We have the TB-Only submodel when𝑆𝐻 = 𝑆𝐷 = 𝐸𝐻 = 𝐸𝐷 = 𝐼𝐻1 = 𝐼𝐻2 = 𝐼𝐷1 = 𝐼𝐷2 = 𝐼𝐻3 = 𝐼𝐷3 = 𝑅𝐻 = 𝑅𝐷 = 0,which is given by
𝑑𝑆𝑇
𝑑𝑡 = 𝑀𝑇 − (𝜇 + 𝛼𝐻 + 𝛼𝐷+ 𝜆𝑇)𝑆𝑇, 𝑑𝐸𝑇
𝑑𝑡 = 𝜆𝑇(𝑆𝑇 + 𝛽1′𝑅𝑇) − (𝛼𝐻 + 𝛼𝐷+ 𝜂 + 𝜇)𝐸𝑇, 𝑑𝐼𝑇1
𝑑𝑡 = (1 − 𝛽∗)𝜂𝐸𝑇 − (𝑙𝑇 + 𝑡𝐻𝛼𝐻 + 𝑡𝐷𝛼𝐷+ 𝜇 + 𝑑𝑇 + 𝜂11)𝐼𝑇1, 𝑑𝐼𝑇2
𝑑𝑡 = (1 − 𝑝𝑇)𝛽∗𝜂𝐸𝑇 + 𝑙𝑇𝐼𝑇1 − (𝑚𝑇 + 𝜇 + 𝑡𝑇′𝑑𝑇+ 𝜂14+ 𝑡𝐻𝛼𝐻 + 𝑡𝐷𝛼𝐷)𝐼𝑇2, 𝑑𝐼𝑇3
𝑑𝑡 = 𝛽∗𝑝𝑇𝜂𝐸𝑇 + 𝜂14𝐼𝑇2 − (𝜂∗11+ 𝜇 + 𝑡𝑇∗𝑑𝑇 + 𝑡𝐻𝛼𝐻 + 𝑡𝐷𝛼𝐷)𝐼𝑇3, 𝑑𝑅𝑇
𝑑𝑡 = 𝑚𝑇𝐼𝑇2 + 𝜂11𝐼𝑇1+ 𝜂∗11𝐼𝑇3 − (𝜇 + 𝛽1′𝜆𝑇 + 𝛼𝐻 + 𝛼𝐷)𝑅𝑇, (2.10) with initial conditions:
𝑆𝑇(0) > 0, 𝐸𝑇(0) > 0, 𝐼𝑇1(0) > 0, 𝐼𝑇2(0) > 0, 𝐼𝑇3(0) > 0and𝑅𝑇(0) > 0.
The TB-infection rate for this submodel is defined as 𝜆𝑇 = 𝛼∗𝐼𝑇1+ 𝐼𝑇2+ 𝐼𝑇3
𝑁𝑇 ,
and the total population is given by
𝑁𝑇 = 𝑆𝑇 + 𝐸𝑇 + 𝐼𝑇1+ 𝐼𝑇2+ 𝐼𝑇3+ 𝑅𝑇.
Due to biological constraints, the system (2.10) is studied in the following region:
𝐷1 = {
(𝑆𝑇, 𝐸𝑇, 𝐼𝑇1, 𝐼𝑇2, 𝐼𝑇3, 𝑅𝑇) ∈ ℝ6+ ∶ 𝑁𝑇(𝑡) ≤ 𝑀𝑇 𝜇
} .
We can show for this submodel (2.10) that the solutions,(𝑆𝑇(𝑡), 𝐸𝑇(𝑡), 𝐼𝑇1(𝑡), 𝐼𝑇2(𝑡), 𝐼𝑇3(𝑡), 𝑅𝑇(𝑡)) are bounded and positively invariant in𝐷1(biologically feasible region).
Disease-Free Equilibrium Point
The disease-free equilibrium point of model (2.10) is given by𝜖0𝑇 =
(𝑆0𝑇, 0, 0, 0, 0, 0 ), where𝑆0𝑇 = 𝑀𝑇
𝜇 + 𝛼𝐻 + 𝛼𝐷
.
The matrices for the new infection terms,𝐹𝑇 and the other terms,𝑉𝑇 for system (2.10) are given by:
𝐹𝑇 =
The basic reproduction number obtained is
ℜ𝑇0 = 𝜌(𝐹𝑇𝑉𝑇−1) = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)(𝑘13𝑘14+ 𝑙𝑇(𝑘14+ 𝜂14)) + (1 − 𝑝𝑇)𝛽∗𝑘12(𝑘14+ 𝜂14) + 𝛽∗𝑝𝑇𝑘12𝑘13) 𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘12𝑘13𝑘14 ,
(2.11) where𝜌(𝐹𝑇𝑉𝑇−1) indicate the spectral radius of 𝐹𝑇𝑉𝑇−1. We have the following theorem [80]:
Lemma 2.3.1. The disease-free equilibrium point𝜖0𝑇 is locally asymptotically stable when ℜ𝑇0 < 1and unstable whenℜ𝑇0 > 1.
Proof. The Jacobian matrix of the submodel (2.10) at𝜖0𝑇 is
𝐽 (𝜖0𝑇) =
2.3 | BASIC REPRODUCTION NUMBER STUDY
Thusℜ𝑇0 < 1,which means that the solution of Det[𝐽 (𝜖0𝑇) − 𝜆𝐼]=0 (𝐼 is the Identity matrix) have negative real parts, implying that𝜖0𝑇 is locally asymptotically stable whenever ℜ𝑇0 < 1and unstable ifℜ𝑇0 > 1.
Now, we list two conditions that if met, also guarantee the global asymptotic stability of the disease-free equilibrium point. Following [38], we rewrite the model (2.10) as
𝑑𝑆
𝑑𝑡 = 𝐹 (𝑆, 𝐼 ), 𝑑𝐼
𝑑𝑡 = 𝐺(𝑆, 𝐼 ), 𝐺(𝑆, 0) = 0, (2.12)
where𝑆 ∈ ℝ2+is the vector whose components are the number of uninfected and recovered individuals(𝑆𝑇, 𝑅𝑇)and𝐼 ∈ ℝ4+denotes the number of infected individuals including the latent and the infectious(𝐼𝑇1, 𝐼𝑇2, 𝐼𝑇3).
The disease-free equilibrium is now denoted by 𝐸𝑇0 = (𝑆0𝑇∗, 0) where 𝑆0𝑇∗ = (𝑆0𝑇, 0), 𝑆0𝑇 =
(
𝑀𝑇 𝜇 + 𝛼𝐻 + 𝛼𝐷, 0
).
The conditions that must be fulfilled to guarantee the global asymptotic stability of𝐸0𝑇 are,
(𝐻1) ∶ For 𝑑𝑆
𝑑𝑡 = 𝐹 (𝑆, 0), 𝑆0𝑇∗ is globally asymptotically stable,
(𝐻2) ∶ 𝐺(𝑆, 𝐼 ) = 𝐴𝐼 − 𝐺∗(𝑆, 𝐼 ), 𝐺∗(𝑆, 𝐼 ) ≥ 0, for (𝑆, 𝐼 ) ∈ 𝐷1, (2.13) where𝐴 = 𝐷𝐼𝐺(𝑆0𝑇∗, 0)(𝐷𝐼𝐺(𝑆0𝑇∗, 0)is the Jacobian of𝐺at(𝑆0𝑇∗, 0)and𝐷1is the region where the model makes biological sense (biologically feasible region).
If model (2.10) satisfies the conditions(𝐻1)and(𝐻2), then the following result holds [80].
Lemma 2.3.2. The fixed point𝐸0𝑇 is a globally asymptotically stable equilibrium of model (2.10) provided thatℜ𝑇0 < 1and that the conditions(𝐻1)and(𝐻2)are satisfied.
Proof. Let
𝐹 (𝑆, 0) = (
𝑀𝑇 − (𝜇 + 𝛼𝐻 + 𝛼𝐷)𝑆𝑇
0 ).
As𝐹 (𝑆, 0)is a linear equation, we have that𝑆0𝑇∗ is globally stable, hence𝐻1is satisfied.
Then,
𝐴 = 𝐷𝐼𝐺(𝑆𝑇∗, 0) =
⎛
⎜
⎜
⎜
⎝
−𝑘11 𝛼∗ 𝛼∗ 𝛼∗ (1 − 𝛽∗)𝜂 −𝑘12 0 0 (1 − 𝑝𝑇)𝛽∗𝜂 𝑙𝑇 −𝑘13 0 𝑝𝑇𝛽∗𝜂 0 𝜂14 −𝑘14
⎞
⎟
⎟
⎟
⎠ ,
𝐼 = (𝐸𝑇, 𝐼𝑇1, 𝐼𝑇2, 𝐼𝑇3) ,
𝐺∗(𝑆, 𝐼 ) = 𝐴𝐼𝑇 − 𝐺(𝑆, 𝐼 ),
The proof of the local and global stability at the infection-free equilibrium point can be found in [80].
Using the threshold quantity,ℜ𝑇0, in (2.11), we want to study the impact of resistance to tuberculosis treatment on the dynamics of the disease in a population and find conditions that characterize these effects. Parameters𝑙𝑇 and𝜂14associated with MDR-TB and XDR-TB are between 0 and 1 by definition. We are now going to study the possible combinations in the behavior of these parameters based on the limits. We have
𝑙lim𝑇→0 𝜂14→0
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)𝑘013𝑘14+ (1 − 𝑝𝑇)𝛽∗𝑘120𝑘14+ 𝛽∗𝑝𝑇𝑘120𝑘130)
𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘120𝑘130𝑘14 , (2.14) where𝑘120 is𝑘12 for𝑙𝑇 = 0and𝑘130 is𝑘13for𝜂14 = 0. Then in practice𝑙𝑇 → 0and𝜂14 → 0 means zero resistance, i.e. elimination of resistance to tuberculosis treatment. If the limit (2.14) is greater than unity, then when𝑙𝑇 → 0and𝜂14 → 0it has a negative impact on TB transmission control. That is, if
𝛼∗𝜂𝑀𝑇 and𝜂14 → 0it means a negative impact on TB transmission control. That is, when
𝛼∗𝑀𝑇
2.3 | BASIC REPRODUCTION NUMBER STUDY
where𝑘131 is𝑘13for𝜂14 = 1and𝑘131 = 𝑘130 + 1. If the limit (2.18) is greater than unity, then when𝑙𝑇 → 0and𝜂14 → 1it has a negative impact on TB transmission control. That is,
if: 𝛼∗𝑀𝑇
𝑁𝑇(𝛼𝐻 + 𝛼𝐷 + 𝜇) > 𝑘11𝑘120𝑘131𝑘14
(1 − 𝛽∗)𝑘14𝑘131 + (1 − 𝑝𝑇)𝛽∗𝑘120(𝑘14+ 1) + 𝑘120𝑘131𝛽∗𝑝𝑇. (2.19) For𝑙𝑇 → 1and𝜂14 → 1, we have
𝑙lim𝑇→1 𝜂14→1
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)(𝑘14𝑘131 + (𝑘14+ 1)) + (1 − 𝑝𝑇)𝛽∗𝑘121(𝑘14+ 1) + 𝑘121𝑘131𝛽∗𝑝𝑇) 𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘121𝑘131𝑘14 .
(2.20) If the limit (2.20) is greater than unity, then when𝑙𝑇 → 1and𝜂14 → 1it has a negative impact on TB transmission control. That is, when
𝛼∗𝑀𝑇
𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇) > 𝑘11𝑘121𝑘131𝑘14
(1 − 𝛽∗)(𝑘14𝑘131 + (𝑘14+ 1)) + (1 − 𝑝𝑇)𝛽∗𝑘12′ (𝑘14+ 1) + 𝑘121𝑘131𝛽∗𝑝𝑇
. (2.21) Let us define the following expressions:
Δ𝑇 = 𝛼∗𝑀𝑇
𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇), (2.22)
Δ𝑇1 = 𝑘11𝑘120𝑘130𝑘14
(1 − 𝛽∗)𝑘130𝑘14+ (1 − 𝑝𝑇)𝛽∗𝑘120𝑘14+ 𝛽∗𝑝𝑇𝑘120𝑘130 , (2.23) Δ𝑇2 = 𝑘11𝑘121𝑘130𝑘14
(1 − 𝛽∗)(𝑘130𝑘14+ 𝑘14) + (1 − 𝑝𝑇)𝛽∗𝑘112𝑘14+ 𝛽∗𝑝𝑇𝑘121𝑘130 , (2.24) Δ𝑇3 = 𝑘11𝑘120𝑘131𝑘14
(1 − 𝛽∗)𝑘14𝑘131 + (1 − 𝑝𝑇)𝛽∗𝑘120(𝑘14+ 1) + 𝑘120𝑘131𝛽∗𝑝𝑇
, (2.25)
Δ𝑇4 = 𝑘11𝑘121𝑘131𝑘14
(1 − 𝛽∗)(𝑘14𝑘131 + (𝑘14+ 1)) + (1 − 𝑝𝑇)𝛽∗𝑘12′ (𝑘14+ 1) + 𝑘121𝑘131𝛽∗𝑝𝑇. (2.26) We have the following result:
Lemma 2.3.3. 1. The impact when𝑙𝑇 → 0 and 𝜂14 → 0 is positive in reducing TB transmission in this subpopulation only if Δ𝑇 < Δ𝑇1, no impact if Δ𝑇 = Δ𝑇1 and a negative impact ifΔ𝑇 > Δ𝑇1.
2. The impact when𝑙𝑇 → 1and 𝜂14 → 0 is positive in reducing TB transmission in this subpopulation only ifΔ𝑇 < Δ𝑇2, no impact ifΔ𝑇 = Δ𝑇2 and a negative impact if Δ𝑇 > Δ𝑇2.
3. The impact when𝑙𝑇 → 0and 𝜂14 → 1 is positive in reducing TB transmission in this subpopulation only ifΔ𝑇 < Δ𝑇3, no impact ifΔ𝑇 = Δ𝑇3 and a negative impact if Δ𝑇 > Δ𝑇3.
4. The impact when𝑙𝑇 → 1and 𝜂14 → 1 is positive in reducing TB transmission in this subpopulation only ifΔ𝑇 < Δ𝑇4, no impact ifΔ𝑇 = Δ𝑇4 and a negative impact if Δ𝑇 > Δ𝑇4.
Now, we study the relationship between resistance and recovery parameters. The treat-ment aims to avoid resistance and for patients to recover. First, we analize the relationship between MDR-TB (𝑙𝑇) and the recovery parameter (𝜂11), because we want to avoid MDR-TB so it is necessary that the patient recovers before having this resistance. We have the following limits.
𝑙lim𝑇→0 𝜂11→1
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)𝑘13𝑘14+ (1 − 𝑝𝑇)𝛽∗𝑘1201(𝑘14+ 𝜂14) + 𝑝𝑇𝛽∗𝑘1201𝑘13)
𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘1201𝑘13𝑘14 , (2.27) where𝑘1201represents𝑘12when𝑙𝑇 → 0and𝜂11→ 1.
𝑙lim𝑇→1 𝜂11→0
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)(𝑘13𝑘14+ (𝑘14+ 𝜂14)) + (1 − 𝑝𝑇)𝛽∗𝑘1210(𝑘14+ 𝜂14) + 𝑝𝑇𝛽∗𝑘1210𝑘13) 𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘1210𝑘13𝑘14
, (2.28) where𝑘1210represents𝑘12when𝑙𝑇 → 1and𝜂11→ 0.
lim
𝑙𝑇→1 𝜂11→1
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)(𝑘13𝑘14+ (𝑘14+ 𝜂14)) + (1 − 𝑝𝑇)𝛽∗𝑘1211(𝑘14+ 𝜂14) + 𝑝𝑇𝛽∗𝑘1211𝑘13) 𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘1211𝑘13𝑘14 ,
(2.29) where𝑘1211represents𝑘12when𝑙𝑇 → 1and𝜂11→ 1.
𝑙lim𝑇→0 𝜂11→0
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)𝑘13𝑘14+ (1 − 𝑝𝑇)𝛽∗𝑘1200(𝑘14+ 𝜂14) + 𝑝𝑇𝛽∗𝑘1200𝑘13)
𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘1200𝑘13𝑘14 , (2.30) where𝑘1200represents𝑘12when𝑙𝑇 → 0and𝜂11→ 0.
Let us denote:
Δ𝑇5 = 𝑘11𝑘1200𝑘13𝑘14
(1 − 𝛽∗)𝑘13𝑘14+ (1 − 𝑝𝑇)𝛽∗𝑘1200(𝑘14+ 𝜂14) + 𝑝𝑇𝛽∗𝑘1200𝑘13, (2.31) Δ𝑇6 = 𝑘11𝑘1201𝑘13𝑘14
(1 − 𝛽∗)𝑘13𝑘14+ (1 − 𝑝𝑇)𝛽∗𝑘1201(𝑘14+ 𝜂14) + 𝑝𝑇𝛽∗𝑘1201𝑘13
, (2.32)
Δ𝑇7 = 𝑘11𝑘1210𝑘13𝑘14
(1 − 𝛽∗)(𝑘13𝑘14+ (𝑘14+ 𝜂14)) + (1 − 𝑝𝑇)𝛽∗𝑘1210(𝑘14+ 𝜂14) + 𝑝𝑇𝛽∗𝑘1210𝑘13, (2.33) Δ𝑇8 = 𝑘11𝑘1211𝑘13𝑘14
(1 − 𝛽∗)(𝑘13𝑘14+ (𝑘14+ 𝜂14)) + (1 − 𝑝𝑇)𝛽∗𝑘1211(𝑘14+ 𝜂14) + 𝑝𝑇𝛽∗𝑘1211𝑘13. (2.34) We have the following results:
Lemma 2.3.4. 1. The impact when 𝑙𝑇 → 0and 𝜂11 → 0is positive in reducing TB transmission in this subpopulation only if Δ𝑇 < Δ𝑇5, no impact if Δ𝑇 = Δ𝑇5 and a negative impact ifΔ𝑇 > Δ𝑇5.
2. The impact when𝑙𝑇 → 0 and 𝜂11 → 1 is positive in reducing TB transmission in this subpopulation only ifΔ𝑇 < Δ𝑇6, no impact ifΔ𝑇 = Δ𝑇6 and a negative impact if Δ𝑇 > Δ𝑇6.
3. The impact when𝑙𝑇 → 1 and 𝜂11 → 0 is positive in reducing TB transmission in
2.3 | BASIC REPRODUCTION NUMBER STUDY
this subpopulation only ifΔ𝑇 < Δ𝑇7, no impact ifΔ𝑇 = Δ𝑇7 and a negative impact if Δ𝑇 > Δ𝑇7.
4. The impact when𝑙𝑇 → 1and 𝜂11 → 1 is positive in reducing TB transmission in this subpopulation only ifΔ𝑇 < Δ𝑇8, no impact ifΔ𝑇 = Δ𝑇8 and a negative impact if Δ𝑇 > Δ𝑇8.
Now, we examine the relationship between the XDR-TB parameter (𝜂14) and recovery after reporting as XDR-TB (𝑚𝑇). The aim, in this case, is to have a patient recover before reporting as XDR-TB. We show the different relationships between these parameters with respect toℜ𝑇0.
𝜂lim14→0 𝑚𝑇→1
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)(𝑘1301𝑘14+ 𝑙𝑇𝑘14) + (1 − 𝑝𝑇)𝛽∗𝑘12𝑘14+ 𝑝𝑇𝛽∗𝑘12𝑘1301)
𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘12𝑘1301𝑘14 (2.35) where𝑘1301 represents𝑘13 when𝜂14 → 0and𝑚𝑇 → 1.
𝜂lim14→1 𝑚𝑇→0
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)(𝑘1310𝑘14+ 𝑙𝑇(𝑘14+ 1)) + (1 − 𝑝𝑇)𝛽∗𝑘12(𝑘14+ 1) + 𝑝𝑇𝛽∗𝑘12𝑘1310) 𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘12𝑘1310𝑘14
(2.36) where𝑘1310 represents𝑘13 when𝜂14 → 1and𝑚𝑇 → 0.
𝜂lim14→1 𝑚𝑇→1
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)(𝑘1311𝑘14+ 𝑙𝑇(𝑘14+ 1)) + (1 − 𝑝𝑇)𝛽∗𝑘12(𝑘14+ 1) + 𝑝𝑇𝛽∗𝑘12𝑘1311) 𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘12𝑘1311𝑘14
(2.37) where𝑘1311 represents𝑘13 when𝜂14 → 1and𝑚𝑇 → 1.
𝜂lim14→0 𝑚𝑇→0
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)(𝑘1300𝑘14+ 𝑙𝑇𝑘14) + (1 − 𝑝𝑇)𝛽∗𝑘12𝑘14+ 𝑝𝑇𝛽∗𝑘12𝑘1300)
𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘12𝑘1300𝑘14 (2.38) where𝑘1300 represents𝑘13 when𝜂14 → 0and𝑚𝑇 → 0.
Let us consider the following expressions:
Δ𝑇9 = 𝑘11𝑘12𝑘1300𝑘14
(1 − 𝛽∗)(𝑘1311𝑘14+ 𝑙𝑇𝑘14) + (1 − 𝑝𝑇)𝛽∗𝑘12(𝑘14+ 1) + 𝑝𝑇𝛽∗𝑘12𝑘1300, (2.39) Δ𝑇10 = 𝑘11𝑘12𝑘1301𝑘14
(1 − 𝛽∗)(𝑘1301𝑘14+ 𝑙𝑇𝑘14) + (1 − 𝑝𝑇)𝛽∗𝑘12𝑘14+ 𝑝𝑇𝛽∗𝑘12𝑘1301, (2.40) Δ𝑇11 = 𝑘11𝑘12𝑘1310𝑘14
(1 − 𝛽∗)(𝑘1310𝑘14+ 𝑙𝑇(𝑘14+ 1)) + (1 − 𝑝𝑇)𝛽∗𝑘12(𝑘14+ 1) + 𝑝𝑇𝛽∗𝑘12𝑘1310, (2.41) Δ𝑇12 = 𝑘11𝑘12𝑘1311𝑘14
(1 − 𝛽∗)(𝑘1311𝑘14+ 𝑙𝑇(𝑘14+ 1)) + (1 − 𝑝𝑇)𝛽∗𝑘12(𝑘14+ 1) + 𝑝𝑇𝛽∗𝑘12𝑘1311. (2.42) We obtain the following result:
Lemma 2.3.5. 1. The impact when𝜂14 → 0and 𝑚𝑇 → 0is positive in reducing TB transmission in this subpopulation only if Δ𝑇 < Δ𝑇9, no impact if Δ𝑇 = Δ𝑇9 and a negative impact ifΔ𝑇 > Δ𝑇9.
2. The impact when𝜂14 → 0and 𝑚𝑇 → 1is positive in reducing TB transmission in this subpopulation only ifΔ𝑇 < Δ𝑇10, no impact ifΔ𝑇 = Δ𝑇10 and a negative impact if Δ𝑇 > Δ𝑇10.
3. The impact when𝜂14 → 1and 𝑚𝑇 → 0is positive in reducing TB transmission in this subpopulation only ifΔ𝑇 < Δ𝑇11, no impact ifΔ𝑇 = Δ𝑇11 and a negative impact if Δ𝑇 > Δ𝑇11.
4. The impact when𝜂14 → 1and 𝑚𝑇 → 1is positive in reducing TB transmission in this subpopulation only ifΔ𝑇 < Δ𝑇12, no impact ifΔ𝑇 = Δ𝑇12 and a negative impact if Δ𝑇 > Δ𝑇12.
Studying the resistance parameters (𝑙𝑇, 𝜂14) in conjunction with the recovery parameters (𝜂11, 𝑚𝑇). We present two cases, (1) when the resistance parameters tend to unity and the recovery parameters tend to zero, (2) the opposite case:
𝑙lim𝑇→1 𝜂14→1 𝜂11→0 𝑚𝑇→0
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)(𝑘1310𝑘14+ (𝑘14+ 1)) + (1 − 𝑝𝑇)𝛽∗𝑘1210(𝑘14+ 1) + 𝑝𝑇𝛽∗𝑘1210𝑘1310) 𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘1012𝑘1310𝑘14
(2.43)
𝑙lim𝑇→0 𝜂14→0 𝜂11→1 𝑚𝑇→1
ℜ𝑇0 = 𝛼∗𝑀𝑇𝜂((1 − 𝛽∗)𝑘1301𝑘14+ (1 − 𝑝𝑇)𝛽∗𝑘1201𝑘14+ 𝑝𝑇𝛽∗𝑘1201𝑘1301)
𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘0112𝑘1301𝑘14 (2.44)
and if we define
Δ𝑇13 = 𝑘11𝑘1210𝑘1310𝑘14
(1 − 𝛽∗)(𝑘1310𝑘14+ (𝑘14+ 1)) + (1 − 𝑝𝑇)𝛽∗𝑘1210(𝑘14+ 1) + 𝑝𝑇𝛽∗𝑘1210𝑘1310, (2.45) Δ𝑇14 = 𝑘11𝑘1201𝑘1301𝑘14
(1 − 𝛽∗)𝑘1301𝑘14+ (1 − 𝑝𝑇)𝛽∗𝑘1201𝑘14+ 𝑝𝑇𝛽∗𝑘1201𝑘1301, (2.46) we obtain the following results:
Lemma 2.3.6. 1. The impact of the resistance parameters when they tend to unity (𝑙𝑇, 𝜂14 → 1) with respect to the recovery parameters when they tend to zero (𝜂11, 𝑚𝑇 → 0) is positive in reducing tuberculosis transmission in this subpopulation only ifΔ𝑇 < Δ𝑇13, no impact ifΔ𝑇 = Δ𝑇13 and a negative impact ifΔ𝑇 > Δ𝑇13.
2. The impact of the recovery parameters recovery parameters when they tend to unity (𝜂11, 𝑚𝑇 → 1) with respect to the recovery parameters when they tend to zero (𝑙𝑇, 𝜂14 → 0) is positive in reducing tuberculosis transmission in this subpopulation only ifΔ𝑇 <
Δ𝑇14, no impact ifΔ𝑇 = Δ𝑇14 and a negative impact ifΔ𝑇 > Δ𝑇14.
Endemic Equilibrium Point
To find the endemic equilibrium point of TB-Only submodel (2.10), we solve the following system of equations,
2.3 | BASIC REPRODUCTION NUMBER STUDY
Substituting equations (2.47) into the TB-infection rate for this submodel
(𝜆𝑇 =
where𝜆∗𝑇 = 0corresponds to the disease-free equilibrium and𝜆𝑇∗ ≠ 0means the existence of endemic equilibrium. For a disease to spread, the force of infection (𝜆∗𝑇) should be positive.
So for𝜆𝑇∗ to be positive, we need the following inequality to be satisfied.
𝛼∗𝑀𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)((1 − 𝛽∗)𝜂(𝑘13𝑘14+ 𝑙𝑇(𝑘14+ 𝜂14) + (1 − 𝑝𝑇)𝑘12𝛽∗𝜂(𝑘14+ 𝜂14) + 𝑘12𝑘13𝛽∗𝜂𝑝𝑇) 𝑁𝑇
− (𝛼𝐻 + 𝛼𝐷+ 𝜇)2𝑘11𝑘12𝑘13𝑘14 > 0,
and
𝛼∗𝑀𝑇((1 − 𝛽∗)𝜂(𝑘13𝑘14+ 𝑙𝑇(𝑘14+ 𝜂14) + (1 − 𝑝𝑇)𝑘12𝛽∗𝜂(𝑘14+ 𝜂14) + 𝑘12𝑘13𝛽∗𝜂𝑝𝑇) 𝑁𝑇(𝛼𝐻 + 𝛼𝐷+ 𝜇)𝑘11𝑘12𝑘13𝑘14 > 1, which implies thatℜ𝑇0 > 1.
Then, we have the following lemma:
Lemma 2.3.7. The TB-Only submodel (2.10) has a unique endemic equilibrium point𝜖∗𝑇, wheneverℜ𝑇0 > 1.