42 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 The Effect of Individuals Asymptomatic (Carrier) on The Dynamical Behavior Of a COVID-19 Virus Abstract In this paper, a novel coronavirus (COVID-19) model is proposed and investigated. In fact, the pandemic spread through a close contact between infected people and other people but sometimes the infected people could show two cases; the first is symptomatic and the other is asymptomatic (carrier) as the source of the risk. The outbreak of Covid-19 virus is described by a mathematical model dividing the population into four classes. The first class represents the susceptible people who are unaware of the disease. The second class refers to the susceptible people who are aware of the epidemic by media coverage. The third class is the carrier individuals (asymptomatic) and the fourth class represents the infected individuals. The existence, uniqueness and bounded-ness of the solutions of the model are discussed. All possible equilibrium points are determined. The locally asymptotically stable of the model is studied. Suitable Lyapunov functions are used to investigate the globally asymptotical stability of the model. Finally, numerical simulation is carried out to confirm the analytical results and to understand the effect of varying the parameters of how the disease spreads. Keywords: n-Coronavirus, Covid-19, Stability, Awareness, Asymptomatic People. 1. Introduction In November 2002, the severe acute respiratory syndrome coronavirus emerged in China causing global anxiety as the outbreak rapidly spreads, and by July 2003, had resulted in over 8000 cases in 26 countries. The SARS epidemic of 2003 reported 8098 cases with 774 deaths, and was eventually brought under control by July 2003, in a period of 8 months [1,2]. Ahmed et al [3] gives a review of China response in comparison with other SARS-affected countries. Riley et al [4] studied the transmission dynamics of SARS in Hong Kong. Ibn Al Haitham Journal for Pure and Applied Science Journal homepage: http://jih.uobaghdad.edu.iq/index.php/j/index Doi: 10.30526/34.2.2624 Article history: Received,15,June,2020, Accepted 20,Julay ,2020, Published in April 2021 Husseiny-Hassan F. AL Hassan.fadhil.r@sc.uobaghdad.edu.iq Department of Mathematics, College of Science, University of Baghdad, Iraq. Ahmed A. Mohsen aamuhseen@gmail.comDepartment Department of Mathematics, College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad, Iraq Xueyong Zhou xueyongzhou@xynu.edu.cn School of Mathematics and Statistics, Xinyang Normal University, Xinyang 464000, Henan, P.R. China mailto:aamuhseen@gmail.comDepartment mailto:xueyongzhou@xynu.edu.cn 43 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 Since December 2019, a novel corona virus, named (COVID-19), emerged in Wuhan, China and the epidemic has globally spread, updated on January 30, 2020, COVID-19 was declared a public health emergency of international concern. It caused more than 200,000 deaths out of 3,177,500 cases in the world among which 1,434,000 in Europe, 1,291,000 in Americas, 148,000 in Western Pacific and other cases in Asia and Africa [5]. The number of the total cases, total death, new confirmed cases and recovery for some countries in the world are shown in Figure 1. Figure 1. The number of total cases, total death, new confirmed cases and recovery of some countries in world from 20 January to 05 May 2020. There are some factors that complicate the infection dynamics of novel coronavirus and add challenges to how to control the epidemic. First, the clinical evidence shows that the incubation period of this epidemic ranges from 7 to 14 days. During this period of time, infected individuals may not develop any symptoms and may not be aware of their infection, yet they are capable of transmitting the disease to other people. Secondly, the virus is new and there are no vaccines currently available for treatment. Thirdly, many people do not take into account prevention measures. Fourthly, the origin of the infection is still uncertain although it is widely speculated that wild animals such as bats, civets and minks are responsible for starting up the epidemic [6]. A number of modeling studies have already been performed for the COVID-19 epidemic. Feng et al. [7] studied a COVID-19 model in UK with the effects of media and quarantine.Yang and Wang [8] have proposed a mathematical model for COVID- 19 epidemic in Wuhan, China. Wu et al. [9] have introduced epidemic model with (SEIR) type to study the transmission dynamic and global spread of the disease depending on reported data from December 31, 2019 to January 28, 2020. Mohsen et al. [10] have introduced a mathematical model for COVID-19 pandemic involving the quarantine strategy and media coverage effects. Abdulkadhim and Alhusseiny [11] have studied the global stability and bifurcation of a COVID-19 virus modeling with possible loss of the immunity. Kucharski et al. [12] have discussed early dynamics of transmission and control of COVID-19 in a mathematical model. 44 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 In this study, the researchers have proposed and studied the mathematical model of the novel coronavirus involving awareness programs. Details about the mathematical modeling of the novel coronavirus are shown in Section 2. Some basic properties (existence equilibrium points and calculated reproduction number ) of the model are discussed in Section 3. The local stability analysis is studied by using Gersgorin theorem in Section 4. The cases of the backward bifurcation occurrence are shown in Section 5. By using Lyapunov functions, the researchers have also studied the global stability of the proposed model at all equilibrium points as shown in Section 6. Finally, the effect of media coverage to raise the people's awareness to the risk of the coronavirus and the effect of the direct contact with carrier individuals on breaking out the COVID-19 among people are shown by numerical simulation of the proposed model. 2. Model Formulation The researchers have formulated the coronavirus mathematical model involving awareness. It is assumed that the total population denoted by N is subdivided into four compartments: unaware susceptible individuals denoted by (𝑆𝑒) , aware susceptible individuals denoted by (π‘†π‘Ž), asymptomatic individuals (Carrier) denoted by (𝐢), and infected individuals (symptomatic) denoted by (𝐼), Recovery individuals denoted by 𝑅(𝑑). Thus, 𝑁 = 𝑆𝑒 + π‘†π‘Ž + 𝐢 + 𝐼 + 𝑅. Let (𝑣) be the coronavirus resources in (host). This model describes COVID-19 virus by the following equations: 𝑆�̇� = πœ“ βˆ’ 𝛼𝑆𝑒 βˆ’ 𝛽𝑆𝑒𝑣 βˆ’ πœŽπ‘†π‘’πΆ βˆ’ πœ‡π‘†π‘’ 𝑆�̇� = 𝛼𝑆𝑒 βˆ’ 𝛽1(1 βˆ’ πœ–)π‘†π‘Žπ‘£ βˆ’ 𝜎1(1 βˆ’ πœ–)π‘†π‘ŽπΆ βˆ’ πœ‡π‘†π‘Ž οΏ½Μ‡οΏ½ = 𝛽𝑆𝑒𝑣 + 𝛽1(1 βˆ’ πœ–)π‘†π‘Žπ‘£ + πœŽπ‘†π‘’πΆ + 𝜎1(1 βˆ’ πœ–)π‘†π‘ŽπΆ βˆ’ (πœ‡ + πœƒ + 𝛾 + 𝛿1)𝐢 𝐼̇ = 𝛾𝐢 βˆ’ (πœ‡ + πœƒ + 𝛿2)𝐼 οΏ½Μ‡οΏ½ = 𝛿1𝐢 + 𝛿2𝐼 βˆ’ πœ‡π‘… οΏ½Μ‡οΏ½ = π‘Ÿπ‘£(1 βˆ’ 𝑣 π‘˜ ) βˆ’ 𝑑𝑣 (1) With initial conditions: 𝑆𝑒 > 0,π‘†π‘Ž β‰₯ 0,𝐢 β‰₯ 0,𝐼 β‰₯ 0,𝑅 β‰₯ 0,𝑣 β‰₯ 0 . In system (1) the birth rate of individuals is given𝛹 > 0. The awareness rate is given𝛼 β‰₯ 0. The parameters 𝛽 > 0,𝛽1 > 0,𝜎 > 0 and 𝜎1 > 0 respectively measure the contact rate between susceptible with carrier, and coronavirus with the prevention of disease that is given a rate of (0 ≀ πœ– ≀ 1). The death rate is πœ‡ > 0. The death rate due to the disease is πœƒ > 0. After 14th days, symptoms of the disease begin to appear on the carrier that is given a rate of 𝛾 > 0. The recovery rates from carrier and infected individuals are 𝛿𝑖 > 0 , 𝑖 = 1,2 respectively. π‘Ÿ > 0,π‘˜ > 0 and 𝑑 > 0 respectively, represent the spread rate of the virus, the carrying capacity of coronavirus and the removal rate of virus. Theorem (1): All solutions of system (1) which initiate in 𝑅+ 5 are uniformly bounded. Proof: Let 𝑀(𝑇) = (𝑁(𝑑), 𝑣(𝑑)) where N(t) = Su(t) + π‘†π‘Ž(t)+ C(t),+I(t) + R(t); Then π‘‘π‘Š 𝑑𝑑 = ( 𝑑𝑁 𝑑𝑑 , 𝑑𝑣 𝑑𝑑 ) = (πœ“ βˆ’ πœ‡Su βˆ’ πœ‡π‘†π‘Ž βˆ’ (πœ‡ + πœƒ)𝐢 βˆ’ (πœ‡ + πœƒ)𝐼 βˆ’ πœ‡π‘…,π‘Ÿπ‘£(1 βˆ’ 𝑣 𝐾 ) βˆ’ 𝑑𝑣) 45 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 We know that 𝑑𝑁 𝑑𝑑 ≀ πœ“ βˆ’ π‘žπ‘, where π‘ž = π‘šπ‘–π‘›{πœ‡ ,πœ‡ + πœƒ,𝑑} Which implies that lim π‘‘β†’βˆž 𝑠𝑒𝑝𝑁(𝑑) ≀ πœ“ π‘ž While, the last equation of system (1) it follows that sup(π‘Ÿπ‘£(1 βˆ’ 𝑣 π‘˜ )) ≀ π‘Ÿπ‘˜ 4 Hence 𝑑𝑣 𝑑𝑑 ≀ π‘Ÿπ‘˜ 4 βˆ’ 𝑑𝑣 So that lim π‘‘β†’βˆž sup𝑣(𝑑) ≀ π‘Ÿπ‘˜ 4𝑑 This completes the proof of theorem β–  3. Existence of equilibrium points It is easy that the infected and recovery population I and R, are related with carrier population only . Hence for fixed value of C, the values of I and R can be determined directly by solving the fourth equation in system (1). Then, we can calculate the values of I and R the following form 𝐼 = π›ΎπΆβˆ— πœ‡+πœƒ (2a) 𝑅 = 𝛿1πΆβˆ—+𝛿2πΌβˆ— πœ‡ (2b) Consequently, for simplifying, system (1) can be reduced to the following system, in which we can determine the values of I and R, by solving it, 𝑆�̇� = πœ“ βˆ’ 𝛼𝑆𝑒 βˆ’ 𝛽𝑆𝑒𝑣 βˆ’ πœŽπ‘†π‘’πΆ βˆ’ πœ‡π‘†π‘’ 𝑆�̇� = 𝛼𝑆𝑒 βˆ’ 𝛽1(1 βˆ’ πœ–)π‘†π‘Žπ‘£ βˆ’ 𝜎1(1 βˆ’ πœ–)π‘†π‘ŽπΆ βˆ’ πœ‡π‘†π‘Ž οΏ½Μ‡οΏ½ = 𝛽𝑆𝑒𝑣 + 𝛽1(1 βˆ’ πœ–)π‘†π‘Žπ‘£ + πœŽπ‘†π‘’πΆ + 𝜎1(1 βˆ’ πœ–)π‘†π‘ŽπΆ βˆ’ (πœ‡ + πœƒ + 𝛾 + 𝛿1)𝐢 οΏ½Μ‡οΏ½ = π‘Ÿπ‘£(1 βˆ’ 𝑣 π‘˜ ) βˆ’ 𝑑𝑣 (3) Now, we can compute the reproduction number for the given system (3) that is denoted by β„›0, such that β„›0 = πœŽπœ“ πœ‡(πœ‡+πœƒ+𝛾+𝛿1) (4) Therefore, system (3) has at most two biologically feasible points, namely, 𝐸𝑖 = (𝑆𝑒𝑖,π‘†π‘Žπ‘–,𝐢𝑖,𝑣𝑖), 𝑖 = 0,1. The existence conditions for each of these equilibrium points are discussed in the following: ο‚· In the absence of COVID-19 virus, that is π‘†π‘Ž = 𝐢 = 𝑣 = 0. Then, system (3) has a unique positive equilibrium point, namely COVID-19 free equilibrium point, which is denoted by 𝐸0 = (𝑆𝑒0,0,0,0) where 𝑆𝑒0 = πœ“ πœ‡ (5a) provided that the following condition holds 𝛼 = 0 (5b) 46 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 ο‚· The endemic equilibrium point or COVID-19 equilibrium point, which is denoted by 𝐸1 = (𝑆𝑒1,π‘†π‘Ž1,𝐢1,𝑣1), where 𝑆𝑒1 = π‘ŸΟˆ π‘Ÿπ›Ό+𝛽(π‘Ÿβˆ’π‘‘)π‘˜+π‘Ÿπœ‡+π‘ŸπœŽπΆ1 ; π‘†π‘Ž1 = π›Όπ‘Ÿ2ψ 𝐴1 𝐢1 2+𝐴2𝐢1+𝐴3 ; 𝑣1 = (π‘Ÿβˆ’π‘‘)π‘˜ π‘Ÿ (6a) here 𝐴1 = 𝜎1 (1 βˆ’ πœ–)π‘Ÿ 2𝜎 𝐴2 = 𝛽1(1 βˆ’ πœ–)(π‘Ÿ βˆ’ 𝑑)π‘˜π‘ŸπœŽ + 𝜎1 (1 βˆ’ πœ–)π‘Ÿ[π‘Ÿπ›Ό + 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜ + π‘Ÿπœ‡] + π‘Ÿ 2 πœ‡πœŽ 𝐴3 = [𝛽1(1 βˆ’ πœ–)(π‘Ÿ βˆ’ 𝑑)π‘˜ + π‘Ÿπœ‡][π‘Ÿπ›Ό + 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜ + π‘Ÿπœ‡] exists under the condition π‘Ÿ > 𝑑 (6b) Now, substituting the equation (6a) in 3rd equation of system (3), and simplifying that result, we get 𝐷1𝐢 4 + 𝐷2𝐢 3 + 𝐷3𝐢 2 + 𝐷4𝐢 + 𝐷5 = 0 (7a) Where 𝐷1 = βˆ’(𝛾 + πœƒ + πœ‡ + 𝛿1)π‘Ÿ 2𝜎𝐴1 < 0 𝐷2 = πœŽπ‘Ÿ 2πœ“π΄1 βˆ’ (𝛾 + πœƒ + πœ‡ + 𝛿1)(π‘Ÿ 2𝛼 + 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜π‘Ÿ + π‘Ÿ2πœ‡)𝐴1 βˆ’(𝛾 + πœƒ + πœ‡ + 𝛿1)π‘Ÿ 2𝜎𝐴2 𝐷3 = 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜π‘Ÿπœ“π΄1 + πœŽπ‘Ÿ 2πœ“π΄2 + 𝜎1(1 βˆ’ πœ–)π›Όπ‘Ÿ 4πœ“πœŽ βˆ’[𝛾 + πœƒ + πœ‡ βˆ’ (𝛾 + πœƒ + πœ‡ + 𝛿1)π‘Ÿ 2𝜎𝐴2](π‘Ÿ 2 𝛼 + 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜π‘Ÿ + π‘Ÿ2πœ‡)𝐴2 βˆ’(𝛾 + πœƒ + πœ‡ + 𝛿1)π‘Ÿ 2𝜎𝐴3 𝐷4 = 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜π‘Ÿπœ“π΄2 + 𝛽1(1 βˆ’ πœ–)(π‘Ÿ βˆ’ 𝑑)π‘˜π›Όπ‘Ÿ 3πœ“πœŽ + πœŽπ‘Ÿ2πœ“π΄3 +𝜎1(1 βˆ’ πœ–)π›Όπ‘Ÿ 3πœ“(π‘Ÿπ›Ό + 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜π‘Ÿ + πœ‡π‘Ÿ) βˆ’(𝛾 + πœƒ + πœ‡ + 𝛿1)[π‘Ÿ 2𝛼 + 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜π‘Ÿ + π‘Ÿ2πœ‡] 𝐷5 = 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜π‘Ÿπœ“π΄3 + 𝛽1π‘˜π›Όπ‘Ÿ 2πœ“(1 βˆ’ πœ–)(π‘Ÿ βˆ’ 𝑑)[π‘Ÿπ›Ό + 𝛽(π‘Ÿ βˆ’ 𝑑)π‘˜ + π‘Ÿπœ‡] > 0 Obviously, the endemic equilibrium point exists unique if and only if the following condition holds 𝐷3 > 0 ,𝐷4 > 0 (7b) or 𝐷2 < 0 ,𝐷3 < 0 (7c) 4. Local stability analysis In this section, the local stability analysis of the all equilibrium points 𝐸𝑖, 𝑖 = 0,1 of system (3) studied as shown in the following theorems. Theorem (2): The COVID-19 free equilibrium point 𝐸0 of the system (3) is locally asymptotically if the following condition is satisfied β„›0 < 1 (8a) π‘Ÿ < 𝑑 (8b) Proof: The Jacobian matrix of system (3) at 𝐸0 can be written as 47 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 𝐽(𝑆𝑒0,0,0,0 ) = ( βˆ’πœ‡ 0 βˆ’πœŽπ‘†π‘’0 βˆ’π›½π‘†π‘’0 0 βˆ’πœ‡ 0 0 0 0 πœŽπ‘†π‘’0 βˆ’ (𝛾 + πœƒ + πœ‡ + 𝛿1) 𝛽𝑆𝑒0 0 0 0 π‘Ÿ βˆ’ 𝑑 ) Thus, the eigenvalues of 𝐽(𝐸0) are given by πœ†1 = πœ†2 = βˆ’πœ‡ < 0 πœ†3 = πœŽπ‘†π‘’0 βˆ’ (𝛾 + πœƒ + πœ‡ + 𝛿1) πœ†4 = π‘Ÿ βˆ’ 𝑑 It is easy from above to verify that conditions (8a) and (8b) guarantee that the eigenvalue πœ†3 and πœ†4 are negative respectively. Then, 𝐸0 is locally asymptotically stable. However, it is a saddle point otherwise. β–  Theorem (3): The endemic equilibrium point 𝐸1 of the system (3) is locally asymptotically if the following conditions are hold 2πœŽπ‘†π‘’1 + 2𝜎1(1 βˆ’ πœ–)π‘†π‘Ž1 < Ξ³ + ΞΈ + ΞΌ + 𝛿1 (9a) π‘˜ [2𝛽𝑆𝑒1 + 2𝛽1(1 βˆ’ πœ–)π‘†π‘Ž1 + π‘Ÿ] < 2π‘Ÿπ‘£1 + π‘˜π‘‘ (9b) Proof: The Jacobian matrix of system (3) at 𝐸1 can be written as 𝐽(𝑆𝑒1,π‘†π‘Ž1,𝐢1,𝑣1) = ( 𝑏11 0 𝑏13 𝑏14 𝑏21 𝑏22 𝑏23 𝑏24 𝑏31 𝑏32 𝑏33 𝑏34 0 0 0 𝑏44 ) Here 𝑏11 = βˆ’(𝛼 + 𝛽𝑣1 + 𝜎𝐢1 + πœ‡) ; 𝑏13 = βˆ’πœŽπ‘†π‘’1 ; 𝑏14 = βˆ’π›½π‘†π‘’1 ; 𝑏21 = 𝛼 ; 𝑏22 = βˆ’[𝛽1(1 βˆ’ πœ–)𝑣1 + 𝜎1(1 βˆ’ πœ–)𝐢1 + πœ‡] ; 𝑏23 = βˆ’πœŽ1 (1 βˆ’ πœ–)π‘†π‘Ž1; 𝑏24 = βˆ’π›½1(1 βˆ’ πœ–)π‘†π‘Ž1 𝑏31 = 𝛽𝑣1 + 𝜎𝐢1 ; 𝑏32 = 𝛽1(1 βˆ’ πœ–)𝑣1 + 𝜎1(1 βˆ’ πœ–)𝐢1 ; 𝑏33 = πœŽπ‘†π‘’1 + 𝜎1(1 βˆ’ πœ–)π‘†π‘Ž1 βˆ’ (𝛾 + πœƒ + πœ‡ + 𝛿1); 𝑏34 = 𝛽𝑆𝑒1 + 𝛽1(1 βˆ’ πœ–)π‘†π‘Ž1 𝑏44 = βˆ’2π‘Ÿπ‘£1 π‘˜ + (π‘Ÿ βˆ’ 𝑑); 𝑏12 = 𝑏41 = 𝑏42 = 𝑏43 = 0 Now, according to Gersgorin theorem [13], if the following condition holds: |𝑏𝑖𝑖| > βˆ‘|𝑏𝑖𝑗| 4 𝑖=1 𝑖≠𝑗 = 𝑃𝑖 Then all eigenvalues of 𝐽(𝐸1) exist in the region: Ω =βˆͺ { π‘ˆβˆ—πœ– 𝐢 ∢ |π‘ˆβˆ— βˆ’ 𝑏𝑖𝑖| < βˆ‘|𝑏𝑖𝑗| 4 𝑖=1 𝑖≠𝑗 } Then, all the eigenvalues of 𝐽(𝐸1) exist in the disc centered at 𝑏𝑖𝑖 with radius 𝑃𝑖. Thus if the diagonal elements are negative and the condition (9a) holds, all the eigenvalues will be exist in the left half plane and the 𝐸1 of system (3) is locally asymptotically stabile. Clearly conditions (9a)-(9b) guarantee the existence of all eigenvalues in the left half plane and the proof follows. β–  48 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 5. Global stability analysis The propose of this section is to investigate the global stability of COVID-19 virus mathematical model given by equations in system (3) near COVID-19 free and COVID-19 points respectively given by 𝐸𝑖, 𝑖 = 0,1. We obtain the result in the following theorems Theorem (4): 𝐸0 is globally asymptotically stable provided that the following conditions hold: β„›0 < 1 (10a) 𝛽𝑆𝑒0 < 𝑑 βˆ’ π‘Ÿ (10b) Proof: Consider the following function 𝑉0(𝑆𝑒,π‘†π‘Ž,𝐢,𝑣) = (𝑆𝑒 βˆ’ 𝑆𝑒0 βˆ’ 𝑆𝑒0 ln 𝑆𝑒 𝑆𝑒0 ) + π‘†π‘Ž + 𝐢 + 𝑣 Clearly, 𝑉0: 𝑅+ 4 β†’ 𝑅 is a continuously differentiable function such that 𝑉0(𝑆𝑒0,0,0,0) = 0 and 𝑉0(𝑆𝑒,π‘†π‘Ž,𝐢,𝑣) > 0, βˆ€ (𝑆𝑒,π‘†π‘Ž,𝐢,𝑣) β‰  (𝑆𝑒0,0,0,0) . Further, we have 𝑑𝑉0 𝑑𝑑 = ( 𝑆𝑒 βˆ’ 𝑆𝑒0 𝑆𝑒 )[ψ βˆ’ 𝛽𝑆𝑒𝑣 βˆ’ πœŽπ‘†π‘’πΆ βˆ’ πœ‡π‘†π‘’] + [βˆ’π›½1(1 βˆ’ πœ–)π‘†π‘Žπ‘£ βˆ’ 𝜎1(1 βˆ’ πœ–)π‘†π‘ŽπΆ βˆ’ πœ‡π‘†π‘Ž] +[𝛽𝑆𝑒𝑣 + 𝛽1(1 βˆ’ πœ–)π‘†π‘Žπ‘£ + πœŽπ‘†π‘’πΆ + 𝜎1(1 βˆ’ πœ–)π‘†π‘ŽπΆ βˆ’ (𝛾 + πœƒ + πœ‡ + 𝛿1)𝐢] +[(π‘Ÿ βˆ’ 𝑑)𝑣 βˆ’ π‘Ÿ π‘˜ 𝑣2] Now, by doing some algebraic manipulation and using the conditions (8), (10a) and (10b), we get 𝑑𝑉0 𝑑𝑑 ≀ βˆ’πœ‡ 𝑆𝑒 (𝑆𝑒 βˆ’ 𝑆𝑒0) 2 βˆ’ πœ‡π‘†π‘Ž βˆ’ [(𝛾 + πœƒ + πœ‡ + 𝛿1) βˆ’ πœŽπ‘†π‘’0]𝐢 βˆ’ π‘Ÿ π‘˜ 𝑣2 βˆ’ [(𝑑 βˆ’ π‘Ÿ) βˆ’ 𝛽𝑆𝑒0]𝑣 Obviously, οΏ½Μ‡οΏ½0 = 0 at 𝐸0 = (𝑆𝑒0,0,0,0), moreover οΏ½Μ‡οΏ½0 < 0 otherwise. Hence, οΏ½Μ‡οΏ½0 is negative definite and then the solution starting from any initial point that satisfies the conditions (8b) with (10a) and (10b) will approach asymptotically to COVID-19 free equilibrium point. Hence, the proof is complete. β–  Theorem (5): 𝐸1 is global asymptotically stable if β„›0 > 1. Proof: For the COVID-19 equilibrium 𝐸1 = (𝑆𝑒1,π‘†π‘Ž1,𝐢1,𝑣1); 𝑆𝑒1,π‘†π‘Ž1,𝐢1 and 𝑣1 satisfies equations πœ“ βˆ’ 𝛼𝑆𝑒 βˆ’ 𝛽𝑆𝑒𝑣 βˆ’ πœŽπ‘†π‘’πΆ βˆ’ πœ‡π‘†π‘’ = 0 𝛼𝑆𝑒 βˆ’ 𝛽1(1 βˆ’ πœ–)π‘†π‘Žπ‘£ βˆ’ 𝜎1(1 βˆ’ πœ–)π‘†π‘ŽπΆ βˆ’ πœ‡π‘†π‘Ž = 0 𝛽𝑆𝑒𝑣 + 𝛽1(1 βˆ’ πœ–)π‘†π‘Žπ‘£ + πœŽπ‘†π‘’πΆ + 𝜎1(1 βˆ’ πœ–)π‘†π‘ŽπΆ βˆ’ (πœ‡ + πœƒ + 𝛾 + 𝛿1)𝐢 = 0 π‘Ÿπ‘£(1 βˆ’ 𝑣 π‘˜ ) βˆ’ 𝑑𝑣 = 0 (11) By applying (11) and putting 𝑆𝑒 𝑆𝑒1 = π‘₯ , π‘†π‘Ž π‘†π‘Ž1 = 𝑦 , 𝐢 𝐢1 = 𝑧 , 𝑣 𝑣1 = 𝑒 49 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 We have π‘₯β€² = π‘₯[ πœ“ 𝑆𝑒1 ( 1 π‘₯ βˆ’ 1) βˆ’ 𝛽𝑣1(𝑒 βˆ’ 1) βˆ’ 𝜎𝐢1(𝑧 βˆ’ 1)] 𝑦′ = 𝑦[𝛼 𝑆𝑒1 π‘ π‘Ž1 (π‘₯ βˆ’ 1) βˆ’ 𝛽1(1 βˆ’ πœ–)𝑣1(𝑒 βˆ’ 1) βˆ’ 𝜎1(1 βˆ’ πœ–)𝐢1(𝑧 βˆ’ 1)] 𝑧′ = 𝑧[ 𝛽𝑆𝑒1𝑣1 𝐢1 ( π‘₯𝑒 𝑧 βˆ’ 1) + 𝛽1(1 βˆ’ πœ–)π‘†π‘Ž1𝑣1 𝑐1 ( 𝑦𝑒 𝑧 βˆ’ 1) + πœŽπ‘†π‘’1(π‘₯ βˆ’ 1) + 𝜎1(1 βˆ’ πœ–)(𝑦 βˆ’ 1)] 𝑒′ = 𝑒[ βˆ’π‘Ÿπ‘£1 π‘˜ (𝑧 βˆ’ 1)] (12) Define the Lyapunov function 𝑉1 = 𝑆𝑒1(π‘₯ βˆ’ 1 βˆ’ lnπ‘₯) + π‘†π‘Ž1(𝑦 βˆ’ 1 βˆ’ ln𝑦) +𝐢1(𝑧 βˆ’ 1 βˆ’ ln𝑧) + 𝑣1(𝑒 βˆ’ 1 βˆ’ ln𝑒) (13) The derivative of 𝑉1 is given by 𝑑𝑉1 𝑑𝑑 = 𝑆𝑒1 π‘₯βˆ’1 π‘₯ π‘₯β€² + π‘†π‘Ž1 π‘¦βˆ’1 𝑦 𝑦′ + 𝐢1 π‘§βˆ’1 𝑧 𝑧′ + 𝑣1 π‘’βˆ’1 𝑒 𝑣′ 𝑑𝑉1 𝑑𝑑 = (π‘₯ βˆ’ 1)[πœ“( 1 π‘₯ βˆ’ 1) βˆ’ 𝛽𝑆𝑒1𝑣1(𝑒 βˆ’ 1) βˆ’ πœŽπ‘†π‘’1𝐢1(𝑧 βˆ’ 1)] +(𝑦 βˆ’ 1)[𝛼𝑆𝑒1(π‘₯ βˆ’ 1) βˆ’ 𝛽1(1 βˆ’ πœ–)π‘†π‘Ž1𝑣1(𝑒 βˆ’ 1) βˆ’ 𝜎1(1 βˆ’ πœ–)π‘†π‘Ž1𝐢1(𝑧 βˆ’ 1)] +(𝑧 βˆ’ 1)[𝛽𝑆𝑒1𝑣1 ( π‘₯𝑒 𝑧 βˆ’ 1) + 𝛽1(1 βˆ’ πœ–)π‘†π‘Ž1𝑣1 ( 𝑦𝑒 𝑧 βˆ’ 1) +πœŽπ‘†π‘’1𝐢1(π‘₯ βˆ’ 1) + 𝜎1(1 βˆ’ πœ–)𝐢1(𝑦 βˆ’ 1)] + (𝑒 βˆ’ 1)[ βˆ’π‘Ÿπ‘£1 2 π‘˜ (𝑧 βˆ’ 1)] Furthermore, by simplifying the resulting terms, we get that 𝑑𝑉1 𝑑𝑑 = πœ“(2 βˆ’ π‘₯ βˆ’ 1 π‘₯ ) + 𝛽𝑆𝑒1𝑣1 (π‘₯ + 𝑒 βˆ’ 𝑧 βˆ’ π‘₯𝑒 𝑧 ) + 𝛼𝑆𝑒1(π‘₯𝑦 βˆ’ π‘₯ βˆ’ 𝑦 + 1) +𝛽1(1 βˆ’ πœ–)π‘†π‘Ž1𝑣1 (𝑦 + 𝑒 βˆ’ 𝑧 βˆ’ 𝑦𝑒 𝑧 ) βˆ’ π‘Ÿπ‘£1 2 π‘˜ (𝑧𝑒 βˆ’ 𝑧 βˆ’ 𝑒 + 1) Since the arithmetical mean is greater than, or equal to the geometrical mean, then, 2 βˆ’ π‘₯ βˆ’ 1 π‘₯ ≀ 0 for π‘₯ > 0 and 2 βˆ’ π‘₯ βˆ’ 1 π‘₯ = 0 if and only if π‘₯ = 1;π‘₯ + 𝑒 βˆ’ 𝑧 βˆ’ π‘₯𝑒 𝑧 ≀ 0 for π‘₯,𝑧,𝑒 > 0 and π‘₯ + 𝑒 βˆ’ 𝑧 βˆ’ π‘₯𝑒 𝑧 = 0 if and only if π‘₯ = 1,𝑧 = 𝑒; π‘₯𝑦 βˆ’ π‘₯ βˆ’ 𝑦 + 1 ≀ 0 for π‘₯,𝑦 > 0 and π‘₯𝑦 βˆ’ π‘₯ βˆ’ 𝑦 + 1 = 0 if and only if π‘₯ = 𝑦 = 1;𝑦 + 𝑒 βˆ’ 𝑧 βˆ’ 𝑦𝑒 𝑧 ≀ 0 for 𝑦,𝑧,𝑒 > 0 and 𝑦 + 𝑒 βˆ’ 𝑧 βˆ’ 𝑦𝑒 𝑧 = 0 if and only if 𝑦 = 1,𝑧 = 𝑒; 𝑧𝑒 βˆ’ 𝑧 βˆ’ 𝑒 + 1 ≀ 0 for 𝑧,𝑒 > 0 and 𝑧𝑒 βˆ’ 𝑧 βˆ’ 𝑒 + 1 = 0 if and only if 𝑧 = 𝑒 = 1. Therefore, 𝑉1 β€² ≀ 0 for π‘₯,𝑦,𝑧,𝑒 > 0 and 𝑑𝑉1 𝑑𝑑⁄ = 0 if and only if π‘₯ = 𝑦 = 1,𝑧 = 𝑒 = 1, the maximum invariant set of system (3) on the set {(π‘₯,𝑦,𝑧,𝑒):𝑑𝑉1 𝑑𝑑⁄ = 0} is the singleton (1,1,1,1). Thus, for system (3), the COVID-19 equilibrium point 𝐸1 is globally asymptotically stable if β„›0 > 1 by LaSalle Principle [14]. 50 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 6. Numerical Simulation In this section, we present the numerical simulation results and effect of the parameters of the analytic results of system (3). Using MATLAB version 8 and C++ software, we begin the simulation with a set outbreak data published daily by WHO and other sources [8, 15-20], as shown in Table 1 which are described in Section 2. Table 1: Definitions and values of parameters in system (3) Parameter Definition Estimated mean values Reference πœ“ 𝛼 𝛽 𝛽1 𝜎 𝜎1 πœ‡ 𝛾 πœ– πœƒ π‘Ÿ π‘˜ 𝑑 𝛿1 Birth rate Awareness rate Contact rate between 𝑆𝑒 π‘Žπ‘›π‘‘ 𝑣 Contact rate between π‘†π‘Ž π‘Žπ‘›π‘‘ 𝑣 Contact rate between 𝑆𝑒 π‘Žπ‘›π‘‘ 𝐢 Contact rate between π‘†π‘Ž π‘Žπ‘›π‘‘ 𝐢 Natural death rate Transmission asymptomatic to symptomatic Prevention rate Death rate due to the disease Spread rate of the virus carrying capacity in environment Removal rate of virus Recovery rate from carrier 271.23 π‘π‘’π‘Ÿ π‘‘π‘Žπ‘¦ 𝛼 β‰₯ 0 π‘π‘’π‘Ÿ π‘‘π‘Žπ‘¦ 1.555 Γ— 10βˆ’8 /π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ 3.1 Γ— 10βˆ’7/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ 1.555 Γ— 10βˆ’8/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ 3.1 Γ— 10βˆ’7/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ 3.01 Γ— 10βˆ’4/π‘π‘’π‘Ÿπ‘ π‘œπ‘›/π‘‘π‘Žπ‘¦ 7 π‘‘π‘Žπ‘¦π‘  0.03/π‘π‘’π‘Ÿπ‘ π‘œπ‘› 0.01 π‘π‘’π‘Ÿ π‘‘π‘Žπ‘¦ 2 0.5 3 π‘π‘’π‘Ÿ π‘‘π‘Žπ‘¦ 0.03 π‘π‘’π‘Ÿπ‘‘π‘Žπ‘¦ [8] - [8,19] [8,19] [8,19] [8,19] [18] - - [8] - - [20] - Clearly, the initial point for 𝐼(0) = 25, we assume that the 𝐢(0) = 10, π‘†π‘Ž = 10, 𝑆𝑒 = 15 and 𝑣 = 20. Using this initial point and simulated the system (1), we get the values of parameters shown in Table 1, whereas the basic reproduction number is estimated β„›0 = 0.914 < 1 with condition (8b), we obtain the dynamical behavior of system (3) converge to COVID-19 free equilibrium point 𝐸0 = (901096,0,0,0,0), illustrating its asymptotical stability that is stated in Theorem (2). This is shown in Figure 2. 51 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 Figure 2: A simulation result for the dynamics of coronavirus model using the parameters from Table 1. The reproduction number β„›0 = 0.07 and the behavior coronavirus converge to COVID-19 free equilibrium point. However, for the same data given by Table 1. with 𝑑 = 1, (i.e. the condition 6b is holds), and 𝜎 = 1.555 Γ— 10βˆ’5 (i.e. the contact rate between 𝑆𝑒 π‘Žπ‘›π‘‘ 𝐢 will increase). it is observed that, although system (3) is solved with the same initial point, the dynamical behavior of system (3) converge to COVID-19 equilibrium point 𝐸1 = (979,2734,1752,24299,0,25), illustrating its asymptotical stability that is stated in Theorem (3), and reproduction number is estimated β„›0 = 7.77 > 1. This is shown in Figure 3. 52 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 Figure 3: A simulation result for the dynamics of coronavirus model using the parameters from Table 1 with 𝑑 = 1 and 𝜎 = 1.555 Γ— 10βˆ’5. The reproduction number β„›0 = 7.77 and the behavior of coronavirus converge to COVID-19 equilibrium point. Similarly, if the increase effect of the contact rate between 𝑆𝑒 π‘Žπ‘›π‘‘ 𝑣 (𝛽), as well as used the values of parameters shown in Table 1 with 𝑑 = 1, (i.e. the condition 6b is holds), and 𝛽 = 1.555 Γ— 10βˆ’4 (i.e. the contact rate between 𝑆𝑒 π‘Žπ‘›π‘‘ 𝑣 will increase)., we obtain the dynamical behavior of system (3) still converge to COVID-19 equilibrium point 𝐸1 = ((61636,170389,25,334,0.25)), illustrating its asymptotical stability that is stated in Theorem (3). This is shown in Figure 4. 53 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 Figure 4: A simulation result for the dynamics of coronavirus model using the parameters from Table 1 with 𝑑 = 1 and 𝛽 = 1.555 Γ— 10βˆ’4. The dynamical behavior of coronavirus model converge to COVID-19 equilibrium point. 7. Discussions and conclusions In this manuscript, a coronavirus model is proposed and discussed. This model has two feasible equilibrium points which are COVID-19 free equilibrium point and epidemic equilibrium point. The results of the proposed work may be used to show and understand the effect of awareness rate, prevention rate of disease and contact rate between asymptomatic people and other. The stability (local and global) of the model has been analyzed at all equilibrium points. It has been obtained that COVID-19 free is locally asymptotically stable if the conditions (8a) and (8b) are hold and it is global stable under the conditions (10a) and (10b) are hold. While, the epidemic point it has been obtained under certain conditions for locally as well as globally asymptotically stable under the conditions (9a-9b) and (11a-11g) are hold respectively. To validate the analytical results, we have executed the numerical simulations for investigating the dynamics of a coronavirus mathematical model. We conclude if the removal rate of virus is increased through used the sterilization materials could us to decrease virus rate and hence, we get the basic reproduction number less than one (i.e. satisfied the condition 10a and 10b). Also caution should be exercised against asymptomatic people who are the source of the risk, so direct contact should be reduced with them. Finally, a rapid health examination to detect the infected is one of the important factors in the control to spread of the epidemic. 54 Ibn Al-Haitham Jour. for Pure & Appl. Sci. 34 (2) 2021 Acknowledgements The authors are thankful to acknowledge the reviewers for their valuable suggestions and comments have contributed to the improvement of the authors work. References 1. Tan, C. C. SARS in Singapore key lessons from an epidemic. Annals Academy Medicine Singapore, 2006, 35, 345-349. 2. Tsang, T.; Lam, T., SARS public health measures in Hong Kong. Respirology Journal, 2003, 8, 46-48. 3. 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