STABILITY ANALYSIS OF MATHEMATICAL MODEL NEW CORONA VIRUS (COVID-19) DISEASE SPREAD IN POPULATION

unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. The New Corona Virus epidemic is the most serious epidemic that the international community had known in 2019 and this is manifested by the deaths it claimed and in a short time. Its risk is much greater than (MERS) disease that emerged in the Republic of Korea and it is spreading largely more than SARS disease which appeared in Saudi Arabia and the Middle East. This deadly disease caught the public’s attention and caused terror in many societies around the world. We are building a dynamic model based on the detailed data of mortality from the World Health Organization (WHO) and the actual spread of the epidemic. By using Routh-Hurwitz criteria and constructing Lyapunov functions, the local and the global stability of the disease-free equilibrium and the disease equilibrium are obtained. We also study the sensitivity analysis of model parameters to know the parameters that have a high impact on the reproduction number R0. Finally, numerical simulations are performed to verify the theoretical analysis using Matlab.

Many studies and research of mathematical models can be used to analyze the spread of infectious diseases or the social behavior of people [5,6,9,10,14,16,19,24,25]. As regard to Corona virus, several different mathematical models have been formulated and studied to help in reducing the number of people who have an infectious desease [2,3,9,10]. M.Tahir et al [2] developed a non-linear mathematical model of MERS-CoV and studied the global stability analysis and also they introduced Lyapunov function. Zhi-Qiang Xia et al [18] constructed two dynamical models to simulate the propagation processes and found out that the basic reproduction number R 0 reaches up to 4.422. They showed that the reasons of the quick spread of the disease are the lack of self-protection and control measures. A. Naheed et al's study [19] of a population model based on the epidemic of (SARS) examined the effect of the diffusion on the spread of the disease and analysed the stability of the numerical solutions.
We will propose a mathematical model that defines and describes the spread of the new Corona virus . The discrete modeling is more realistic but since the data on (COVID-19) is collected daily, we rely on a continuous model in particular because it is less complicated to be processed. Majority cases of (COVID-19) virus spread from human-tohuman connection. The virus is transmitted by direct contact with an infected person. So, the use of a boiler model for an infectious disease is very appropriate in this case. We will first test the local stability of the model in both disease-free model and in endemic equilibrium, then we will test the global stability of the model.   We consider the following system of five non-linear differential equations: (1) and R(0) ≥ 0 are the given initial states.
Where Λ represents new birth rate in susceptible human population, β represents the transmission coefficient from susceptible individuals to asymptomatic infected cases or cases with mild symptoms due to the movement and contact that occur among them.µ represents the natural death rate in all compartments. α represents the rate of transmission of asymptomatic infected cases or cases with mild symptoms to infected individuals with symptoms. λ is the transmission coefficient of the infected people with symptoms to the hospitalized cases. γ is the transmission coefficient of the hospitalized cases to the recovered cases. θ represents the rate of transmission of asymptomatic infected cases or cases with mild symptoms to the recovered cases due the strong immunity of these individuals. δ 1 and δ 2 respectively represent the death rate of the infected individuals and the death rate of the hospitalized cases.
1.2. Basic Properties of the model.

Invariant Region.
It is necessary to prove that all solutions of system (1) with positive initial data will remain positive for all times t > 0. This will be established by the following lemma.

Lemma 1. All feasible solutions S(t), E(t), I(t), H(t) and R(t) of system equation (1) are bounded by the region
Proof. From the system equation (1) It follows that Where N(0) is the initial value of total number of people, thus Hence, for the analysis of model (1), we get the region which is given by the set:  Proof. From the first equation of the system (1), we have Implies that Taking integral with respect to s from 0 to t, we get Multiplying the equation (11) by exp(− t 0 A(s)ds), we get Then, So, the solution P(t) is positive.
Similarly, from the others equations of system (1), we have Therefore, we can see that the solutions S(t), E(t), I(t), H(t) and R(t) of the system (1) are positive for all t ≥ 0. This completes the proof.
The first three equations in system (1) are independents of the variables H and R. Hence, the dynamics of equation system (1) is equivalent to the dynamics of equation system: Where:

STABILITY ANALYSIS
. 2.2. Local stability analysis. Now we proceed to study the stability behavior of equilibria E 0 eq and E * eq .
2.2.1. The disease-free equilibrium. In this section, we analyze the local stability of the COVID-19 disease-free equilibrium.
Proof. The Jacobian matrix at E eq is given by (23) J The Jacobian matrix for the disease-free equilibrium is given by The characteristic equation of this matrix is given by det (J(E 0 eq ) − λ I 3 ) = 0, where I 3 is a square identity matrix of order 3.
Therefore, we see that the characteristic equation ϕ(ζ ) of J(E 0 eq ) are: Where, eigenvalues of the characteristic equation of J(E 0 eq ) are: Therefore, all the eigenvalues of the characteristic equation J(E 0 eq ) are clearly real and negative if R 0 < 1.
We conclude that the desease-free equilibrium is locally asymptotically stable if R 0 < 1 and unstable if R 0 > 1.

Disease present equilibrium.
In this section, we analyze the local stability of the disease present equilibrium. We consider dS(t) dt = 0, dE(t) dt = 0 and dI(t) dt = 0 . We have From the second equation in the system (18), we have Also, the third equation in the system (18) gives Let the following theorem analysis the local stability of the disease present equilibrium.
Thus, the present equilibrium E * eq of system (18) is locally asymptotically stable if R 0 > 1.

GLOBAL STABILITY
3.1. Global stabilty of the COVID-19 disease-free equilibrium. To show that the system (18) is globally asymptotically stable, we use the Lyapunov function theory for both the COVID-19 disease free equilibrium and the COVID-19 disease present equilibrium. First, we present the global stability of the COVID-19 disease-free equilibrium E 0 eq . Theorem 5. The COVID-19 disease free equilibruim E 0 eq is globally asymptotically stable Ω If R 0 ≤ 1 and unstable otherwise.
Proof. Let the following Lyapunov function: Then, the time derivative of the Lyapunov function is given by: Using Λ = µS 0 to rewrite this, we get Note that dV dt = 0 if and only if S = S 0 and E = 0. Hence, by Lasalle's invariance principle [17], E 0 eq is globally asymptotically stable in Ω.

3.2.
Global stability of the COVID-19 disease present equilibrium. The final result of the global stability of E * eq in this section is as follows: Theorem 6. The disease of COVID-19 disease present equilibrium point E * eq is globally asymptotically stable if R 0 > 1.
Proof. Let the Lyapunov function V : Then, the time derivative of the Lyapunov function is Also, we obtain Hence, by LaSalle's invariance principle [17] the COVID-19 disease present equilibrium point E * eq is globally asymptotically stable on Γ.

SENSITIVITY ANALYSIS OF R 0
Sensitivity analysis is commonly used to determine the model robustness to parameter values, that is, to help us know the parameters that have a high impact on the reproduction number R 0 (because there are usually errors in data collection and assumed parameter values).
Using the approach in Chitnis et al [5], we calculate the normalized forward sensitivity indices Denote the sensitivity index of R 0 with respect to the parameter n, we get From the above discussion we observe that the basic reproduction number R 0 is most sensitive to changes in β . If β increases R 0 will also increase with the same proportion and if β decreases in the same proportion, µ, α and θ will have an inversely proportional relationship with R 0 . So, an increase in any one of them will bring about a decrease in R 0 . However, the size of the decrease will be proportionally smaller. Recall that µ is the natural death rate of the population. Given R 0 's sensitivity to β , it seems sensible to focus efforts on the reduction of β . In other words, this sensitivity analysis tells us that prevention is better than cure. Efforts to increase prevention are more effective in controlling the spread of habitual COVID-19 disease than efforts to increase the numbers of individuals accessing quarantine.

NUMERICAL SIMULATIONS :
In this section, we illustrate some numerical solutions of model (1) for different values of the parameters. The resolution of the system (1) was created using the Gauss-Seidel-like implicit finite-difference method developed by Gumel et al [9] presented in [15] and denoted GSS1 method. We use the following different initial values such that S + E + I + H + R = 1000.
We use and present some numerical simulations of the system (1) to illustrate our results.By We start by a graphic representation of the COVID-19 disease-free equilibrium E 0 and we use the same parameters and different initial values given in table1. R 0 = 0.437 and R 0 < 1.
From these figures, using the different values of initial variables S 0 , E 0 , I 0 , H 0 and R 0 , we obtained the following remarks: The number of potential individuals increases and approaches the number S 0 = 1536 (see Figure 2 (a)). Also, the number of the asymptomatic infected cases or cases with mild symptoms decreases and converges to zero (see Figure 2 (b)). The number of the infected people with symptoms and carriers of the virus increases at first, after that it decreases and approaches zero (see Figure 2 (c)). The number of the hospitalized cases decreases and approaches zero (see Figure 2 (d)).The number of the recovered cases decreases and approaches zero (see Figure 2 (e)). Therefore, the solution curves to the equilibrium E 0 eq (S 0 , 0, 0, 0, 0) when R 0 < 1. Hence, model (1) is globally asymptotically stable.
From these figures, using the different values of initial variables S 0 , E 0 , I 0 , H 0 , and R 0 , we obtained the following remarks: The number of potential individuals increases at first, then it decreases slightly and approaches the value S * = 352 (see Figure 3 (a)).Concerning the number of the asymptomatic infected cases or cases with mild symptoms, it decreases rapidly at first, then it increases slightly and converges the value E * = 876 (s2ee Figure 3 (b)). The number of the infected people with symptoms and carriers of the virus increases and converges the value I * = 219 (see Figure 3 (c)). Also, the number of the hospitalized cases decreases and converges the value H * = 62 (see Figure 3 (d)). The number of the recovered cases decreases and converges the value R * = 28 (see Figure 3 (e)). Therefore, the solution curves to the equilibrium E * eq (S * , E * , I * , H * , R * ) when R 0 > 1. Hence, the model (1) is globally asymptotically stable.

CONCLUSION
In this work, we formulated and presented a continuous mathematical model SEIHR of COVID-19 disease that describes the dynamics of citizens who were infected by this desease.
We have found R 0 = Λβ µN(µ+α+θ ) , as basic reproduction number of the system (1), which helps us to determine the dynamical of the system. We also studied the sensitivity analysis of model parameters to know the parameters that have a high impact on the reproduction number R 0 . We used the stability analysis theory for nonlinear systems to analyze the mathematical COVID-19 disease model and to study both the local and global stability of COVID-19 disease. Local asymptotic stability for the COVID-19 disease-free equilibrium E 0 eq can be obtained if the threshold quantity R 0 ≤ 1. On the other hand, if R 0 > 1, then the COVID-19 disease present equilibrium E * eq is locally asymptotically stable. A Lyapunov function was used to show that E 0 eq is globally asymptotically stable if R 0 ≤ 1. Also, a Lyapunov function was used to show that E * eq is globally asymptotically stable if R 0 > 1.