A Mathematical Study of a Generalized SEIR Model of COVID-19

In this work (Part I), we reinvestigate the study of the stability of the Covid-19 mathematical model constructed by Shah et al. (2020) [1]. In their paper, the transmission of the virus under different control strategies is modeled thanks to a generalized SEIR model. This model is characterized by a five dimensional nonlinear dynamical system, where the basic reproduction number R0can be established by using the next generation matrix method. In this work (Part I), it is established that the disease free equilibrium point is locally as well as globally asymptotically stable when R0 < 1. When R0 > 1, the local and global asymptotic stability of the equilibrium are determined employing the second additive compound matrix approach and the Li-Wang’s (1998) stability criterion for real matrices [2]. In the second paper (Part II), some control parameters with uncertainties will be added to stabilize the five-dimensional Covid-19 system studied here, in order to force the trajectories to go to the equilibria. The stability of the Covid-19 system with these new parameters will also be assessed in Intissar (2020) [3] applying the Li-Wang criterion and compound matrices theory. All sophisticated technical calculations including those in part I will be provided in appendices of the part II.


Introduction
The evolution of epidemics is one of the most dangerous problems for a society. As mankind already faced severe pandemics such as the Spanish flu in 1917, the Honk Kong flu (H3N2) in 1968 and the swine flu (H1N1) in 2009, the forecast of epidemics evolution appears to be one of the most critical topics for our societies. On January 7, 2020, the isolation of a new coronavirus by a team of Chinese scientists, causing severe acute respiratory syndrome for the patients infected with this virus [4,5] (later designated coronavirus disease 2019 (Covid-19) by the World Health Organization), shed a new light on this issue.
Several efforts were done since the 1970's in order to understand the spread of diseases and to forecast their evolution through mathematical models. Amongst the various papers and preprints published to better understand the properties of the Covid-19 and model its evolution in different countries, a mathematical Covid-19 model was constructed by Shah et al. (2020) [1] to study the human to human transmission of the Covid-19. This model is reinvestigated in this paper.
The work is organized as follows:  In section 1, the mathematical Covid-19 model and its parameters are presented, alongside with some preliminary results on linear stability analysis for systems of ordinary differential equations.
 In section 2, the Li-Wang's stability criterion for real matrices (used to study the stability of an epidemic model of SEIR type with varying total population) is presented and some spectral properties of M-matrices are recalled.
 In section 3, some preliminary definitions and some lemmas for linear stability of our (covid- 19) system are provided.
 To that regard, a very important threshold quantity is the basic reproduction number, sometimes called the basic reproductive number or basic reproductive ratio (Heffernan et al. 2005 [6]), which is usually denoted by 0 .
From an epidemiological perspective, 0 refers to the average number of secondary cases produced by one infected individual introduced into a population of susceptible individuals, where an infected individual has acquired the disease, and susceptible individuals are healthy but can acquire the disease. In reality, the value of 0 for a specific disease depends on many variables, such as location and density of population.
 In section 4, the study of the stability of equilibrium points of the (Covid- 19) system is performed using the 0 criterion and Li-Wang criterion on second additive compound matrix associated to Jacobian matrix of the (covid- 19) system.
The study of the stability of Jacobian matrices of an order less than three of a dynamic system yields a reasonable 0 , but for more complex compartmental models, especially those with more infected compartments, the study of the stability is difficult as it relies on the algebraic Routh-Hurwitz conditions for stability of the Jacobian matrix. An alternative method proposed by Diekmann et al. (1990) [7] and elaborated by van den Driessche and Watmough (2002) [8] gives a way of determining 0 for a compartmental model by using the next generation matrix.
The main part of the section 4 is the determination of equilibrium points of our Covid-19 system and the explicit calculation of additive compound matrix of Jacobian matrix associated to this system. In this work, it is the first time that the explicit calculation of a second additive compound matrix associated with a square matrix of order 5 is given. 1 : Transmission rate of individuals moving from exposed to infected class: 0.55 Calculated : Rate at which hospitalised individuals get recovered and become exposed again : 0.35 Assumed 10 : Rate at which infected individuals recovered themselves due to strong immunity and again become exposed Using the above representation, dynamical system of set of nonlinear differential for the model is formulated as follow: (v) For other mathematical systems of epidemic models, we can consult these references [9][10][11]. One can turn the Cauchy problem (1.3) into an integral equation by using the following so called Duhamel formula: Therefore ( ) > 0; ∀ ≥ 0. and ( ) > 0; ∀ ≥ 0.
We can observe also that is a Metzler matrix (a matrix = ( 1 ≤ , ≤ is a Metzler matrix if all of its elements are non-negative except for those on the main diagonal, which are unconstrained.) That is, a Metzler matrix is any matrix which satisfies = ( ); ≥ 0, ≠ .
Thus, (1.3) is a monotone system. It follows that, ℝ + 2 is invariant under the flow of (1.3).
• We will say that an equilibrium point * is stable if: where ( ) is a solution of (1.6) • We will say that an equilibrium point * is asymptotically stable if for each neighborhood of * there exists a neighborhood such that * ∈ ⊂ and (0) ∈ implies that the solution ( ) satisfies ( ) ∈ for all > 0, and that ( ) ⟶ * as ⟶ +∞.
In particular, a system is called asymptotically stable around its equilibrium point at the origin if it satisfies the following two conditions: The first condition requires that the state trajectory can be confined to an arbitrarily small "ball" centered at the equilibrium point and of radius , when released from an arbitrary initial condition in a ball of sufficiently small (but positive) radius 1 . This is called stability in the sense of Lyapunov (i.s.L.).
It is possible to have stability in the sense of Lyapunov without having asymptotic stability, in which case we refer to the equilibrium point as marginally stable. Nonlinear systems also exist that satisfy the second requirement without being stable i.s.L. An equilibrium point that is not stable i.s.L. is termed unstable.
The question of interest is whether the steady state is stable or unstable. Consider a small perturbation from the steady state by letting = * + , 1 ≤ ≤ where both , 1 ≤ are understood to be small. The question of interest translates into the following: will , 1 ≤ where both grow (so that , 1 ≤ ≤ move away from the steady state), or will they decay to zero (so that , ,1 ≤ ≤ move towards the steady state)?
In the former case, we say that the steady state is unstable, in the latter it is stable.To see whether the perturbation grows or decays, we need to derive differential equations for , 1 ≤ We do so as follows: The . . .. denote higher order terms, Since ; 1 ≤ ≤ are assumed to be small, these higher order terms are extremely small. The above linear system for ; 1 ≤ ≤ has the trivial steady state = 0; 1 ≤ ≤ , and the stability of this trivial steady state is determined by the eigenvalues of the matrix, as follows: If we can safely neglect the higher order terms, we obtain the following linear system of equations governing the evolution of the perturbations , 1 ≤ ≤ :

)
We refer to the matrix as the Jacobian matrix of the original system at the steady state * .
if the eigenvalues of the Jacobian matrix all have real parts less than zero, then the steady state is stable.
if the eigenvalues of the Jacobian matrix all have real parts < 0, then the steady state is asymptotically stable.
If at least one of the eigenvalues of the Jacobian matrix has real part greater than zero, then the steady state is unstable.
Otherwise there is no conclusion (then we have a borderline case between stability and instability; such cases require an investigation of the higher order terms we neglected, and this requires more sophisticated mathematical machinery discussed in advanced courses on ordinary differential equations). ⧫

Definition 1.6
An equilibrium point * is said hyperbolic if all eigenvalues of the Jacobian matrix have real parts ≠ 0.

Remark 1.7
A hyperbolic equilibrium point * is asymptotically stable if the eigenvalues of the Jacobian matrix all have real parts < 0 or otherwise it is unstable.
Let be the Jacobian matrix, assume that it is a real hyperbolic matrix, i.e. ℜ ≠ 0 for for all eigenvalues of , then There is a linear change of variables [good coordinates ( , )] that induces a splitting into stable and unstable spaces ℝ = ℰ ⊕ ℰ so that in the new variables Last but not least, there is a theorem (the Hartman-Grobman Theorem) that guarantees that the stability of the steady state * of the original system is the same as the stability of the trivial steady state 0 of the linearized system. Let * be an equilibrium point of nonlinear system (1.6) then by applying a translation, we can always assume 0 is a equilibrium point of (1.6).
• Poincaré in his dissertation showed that if is analytic at the equilibrium point * , and the eigenvalues of * are nonresonant, then there is a formal power series of change of variable to change (1.6) to a linear system [4,12] .
• Hartman and Grobman showed that if is continuously differentiable, then there is a neighborhood of a hyperbolic equilibrium point and a homeomorphism on this neighborhood, such that the system in this neighborhood is changed to a linear system under such a homeomorphism [13][14][15][16][17].
Thus, the procedure to determine stability of * is as follows: 1. Compute all partial derivatives of the right-hand-side of the original system of differential equations, and construct the Jacobian matrix.

Theorem 1.11
If (1.6) admits a Liapunov function at an equilibrium point * , then * is stable and if the Liapunov function is strictly decreasing then * is asymptotically stable. ⧫ We outline in the next section the Li-Wang's stability criterion for real matrices and we recall of some spectral properties of −matrices.

On Li-Wang's Stability Criterion of Real Matrix Definition 2.1
Let be an × matrix and let ( ) be its spectrum. The stability modulus of is defined by ( ) = {ℛ ; ∈ ( )} i.e. s( ) is the maximum real part of the eigenvalues of called also the spectral abscissa.
is said to be stable if ( ) < 0. ⧫ The stability of a matrix is related to the Routh-Hurwitz problem on the number of zeros of a polynomial that have negative real parts. Routh-Hurwitz discovered necessary and sufficient conditions for all of the zeros to have negative real parts, which are known today as the Routh-Hurwitz conditions. A good and concise account of the Routh-Hurwitz problem can be found in Banks et al. (1992) [5].
The Li-Wang criterion offer an alternative to the well-known Routh-Hurwitz. It based on Lozinskĭ measures and second additive compound matrix. For detailed discussions on compound matrices, the reader is referred to Li-Wang Li-Wang [2] and for additive compound matrices to Fiedler (1974) [18].
• In Li-Wang [2] a necessary and sufficient condition for the stability of an × matrix with real entries is derived (Li-Wang criterion) by using a simple spectral property of additive compound matrices.
• A survey is given of a connection between compound matrices and ordinary differential equations by Muldowney (1990) [19].

Now, let
( ) be the linear space of × matrices with entries in , where = ℝ or ℂ.

Definition 2.2
• Let ∧ denote the exterior product in , and let 1 ≤ ≤ be an integer. With respect to the canonical basis in the ℎ exterior product space ∧ , the ℎ additive compound matrix [ ] of is a linear operator on ∧ whose definition on a decomposable element 1
• The logarithmic norm of a matrix for an arbitrary norm was introduced by the Leningrad mathematician Lozinskii (1958) [25] and the Swedish mathematician Dahlquist (1959) [26] in their papers on the numerical integration of ordinary differential equations. For linear bounded operators in Banach spaces, a similar notion was introduced Daletskii and Krein (1970) [27], Problems and supplement to Chap. I. ⧫ • Let = ( ) be a real or complex square × matrix, and let 1 , 2 , . . . . , be the complete set of its eigenvalues denoted by ( ) (the spectrum of the matrix ). The maximal real part of these eigenvalues is denoted by ( ) i.e. ( ) = 1≤ ≤ ℜ . (spectral abscissa). The term "spectral abscissa" (by analogy with the spectral radius ( ) = lim || || 1 as ⟶ +∞ of a matrix ) and the notation for it were proposed in Perov (2002) [28].
It follows from the definition of that − || || ≤ || || ≤ || || , 0 ≤ < +∞ but −|| || and || || are not the best constants. Now, let and the best constants in the estimate : the existence of such constants is beyond doubt. We see from the last equality in (2.5) that is the spectral abscissa of the matrix : = ( ). Let us stress that the spectral abscissa is independent of the choice of the norm.
(ii) We see from the last equality in (2.6) that is the logarithmic norm of the matrix : = || || .

7)
Provided that at least one of the limits in (2.7) exists. Provided that at least one of the limits in (2.8) exists. As we have already said above, the last limit exists and serves to define the logarithmic norm. It remains to prove that In (2.9), the quantity on the left exists, is finite and is equal to ; as proved above, the limit on the right exists, is finite and will be denoted by . The definition of the number implies the inequality ≥ .
Suppose for the time being that the written inequality is strict: > . For a sufficiently small > 0, we can write − ≥ + . From the obtained > 0, we then find a = such that After this, consider an arbitrary fixed > 0. Let us choose a natural number so that 0 < ≤ .
After this, we estimate Thus, and this explicitly contradicts the definition of the number .
This Theorem implies the important inequality: For every , ∈ (ℂ), ≥ 0 , and ∈ ℂ the following relations hold: In the partial case for the Holder vector -norm defined by || || = (∑ =1 | | ) 1 and || || ∞ = 1≤ ≤ {| |} then the corresponding matrix measure can be calculated explicitly in the cases:  If = ( ) 1≤ , ≤ has elements satisfying (2.4), it is possible to define and , such that : Then, for any choice of and , satisfying (2.5) we have where an empty product is defined to be 1 and denotes determinant of .
However, the above theorem is not implied by any of their results. ⧫

 Bounds on norms of compound matrices
Let be a matrix in (ℂ), For subsets and of {1, . . . , } we denote by ( | ) the sub-matrix of whose rows are indexed by and whose columns are indexed by in their natural order.

Lemma 2.13
Let be a square real or complex matrix such that : Then is invertible and the set of its eigenvalues is included in ⋃ =1 { ∈ ℂ; | − | ≤ ∑ ≠ | |}.
The 0 ℎ equation of te system = 0 can be written as follow : (ii) ( [2] ) < 0 can be interpreted as ̂, < 0 for = 1, . . . , (  Where denotes the transpose. ⧫ • Positive definite matrices are of both theoretical and computational importance in a wide variety of applications. They are used, for example, in optimization algorithms and in the construction of various linear regression models (Johnson 1970) [42].
A positive definite matrix has at least one matrix square root. Furthermore, exactly one of its matrix square roots is itself positive definite.
A necessary and sufficient condition for a complex matrix to be positive definite is that the Hermitian part where denotes the conjugate transpose, be positive definite.
This means that a real matrix is positive definite iff the symmetric part where is the transpose, is positive definite (Johnson 1970 [42]).
• Confusingly, the discussion of positive definite matrices is often restricted to only Hermitian matrices, or symmetric matrices .
A Hermitian (or symmetric) matrix is positive definite iff all its eigenvalues are positive. Therefore, a general complex (respectively, real) matrix is positive definite iff its Hermitian (or symmetric) part has all positive eigenvalues.
The determinant of a positive definite matrix is always positive, so a positive definite matrix is always nonsingular. ⧫

Definition 2.16
(1) An real square matrix is said Z-matrix if their of diagonal elements are all non-positive.
(1) An real square matrix is said −matrix if it is −matrix and fulfilling one of the conditions of the following theorem of Fiedler and Ptàk [47]. ⧫

Theorem 2.17 (Fiedler-Ptàk)
Let be a Z-matrix. Then the following conditions are equivalent to each other : 1 There exists a vector ≥ 0 such that > 0; 2 there exists a vector > 0 such that > 0; 3 there exists a diagonal matrix with positive diagonal elements such that > 0 (here is the vector whose all coordinates are 1); 4 there exists a diagonal matrix with positive diagonal elements such that the matrix = is a matrix with dominant positive principal diagonal; Consequently and * are strictly dominant diagonal matrices. Then + * is also a strictly dominant diagonal matrix and it is a definite positive matrix because it is symmetric. Then there exists > 0 such that < , >≥ || || 2 . Now, let = and are the elements the diagonal matrix then we have
Consequently, there exists such that > 0 which entails ( ) by applying the property 13 of above theorem on the M-matrices. If the matrix of the system (2.9) is Schur stable, then the fixed point * of the system (2.9) is asymptotically stable. that is the eigenvalues of have strictly negative real part. In next section we give some preliminary definitions and lemmas for linear stability of above system.

Some Preliminary Definitions and Lemmas
• Writting the above five-dimensional system as follow:

• Basic reproduction number
Mathematical modeling can play an important role in helping to quantify possible disease control strategies by focusing on the important aspects of a disease, determining threshold quantities for disease survival, and evaluating the effect of particular control strategies.
A very important threshold quantity is the basic reproduction number, sometimes called the basic reproductive number or basic reproductive ratio (Heffernan et al. 2005 [6]), which is usually denoted by ℛ 0 .
The epidemiological definition of ℛ 0 is the average number of secondary cases produced by one infected individual introduced into a population of susceptible individuals, where an infected individual has acquired the disease, and susceptible individuals are healthy but can acquire the disease.
In reality, the value of ℛ 0 for a specific disease depends on many variables, such as location and density of population.
The study of the stability of jacobian matrices of order less than three of a dynamic system yields a reasonable ℛ 0 , but for more complex compartmental models, especially those with more infected compartments, the study of the stability is difficult as it relies on the algebraic Routh-Hurwitz conditions for stability of the Jacobian matrix.
An alternative method proposed by Diekmann et al. (1990) [7] and elaborated by van den Driessche and Watmough (2002) [8] gives a way of determining ℛ 0 for a compartmental model by using the next generation matrix.
Here an outline of this method is given, the proofs and further details can be found in van den Driessche and Watmough (2002) and van den Driessche and Watmough (2008) [50]. Let = ( 1 , 2 , . . . . , , . . . , ) be the number of individuals in each compartment, where the first < compartments contain infected individuals.
Assume that the equilibrium point * exists and is stable in the absence of disease, and that the linearized equations for 1 , . . . , at the * decouple from the other equations. The assumptions are given in more details in the references cited above.
Consider these equations written in the form: For the non-singularity of , according to the Perron-Frobenius theorem, it must be the case that > ( ). Also, for a non-singular M-matrix, the diagonal elements of must be positive. Here we will further characterize only the class of non-singular M-matrices. ⧫

Definition 3.3 (Metzler matrix)
In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasi-positive) or essentially nonnegative if all of its elements are non-negative except for those on the main diagonal, which are unconstrained. That is, a Metzler matrix is any matrix A which satisfies = ( ); ≥ 0, ≠ . ⧫ -matrices are very useful. We can found some of their applications to ecology, numerical analysis, probability, mathematical programming, game theory, control theory, and matrix theory. There are many definitions of M-matrices equivalent to the above. For example, if a matrix has the sign pattern and ( ) > 0, then is a non-singular M-matrix [5].
A matrix of the form − , ≥ 0 is called a −matrix.
• Observe that a −matrix is an −matrix if and only if + is nonsingular for all > 0.
We said that a matrix = ( ) of order has the sign pattern if ≤ 0 for all ≠ .
If a matrix has the sign pattern and ( ) > 0, then is a non-singular M-matrix [52].

Lemma 3.4
Let be a non-singular M-matrix and suppose and −1 have the sign pattern.Then is a non-singular Mmatrix if and only if −1 is a non-singular M-matrix. ⧫ In general, this lemma does not hold if a singular M-matrix. It can be shown to hold if is singular and irreducible. However, this is not sufficient for our needs in part II of this work. we shall need of the following lemma : Hence, the above lemma implies statement (i). A separate continuity argument can be constructed for each implication in the singular case.
The following theorem collects conditions that characterize nonsingular −matrices.

Proof
Conditions (b), (c), and (e) are due to Ostrowski [30], who introduced the concept of −matrices. Condition (e) is known in the economics literature as the Hawkins-Simon condition [54].
Many additional characterizations of nonsingular (and of singular) M-matrices are given in Berman and Plemmons (1979) [52].
A subset of the set of all M-matrices that contains the nonsingular M-matrices and whose matrices share many of their properties is the set of group-invertible M-matrices (M-matrices with "property c").
Basic reproduction number ℛ 0 for the model can be established using the next generation matrix method [56] and [7].

Definition 3.7
The basic reproduction number ℛ 0 is obtained as the spectral radius of matrix −1 at disease free equilibrium point. Where and are constructed as below: ) and = ( ( * ) ) for 1 ≤ , ≤ .
For our system the graph of ℛ 0 with respect 1 is : Basic reproduction number of infections ℛ 0 as a function of 1 . All other parameters are fixed.
From the above functions ( ),1 ≤ ≤ 5 of our system, we consider the associated functions (̂),1 ≤ ≤ 5 where we delete the linear elements and the negative nonlinear elements, i.e : In next section, we apply the corollary 2.11 to stability of Covid-19 system.

Theorem 4.4
If ℛ 0 < 1, the DFE is locally asymptotically stable. If ℛ 0 > 1, the DFE is unstable. ⧫ The epidemiological interpretation of Theorem 4.4 is that, (covid-19) can be eliminated in the population when ℛ 0 < 1 if the initial conditions of the dynamical system (covid-19) are in the basin of attraction of the DFE 0 .
A necessary condition for all the roots of the characteristic polynomial to admit a negative real part, all the coefficients must be positive, that is to say: 1 > 0, 2 > 0, . . . , 3 > 0.
Now, if we consider the discriminant of which is given by : we observe that : • 4 if Δ > 0, then a necessary and sufficient condition for an equilibrium point to be locally asymptotically stable is 1 > 0, 3 > 0, 1 2 − 3 > 0.
• 7 A necessary condition for an equilibrium point to be locally asymptotically stable is 3 > 0.
We remark that • 4 , • 6 and • 4 are not satisfy by the coefficients of ( ). So we have to solve the cubic equation ( ) = 0 by the Cardan's method which is ingenious and effective, but quite non-intuitive. The expression Δ = 3 + 2 is called the discriminant of the equation See for example Nickalls (1993) [57] for a brief description of Cardan's method.
• Substantial technical difficulties for explicit expression of * [2] In Appendix of Li-Wang [2], we found that for = 2,3, and 4, an explicit expression of second additive compound matrices [2] of × matrices = ( ) 1≤ , ≤ which are given respectively by: In the same way that the section 5 of Li-Wang [2] where they have studied the stability of an epidemic model of SEIR type, we apply their criterion to the following epidemic model: where (Λ, 1 , ( 2 , , , ) are given parameters.
• Determination of equilibrium points of the system (* ) and calculation of basic reproduction number ℛ 0 Let = ( , 1 , 2 ) then the Jacobian matrix of above system is : Now, we consider the following equations: then the endemic equilibrium point of (*) is asymptotically stable.
In order to apply the corollary of the Li-Wang criterion, it remains to calculate the determinant of * * = | In next lemma, we give the explicit entries of second additive compound matrix of × matrix = ( ) where = 5

Conclusions
• In this part I, a generalized SEIR model of COVID-19 was discussed. After a glance on basic properties of the model including, the basic reproduction number and the equilibria of the model, we turned on the stability of these states. It was proved that the free equilibrium state is locally as well as globally asymptotically stable when ℛ 0 < 1. Furthermore, the second additive compound matrix approach was used to establish the local asymptotic stability of free equilibrium state when ℛ 0 > 1.
• In second paper (Part II), In order to control the Covid-19 system, i.e., force the trajectories to go to the equilibria we will add some control parameters with uncertain parameters to stabilize the five-dimensional Covid-19 system studied in this paper.

Acknowledgements
I thank my soon Jean-Karim Intissar for several important suggestions and comments on this work.