key: cord-0152988-kbhfzrl8
authors: Intissar, Abdelkader
title: Application of the criterion of Li-Wang to a five dimensional epidemic model of COVID-19. Part I
date: 2020-07-12
journal: nan
DOI: nan
sha: e3364322f359adbe53ed9268ce21d872349ebcd6
doc_id: 152988
cord_uid: kbhfzrl8

The dynamics of many epidemic models for infectious diseases that spread in a single host population demonstrate a threshold phenomenon. If the basic reproduction number R0 is below unity, the disease-free equilibrium P0 is globally stable in the feasible region and the disease always dies out. If R0>1, a unique endemic equilibrium P? is globally asymptotically stable in the interior of the feasible region and the disease will persist at the endemic equilibrium if it is initially present. In this paper (Part I), we reinvestigate the study of the stability or the non stability of a mathematical Covid-19 model constructed by Nita H. Shah, Ankush H. Suthar and Ekta N. Jayswal. We use a criterion of Li-Wang for stability of matrices [Li-Wang] on the second additive compound matrix associated to their model. 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. Based on compound matrices theory, we apply in [Intissar] again the criterion of Li-Wang to study the stability of equilibrium points of Covid-19 system with uncertain parameters. In this part II, all sophisticated technical calculations including those in part I are given in appendices.

A mathematical Covid-19 model is constructed in [Shah et al] by Nita H. Shah, Ankush H. Suthar and Ekta N. Jayswal to study human to human transmission of the Covid-19.

The model consists all possible human to human transmission of the virus.

The Covid-2019 is highly contagious in nature and infected cases are seen in most of the countries around the world, hence in the model the susceptible population class is ignored and whole population is divided in five compartments :

(1) class of exposed individuals E(t) (individuals surrounded by infection by not yet infected),

(2) class of infected individuals by Covid-19 I(t),

(3) class of critically infected individuals by Covid-19 C(t),

(4) class of hospitalised individuals H(t), and (5) class of dead individuals due to .

Human to human transmission dynamics of Covid-19 is describe graphically in • Table of parameters used in the model is described as follow :

B : Birth rate of class of exposed individuals : 0.80 Calculated µ Natural death rate : 0.01 Assumed β 1 : Transmission rate of individuals moving from exposed to infected class : 0.55 Calculated β 2 : Rate at which infected individuals goes into sever condition or in critical condition : 0.40 Calculated : Rate at which hospitalised individuals get recovered and become exposed again : 0.35 Assumed β 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 :

= B − β 1 EI + β 7 ED + β 9 H + β 10 EI − µE dI dt = β 1 EI − β 2 I − β 6 I − β 8 I − β 10 EI − µI

Remark 1.1 (i) All of the parameters in (covid-19) are assumed to be nonnegative. (v) For other mathematical systems of epidemic models, we can consult these references [Li et al] , [Beretta et al] and [Sun et al] . Proof (i) • Let's suppose I(0) > 0, then from the second equation of (covid-19)), if χ(t) = (β 10 − β 1 )E + β 2 + β 6 + β 8 + µ then the integration from 0 to t > 0 gives :

Therefore I(t) > 0; ∀t ≥ 0.

• Consider the following sub-equations related to the variables C and H : Therefore C(t) > 0; ∀t ≥ 0. and H(t) > 0; ∀t ≥ 0.

We can observe also that M is a Metzler matrix (a matrix A = (a ij 1 ≤ i, j ≤ n 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 A which satisfies A = (a ij ); a ij ≥ 0, i = j.

Thus, (1.3) is a monotone system. It follows that, R 2 + is invariant under the flow of (1.3).

• Let's suppose E(0) > 0, then from the first equation of (covid-19)), if χ(t) = (β 1 − β 10 )I − β 7 D + µ and π(t) = β 9 H(t) + B which is > 0.

the integration from 0 to t > 0 gives :

Therefore E(t) > 0; ∀t ≥ 0.

• Let's suppose D(0) > 0, then from the 5 th equation of (covid-19)), if χ(t) = β 7 E and π(t) = β 6 I(t) + β 5 C(t) which is > 0.

the integration from 0 to t > 0 gives :

Therefore D(t) > 0; ∀t ≥ 0. Consider the following n-dimensional system :

where f : Ω ⊂ R n −→ R n is C 1 -function.

Definition 1.4

• We say that x * is an equilibrium point of (1.6) if f (x * ) = 0.

• We will say that an equilibrium point x * is stable if

where φ t (x) is a solution of (1.6)

• We will say that an equilibrium point x * is asymptotically stable if for each neighborhood U of x * there exists a neighborhood W such that x * ∈ W ⊂ U and x(0) ∈ W implies that the solution φ t (x) satisfies φ t (x) ∈ U for all t > 0, and that φ t (x) −→ x * as t −→ +∞.

In particular, a system is called asymptotically stable around its equilibrium point at the origin if it satisfies the following two conditions :

1. Given any > 0; ∃δ 1 > 0 such that if || x(0) ||< δ 1 , then

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 δ 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.

• Linear stability analysis for systems of ordinary differential equations

Consider the n-dimensional dynamical system (1.6) written in the following form :

(1.7)

x(t) = (x 1 (t), .., x i (t), .., x n (t), 1 ≤ i ≤ n and 0 ≤ t < +∞ where x(0) = (x 1 (0), .., x i (0), .., x n (0)) = x 0 is fixed and f i : R n −→ R are C 1 -functions which are given.

The question of interest is whether the steady state is stable or unstable. Consider a small perturbation from the steady state by letting x i = x * i + u i , 1 ≤ i ≤ n where both u i , 1 ≤ i are understood to be small. The question of interest translates into the following : will u i , 1 ≤ i where both grow (so that x i , 1 ≤ i ≤ n move away from the steady state), or will they decay to zero (so that x i , , 1 ≤ i ≤ n 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 u i , 1 ≤ i We do so as follows :

The .... denote higher order terms, Since u i ; 1 ≤ i ≤ n are assumed to be small, these higher order terms are extremely small. The above linear system for u i ; 1 ≤ i ≤ n has the trivial steady state u i = 0; 1 ≤ i ≤ n, 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 u i , 1 ≤ i ≤ n :

. .

We refer to the matrix as the Jacobian matrix of the original system at the steady state x * .

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).

An equilibrium point x * is said hyperbolic if all eigenvalues of the Jacobian matrix have real parts = 0.

A hyperbolic equilibrium point x * is asymptotically stable if the eigenvalues of the Jacobian matrix all have real parts < 0 or otherwise it is unstable.

Let A be the Jacobian matrix, assume that it is a real hyperbolic matrix, i.e. eλ = 0 for for all eigenvalues λ of A, then

There is a linear change of variables [good coordinates (x s , x u )] that induces a splitting into stable and unstable spaces R n = E s ⊕ E u so that in the new variables

We have written x s = P s x, x u = P u x where P s : R n −→ E s and P u : R n −→ E u are the orthogonal projections.

Last but not least, there is a theorem (the Hartman-Grobman Theorem) that guarantees that the stability of the steady state x * of the original system is the same as the stability of the trivial steady state 0 of the linearized system. Let x * 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 f is analytic at the equilibrium point x * , and the eigenvalues of J x * are nonresonant, then there is a formal power series of change of variable to change (1.6) to a linear system [Poincaré] and [Arnold] .

• Hartman and Grobman showed that if f 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 [Grobman] , [Hartman1] , [Hartman 2] , [Perko,] and [X-Wang] .

Theorem 1.8 (Hartman-Grobman theorem) Let Ω be an open set of R n containing the origin, f : Ω −→ R n be a C 1 -function on Ω, 0 be a hyperbolic equilibrium point of the system (1.6), and U r = {x; || x ||< r} be the neighborhood of the origin of radius r. For any r, > 0 such that U r+ ⊂ Ω, there exists a transformation y = H(x), H(0) = 0 and H is a homeomorphism in a neighborhood of 0, such that the system (1.6) is changed into the linear system

Proof see http://www.math.utah.edu/∼treiberg/M6414HartmanGrobman.pdf.

Thus, the procedure to determine stability of x * 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.

2. Evaluate the Jacobian matrix at the steady state.

3. Compute eigenvalues.

4. Conclude stability or instability based on the real parts of the eigenvalues.

Definition 1.9 (Liapunov function) Let x * be an equilibrium point of (1.6), U ⊂ Ω be a neighborhood of x * and L : U −→ R be a continuous function. We say that L is Liapunov function for (1.6) at x * if

(1) L(x * ) = 0 and for every x = x * we have L(x) > 0 ;

(2) The function t −→ L(φ t (x)) is decreasing.

We say that L is strictly Liapunov function for (1.6) at x * if L satisfy (1) and

(3) the function t −→ L(φ t (x)) is strictly decreasing.

If L is C 1 function then we can replace :

-The condition (2) by ∀ x ∈ U, < ∇L(x), f (x) >≤ 0.

and -The condition (3) by ∀ x ∈ U, < ∇L(x), f (x) >< 0.

If (1.6) admits a Liapunov function at an equilibrium point x * , then x * is stable and if the Liapunov function is strictly decreasing then x * is asymptotically stable.

We outline in the next section the Li-Wang's stability criterion [Li-Wang] for real matrices and we recall of some spectral properties of M-matrices. In section 3 we give some preliminary definitions and some lemmas for linear stability of (covid-19) system. In section 4, we present a study of stability of equilibrium points of (covid-19) system by using the R 0 criterion and Li-Wang criterion on second additive compound matrix associated to Jacobian matrix of system (covid-19). § 2 On Li-Wang's stability criterion of real matrix.

Let A be an n × n matrix and let σ(A) be its spectrum. The stability modulus of A is defined by s(A) = M ax{Reλ; λ ∈ σ(A)} i.e. s(A) is the maximum real part of the eigenvalues of A called also the spectral abscissa.

A is said to be stable if s(A) < 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].

The Li-Wang criterion offer an alternative to the well-known Routh-Hurwitz. It based on Lozinskiǐ measures and seconde additive compound matrix. For detailed discussions on compound matrices, the reader is referred to [Li-Wang] and for additive compound matrices to [Fiedle] .

• In [Li-Wang] a necessary and sufficient condition for the stability of an n × n 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 James S ; Muldowney in [Muldowney] And for an application of Li-Wang criterion, we can consult [Diekmann et al] , [Intissar et al] and [Khanh] . Now, let M n (K) be the linear space of n × n matrices with entries in K, where K = R or C.

• Let ∧ denote the exterior product in K n , and let 1 ≤ k ≤ n be an integer. With respect to the canonical basis in the kth exterior product space ∧ k K n , the kth additive compound matrix A [k] of A is a linear operator on ∧ k K n whose definition on a decomposable element

1≤i,j≤n and for any integer i = 1, ..., C k n , let ((i)) = (i 1 , i 2 , ...., i k ) be the ith member in the lexicographic ordering of integer k-tuples such that

. Then

• The entry in the ith row and the jth column of

(−1) r+s a jr,is if exactly one entry of i s does not occur in ((j)) and j r does not occur in ((i)), 0 if ((i)) dif f ers f rom ((j)) in two or more entries.

(2..2)

.

• Let ||.|| denote a vector norm in K n and the operator norm it induces in M n (K).

• 1 The Lozinskiȋ measure µ (also known as logarithmic norm || . || log ) on M n (K) with respect to ||.|| is defined by (see [Coppel] , p. 41)

• 2 By the logarithmic norm of a matrix A we mean the real number defined by the formula :

• The existence of a limit in (2.3) bis is established on the basis of the convexity of the function I + tA (see [ [Bylov et al ] , Supplement I, Sec. 2], whence we also borrow the notation for the logarithmic norm).

• The logarithmic norm of a matrix for an arbitrary norm was introduced by the Leningrad mathematician Lozinskii [Lozinski] and the Swedish mathematician Dahlquist [Dahlquist] 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 in their book [[Daletskii-Krein] , Problems and supplement to Chap. I].

• Let A = (a ij ) be a real or complex square n × n matrix, and let λ 1 , λ 2 , ...., λ n be the complete set of its eigenvalues denoted by σ(A) (the spectrum of the matrix A). The maximal real part of these eigenvalues is denoted by s(A) i.e. s(A) = max 1≤i≤n eλ i . (spectral abscissa). The term "spectral abscissa" (by analogy with the spectral radius ρ(A) = lim || A n || 1 n as n −→ +∞ of a matrix A) and the notation for it were proposed in [ [Perov] , p. 23].

• What the best upper and lower bounds for || e tA ||?, 0 ≤ t < +∞ where e tA = I + n k=1 t k A k k! .

It follows from the definition of e tA that e −t||A|| ≤|| e tA ||≤ e t||A|| , 0 ≤ t < +∞ but − || A || and || A || are not the best constants. Now, let α and β the best constants in the estimate :

the existence of such constants is beyond doubt.

(i) Let α be the best constant in estimate (2.4) from below. Then

(2.5) (ii) Let β be the best constant in estimate (2. 4) from below. Then

We see from the last equality in (2.5) that α is the spectral abscissa of the matrix A : α = s(A). 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 β is the logarithmic norm of the matrix A :

Consider the logarithmic function on the positive semi-axis. In view of its continuous differentiability, it locally satisfies the Lipschitz condition. Therefore, for any > 0, we can indicate a δ = δ > 0such that

Therefore, under the conditions ||| e tA || −1 |< δ and ||| I + tA || −1 |< δ, , we have

Therefore,

provided that at least one of the limits in (2.7) exists.

Further, setting (t) =|| I + tA || − || I ||, we can write

whence, using the well-known relation

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 β ; as proved above, the limit on the right exists, is finite and will be denoted by b. The definition of the number β implies the inequality β ≥ b.

Suppose for the time being that the written inequality is strict : β > b. For a sufficiently small > 0, we can write β − ≥ b + . From the obtained > 0, we then find a δ = δ such that || e tA ||≤|| e t(b+ ) || for 0 < t ≤ δ.

After this, consider an arbitrary fixed t > 0. Let us choose a natural number k so that 0 < t k ≤ δ. After this, we estimate

Thus,

and this explicitly contradicts the definition of the number β.

This Theorem implies the important inequality :

For every A, B ∈ M n (C), α ≥ 0 , and ξ ∈ C the following relations hold :

In the partial case for the Holder vector p-norm defined by

then the corresponding matrix measure can be calculated explicitly in the cases :

respectively , where A * denotes the Hermitian adjoint of A.

If A is real symmetric, then µ 2 (A) = s(A).

For a real matrix A, conditions µ ∞ (A) < 0 or µ 1 (A) < 0 can be interpreted as a ii < 0 for i = 1, ..., n, and A is diagonally dominant in rows or in columns, respectively.

• Some upper and lower bounds for the determinant of n × n matrix A with positive diagonal elements

Let A = (a ij ) 1≤i,j≤n be a real matrix satisfying :

Then we have the following result :

Theorem 2.4

If A = (a ij ) 1≤i,j≤n has elements satisfying (2.4), it is possible to define l i and r i , such that :

Then, for any choice of l i and r i , satisfying (2.5) we have

where an empty product is defined to be 1 and detA denotes determinant of A.

To prove this result, we need the following bound given by Price [Price] :

Let D n represent det A then we procceed by induction on n :

(a) For n = 2, let A =   a 11 a 12 a 21 a 22   has elements satisfying (2.4) then r 1 =| a 12 | and r 2 = 0 then by observing that :

• 1 | a 12 | a 22 ≥| a 12 || a 21 |=| a 12 a 21 |≥ −a 12 a 21 . and • 2 | a 12 | a 22 ≥| a 12 || a 21 |=| a 12 a 21 |≥ a 12 a 21 .

we deduce that the Price's theorem holds.

a 21 a 22 = l 1 + r 1 a 12 a 21 l 2 + r 2 and expanding it by diagonal elements in the following form :

Therefore l 1 l 2 + l 1 r 2 + r 1 r 2 ≤ D 2 ≤ r 1 r 2 + l 1 r 2 + (l 1 + 2r 1 )l 2 , since 0 ≤ r 1 a 12 a 21 l 2 ≤ (r 1 + a 12 )l 2 < 2r 1 l 2 by (2.5) and (2.7) .

(b) Assume that for any matrix of order n − 1 with elements satisfying (2.5),

and the elements a ij satisfy (2.5), partition D n as follows :

row vector with components a nj 1 ≤ j ≤ n − 1, and as in (2.5) , l n + r n = a nn , l n ≥ n−1 j=1 | a nj , r n ≥ 0.

Then we can write D n as the sum of two determinants, i.e.,

But the elements of ∆ satisfy (2.4), hence, by (2.5) and (2.7) we deduce that

Also, by inductive assumption, since A 1 , is of order n − 1, and, by (2.5),

We have , using (2.10), (2.9) and (2.8), [Brenner2] and H. Schneider [Schneider] have given lower and upper bounds for the absolute value of determinants satisfying more general condition than (2.4).

However, the above theorem is not implied by any of their results.

• Bounds on norms of compound matrices

Let A be a matrix in M n (C), For subsets α and β of {1, ..., n} we denote by A(α | β) the sub-matrix of A whose rows are indexed by α and whose columns are indexed by β in their natural order.

Let k be a positive integer, k ≤ n. we denote by C k (A) the k th of the matrix A, that is,

• Example 2 if A ∈ M 3 (R) and k = 2 then :

and k = 3 then :

The most important property of the compound mapping is that it is multiplicative.

Let A and B be n × n matrices and let 1 ≤ k ≤ n, then

This property is equivalent to the Binet-Cauchy theorem :

Theorem 2.7 (Binet-Cauchy Theorem)

Let A be a n × m complex matrix, B be a m × l complex matrix and p ≤ min{n, m, l} then

Some other principal properties of compound matrices are given in [ [Aitken] , [Marcus] , [Mitrouli et al], [Kravvaritis et al] ] for A ∈ M n (C) and p an integer, 1 ≤ p ≤ n :

in particular, let A ∈ M n (C) and k ≤ n then we have :

• 6 if {λ i , i = 1....., n} are eigenvalues of A then the eigenvalues of A [k] are of the following form :

• 7 if {λ i , i = 1....., n} are eigenvalues of A then the eigenvalues of C k (A) are of the following form :

The main use of compound matrices are their spectral properties which follow from the previous lemma together with the Jordan Canonical Form.

The compounds of companion matrices can be used to study products of roots of polynomials. An excellent example of this can be seen in [Hong et al] where there is an extensive description of the compounds of companion matrices as well as some applications. Now, let ν be a vector norm on C n , and for a positive integer k, k ≤ n, let µ be a norm on

with col i (B) denotes the i th column of B.

• Some criteria of stability on matrices given by Li and Wang using the compound matrix and Lozinskii measure. 

Theorem 2.10 (see [Li-Wang] )

Assume that A ∈ M n (R) and (−1) n det(A) > 0. Then A is stable if the following conditions are verified :

are the entries of seconde additive compound matrix A [2] .

If we take as Lozinskiȋ's measure µ(

then by applying the above theorem µ(A [2] ) < 0 and A is stable.

A matrix A = (a ij ); 1 ≤ i, j ≤ n is said to have dominant principal diagonal if | a ii |> n k =i | a ik | for each 1 ≤ i ≤ n. ( )

Let A be a square real or complex matrix such that :

Then A is invertible and the set of its eigenvalues is included in

The i th 0 equation of te system Ax = 0 can be written as follow :

It follows that it is not dominant principal diagonal in particular there exists i such that

) < 0 can be interpreted asâ j,j < 0 for j = 1, ..., n(n−1)

, and A [2] is diagonally dominant in columns.

• Positive Definite Matrix Definition 2.15

for all nonzero complex vectors x ∈ C n , where x * denotes the conjugate transpose of the vector x.

In the case of a real matrix A, equation (2.4) reduces to

where x T 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) . 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 A to be positive definite is that the Hermitian part

where A H denotes the conjugate transpose, be positive definite.

This means that a real matrix A is positive definite iff the symmetric part

where A T is the transpose, is positive definite (Johnson 1970) .

• Confusingly, the discussion of positive definite matrices is often restricted to only Hermitian matrices, or symmetric matrices .

in the case of real matrices (Pease 1965 [ Pease] , Johnson 1970 [Johnson] , Marcus and Minc 1988, p 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.

(1) An real square matrix A is said Z-matrix if their of diagonal elements are all non-positive.

(1) An real square matrix A is said M-matrix if it is Z-matrix and fulfilling one of the conditions of the following theorem of Fiedler and Ptàk [Fiedler-Ptàk] .

Theorem 2.12 (Fiedler-Ptàk) Let A be a Z-matrix. Then the following conditions are equivalent to each other :

1 o There exists a vector x ≥ 0 such that Ax > 0 ;

2 o there exists a vector x > 0 such that Ax > 0 ;

3 o there exists a diagonal matrix D with positive diagonal elements such that ADe > 0 (here e is the vector whose all coordinates are 1) ;

4 o there exists a diagonal matrix D with positive diagonal elements such that the matrix W = AD is a matrix with dominant positive principal diagonal ;

5 o for each diagonal matrix R such that R ≥ A the inverse R −1 exists and ρ(R −1 (P − A)) < 1, where P is the diagonal of A ; 10 o there exists a permutation matrix P such that PAP −1 may be written in the form RS where R is a lower triangular matrix positive diagonal elements such that R is a Z-matrix and S is an upper triangular matrix with positive diagonal elements such that S is a Z-matrix ; (ii) There exists a diagonal matrix D with elements > 0 such that DA is definite positive matrix.

Let e be the vector having all its components equal to 1. We define x and y by Ax = e and A * y = e. Then we have x i > 0 and y i > 0 for all i.

Consequently B and B * are strictly dominant diagonal matrices. Then B + B * is also a strictly dominant diagonal matrix and it is a definite positive matrix because it is symmetric. Then there exists α > 0 such that < Bu, u >≥ α || u || 2 . Now, let d i = y i x i and d i are the elements the diagonal matrix D then we have

Consequently, there exists i such that u i v i > 0 which entails (i) by applying the property 13 o of above theorem on the M-matrices.

• The Schur stability criteria of matrices using the additive compound matrix Definition 2.18 (Shur stability)

A matrix A is said to be Schur stable if ρ(A) < 1, where ρ(A) = max{| λ |; λ ∈ σ(A)} (the spectral radius of A).

Consider the C r ; r ≥ 1 map :

x −→ g(x); x ∈ R n (2.8)

If (2.8) has a fixed point x = x * , that is, x * = g(x * ), then the linear map corresponding to ((2.8) is

where A = Dg(x * ), the Jacobian matrix of g at x * .

Lemma 2.19 (see [Liao] Theorem 2.1).

If the matrix A of the system (2.9) is Schur stable, then the fixed point x * of the system (2.9) is asymptotically stable. that is the eigenvalues of A have strictly negative real part.

.

2 Let A ∈ M n (R), then ρ(A) < 1 ⇐⇒ ρ(C 2 (A)) < 1 and det(I − A 2 ) > 0.

3 Let A ∈ M n (R), then ρ(A) < 1 ⇐⇒ σ 1 σ 2 < 1 and det(I − A 2 ) > 0.

where {σ 1 , σ 2 , ....., σ n } are the singular values of A, i.e the eigenvalues of the symmetric matrix √ A * A such that σ 1 ≥ σ 2 ≥ ...... ≥ σ n ≥ 0.

In next section we give some preliminary definitions and lemmas for linear stability of above system and in section 4, we apply last corollary to stability of Covid 19 system. § 3 Some preliminary definitions and lemmas

• Writing the above five-dimensional system as follow :

where the « T » denotes transpose.

• 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, Smith, & Wahl, 2005) [Heffernan et al], 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. Here an outline of this method is given, the proofs and further details can be found in van den Driessche and Watmough (2002) Assume that the equilibrium point x * exists and is stable in the absence of disease, and that the linearized equations for x 1 , ..., x m at the x * decouple from the other equations. The assumptions are given in more details in the references cited above.

Consider these equations written in the form :

In this splitting, It is assumed that F i , V i ∈ C 2 and F i = 0, m + 1 ≤ i ≤ n Remark 3.1

Let n = 5 and (x 1 , x 2 , ...., x 5 ) T = (E, I, C, H, D) T the compments of our system Covid-19 then we have :

The Jacobian matrix associated to 

and for m = 2 we have The Jacobian matrix associated to 

and for m = 2 we have

• Important case Let E = B µ , I = C = H = D = 0 and α = β 2 + β 6 + β 8 + µ then we have :

which is called effective basic reproduction number.

These following figures give the curves of R 0 -evolution with respect µ as abscissa of step ∆µ = 0.015 and parameter β 10 but the other parameters are fixed as in above table. Let A = (a ij ) be a n × n real matrix such that (a ij ) ≤ 0 for all i = j, 1 ≤ i, j ≤ n. Then matrix A is also an M-matrix if it can be expressed in the form

where s is at least as large as the maximum of the moduli of the eigenvalues of B, and I is an identity matrix.

For the non-singularity of A, according to the Perron-Frobenius theorem, it must be the case that s > ρ(B). Also, for a non-singular M-matrix, the diagonal elements a ii of A must be positive. Here we will further characterize only the class of non-singular M-matrices.

In mathematics, especially linear algebra, a matrix is called Metzler, quasipositive (or quasipositive) 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 A = (a ij ); a ij ≥ 0, i = j.

sis, probability, mathematical programming, game theory, control theory, and matrix theory.

• Some fondamental properties of M-matrices A matrix of the form αI − B, B ≥ 0 is called a Z−matrix.

• Observe that a Z−matrix A is an M −matrix if and only if A + l is nonsingular for all > 0.

We said that a matrix A = (a ij ) of order n has the Z sign pattern if a ij ≤ 0 for all i = j. In general, this lemma does not hold if B a singular M-matrix. It can be shown to hold if B 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 M −matrices. Condition (d) is due to Schneider [Schneider] and Ky-Fan in [Fan] and the condition (f) to Fiedler and Ptak [Fiedler-Ptàk] .

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 [Cheng-Shan] and [Diekmann et al] .

Definition 3.7

The basic reproduction number R 0 is obtained as the spectral radius of matrix FV −1 at disease free equilibrium point. Where F and V are constructed as below :

For our system the graph of R 0 with respect 1 µ is :

Basic reproduction number of infections R 0 as a function of 1 µ . All other parameters are fixed.

From the above functions (f i ), 1 ≤ i ≤ 5 of our system, we consider the associated functions (f i ), 1 ≤ i ≤ 5 where we delete the linear elements and the negative nonlinear elements, i.e :

and the associated functions (ĝ i ), 1 ≤ i ≤ 5 where we delete the non negative nonlinear elements and we take the opposite of the obtained expression , i.e :

We define the matrices F and V as follow : 

Remark 3.8

(i) The explicit matrix F is : 

where α = β 10 E + β 8 + β 6 + β 2 + µ β = β 3 + β 5 + µ γ = β 9 + β 4 + µ

As the form of the matrix F is simple F=

then the matrix FV −1 has the following form :

Lemma 3.10

In order to simplify the notations and avoid lengthy expressions, we define the parameters :

then the eigenvalues of FV −1 are λ i ; 1 ≤ i ≤ 5 where λ 1 and λ 2 are the zeros of

and λ 3 = λ 4 = λ 5 = 0.

Lemma 3.11

where ∆ = (a + d) 2 − 4(ad − bc) § 4. Determination of equilibrium points Theorem 4.1

If the control reproduction number R 0 is is less than 1, model (covid-19) has a unique equilibrium : the disease-free equilibrium (DFE) P 0 = ( B µ , 0, 0, 0, 0).

Conversely, if R 0 > 1 , model (covid-19) has two equilibria : the DFE and a unique endemic equilibrium P * = (E * , I * , C * , H * , D * ) = (E * ,αH * ,βH * ,

and E * = µ + β 2 + β 6 + β 8 β 1 − β 10 = α β 1 − β 10 with α = β 2 + β 6 + β 8 + µ ; α = β 3 β 4 + (β 4 + β 3 + µ)(β 5 + β 3 + µ) β 2 β 3 + β 8 (β 5 + β 3 + µ) ; β = β 8 β 4 + β 2 (β 4 + β 9 + µ) β 2 β 3 + β 8 (β 5 + β 3 + µ) ; and γ = β 6 β 7 E * α +

(Equilibrium points) (i) We observe that P 0 = ( B µ , 0, 0, 0, 0) is an equilibrium point which is called disease free equilibrium point.

(ii) A second equilibrium point P * is given by P * = (E * , I * , C * , H * , D * ) = (E * ,αH * ,βH * , H * ,γH * )

and E * = µ + β 2 + β 6 + β 8 β 1 − β 10 which is called Endemic equilibrium point.

In fact, If I = 0 then from equation (2), we deduce that E * = µ + β 2 + β 6 + β 8 β 1 − β 10 .

Writing the equations (3) and (4) to deduce that

it follows from equation (5) 

and from equation (1),

Then we get P * = (E * , I * , C * , H * , D * ) = (E * ,αH * ,βH * , H * ,γH * )

and E * = µ + β 2 + β 6 + β 8 β 1 − β 10 which is called Endemic equilibrium point.

If the parameters (β 1 , β 2 , ....., β 10 ) satisfy one of the following conditions :

(i) β 1 < β 10 ;

(ii) (β 2 + β 6 + β 8 + µ)α < β 5β + β 9 .

Then the model (covid-19) has a unique equilibrium :

The disease-free equilibrium (DFE) P 0 = ( B µ , 0, 0, 0, 0).

The equilibrium P * = (E * , I * , C * , H * , D * ) is called feasible if its components are positive.

Thanks to [Driessche et al 1]., the following result is straightforward.

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 P 0 .

The theorem 4.4 shows also that, R 0 is a threshold which can determine if the disease will be spread or not. Thus, reducing its value, is a means to mitigate or even eliminate the (covid-19) . It can be therefore important to determine among model parameters those who mostly influence its value.

then it is easy to verify that e, i, c, h and d satisfy the following system of differential equations :

the point p * = (e * , i * , c * , h * , d * ) = (0, 0, 0, 0) is an equilibrium point of the system (4.1).

The jacobian matrix of the system (4.1) is given by : 

In particular we deduce that

Now we recall some technic calculations of determinant of a matrix in the following form : 1 a 1,2 a 1,3 a 1,4 a 2,1 a 2,2 a 2,3 a 2,4 a 3,1 a 3,2 a 3,3 a 3,4 i.e.

In particular we have : Let

where a = β 1 − β 10 , α = β 2 + β 6 + β 8 + µ, β = β 2 + β 5 + µ and γ = β 4 + β 9 + µ (ii) Let β 10 < β 1 and 2β 8 aE * + αγ + β 8 β 9 > β 8 α then detJ p 0 < 0.

(iii) Under the conditions of (ii) we observe that a assymption of Li-Wang criterion is satisfied. Now, if β 10 < β 1 , we write J p 0 in the following form

we deduce that : 

Setting

and χ(λ) = λ 3 + a 1 λ 2 + a 2 λ + a 3 . If λ 1 , λ 1 and λ 3 are the zeros of χ(λ) = 0

Then we have

• λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 = a 2

• λ 1 λ 2 λ 3 = −a 3 . and Proposition 4.7

Let χ(λ) = λ 3 + a 1 λ 2 + a 2 λ + a 3 , so that χ is uniformly asymptotically stable (uas), it is necessary that it suffices that ∆ 1 = a 1 > 0, ∆ 2 = a 1 a 2 − a 3 > 0 and ∆ 3 = a 3 ∆ 2 > 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 : a 1 > 0, a 2 > 0, ..., a 3 > 0.

• As a 1 = β 9 β 8 γ u(av + 1) − av − β 8 γ ≤ 0 then we can not apply this above proposition for χ(λ).

Now, if we consider the discriminant of χ which is given by :

we observe that :

• 1 If ∆ χ > 0, 3 different real roots of the equation χ(λ) = 0.

• 2 If ∆ χ = 0, one double or triple root of the equation χ(λ) = 0.

• 3 If ∆ χ < 0, one real root and two complex roots of the equation χ(λ) = 0.

• 4 if ∆ χ > 0, then a necessary and sufficient condition for an equilibrium point to be locally asymptotically stable is a 1 > 0, a 3 > 0, a 1 a 2 − a 3 > 0.

• 5 if ∆ χ < 0, a 1 < 0, a 2 < 0, then all roots of χ(λ) = 0 satisfy the condition |arg(λ)| < π 2 .

• 6 if ∆ χ > 0, a 1 > 0, a 2 > 0, a 1 a 2 − a 3 = 0, then an equilibrium point is locally asymptotically stable.

• 7 A necessary condition for an equilibrium point to be locally asymptotically stable is a 3 > 0.

• 8 if the conditions ∆ χ < 0, a 1 > 0, a 2 > 0, a 1 a 2 − a 3 = 0 are satisfied, then an equilibrium point is not locally asymptotically stable.

We remark that • 4 , • 6 and • 4 are not satisfy by the coefficients of χ(λ). So we have to solve the cubic equation χ(λ) = 0 by the Cardan's method which is ingenious and effective, but quite non-intuitive. Let P the general cubic equation :

Then P has solutions :

The expression ∆ = Q 3 + R 2 is called the discriminant of the equation

See for example Nickalls in [Nickalls] for a brief description of Cardan's method.

• Substantial technical difficulties for explicit expression of J

In Appendix of [Wang-Li] , we found that for n = 2, 3, and 4, an explicit expression of second additive compound matrices A [2] of n×n matrices A = (a ij ) 1≤i,j≤n which are given respectively by : in the same way that the section 5 of [Wang-Li] where Li and Wang studied the stability of an epidemic model of SEIR type, we apply their criterion to the following epidemic model :

where (Λ, β 1 , (β 2 , µ, γ, d) are given parameters.

• Determination of equilibrium points of the system (* 1 ) and calculation of basic reproduction number R 0

Let E = (S, I 1 , I 2 ) then the Jacobian matrix of above system is :

Now, we consider the following equations :

then we observe that E 0 = ( Λ µ , 0, 0) is a trivial equilibrium point of (* 3 ) (Disease free equilibrium point) and so

By using the next generation matrix method, the basic reproduction number R 0 is obtained as the spectral radius of matrix (−FV −1 ) at disease free equilibrium point where F and V are as below :

It follows that :

Evolution of R 0 with respect µ where β 1 = 0, 3, β 2 = 0, 8, γ = 0, 1 , Λ = 0, 7 and d = 0, 04

Now let I 1 = 0 then from (3) we deduce that :

From (* 7 ) and (2) we deduce that :

Now from (1) + (2), we deduce that :

Let E * = (S * , I * 1 , I * 2 ) where S * = µ + γ β 1 + β 2 δ , I * 1 = Λ − µS * µ + d and I * 2 = δI * 1 where δ = γ µ + d .

• The Jacobian matrix at the endemic equilibrium point E * = (S * , I * 1 , I * 2 ) of the system (*) is :

• the second additive compound matrix associated to J E * is :

Proposition 4.9

• 1 β 2 < γ δ 2

• 2 (µ + γ)(µ + d)(β 1 + β 2 δ) Λ(β 1 + β 2 δ) − µ(µ + γ) + β 1 (µ + γ) β 1 + β 2 δ < d + γ + 2µ

• 3 β 2 δΛ + µ(γ + µ) < β 1 Λ.

then the endemic equilibrium point of (*) is asymptotically stable. E * is similar to matrix A = PJ

[2] E * P −1 = (a ij ) 1≤i,j≤3 which is given by :

Under the conditions • 1 and • 2 , we observe that the diagonal elements of A are negative and

(1) a 11 + | a 12 | + | a 13 |< 0

(2) a 22 + | a 21 | + | a 23 |< 0

(1) a 33 + | a 32 |< 0

i.e A is diagonally dominant in rows.

In order to apply the corollary of the Li-Wang criterion, it remains to calculate the determinant of J E * detJ E * = −β 1 I * 1 − β 2 I * 2 − µ −β 1 S * −β 2 S *

Under condition • 3 we deduce that detJ E * < 0 In next lemma, we give the explicit entries of second additive compound matrix of n × n matrix A = (a ij ) where n = 5

Lemma 4.10

For n = 5, an explicit expression of second additive compound matrix A [2] is given by :

• As a 13 = 0, a 23 = 0, a 24 = 0, a 25 = 0, a 31 = 0, a 35 = 0, a 41 = 0, a 45 = 0 and a 54 = 0 in J [2] p * , we deduce that the explicit expression of second additive compound matrix J

[2] p * where P * = ( B µ , 0, 0, 0, 0) is :

where a = β 1 − β 10 , α = β 2 + β 6 + β 8 + µ, β = β 2 + β 5 + µ and γ = β 4 + β 9 + µ. (i) If β 10 < β 1 and 2β 8 aE * + αγ + β 8 β 9 > β 8 α then detJ p 0 < 0.

(ii) if we have :

p 0 is diagonally dominant in columns.

(iii) the equilibrium point of (4.1) is asymptotically stable.

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. Based on compound matrices theory, we have constructed in [Intissar] the controllers : i.e. U = (u ij ) where u ij = 0 except (u 12 , u 21 , u 51 ) ∈ R 3 ; 1 ≤ i, j ≤ 5 to stabilize the system (4.1), in particular to study the stability of following matrix : J p * ,u1,u2,u3 = J p * + U and its second additive compound matrix (J p * + U) [2] , by applying again the criterion of Li-Wang on second compound matrix associated to the system (4.1) with these controllers.

We have constructed a Lyapunov function L of the system (4.1) for apply the classical Lyapunov theorem and to get : (ii) V(e, i, c, h, d) < 0, 0 <|| (e, i, c, h, d) ||< r 1 for some r 1 , i.e. if L is lnd.

(iii) (0, 0, 0, 0, 0) is an asymptotically stable equilibrium point.

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