.1. Graphs for General Population Dynamics

In this section, to verify the theoretical results of the model, numerical simulations are carried out by using MATLAB ode45 solver with the following initial conditions:Sh0=120,000,Ah0=30,000, Uh0=50,000,Eh0=25,000,Ih0=35,000,Rh0=20,000. It is important to note that parameter values and initial number of populations are taken for illustrative purpose.Figure2(a)reflects that when the value of Λh=7.99, θ=0.47, α1=0.19, α2=0.1399, αh=0.22, ϕh=0.59, ωh=0.1, τ=0.09, β=0.389, δh=0.6, ρh=0.99, γh=0.1, and ηh=0.0898, then R0=0.9785<1. It is shown that all trajectories of the solutions of the model system (1) converge towards disease-free equilibrium point components, or noninfective class of human population tends to nonzero components, and the infective class of human population tends to zero component. In this case, the basic reproduction number is less than unity (R0=0.9785<1) which confirms with our local stability analysis of disease (COVID-19)-free equilibrium point for model system (1) whenever R0<1 stated from Theorem 4.Figure 2
Change of human populations at disease free and endemic equilibrium points.
(a)
R0=0.9785<1(b)
R0=21.2443>1Figure2(b)reflects that when parameter values changed as Λh=12.4, θ=0.99, α1=0.2, α2=0.14, αh=0.0701, ϕh=0.79, ωh=0.1, τ=0.79, β=0.39, δh=0.5, ρh=0.75, γh=0.0391, and ηh=0.0899, then R0=21.2443>1. It is shown that all trajectories of solutions of model system (1) converge towards the endemic (COVID-19 persistent) equilibrium point components of model (1), or all distinct classes of human populations coexist. In this case, the basic reproduction number is greater than unity (R0=21.2443>1) which supports our analytical result about local stability of endemic equilibrium point E∗ for model system (1) whenever R0>1 stated from Theorem 7.


## 4.2. Graphs on Newly Included Parameters in the Model

As we observe from Figure3(a), when the values of ωh and γh increase and the values of θ decreases while other parameter values remain constant, then the value of the secondary infection, R0, decreases and this leads to increase the number of aware susceptible human population. If we implement effective awareness creation mechanisms between individuals, then the transmission rate and the spread of (COVID-19) will be eliminated. As we observe from Figure 3(b), when the values of ηh, θ, and β increase, then the values of secondary infection, R0, increases. If we implement effective mechanisms to create awareness between humans to avoid contacts and reduce the immigration rate of COVID-19 asymptomatic humans, then the number of exposed human population will be reduced and also the transmission of COVID-19 pandemic will be reduced.Figure 3
The impacts of some parameters on aware susceptible and exposed human populations.
(a)
The impact ofωh, γh, and θ on aware susceptible humans(b)
The impact ofηh, θ, and β on exposed humans


## 4.3. Graphs for Sensitive Analysis of the Model

The authors can find some significant results which have shown in Figures4(a) and 4(b), and one can observe that the large value of θ or ηhcan lead to the large value of secondary infection R0. This implies that high proportion rate from susceptible human into unaware susceptible human or high rate of COVID-19 asymptomatic human immigrants can increase the opportunity of COVID-19 outbreak. Generally, from Figure 4, we found that R0 is more sensitive to the proportion rate (θ) of susceptible humans into unaware susceptible human than the rate of COVID-19 asymptomatic human immigrant (ηh). This supports the idea that θ is the most sensitive parameter in the transmission of COVID-19. Therefore, decreasing the proportion rate of humans into unaware human class by creating unlimited awareness between individuals helps to reduce COVID-19 outbreak. From Figures 5(a) and 5(b), we can see that the large value of δh or γhcan lead to the small value of secondary infection, R0. This is to say that the high progression rate from exposed into infectious or recovered class and the recovery rate from infectious human class into recovered human class can decrease the opportunity of COVID-19 outbreak. Generally, from Figure 5, we found that R0 is less sensitive to the progression rate (δh) of exposed humans into infectious and recovered human class than the recovery rate γh of individuals from their infection. This supports the idea of sensitivity analysis result that δh is the most sensitive parameter in the control of COVID-19 transmission than γh.Figure 4
The impact ofθ and ηh on basic reproduction number R0.
(a)
Basic reproduction numberR0 vs. θ(b)
Basic reproduction numberR0 vs. ηhFigure 5
The impact ofδh and γh on basic reproduction number R0.
(a)
Basic reproduction numberR0 vs. δh(b)
Basic reproduction numberR0 vs. γhIn Figure6(a), it can be seen that large value of δh or γh and small value of ηh in the presence of β can lead to small value of R0. That is to say, if we increase the progression rate of exposed into infectious by diagnosis to be quarantined or recovered human class by treatment and the rate of COVID-19 asymptomatic humans into exposed human class by prediagnosis in the presence of human to human contact, the transmission of COVID-19 pandemic will decrease in the population. From Figure 6(b), it can be seen that large value of θ or ηh or τ can lead to the large value of R0. If we reduce the proportion rate from susceptible human class into unaware susceptible human class by creating awareness and the rate of infective COVID-19 asymptomatic human immigrants by effective prediagnosis, then the disease (COVID-19) outbreak will end.Figure 6
The impact of onηh, γh, δh, θ, and τ on R0in the presence of contact rate β.
(a)
The impact ofηh, γh, and δh on R0 in the presence of contact rate β(b)
The impact ofηh, θ, and τ on R0 in the presence of contact rate β


## 5. Conclusion

In this study, a nonlinear deterministic mathematical model of COVID-19 pandemic is developed and analyzed to investigate the impact of awareness and COVID-19 asymptomatic human immigrants in the transmission of COVID-19. We first obtained the domain where the model gives epidemiologically meaningful and mathematically well-posed by the fundamental existence and uniqueness theorem. Both positivity and invariant region of the model solutions are shown analytically. The basic reproduction number,R0, is computed using next generation matrix approach. The analytical analysis showed that the disease (COVID-19) free and endemic (COVID-19 persistent) equilibrium points of the model exist under certain conditions. We analyzed both the local and global stability of disease-free equilibrium point based on R0. The disease-free equilibrium point of the model is locally as well as globally asymptotically stable whenever R0<1 and unstable whenever R0>1. From epidemiological point of view, the disease (COVID-19) will die out in the population whenever <1 and persists in the population whenever R0>1. Positive endemic equilibrium point of the model exists, and it is locally asymptotically stable whenever R0>1, so that the reinvasion of COVID-19 may possible in the population.We performed sensitivity analysis of the basic reproduction number with respect to model parameters to identify which parameters have a strong influence on COVID-19 transmission dynamical system. Both analytical analysis and numerical simulation results of the model ensured that the most sensitive parameters for the transmission of COVID-19 areθ in which susceptible individuals will join unaware human class and contact rate β of those unaware susceptible humans with exposed and infectious human population, while the most sensitive parameter to control COVID-19 transmission is the progression rate δh followed by the recovery rate γh.


## 6. Concluding Remarks and Suggestions

It is necessary to achieve a better understanding on the COVID-19 pandemic, taking into account awareness campaign between humans and control of COVID-19 asymptomatic human immigrants in order to reduce the number of infections and mortality rates. The model developed in the present manuscript has the advantage of describing the best way of controlling the COVID-19 outbreak. As we demonstrated in the theoretical analysis and numerical results, reducing the values ofθ (the proportion rate of susceptible into unaware human class) and ηh (the rate of COVID-19 asymptomatic human immigrants) helps to reduce exposed and infectious individuals. These help to control COVID-19 outbreak. Therefore, in order to control COVID-19 outbreak, policy makers or health workers must give great emphasis on how to further create awareness between humans and effective mechanisms to reduce infective COVID-19 asymptomatic human immigrants.



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*Source: 6260262-2022-05-28.xml*