The endemic equilibria E 1 ∗ and E ∗ 2

We now look at the two endemic equilibria corresponding to the cases when b=0 and b=bmax.

3.4.3.1 The endemic equilibrium E^∗₁

When b = 0, the endemic equilibrium for the EVD model in terms of I^∗ is given by E^∗₁ = ^S^∗, I^∗, H^∗, D^∗, 0

D^∗ = (Q2δ2+µ0δ2)I^∗

ρ Q2 with R⁰_h =S⁰β c

(ρ+η2δ1)Q2+ (ρ η1+η2δ2)µ0

ρ Q2(Q₁+µ0)

 . At equilibrium 

S˙(t) = 0, ˙I(t) = 0, ˙H(t) = 0, ˙D(t) = 0

, we solve equa-tion (3.2.9) for H to obtain H^∗. Then we set H= H^∗in equation (3.2.10) and solve for D to get D^∗. We set D = D^∗ in equation (3.2.7) and solve for S to get S^∗ and finally we set S= S^∗ in equation (3.2.8) and solve for I to obtain I^∗. We replace the obtained value of I^∗ in (3.2.7) and simplify to obtain the final results.

At the endemic equilibrium,

(3.2.9) ⇐⇒ H^˙(t) = 0=⇒ H = H^∗, (3.2.10) ⇐⇒ D^˙(t) =0=⇒ D =D^∗,

(3.2.7) ⇐⇒S^˙(t) =0 =⇒S =S^∗, (3.2.8) ⇐⇒ ˙I(t) =₀=⇒ I = I^∗.

I^∗ > 0 when R⁰_h > 1. In this case, the existence of E^∗₁ is driven by R⁰_h which is considered as the reproduction number at minimum hospitalisa-tion rate µ0. In the absence of beds for hospitalisation of EVD patients, the existence of endemic equilibrium points depends on the reproduction num-ber at minimum hospitalisation rate. This is reasonable since admission into ETU is minimal or even stopped when there are no new beds supplied to ETU.

Theorem 3.4.2. The endemic equilibrium E₁^∗ is globally asymptotically stable for R_h >1.

The global stability of the endemic equilibrium, stated by Theo-rem 3.4.2, is proven using LaSalle’s invariance principle, following [116,117,118,119,120].

Proof. We first set the Lyapunov function as L =^S−S^∗−S^∗ln( ^S

S^∗)^+A

I−I^∗−I^∗ln( ^I

I^∗)^+B

H−H^∗−H^∗ln( ^H H^∗)^ +G

D−D^∗−D^∗ln( ^D D^∗)^,

where A, B and G are positive constants to be determined. At the en-demic point E^∗ = (S^∗, I^∗, H^∗, D^∗), we have L(E^∗) = 0. Besides, the partial derivatives of L with respect to each state variable are

∂ L So the endemic state E^∗is a critical point of L and the second derivative is

∂²L The second derivative of L being positive at any point ofΩ, the Lyapunov function L is concave up and the unique endemic equilibrium point is a minimum of F. Let us prove that ˙L ≤ 0. The derivative of L with respect to time t, denoted ˙L is

˙L =

First we take into account the following inequality

γ(b(t)_{, I}(t)) ≥ _{µ0 for all} t≥ _{0 (3.4.3)} and particularly γ(b(t), I(t)) = µ0 in this case since we consider b = 0 at equilibrium. Considering inequality (3.4.3) and the system of equations (3.2.7)-(3.2.11), we obtain At equilibrium, the system of equations (3.2.7)-(3.2.11) yields

Λ =^c β(I^∗+η1H^∗+η2D^∗) +µ

We set

Replacing expressions from the system (3.4.5) into inequality (3.4.4) yields

˙L ≤ −_µ(S−S^∗)²

Expanding the expression of F(x, y, z, u)from system (3.4.6) and group-ing the coefficients of the same variable give

F =^{B µ0}

0 from the expression of ˙L, it remains to prove that F ≤ 0. From the expres-sion of F in (3.4.7), we will first set the terms containing variables and with non negative coefficients to zero in order to get rid of the positive and non constant part of F. The coefficients of y, z, u, x y and u x are thus set to zero

Then Apply the arithmetic mean-geometric mean inequality [117]

a1+a2+. . .+an ≥n√ⁿ

according to the arithmetic mean-geometric mean inequality.

p₁p2p3 =1 and p₁+p2+p3≥3.

Similarly, we prove that other terms composing F(x, y, z, u) in (3.4.8) are negative.

F is then negative and will be equal to zero if x =y=z =u=_{1. So L is} positive definite at the endemic equilibrium and ˙L ≤0 with equality in the set

C =ⁿ(S, I, H, D) : S=S^∗, I= I^∗, H = H^∗, D= D^∗o . The singleton E₁^∗ = ^S^∗, I^∗, H^∗, D^∗, 0

is the unique endemic equilibrium point considered in this case. So any solution which intersectsR⁵+ limits to the endemic equilibrium E^∗₁. By LaSalle’s invariance principle [116], E^∗₁ is globally asymptotically stable onΩ.

3.4.3.2 The endemic equilibrium E^∗₂ equation (3.2.7) at equilibrium. We obtain the following polynomial:

a0(I^∗)³+a₁(I^∗)²+a2I^∗+a3=0 (3.4.10)

Descartes’ law of signs is used to determine the possible number of positive roots of the polynomial (3.4.10) that are summarised in Table 3.1.

a₀ >0 a₁ >0

a2 >0 a2<0 a₃>0

(R_h<1)

a₃<0 (R_h>1)

a₃>0 (R_h <1)

a₃ <0 (R_h >1)

I^∗ 0 1 2 1

Table 3.1: Number of possible endemic fixed points.

Since bmax ≥ 1, the coefficient a1 will always be positive and the pos-sibility of having 3 positive roots for polynomial (3.4.10) is completely dis-carded. Conditions of existence of endemic fixed points of system (3.2.7)-(3.2.11) are summarized in Theorem 3.4.3.

Theorem 3.4.3.

• If all the coefficients of the polynomial (3.4.10) are positive, then system (3.2.7)-(3.2.11) has no endemic steady state.

• If R_h > 1 then the system (3.2.7)-(3.2.11) has a unique endemic equilibrium point.

• If R_h < 1 then the system (3.2.7)-(3.2.11) has two endemic equilibrium points.

Although polynomial (3.4.10) is of order 3, it admits at most two positive roots. So, multiple endemic equilibria exist and these equilibrium points are locally stable. The local stability of E^∗₂ is proven using a theorem based on the center manifold theory [121]. In this case, the threshold value of the reproduction number, R^c_h, below which the DFE is globally stable is a critical point of the polynomial (3.4.10). Besides, the discriminant∆ of the polynomial (3.4.10) is equal to zero at this point. We have

∆ =ξ1a²₃+ξ2a3+ξ3

and

ξ1a²₃+ξ2a3+ξ3 =0 (3.4.11) with ξ1 = −27 a²₀, ξ2=18 a0a1a2−4 a³₁and ξ3 =a²₁a²₂−4 a0a³₂.

Finding the positive roots (a3) of the polynomial in (3.4.11) and setting R^c_h=1−^{ a3}

µ ρ Q2b²_max(Q1+µ1)



will give the analytical expression of a threshold value for the reproduction number. If polynomial in (3.4.11) has two positive roots (∆ > 0), then the largest should be considered.

The following theorem is stated and proved:

Theorem 3.4.4. E⁰₂ is locally asymptotically stable when A > 0 and there exists a positive unstable endemic equilibrium E₂^∗ for R^c_h < R_h < 1. E₂⁰ is globally asymptotically stable when R_h < R^c_h < _{1 and} A > _{0. When} A < 0, E₂⁰ becomes unstable and a negative unstable endemic equilibrium E^∗₂ becomes positive and locally asymptotically stable for R_h > 1. A = 2 ρ Q2

Proof. We set φ =β cas our bifurcation parameter, so that for R_h =1, φ=φ^∗ = ^ρ(Q1+µ1) Q2

S⁰(ρ η₁+δ₂η₂)µ₁+ (δ₁η₂+ρ)Q₂. In order to linearise system (3.2.7)-(3.2.11), we set

S=x1, I =x2, H =x3, D =x4, b=x5

and

S˙ = f1, ˙I = f2, ˙H = f3, ˙D = f4, ˙b= f5.

The Jacobian matrixJ of the linearised system (3.2.7)-(3.2.11) at the DFE E⁰₂and for φ=φ^∗ is given by

Zero is a simple eigenvalue ofJ. The left eigenvector ofJ,

V = (v₁, v₂, v₃, v₄, v₅) and the right eigenvector W = (w₁, w₂, w₃, w₄, w₅)⁰, both associated to the eigenvalue zero are solutions of the system



and we obtain after some tedious algebraic manipulations w_j. So Remark 4 in [121] is verified. Two constantsAand B are defined in [121] as We compute these constants and find

A = 2 ρ Q₂

The direction of the bifurcation is determined by the signs ofAand B. Ob-viouslyB >0 and the sign ofAdetermines the following:











if A >0 then the bifurcation is backward, or

if A <0 then the bifurcation is forward.