The endemic equilibrium is given by:
E∗ =S∗1, S∗2, E∗1, E2∗, I1∗, I2∗, D∗1, D2∗, M1∗, M2∗
To obtain the endemic equilibrium points, we solve equation (6.2.10) for M2 to obtain M∗2, we solve equation (6.2.9) for M1 to obtain M1∗. We solve
(6.2.10)⇐⇒ dS1 dt =0,
=⇒ M2 = M2∗, (6.2.9)⇐⇒ dM1
dt =0,
=⇒ M1 = M1∗, (6.2.7)⇐⇒ dD1
dt =0,
=⇒ D1= D∗1, (6.2.8)⇐⇒ dD2
dt =0,
=⇒ D2= D∗2, (6.2.5)⇐⇒ dI1
dt =0,
=⇒ E1 =E1∗, (6.2.6)⇐⇒ dI2
dt =0,
=⇒ E2 =E2∗, (6.2.3)⇐⇒ dE1
dt =0,
=⇒S1 =S∗1, (6.2.4)⇐⇒ dE2
dt =0,
=⇒S2 =S∗2, (6.2.2)⇐⇒ dS2
dt =0,
=⇒ I2 = I2∗.
I1∗ is a positive solution of the polynomial (6.4.2). By replacing all the state variables in equation (6.2.1) by their expressions at equilibrium from system (6.4.1), we obtain,
P(I1∗) = a0I1∗2+a1I1∗+a2 (6.4.2)
where Theorem 6.4.1. Polynomial(6.4.2) admits:
(i) one positive root if R2 >1,
(ii) zero or two positive roots if R2<1.
Proof. Descartes’ Law of signs helps to determine the number of positive roots of polynomial (6.4.2). We count the number of sign changes of the co-efficients of the polynomial and the value obtained is the maximum number of positive roots of the polynomial. The coefficient a0is always positive. The coefficient a2is positive when R2 <1 and negative when R2 >1. The maxi-mum number of sign changes of the coefficients of the polynomial (6.4.2) is 2 if a1 < 0 and zero if a1 > 0 when R2 < 1. When R2 > 1, the maximum number of sign changes of the polynomial (6.4.2) is one.
From equations in (6.4.1), we can say that R2 > 1 guarantees the exis-tence of a unique endemic equilibrium point in patch 2 so that I1∗ and I2∗ co-exist in this case. When R2 <1 there is no endemic equilibrium point in patch 2 whereas up to two positive values of I1∗ can exist. In this case, con-trolling EVD in patch 1 is then more complex as multiple endemic equilibria could exist. So, the existence of positive values of I1∗relies more on values of R2and not on values of R1 as we could have expected. In fact, movements of exposed individuals from patch 2 to patch 1 continuously feed patch 1 with exposed individuals that later become infectious, so that irrespective of the value of R1, the number of infected individuals in patch 1 is positive.
So, whether there are secondary infections in patch 1 or not, movements of exposed individuals into this patch guarantees always the existence of an endemic equilibrium. EVD control in patch 1 should then first target im-migration’s control and its consequences We notice that in the absence of movements of exposed individuals (σ12 =0), polynomial (6.4.2) becomes
P(I1∗) = b0I1∗+b1 I1∗ and admits a unique positive root with
b0= R1R2
S10S02 ρ1ρ2Q21Q22Q23Q4, b1=ψ2Q1Q2Q3µ ρ1Rb−1, Rb = Q2ψ1Q4ρ2ψ2Λ1+ (1−m)θ12
ψ2Q1Q2Q3µ ρ1
.
In fact, movements of exposed individuals from patch 2 to patch 1 contin-uously feed patch 1 with exposed individuals that later become infectious, so that irrespective of the value of R1, the number of infected individuals in patch 1 is positive. So, whether there are secondary infections in patch 1 or not, movements of exposed individuals into this patch guarantees always the existence of an endemic equilibrium. EVD control in patch 1 should then first target immigration’s control and its consequences.
Theorem 6.4.2. The endemic equilibrium E∗ exists when R2 > 1 and is locally asymptotically stable.
Proof. The jacobian matrix JE∗ of system (6.2.1)-(6.2.10) at the endemic equi-librium is given by
J
E∗=
"
U V 0
6×20
4×4W A
#
where
U =
From equations (6.2.9) and (6.2.10), we obtain at equilibrium M∗1 =π1
Using equations (6.4.3) and (6.4.4) to simplify ψ7and ψ9yields ψ7 = −r1M1∗ <0 and ψ9 = −r2M2∗ <0.
Since JE∗ is a block triangular matrix, its diagonal entries are its eigenvalues.
Since all the diagonal entries of the matrix JE∗ are negative, we can conclude that all the eigenvalues of JE∗ are negative and the endemic equilibrium E∗ is locally asymptotically stable.
6.5 Bifurcation analysis
A change of the topological structure of a system is called bifurcation [126].
A bifurcation driven by R2is illustrated in Figure6.4.
0.2 0.4 0.6 0.8 1 1.2 1.4
Infected population size (I1)
Stable DFE
Infected population size (I1) 10-3
Figures6.4(a) and 6.4(b) illustrate the possible number of positive roots of the polynomial (6.4.2) when R2 is varied. A backward bifurcation is ob-served in Figure6.4(a) when R2 < 1 and a forward bifurcation is observed in Figure 6.4(b) when R2 > 1. The complexity of the system of equations (6.2.1)-(6.2.10) makes it difficult to prove the existence of The backward bi-furcation but we can observe it thanks to numerical simulations.
The backward bifurcation shows a locally stable DFE and an unstable EE.
In any disease control, reaching a globally stable DFE is the main target, but
this is more difficult to realise in the case of a backward bifurcation since the DFE is globally stable only below a threshold Rt =1− a
2 1
4 a0ψ0, whose value might be far from 1. The expression of Rt is obtained by setting to zero the discriminant of the polynomial (6.4.2). The forward bifurcation represented in Figure 6.4(b) offers an easier possibility because reducing R2 to values less than one is enough in this case to reach a globally stable DFE and is then the preferable scenario for EVD control.
The income-ratio describes the gap between two economies and this gap is considered as the main driving factor of migration in this work. We ob-serve in Figure6.4that a backward bifurcation is changed into a forward bi-furcation for the chosen parameter values when the value of m is increased.
Increasing the value of m corresponds to bringing closer the two economies considered and this can be done by increasing the per capita income in patch 2 for example. This increase of income certainly improves the living condition of people in this patch and reduces their chances of migrating.
EVD control becomes easier in this case as exposed individuals can be eas-ily tracked and isolated in both patches. Improving the economy of poor countries exposed to EVD in order to stop or limit the migration of individ-uals exposed to the disease is a useful measure to be implemented by all governments, especially governments of rich countries because migration of infected individuals might cause an EVD outbreak on their sol.