EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS
Vol. 15, No. 4, 2022, 2054-2073
ISSN 1307-5543 – ejpam.com
Published by New York Business Global

Mathematical Analysis and Simulation of an
Age-Structured Model with Two-Patch and an

Uncontrolled Migration: Application to Tuberculosis

Badjo Kimba Abdoul Wahid1,∗, Djibo Moustapha1, Saley Bisso2

1 Department of Applied Mathematics and Computer Science,
Faculty of Sciences and Techniques, Dosso University, Dosso, Niger
2 Department of Mathematics and Computer Science, Faculty of Sciences and Techniques,
Abdou Moumouni University, Niamey, Niger

Abstract. In this paper, we studied a two-patch age-structured model of tuberculosis in a context
where the migration is not controlled. Motivated by the fact that no author has highlighted
the impact of migration on the dynamics of transmission of tuberculosis. Each subpopulation is
subdivided into five compartments: susceptible; latent, vaccinated, infective and treated. After the
determination of the reproductive numbers ℜ(ψ, ρ) and ℜ0(ρ), we established the conditions of the
global and local stability of the equilibrium point without disease. It has been shown that there is
only one point of endemic equilibrium. Numerical simulations show that uncontrolled migration
negatively influences the dynamics of tuberculosis.

2020 Mathematics Subject Classifications: 92D25, 92D30, 65P40, 93A10

Key Words and Phrases: Age-structured, reproductive number, two-patch, tuberculosis, sta-
bility, migration, simulation, uncontrolled, endemic

1. Introduction

Tuberculosis (TB) is one of the oldest diseases humanity has known, which is still
relevant today; archaeological excavations in Pharaonic Egypt, ancient India and the Far
East show that the date of its first “official” recognition was 2400 BC [8, 10, 16]. Despite
enormous scientific advances and the availability of effective treatment, TB remains one
of the top five killers worldwide. Understanding its propagation dynamics has become a
concern for the entire scientific community. It is an infection caused by a bacterium called
Mycobacterium tuberculosis. Tuberculosis most commonly affects the lungs, but can also
affect other parts of the body, such as the skin, bones, lymph nodes, liver, digestive tract,
and central nervous system [11]. Tuberculosis is transmitted from one person with active

∗Corresponding author.
DOI: https://doi.org/10.29020/nybg.ejpam.v15i4.4556

Email addresses: abadjokimba@yahoo.fr (B. K. Abdoul Wahid),
moustaphad530@gmail.com (D. Moustapha), bsaley@yahoo.fr (S. Bisso)

https://www.ejpam.com 2054 © 2022 EJPAM All rights reserved.



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2055

pulmonary tuberculosis to another when the latter is exposed to M. tuberculosis. Living
conditions that contribute to a high risk of exposure to TB include overcrowded hous-
ing, homeless shelters and correctional facilities. According to WHO reports, the People
at increased risk of exposure to tuberculosis include those with a history of alcohol and
drug abuse and those from areas where TB is prevalent, including many countries in the
Caribbean, Africa and Asia. Until the emergence of the COVID-19 pandemic (widely
studied, among recent works we can cite [1, 17]), tuberculosis was the leading cause of
death attributable to a single infectious agent, ahead of HIV/AIDS. According to the
2021 WHO report, most people (nearly 90%) who develop the disease are adults and
more often men than women. Nearly a quarter of the world’s population is infected with
M.tuberculosis. Despite the fact tuberculosis is a preventable disease that can be cured.
Several mathematical models have been developed in order to make contributions in the
decision making of the strategies to carry out the control of this disease. We must, the
first mathematical model dealing with the case of tuberculosis to Frost, He predicted that
with the low and falling rate of transmission of tuberculous infection in the United States,
it was most likely that the tubercle bacilli were fighting a losing battle, since they would
be unable to transmit sufficient infection to maintain the balance in their own favor. (See
[21, 22]. Nowadays, we have a fairly broad literature on the mathematical epidemiology of
tuberculosis (We refer readers to the documents [2, 4, 18] and articles cited in these papers
for a good overview of the work done in this field as the list is by no means exhaustive),
but this work neglects a number of factors (either all or some of its parameters) such as the
vaccination program, the compartment of the treaties, individuals with rapid progression
to the state of the disease, the mortality rate related to the disease, the birth rate of the
population in addition, very few of these models are structured in age and the migration is
considered in very few works. On this subject Tewa J.J has dedicated two articles modeled
by an EDO [13, 14]. These factors are far from negligible in African countries. Recently
Marwin Ramli and al, studied the stability of the endemic equilibrium of a SIR model of
tuberculosis where the compartment of the susceptible is subdivided into two subgroups,
that of susceptible vaccinated and those of susceptible unvaccinated, unstructured by age
[12],they made an extinction of this model by adding the latent compartment[20]; Rui Xu
and al Motivated by the work of Yang and al [24] and Mc Cluskey [5], in their paper, they
investigated the effects of incomplete treatment and age structure in latently infected and
infectious individuals the dynamics of tuberculosis [23].Yu Zhao and al. analyzed a model
type SEIR where the compartment of susceptible is subdivided into three age groups:
childhood, middle-age and senior. Our model is an extension of the model proposed in
[19], by the fact that migration is allowed without distinction of the epidemiological status
of individuals.

This type of migration is undoubtedly one of the factors, delaying the process of
eradicating this disease. To evaluate the impact of migration on the propagation dynamics
of this disease, we expressed the reproduction numbers as a function of the migration rate
ℜ(ψ, ρ) and ℜ0(ρ). After establishing the conditions of existence and uniqueness of the
positive solutions of our model by the semi-group method. We studied the global and



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2056

local stability of the equilibrium point without disease, showed the existence of an endemic
equilibrium point and finally the results of the numerical simulations confirm the negative
effects of the uncontrolled migration that would cause tuberculosis in our community.This
paper is organized as follows: Section 1 introduces the two-patch model structured in
age to study the dynamics of TB transmission case of a non controlled migration .The
mathematical model formulation is given in the section 2. The existence of positive and
unique solutions is demonstrated in Section 3. The point of equilibrium without disease,
reproductive numbers ℜ(ψ, ρ) and ℜ0(ρ) are defined in the section 4 and the local stability
of the disease-free equilibrium point in the presence of vaccine. The global stability of the
disease-free equilibrium point to the absence of the vaccine is established in Section 5. The
existence of an endemic equilibrium is proven in Section 6. Some numerical simulation
results are given in Section 7. In Section 8, we have a discussion, conclusion and further
work.

2. Mathematical model formulation

Two-patch age structured model of tuberculosis was considered. The model is to split
the population into two subpopulations. The recruitment is only possible in the class of
susceptible and the vaccinated individuals were able to migrate between the two subpop-
ulations. Each subpopulation is divided into five classes based on their epidemiological
status: susceptible, vaccinated, latent, infectious or treated. We denote these subgroups
Si(t, a), vi(t, a), Li(t, a), Ii(t, a) and Ji(t, a) respectively. The birth rate of the patch i is
bi(a); µi(a) and µ(a) denote the mortality rate related to the disease relative to the patch
i and the rate of natural mortality. The time and age depended of the force of infection of
the subpopulation i is λi(t, a) and vaccination rate is ψi(a); pi(a, a

′) is the probability that
an infective individual of age a′ will have contact with and successfully infect a susceptible
individual of age a, ci(a) is the age-specic per-capita contact/activity rate (all of these
functions are assumed to be continuous and to be zero beyond some maximum age). A
fraction ϕi of newly infected individuals of the sub-population i is assumed to undergo a
fast progression directly to the infectious class Ii. Rate of susceptible passage to latent
infectious state and treatment are respectively ki and ri. σi and δi are the reductions
in risk due to prior exposure to TB and vaccination, respectively , 0 ≤ σi ≤ (1 − ϕi),
0 ≤ δi ≤ (1 − ϕi). The migration rates of the susceptible, latently infected, infectious,
treated, and vaccinated between the two populations are respectively ρi1, ρi2, ρi3, ρi4 and
ρi5, in this paper i = 1, 2.



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2057

fig.1. Flow chart of the two-patch model for tuberculosis disease
transmission.

The age-structured model for the transmission of TB is described by the following
system of partial differential equations:


(
∂
∂t

+ ∂
∂a

)
S1(t, a) = b1(a)N1(t, a) − [λ1(t, a) + ψ1(a) + µ(a) + ρ11]S1(t, a) + ρ21S2(t, a)(

∂
∂t

+ ∂
∂a

)
L1(t, a) = λ1(t, a)[(1 − ϕ1)S1(t, a) + σ1J1(t, a) + δ1V1(t, a)] − (k1 + µ(a) + ρ12)L1(t, a) + ρ22L2(t, a)(

∂
∂t

+ ∂
∂a

)
I1(t, a) = k1L1(t, a) − (r1 + µ(a) + µ1(a) + ρ13)I1(t, a) + ϕ1λ1(t, a)S1(t, a) + ρ23I2(t, a)(

∂
∂t

+ ∂
∂a

)
J1(t, a) = r1I1(t, a) − (σ1λ1(t, a) + µ(a) + ρ14)J1(t, a) + ρ24J2(t, a)(

∂
∂t

+ ∂
∂a

)
V1(t, a) = ψ1(a)S1(t, a) + ρ25V2(t, a) − (ρ15 + µ(a) + δ1λ1(t, a))V1(t, a)(

∂
∂t

+ ∂
∂a

)
S2(t, a) = b2(a)N2(t, a) − [λ2(t, a) + ψ2(a) + µ(a) + ρ21]S2(t, a) + ρ11S1(t, a)(

∂
∂t

+ ∂
∂a

)
L2(t, a) = λ2(t, a)[(1 − ϕ2)S2(t, a) + σ2J2(t, a) + δ2V2(t, a)] − (k2 + µ(a) + ρ22)L2(t, a) + ρ12L1(t, a)(

∂
∂t

+ ∂
∂a

)
I2(t, a) = k2L2(t, a) − (r2 + µ(a) + µ2(a) + ρ23)I2(t, a) + ρ13I1(t, a) + ϕ2λ2(t, a)S2(t, a)(

∂
∂t

+ ∂
∂a

)
J2(t, a) = r2I2(t, a) − (σ2λ2(t, a) + µ(a) + ρ24)J2(t, a) + ρ14J2(t, a)(

∂
∂t

+ ∂
∂a

)
V2(t, a) = ψ2(a)S2(t, a) + ρ15V1(t, a) − (ρ25 + µ(a) + δ2λ2(t, a))V2(t, a)

(1)

with initial and boundary conditions :

Si(t, 0) =
∫ a2
a1
bi(a)Ni(t, a)da

Li(t, 0) = Vi(t, 0) = Ii(t, 0) = Ji(t, 0) = 0

Si(0, a) = S0i(a);Li(0, a) = L0i(a);Vi(0, a) = V0i(a)

Ii(0, a) = I0i(a); Ji(0, a) = J0i(a).



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2058

And λi(t, a) = βi(a)ci(a)
∫ a+
0

Ii(t,a
′)

Ni(t,a′)
pi(a, a

′)da′, assume that

pi(a, a
′) = gi(a)β̂i(a

′) (2)

(see Dietz and Schenzle 1985 [9] ; Greenhalgh 1988 [7]).
Let N(t, a) = N1(t, a) +N2(t, a) and Ni(t, a) = αiN(t, a), with α1 + α2 = 1
N(t, a) = S1(t, a) + L1(t, a) + I1(t, a) + J1(t, a) + V1(t, a) + S2(t, a) + L2(t, a) + I2(t, a) +
J2(t, a) + V2(t, a).
By summing equations of system (1) and (2), we obtain the following equations for the
total population N(t, a):

(
∂
∂t +

∂
∂a

)
N(t, a) = (b(a)− µ(a))N(t, a)− µ1(a)I1(t, a)− µ2(a)I2(t, a)

N(t, 0) =
∫ a2
a1
b(a)N(t, a)da.

(3)

Where b(a) = α1b1(a)+α2b2(a); a1 and a2 are respectively the minimum and maximum
age of procreation and a+ is the maximum age of an individual, with a+ < +∞.
Let 

si(t, a) =
Si(t,a)
N(t,a) ; li(t, a) =

Li(t,a)
N(t,a) ; ii(t, a) =

Ii(t,a)
N(t,a)

ji(t, a) =
Ji(t,a)
N(t,a) ; vi(t, a) =

Vi(t,a)
N(t,a) .

(4)

The system (1) can be normalized as the following system:



(
∂
∂t

+ ∂
∂a

)
s1(t, a) = α1b1(a) − [λ1(t, a) + ψ1(a) + b(a) − µ1(a)i1(t, a) − µ2(a)i2(t, a) + ρ11]s1(t, a) + ρ21s2(t, a)(

∂
∂t

+ ∂
∂a

)
l1(t, a) = λ1(t, a)[(1 − ϕ1)s1(t, a) + σ1j1(t, a) + δ1v1(t, a)]

−(k1 + b(a) − µ1(a)i1(t, a) − µ2(a)i2(t, a) + ρ12)l1(t, a) + ρ22l2(t, a)(
∂
∂t

+ ∂
∂a

)
i1(t, a) = −(r1 + µ1(a) + b(a) − µ1(a)i1(t, a) − µ2(a)i2(t, a) + ρ13)i1(t, a) + ρ23i2(t, a)

+ϕ1λ1(t, a)s1(t, a) + k1l1(t, a)(
∂
∂t

+ ∂
∂a

)
j1(t, a) = r1i1(t, a) − (σ1λ1(t, a) + b(a) − µ1(a)i1(t, a) − µ2(a)i2(t, a) + ρ14)j1(t, a) + ρ24j2(t, a)(

∂
∂t

+ ∂
∂a

)
v1(t, a) = ψ1(a)s1(t, a) + ρ25v2(t, a) − (b(a) + ρ15 + δ1λ1(t, a) − µ1(a)i1(t, a) − µ2(a)i2(t, a))v1(t, a)(

∂
∂t

+ ∂
∂a

)
s2(t, a) = α2b2(a) − [λ2(t, a) + ψ2(a) + b(a) − µ1(a)i1(t, a) − µ2(a)i2(t, a) + ρ21]s2(t, a) + ρ11s1(t, a)(

∂
∂t

+ ∂
∂a

)
l2(t, a) = λ2(t, a)[(1 − ϕ2)s2(t, a) + σ2j2(t, a) + δ2v2(t, a))

−(k2 + b(a) − µ1(a)i1(t, a) − µ2(a)i2(t, a) + ρ22)l2(t, a) + ρ12l1(t, a)(
∂
∂t

+ ∂
∂a

)
i2(t, a) = −(r2 + µ2(a) + b(a) − µ1(a)i1(t, a) − µ2(a)i2(t, a) + ρ23)i2(t, a) + ρ13i1(t, a)

+ϕ2λ2(t, a)s2(t, a) + k2l2(t, a)(
∂
∂t

+ ∂
∂a

)
j2(t, a) = r2i2(t, a) − (σ2λ2(t, a) + b(a) − µ1(a)i1(t, a) − µ2(a)i2(t, a) + ρ24)j2(t, a) + ρ14j1(t, a)(

∂
∂t

+ ∂
∂a

)
v2(t, a) = ψ2(a)s2(t, a) + ρ15v1(t, a) − (b(a) + ρ25 + δ2λ2(t, a) − µ1(a)i1(t, a) − µ2(a)i2(t, a))v2(t, a)

(5)

with boundary conditions

si(t, 0) = Λi; vi(t, 0) = li(t, 0) = ii(t, 0) = ji(t, 0) = 0

with Λ1 + Λ2 = 1.

The problem is well possed, the methode of proof is the same used in [3].



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2059

3. Existence of positive solutions

In this section we will prove that the system (5) has a unique positive solution, and
to achieve this we will write the system (5) in compact form (abstract Cauchy problem).

Consider the Banach space X defined by X = (L1(0, a+))
10, endowed with the norm

∥φ∥ =
2∑
i=1

5∑
j=1

∥φij∥ (6)

Where φ(a) = (φ11(a), φ12(a), φ13(a), φ14(a), φ15(a), φ21(a), φ22(a), φ23(a), φ24(a), φ25(a))
T ∈

X and ∥.∥ is the norm of L1(0, a+) . Let denote by

Ω = {(s1, l1, i1, j1, v1, s2, l2, i2, j2, v2) ∈ X+\0 ≤ s1 + l1 + i1 + j1, v1 + s2 + l2 + i2 + j2 + v2 ≤ 1} (7)

The state space of system (5). Where X+ = (L1
+(0, a+))

10, et L1
+(0, a+) denotes the

positive cone of L1(0, a+). Let A be a linear operator defined by

(Aφ)(a) = (A11, A12, A13, A14, A15, A21, A22, A23, A24, A25)
T (8)



A11 = (− d
daφ11 − (ψ1(a) + b(a) + ρ11)φ11, 0, 0, 0, 0, ρ21φ21, 0, 0, 0, 0)

A12 = (0,− d
daφ12 − (b(a) + k1 + ρ12)φ12, 0, 0, 0, 0, ρ22φ22, 0, 0, 0)

A13 = (0, k1φ12,− d
daφ13 − (r1 + µ1(a) + b(a) + ρ13)φ13, 0, 0, 0, 0, ρ23φ23, 0, 0)

A14 = (0, 0, r1φ13,− d
daφ14 − (b(a) + ρ14)φ14, 0, 0, 0, 0, ρ24φ24, 0)

A15 = (ψ1(a)φ11, 0, 0, 0,− d
daφ15 − (ρ15 + b(a))φ15, 0, 0, 0, 0, ρ25φ25)

A21 = (ρ11φ11, 0, 0, 0, 0,− d
daφ21 − (ψ2(a) + b(a) + ρ21)φ21, 0, 0, 0, 0)

A22 = (0, ρ12φ12, 0, 0, 0, 0,− d
daφ22 − (b(a) + k2 + ρ22)φ22, 0, 0, 0)

A23 = (0, 0, ρ13φ13, 0, 0, 0, k2φ22,− d
daφ23 − (r2 + µ2(a) + b(a) + ρ23)φ23, 0, 0)

A24 = (0, 0, 0, ρ14φ14, 0, 0, 0, r2φ23,− d
daφ24 − (b(a) + ρ24)φ24, 0)

A25 = (0, 0, 0, 0, ρ15φ15, ψ2(a)φ21, 0, 0, 0,− d
daφ25 − (ρ25 + b(a))φ25)

(9)

With

φ(a) = (φ11(a), φ12(a), φ13(a), φ14(a), φ15(a), φ21(a), φ22(a), φ23(a), φ24(a), φ25(a))
T ∈ D(A)



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2060

where D(A) is the domain given by:

D(A) =
{
φ ∈ X\φij ∈ AC[0, a+), φ(0) = (Λ1, 0, 0, 0, 0, 0,Λ2, 0, 0, 0, 0)

T
}

And AC[0, a+) denotes the set of absolutely continuous functions on [0, a+). We also
define a nonlinear operator F : X −→ X by :

(Fφ)(a) =



α1b1(a)− ((Q1φ13)(a))φ11 + (µ1(a)φ13 + µ2(a)φ23)φ11

((Q1φ13)(a))((1− ϕ1)φ11 + σ1φ14 + δ1φ15) + (µ1(a)φ13 + µ2(a)φ23)φ12

ϕ1((Q1φ13)(a))φ11 + (µ1(a)φ13 + µ2(a)φ23)φ13

(µ1(a)φ13 + µ2(a)φ23)− δ1((Q1ϕ13)(a))φ14

(µ1(a)φ13 + µ2(a)φ23)− σ1((Q1φ13)(a))φ15

α2b2(a)− ((Q2φ23)(a))φ21 + (µ1(a)φ13 + µ2(a)φ23)φ21

((Q2φ23)(a))((1− ϕ2)φ21 + σ2φ24 + δ2φ25) + (µ1(a)φ13 + µ2(a)φ23)φ22

ϕ2((Q2φ23)(a))φ21 + (µ1(a)φ13 + µ2(a)φ23)φ23

(µ1(a)φ13 + µ2(a)φ23)− δ2((Q2ϕ23)(a))φ24

(µ1(a)φ13 + µ2(a)φ23)− σ2((Q2φ23)(a))φ25



(10)

where Qi is a bounded linear operator on L1(0, a+) given by

(Qif)(a) = ci(a)βi(a)gi(a)

∫ a+

0
β̂i(a

′)f(a′)da′ (11)

Let

u(t) = (s1(., t), l1(., t), i1(., t), j1(., t), v1(., t), s2(., t), l2(., t), i2(., t), j2(., t), v2(., t))

Thus, we can rewrite the system as an abstract Cauchy problem{
d
dtu(t) = Au(t) + F (u(t))
u(0) = u0

(12)

where

u0(a) = (s01(a), l01(a), i01(a), j01(a), v01(a), s02(a), l02(a), i02(a), j02(a), v02(a))
T .

According to these results we have the following results, ( see [6, 10]).



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2061

Lemma 1. The operator F is continuously Fréchet differentiable on X.

Lemma 2. The operator A genrates a C0-semigroup of the bounded linear operators
{T (t)}t≥0 and the space Ω is positively invariant by {T (t)}t≥0.

Theorem 1. For each u0 ∈ X+ there are a maximal interval of existence [0, tmax) and a
unique continous mild solution for (12) such that

u(t) = u0e
tA +

∫ t

0
eA(t−ξ)F (u(ξ))dξ

4. The disease-free steady state

4.1. Determination of the point of disease-free equilibrum

A steady state (s1(a), l1(a), i1(a), j1(a), v1(a), s2(a), l2(a), i2(a), j2(a), v2(a)) of system
(5) must satisfy the following time-independent system of ordinary differential equations:



d
da
s1(a) = α1b1(a) − [β1(a)c1(a)g1(a)Γ1 + ψ1(a) + b(a) − µ1(a)i1(a) − µ2(a)i2(a) + ρ11]s1(a) + ρ21s2(a)

d
da
l1(a) = β1(a)c1(a)g1(a)Γ1[(1 − ϕ1)s1(a) + δ1v1(a) + σ1j1(a)]

−(b(a) + k1 − µ1(a)i1(a) − µ2(a)i2(a) + ρ12)l1(a) + ρ22l2(a)

d
da
i1(a) = k1l1(a) + ϕ1β1(a)c1(a)g1(a)Γ1s1(a) − [r1 + b(a) + µ1(a) − µ1(a)i1(a) − µ2(a)i2(a) + ρ13]i1(a) + ρ23i2(a)

d
da
j1(a) = r1i1(a) − (σ1β1(a)c1(a)g1(a)Γ1 + b(a) − µ1(a)i1(a) − µ2(a)i2(a) + ρ14)j1(a) + ρ24j2(a)

d
da
v1(a) = ψ1(a)s1(a) − (δ1β1(a)c1(a)g1(a)Γ1 + b(a) − µ1(a)i1(a) − µ2(a)i2(a) + ρ15)v1(a) + ρ25v2(a)

d
da
s2(a) = b2(a) − [β2(a)c2(a)g2(a)Γ2 + ψ2(a) + b(a) − µ1(a)i1(a) − µ2(a)i2(a) + ρ21]s2(a) + ρ11s1(a)

d
da
l2(a) = β2(a)c2(a)g2(a)Γ2[(1 − ϕ2)s2(a) + δ2v2(a) + σ2j2(a)] − (b(a) + k2 − µ1(a)i1(a) − µ2(a)i2(a) + ρ22)l2(a) + ρ12l1(a)

d
da
i2(a) = k2l2(a) + ϕ2β2(a)c2(a)g2(a)Γ2s2(a) − [r2 + b(a) + µ2(a) − µ1(a)i1(a) − µ2(a)i2(a) + ρ23]i2(a) + ρ13i2(a)

d
da
j2(a) = r2i2(a) − (σ2β2(a)c2(a)g2(a)Γ2 + b(a) − µ1(a)i1(a) − µ2(a)i2(a) + ρ24)j2(a) + ρ14j1(a)

d
da
v2(a) = ψ2(a)s2(a) + ρ15v1(a) − (δ2β2(a)c2(a)g2(a)Γ2 + b(a) − µ1(a)i1(a) − µ2(a)i2(a) + ρ25)v2(a)

Γi = 1
αi

∫ a+
0 β̂i(a)ii(a)da

(13)

with initial value conditions

si(0) = Λi; li(0) = ii(0) = ji(0) = vi(0) = 0

Therefore, we obtain the disease-free steady state
s0i (a) = Λie

−
∫ a
0 (b(τ)+ψi(τ)+ρi1)dτ +

∫ a
0 e

−
∫ a
η (b(τ)+ψi(τ)+ρi1)dτ (αibi(η) + ρj1s

0
j (a))dη

v0i (a) =
Λi−s0i (a)+ρj5v0j (a)

1−ρi5 ; l0i (a) = i0i (a) = j0i (a) = 0

(14)



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2062

4.2. Calculation of ℜ(ψ, ρ)-ℜ0(ρ) and stability of the infection-free state

To study the stability of the disease-free steady state, we denote the perturbations of
system by 

si(t, a) = si(t, a) + s0i (a)

li(t, a) = li(t, a) + l0i (a)

ii(t, a) = ii(t, a) + i0i (a)

ji(t, a) = ji(t, a) + j0i (a)

vi(t, a) = vi(t, a) + v0i (a)

(15)

The perturbations satisfy the following equations:



(
∂
∂t

+ ∂
∂a

)
s1(t, a) = ρ21s2(a) − (b(a) + ψ1(a) + ρ11)s1(t, a) − [γ1(t)β1(a)c1(a)g1(a) − µ1(a)i1(t, a) − µ2i2(t, a)]s

0
1(a)(

∂
∂t

+ ∂
∂a

)
l1(t, a) = ρ22l2(a) − (b(a) + k1 + ρ12)l1(t, a) + γ1(t)β1(a)c1(a)g1(a)[(1 − ϕ1)s

0
1(a) + δ1v

0
1(a)](

∂
∂t

+ ∂
∂a

)
i1(t, a) = ρ23i2(a) + k1l1(t, a) + ϕ1β1(a)c1(a)g1(a)γ1(t)s

0
1(a) − (r1 + µ1(a) + b(a) + ρ13)i1(t, a)(

∂
∂t

+ ∂
∂a

)
j1(t, a) = r1i1(t, a) + ρ24j2(a) − (ρ14 + b(a))j1(t, a)(

∂
∂t

+ ∂
∂a

)
v1(t, a) = ψ1(a)s1(t, a) + ρ25v2(t, a) − (ρ15 + b(a))v1(t, a)

−(δ1β1(a)c1(a)g1(a)γ(t) − µ1(a)i1(t, a) − µ2(a)i2(t, a))v
0
1(a)(

∂
∂t

+ ∂
∂a

)
s2(t, a) = ρ11s1(a) − (b(a) + ψ2(a) + ρ21)s2(t, a) − [γ2(t)β2(a)c2(a)g2(a) − µ1(a)i1(t, a) − µ2i2(t, a)]s

0
2(a)(

∂
∂t

+ ∂
∂a

)
l2(t, a) = ρ12l1(a) − (b(a) + k2 + ρ22)l2(t, a) + γ2(t)β2(a)c2(a)g2(a)[(1 − ϕ2)s

0
2(a) + δ2v

0
2(a)](

∂
∂t

+ ∂
∂a

)
i2(t, a) = ρ13i1(a) + k2l2(t, a) + ϕ2β2(a)c2(a)g2(a)γ2(t)s

0
2(a) − (r2 + µ2(a) + b(a) + ρ23)i2(t, a)(

∂
∂t

+ ∂
∂a

)
j2(t, a) = ρ14j1(a) + r2i2(t, a) − (ρ24 + b(a))j2(t, a)(

∂
∂t

+ ∂
∂a

)
v2(t, a) = ψ2(a)s2(t, a) + ρ15v1(t, a) − (ρ25 + b(a))v2(t, a)

−(δ2β2(a)c2(a)g2(a)γ2(t) − µ1(a)i1(t, a) − µ2(a)i2(t, a))v
0
2(a)

γi(t) = 1
αi

∫ a+
0 β̂i(a)ii(t, a)da

(16)

with boundary conditions :

si(t, 0) = li(t, 0) = ii(t, 0) = ji(t, 0) = vi(t, 0) = 0

we consider the exponential solutions of system (16) of the form:

 si(t, a) = si(a)e
λt; li(t, a) = li(a)e

λt; vi(t, a) = vi(a)e
λt

ii(t, a) = ii(a)e
λt; ji(t, a) = ji(a)e

λt
(17)



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2063

The system (16) becomes:



d
da
s1(a) = ρ21s2(a) − (b(a) + ψ1(a) + λ + ρ11)s1(a) − [Γ1β1(a)c1(a)g1(a) − µ1(a)i1(a) − µ2(a)i2(a)]s

0
1(a)

d
da
l1(a) = ρ22l2(a) − (b(a) + k1 + λ + ρ12)l1(a) + Γ1β1(a)c1(a)g1(a)[(1 − ϕ1)s

0
1(a) + δ1v

0
1(a)]

d
da
i1(a) = ρ23i2(a) + ϕ1β1(a)c1(a)g1(a)Γ1 + k1l1(a) − (r1 + µ1(a) + b(a) + λ + ρ13)i1(a)

d
da
j1(a) = ρ24j2(a) + r1i1(a) − (b(a) + λ + ρ14)j1(a)

d
da
v1(a) = ψ1(a)s(a) + ρ25v2(a) − (ρ15 + b(a) + λ)v1(a) − (δ1Γβ1(a)c1(a)g1(a) − µ1(a)i1(a) − µ2(a)i2(a))v

0
1(a)

d
da
s2(a) = ρ11s1(a) − (b(a) + ψ2(a) + λ + ρ21)s2(a) − [Γ2β2(a)c2(a)g2(a) − µ1(a)i1(a) − µ2(a)i2(a)]s

0
2(a)

d
da
l2(a) = ρ12l1(a) − (b(a) + k2 + λ + ρ22)l2(a) + Γ2β2(a)c2(a)g2(a)[(1 − ϕ2)s

0
2(a) + δ2v

0
2(a)]

d
da
i2(a) = ρ13i1(a) + ϕ2β2(a)c2(a)g2(a)Γ2 + k2l2(a) − (r2 + µ2(a) + b(a) + λ + ρ13)i2(a)

d
da
j2(a) = ρ14j1(a) + r2i2(a) − (b(a) + λ + ρ14)j2(a)

d
da
v2(a) = ψ2(a)s2(a) + ρ15v1(a) − (ρ25 + b(a) + λ)v2(a) − (δ2Γβ2(a)c2(a)g2(a) − µ1(a)i1(a) − µ2(a)i2(a))v

0
2(a)

Γi = 1
αi

∫ a+
0 β̂i(a)ii(a)da

(18)

with boundary conditions :

si(0) = li(0) = ii(0) = ji(0) = vi(0) = 0

Let

Nψi(a) = (1− ϕi)s
0
i (a) + δiv

0
i (a) (19)

ρj2lj(a) = ρ̂j2li(a) and ρj3ij(a) = ρ̂j3ii(a) (20)

From equation (18) and (19), we obtain:

li(a) = Γi

∫ a

0
e−

∫ a
η (b(τ)+ki+ρi2−ρ̂j2+λ)dτβi(η)ci(η)gi(η)Nψi(η)dη (21)

ii(a) =

∫ a

0
e−

∫ a
η (b(τ)+ri+λ+µi(a)+ρi3−ρ̂j3)dτ (kili(η) + ϕiΓiβi(η)ci(η)gi(η))dη (22)

Hence, by equations (21) and (22) after changing order of integration, we obtain:

ii(a) = Γi
∫ a
0 e

−
∫ a
η (b(τ)+µi(τ)+ri+λ+ρi3−ρ̂j3)dτβi(η)ci(η)gi(η)

[ϕis
0
i (η) + kiNψi(η)

∫ a

η
e−

∫ η
x (µi(τ)+ri−ki−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdx]dη (23)



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2064

Injecting (23) in the expression of Γi, and dividing both sides the expression by Γi
(since Γi ̸= 0), we get the characteristic equation:

1 = 1
αi

∫ a+
0 β̂i(a)

∫ a
0 e

−
∫ a
η (b(τ)+µi(τ)+ri+λ+ρi3−ρ̂j3)dτβi(η)ci(η)gi(η)

[ϕis
0
i (η) + kiNψi(η)

∫ a

η
e−

∫ η
x (µi(τ)+ri−ki−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdx]dηda (24)

Denote the right-hand side of equation (24) by Gij(λ) i.e :

Gij(λ) =
1
αi

∫ a+
0 β̂i(a)

∫ a
0 e

−
∫ a
η (b(τ)+µi(τ)+ri+λ+ρi3−ρ̂j3)dτβi(η)ci(η)gi(η)

[ϕis
0
i (η) + kiNψi(η)

∫ a

η
e−

∫ η
x (µi(τ)+ri−ki−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdx]dηda (25)

We define the net reproductive number as ℜi(ψ, ρij) = Gi(0), i.e

ℜi(ψi, ρij) = 1
αi

∫ a+
0 β̂i(a)

∫ a
0 e

−
∫ a
η (b(τ)+µi(τ)+ri+ρi3−ρ̂j3)dτβi(η)ci(η)gi(η)

[ϕis
0
i (η) + kiNψi(η)

∫ a

η
e−

∫ η
x (µi(τ)+ri−ki−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdx]dηda (26)

Where
ρij = (ρi2, ρ̂j2, ρi3, ρ̂j3)

We can obtain an expression for ℜi0(ψi, ρij) in a similar way as the derivation of
ℜi(ψ, ρij) by considering Equation (1) without vaccination; i.e., by assuming that ψi(a) ≡
0 and neglecting the equation of vaccinated. It can be shown that ℜi0(ρij) = ℜi(0, ρij)
which is called the basic reproductive number (when a purely susceptible population is
considered) (see [3]).

ℜi0(ρij) =
Λi
αi

∫ a+
0 β̂i(a)

∫ a
0 e

−
∫ a
η (b(τ)+µi(τ)+ri+ρi3−ρ̂j3)dτβi(η)ci(η)gi(η)

[ϕi + ki(1− ϕi)

∫ a

η
e−

∫ η
x (µi(τ)+ri−ki−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdx]dηda (27)

Let
ℜ(ψ, ρ) = max

i,j
ℜi(ψi, ρij) and ℜ0(ρ) = max

i,j
ℜi0(ρij)

The infectious and latent individuals’ migration has a meaningful influence on the spread-
ing of the disease because of the expression of the so many reproduction.

4.3. Local stability of the infection-free equilibrium

Theorem 2. The infection-free steady-state (5) is locally asymptotically stable (l.a.s.) if
ℜ(ψ, ρ) < 1 and unstable if ℜ(ψ, ρ) > 1.



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2065

Proof. Noticing that

G′
ij(λ) < 0; lim

λ→+∞
Gij(λ) = 0; lim

λ→−∞
Gij(λ) = +∞.

We know that equation (23) has a unique negative real solution λ∗ if, and only if, Gij(0) <
1, hence, ℜi(ψi, ρij) < 1 ( Also, equation (23) has a unique positive (zero) real solution
if ℜi(ψi, ρij) > 1 (ℜi(ψi, ρij) = 1). To show that λ∗ is the dominant real part of roots of
Gij(λ), we let λ = x+ iy be an arbitrary complex solution to equation (23). Note that

1 = Gij(λ) =| Gij(x+ iy) |≤ Gij(x),

indicating that Re(λ) ≤ λ∗ . It follows that the infection-free steady state is l.a.s. if
ℜ(ψ, ρ) < 1, and unstable if ℜ(ψ, ρ) > 1.

5. Global stability of the infection-free state

Since µi(a) and ii(t, a) are bounded, there exists a positif constant Rc that satifies

0 ≤
∫ a

η

2∑
i=1

µi(τ)ii(t− a+ τ, τ)dτ ≤ Rc (28)

Theorem 3. The disease-free equilibrium of system (5) is globally asymptotically stable
if ℜ0(ρ) < 1 and Rc < ln( 1

ℜ0(ρ)
).

Proof.
The proof consist to show that

ii(t, a) −→ 0; ji(t, a) −→ 0; li(t, a) −→ 0;

si(t, a) −→ s0i (a) and vi(t, a) −→ Λi − s0i (a), when t→ +∞

Integrating system (5) along characteristic lines we get

li(t, a) =

∫ a

0
e−

∫ a
η (b(τ)−µi(τ)ii(t−a+τ,τ)−µj(τ)ij(t−a+τ,τ)dτ+ki+ρi2−ρ̂j2)dτβi(η)ci(η)gi(η)λi(t−a+η)×

[σiji(t− a+ η, η) + δivi(t− η + a, η) + (1− ϕi)si(t− a+ η, η)]dη, a < t (29)

ii(t, a) =
∫ a
0 e

−
∫ a
ξ (b(τ)−µi(τ)ii(t−a+τ,τ)−µj(τ)ij(t−a+τ,τ)+ri+µi(τ)+ρi3−ρ̂j3)dτ

[ϕiβi(ξ)ci(ξ)gi(ξ)λi(t− a+ ξ) + kil(t− a+ ξ, ξ)]dξ, a < t (30)



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2066

Injecting (29) in (30), and changing order of integration, we obtain:

ii(t, a) =

∫ a

0
e−

∫ a
η (µi(τ)−µi(τ)ii(t−a+τ,τ)−µj(τ)ij(t−a+τ,τ)ri+ρi2−ρ̂j2+ρi3−ρ̂j3)dτβi(η)ci(η)gi(η)λi(t−a+η)×

[ϕisi(t− a+ η, η) + ki(σiji(t− a+ η, η) + δivi(t− η + a, η)

+(1− ϕi)si(t− a+ η, η)

∫ a

η
e−

∫ η
ξ (ri+µi(τ)−ki−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdξ]dη (31)

Injecting (31) in λi(t), and changing order of integration, we obtain:

λi(t) =

∫ a

0
e−

∫ a
η (µi(τ)−µi(τ)ii(t−a+τ,τ)−µj(τ)ij(t−a+τ,τ)ri+ρi2−ρ̂j2+ρi3−ρ̂j3)dτβi(η)ci(η)gi(η)λi(t−a+η)×

[ϕisi(t− a+ η, η) + ki(σiji(t− a+ η, η) + δivi(t− η + a, η)

+(1− ϕi)si(t− a+ η, η)

∫ a

η
e−

∫ η
ξ (ri+µi(τ)−ki−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdξ]dη (32)

It eassy see that

ϕisi(t−a+η, η)+ki(σiji(t−a+η, η)+δivi(t−η+a, η)+(1−ϕi)si(t−a+η, η) ≤ Λi[ϕi+ki(1−ϕi)]

By using inequality (28) and Fatou’s lemma, we have

lim
t→+∞

λi(t) ≤ eRcℜi0(ρi) lim sup
t→+∞

λi(t).

Since eRcℜi0(ρi) < 1, ⇒ lim supt→+∞ λi(t) = 0 ⇒
limt→+∞ ii(t, a) = limt→+∞ji(t, a) = limt→+∞ li(t, a) = 0

limt→+∞ si(t, a) = s0i (a) limt→+∞ vi(t, a) =
Λi−s0i (a)+ρj5v0j (a)

1−ρi5

For this disease can disappear without any form of intervention, according to this
result we must ensure that there is no new infected and the infectious rate does not reach
a certain spread.



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2067

6. Existence of an endemic state

Theorem 4. An interior endemic equilibrium of the form
E∗ = (s∗1(a), l

∗
1(a), i

∗
1(a), j

∗
1(a), v

∗
1(a), s

∗
2(a), l

∗
2(a), i

∗
2(a), j

∗
2(a), v

∗
2(a)) whenever ℜ1(ψ1) > 1

and ℜ2(ψ2) > 1. which corresponds to case when the disease persists in the two sub-
populations.

Proof. E∗ = (s∗1(a), l
∗
1(a), i

∗
1(a), j

∗
1(a), v

∗
1(a), s

∗
2(a), l

∗
2(a), i

∗
2(a), j

∗
2(a), v

∗
2(a)) satifies the

following equations



d
da
s∗1(a) = αib1(a) − [β1(a)c1(a)g1(a)Γ

∗
1 + ψ1(a) + b(a) − µ1(a)i

∗
1(a) − µ2(a)i

∗
2(a) + ρ11]s

∗
1(a) + ρ21s

∗
2(a)

d
da
l∗1(a) = β1(a)c1(a)g1(a)Γ

∗
1 [(1 − ϕ1)s

∗
1(a) + δ1v

∗
1 (a) + σ1j

∗
1 (a)]

−(b(a) + k1 − µ1(a)i
∗
1(a) − µ2(a)i

∗
2(a) + ρ12)l

∗
1(a) + ρ22l

∗
2(a)

d
da
i∗1(a) = k1l

∗
1(a) + ϕ1β1(a)c1(a)g1(a)Γ

∗
1s

∗
1(a) − [r1 + b(a) + µ1(a) − µ1(a)i

∗
1(a) − µ1(a)i

∗
2(a) + ρ13]i

∗
1(a) + ρ23i

∗
2(a)

d
da
j∗1 (a) = r1i

∗
1(a) − (σ1β1(a)c1(a)g1(a)Γ

∗
1 + b(a) − µ1(a)i

∗
1(a) − µ2(a)i

∗
2(a) + ρ14)j

∗
1 (a) + ρ24j

∗(a)

d
da
v∗1 (a) = ψ1(a)s

∗
1(a) + ρ25v

∗
2 (a) − (δ1β1(a)c1(a)g1(a)Γ

∗
1 + b(a) − µ1(a)i

∗
1(a) − µ2(a)i

∗
2(a) + ρ15)v

∗
1 (a)

d
da
s∗2(a) = α2b2(a) − [β2(a)c2(a)g2(a)Γ

∗
2 + ψ2(a) + b(a) − µ1(a)i

∗
1(a) − µ2(a)i

∗
2(a) + ρ21]s

∗
2(a) + ρ11s

∗
1(a)

d
da
l∗2(a) = β2(a)c2(a)g2(a)Γ

∗
2 [(1 − ϕ2)s

∗
2(a) + δ2v

∗
2 (a) + σ2j

∗
2 (a)]

−(b(a) + k2 − µ1(a)i
∗
1(a) − µ2(a)i

∗
2(a) + ρ22)l

∗
2(a) + ρ12l

∗
1(a)

d
da
i∗2(a) = k2l

∗
2(a) + ϕ2β2(a)c2(a)g2(a)Γ

∗
2s

∗
2(a) − [r2 + b(a) + µ2(a) − µ1(a)i

∗
1(a) − µ1(a)i

∗
2(a) + ρ23]i

∗
2(a) + ρ13i

∗
1(a)

d
da
j∗2 (a) = r2i

∗
2(a) − (σ2β2(a)c2(a)g2(a)Γ

∗
2 + b(a) − µ1(a)i

∗
1(a) − µ2(a)i

∗
2(a) + ρ24)j

∗
2 (a) + ρ14j

∗
1 (a)

d
da
v∗2 (a) = ψ2(a)s

∗
2(a) + ρ15v

∗
1 (a) − (δ2β2(a)c2(a)g2(a)Γ

∗
2 + b(a) − µ1(a)i

∗
1(a) − µ2(a)i

∗
2(a) + ρ25)v

∗
2 (a)

Γ∗
i =

∫ a+
0 β̂i(a)i

∗
i (a)da

(33)

s∗i (0) = Λi; l∗i (0) = i∗i (0) = j∗i (0) = v∗i (0) = 0 (34)

s∗i (a) = Λie
−

∫ a
0 [βi(τ)ci(τ)gi(τ)Γ

∗+b(τ)−µ1(τ)i∗1(τ)−µ2(τ)i∗2(τ)+ρi1−ρ̂j1]dτ

+αi

∫ a

0
bi(η)e

−
∫ a
η [βi(τ)ci(τ)gi(τ)Γ

∗+b(τ)−µ1(τ)i∗1(τ)−µ2(τ)i∗2(τ)+ρi1−ρ̂j1]dτdη (35)

Let

h∗i (η,Γ
∗
i ) = (1− ϕi)s

∗
i (η) + δiv

∗
i (η) + σij

∗
i (η) (36)

l∗i (a) = Γ∗
i

∫ a

0
βi(η)ci(η)gi(η)hi(η,Γ

∗
i )e

−
∫ a
η (b(τ)−µ1(τ)i

∗
1(τ)−µ2(τ)i∗2(τ)+ρi2−ρ̂j2)dτdη (37)

i∗i (a) =

∫ a

0
[kil

∗
i (η) + ϕiβi(η)ci(η)gi(η)Γ

∗
i s

∗
i (η)]e

−
∫ a
η (b(τ)+ri+µi(τ)−µ1(τ)i

∗
1(τ)−µ2(τ)i∗2(τ)+ρi3−ρ̂j3)dτdη(38)

j∗i (a) = ri

∫ a

0
i∗i (η)e

−
∫ a
η (σiβi(τ)ci(τ)gi(τ)Γ

∗
i+b(τ)−µ1(τ)i∗1(τ)−µ2(τ)i∗2(η)+ρi4−ρ̂j4)dτdη (39)



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2068

v∗i (a) =

∫ a

0
(ψi(η)s

∗
i (η))e

−
∫ a
η (δiβi(τ)ci(τ)gi(τ)Γ

∗
i+b(τ)−µi(τ)i∗i (τ)−µj(τ)i∗j (τ)+ρi5−ρ̂j5)dτdη (40)

By injecting (37) in (38), we obtain :

i∗i (a) = Γ∗
i

∫ a

0
βi(η)ci(η)gi(η)e

−
∫ a
η (b(τ)−µ1(τ)i

∗
1(τ)−µ2(τ)i∗2(τ)−ri+µi(τ)+ρi3−ρ̂j3)dτ×

[ϕis
∗
i (η) + kihi(η,Γ

∗
i )

∫ a

η
e−

∫ η
ξ (ri+µi(τ)−ki)−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdξ]dη (41)

By injecting (41) in the expression of Γ∗
i , and dividing by Γ∗

i (since Γ∗
i ̸= 0) we obtain :

1 =

∫ a+

0
β̂i(a)

∫ a

0
βi(η)ci(η)gi(η)e

−
∫ a
η (b(τ)−µ1(τ)i

∗
1(τ)−µ2(τ)i∗2(τ)−ri+µi(τ)+ρi3−ρ̂j3)dτ×

[ϕis
∗
i (η) + kihi(η,Γ

∗
i )

∫ a

η
e−

∫ η
ξ (ri+µi(τ)−ki)−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdξ]dηda (42)

Let Hij , the fonction define by :

Hij(Γ
∗
i ) =

∫ a+

0
β̂i(a)

∫ a

0
βi(η)ci(η)gi(η)e

−
∫ a
η (b(τ)−µ1(τ)i

∗
1(τ)−µ2(τ)i∗2(τ)−ri+µi(τ)+ρi3−ρ̂j3)dτ×

[ϕis
∗
i (η) + kihi(η,Γ

∗
i )

∫ a

η
e−

∫ η
ξ (ri+µi(τ)−ki)−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdξ]dηda (43)

Since hi(η, 0) = Nψi(η) i.e when Γ∗
i = 0, s∗i (a) = s0i (a) and v∗i (a) = v0i (a), so the net

reproductive number is given by
Hij(0) = ℜi(ψi, ρij), i.e

ℜi(ψi, ρij) =
∫ a+

0
β̂i(a)

∫ a

0
e−

∫ a
η (b(τ)+µi(τ)+ri+ρi3−ρ̂j3)dτβi(η)ci(η)gi(η)

[ϕis
0
i (η) + kiNψi(η)

∫ a

η
e−

∫ η
x (µi(τ)+ri−ki−ρi2+ρ̂j2+ρi3−ρ̂j3)dτdx]dηda (44)

We now see that an endemic steady state exists if equation (42) has a positive solution.
Since
Hi(0) = ℜi(ψi), hence Hi(0) > 1. We know that s∗i (a) + l∗i (a) + i∗i (a) + v∗i (a) + j∗i (a) =
Λi < 1. Hence

i∗i (a) < 1 (45)



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2069

Since Γ∗
i > 0, from (43) and (45) we obtain :

Γ∗
iHi(Γ

∗
i ) =

∫ a+

0
β̂i(a)

∫ a

0
Γ∗
iβi(η)ci(η)gi(η)e

−
∫ a
η (β(τ)−µ1(τ)i

∗
1(τ)−µ2(τ)i2(τ)+ri+µi(τ))dτ×

[ϕis
∗
i (η) + kihi(η,Γ

∗
i )

∫ a

η
e−

∫ η
ξ ((ri+µi(τ)−ki)dτdξ]dηda <

∫ a+

0
β̂i(a)da = β+i .

In particular, for Γ∗
i = β+i , we have Hi(β

+
i ) < 1, but Hi(0) > 1. Since Hi is continous fonc-

tion of Γ∗
i , we conclude that H(Γ∗

i ) = 1, has a positive solution Γ̂∗
i on ]0;β+i [. This solution

may not be unique since Hi may not be monotone (Hi(Γ
∗
i ) depends on h(α,Γ

∗
i ) which is

defined implicitly). It follows that when ℜi(ψi) > 1, there exists an endemic steady state
distribution which is given by the unique solution of equation (42) corresponding to Γ̂∗

i

7. Simulation

In this section, the numerical method used in our simulations is based on the finite
difference method. Forward in time-Backward in age numerical scheme is used as in[36].
Each equation is system (5) can be rewritten as(

∂

∂t
+

∂

∂a

)
f(t, a) = g(t, a)

can be approximated by

f(tk+1, ai)− f(tk, ai)

∆t
+
f(tk, ai)− f(tk, ai−1)

∆a
= g(tk, ai)

In view of the influence of migration on the dynamics of tuberculosis modeled by
(5), we considered patch2 in an endemic situation fig2 and patch1 under global stability
conditions. In absence of migration, we obtain the fig 3a; 4a; 5a, which translate the
overall stability of the patch. By varying the migration rates, we obtain the figure of the
type fig 3b; 4b; 5b, which indicate that the migration disrupts the stability and could lead
the patch a dramatic situation.



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2070

fig.2. Infectious individuals from path 2 ℜ2(ψ, ρ) > 1

Treated PATCH1

-1

-0.5

0

0.5

1

age0

50

100

150

time(month) 0

2

4

6

fig.3a. Evolution of J1 when ℜ1
0(ρ) < 1, with ρ = 0 fig.3b. Evolution of J1 when

ℜ1
0(ρ) < 1 ,with ρ ̸= 0



B. K. Abdoul Wahid, D.Moustapha, S. Bisso / Eur. J. Pure Appl. Math, 15 (4) (2022), 2054-2073 2071

Infectious PATCH1

-1

-0.5

0

0.5

1

age0

50

100

150

time(month) 0

2

4

6

fig.4a. Evolution of I1 when ℜ1
0(ρ) < 1, with ρ = 0 fig.4b. Evolution of I1 when

ℜ1
0(ρ) < 1, with ρ ̸= 0

Latent PATCH1

-1

-0.5

0

0.5

1

age0

50

100

150

time(month) 0

2

4

6

fig.5a. Evolution of L1 when ℜ1
0(ρ) < 1, with ρ = 0 fig.5b. Evolution of L1 when

ℜ1
0(ρ) < 1, with ρ ̸= 0

8. Conclusion

In this paper, we analyzed a two-patch age-structured model applied to tuberculosis
in a context where migration is not controlled. The aim is to see the impact of migration
on the spread of tuberculosis. Thus, we allowed migration to all individuals regardless of
their epidemiological status. This study shows that migration has a negative impact on
the spread of TB. The results of this analysis compared to those obtained in [19], allow us
to affirm that the control of the migration makes it possible to avoid the propagation of



REFERENCES 2072

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the resistant to all the forms of antituberculous drugs that we had, we also envisage to
study the stability of the endemic equilibrium as well as the possibilities of existence of
Hopf bifurcation using the method developed by P. Magal, S. Ruan [15].

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