TB-HIV/AIDS Submodel

(𝛼_𝐻 + 𝛼_𝐷+ 𝜇)²𝑘₁₁𝑘₁₂𝑘₁₃𝑘₁₄ > 0,

and

𝛼^∗𝑀𝑇((1 − 𝛽^∗)𝜂(𝑘13𝑘14+ 𝑙𝑇(𝑘14+ 𝜂14) + (1 − 𝑝𝑇)𝑘12𝛽^∗𝜂(𝑘14+ 𝜂14) + 𝑘12𝑘13𝛽^∗𝜂𝑝𝑇) 𝑁_𝑇(𝛼_𝐻 + 𝛼_𝐷+ 𝜇)𝑘₁₁𝑘₁₂𝑘₁₃𝑘₁₄ > 1, which implies thatℜ^𝑇₀ > 1.

Then, we have the following lemma:

Lemma 2.3.7. The TB-Only submodel (2.10) has a unique endemic equilibrium point𝜖_∗^𝑇, wheneverℜ^𝑇₀ > 1.

The submodel that relates TB to HIV/AIDS is when𝑆𝐷 = 𝑆𝑇 = 𝐸𝐷 = 𝐸𝑇 = 𝐼𝐷1 = 𝐼𝐷2 = 𝐼_𝑇₁ = 𝐼_𝑇₂ = 𝐼_𝑇₃ = 𝑅_𝑇 = 𝐼_𝐷₃ = 𝑅_𝐷 = 0and is given by

𝑑𝑆𝐻

𝑑𝑡 = 𝑀𝐻 − (𝛼𝐻 𝐷+ 𝜇 + 𝜇𝐻 + 𝜔𝐻𝜆𝐻)𝑆𝐻, 𝑑𝐸_𝐻

𝑑𝑡 = 𝜔𝐻𝜆𝐻(𝑆𝐻 + 𝛽₁^′𝑅𝐻) − (𝜖_𝐻^∗𝜂 + 𝜇 + 𝜇𝐻 + 𝛼𝐻 𝐷)𝐸𝐻, 𝑑𝐼_𝐻₁

𝑑𝑡 = (1 − 𝛽^∗)𝜖_𝐻^∗ 𝜂𝐸_𝐻 − (𝑙_𝐻 + 𝜇 + 𝜇_𝐻 + 𝑑_{𝑇 𝐻} + 𝜂₁₂+ 𝑡_{𝐻 𝐷}𝛼_{𝐻 𝐷})𝐼_𝐻₁, 𝑑𝐼_𝐻₂

𝑑𝑡 = (1 − 𝑝_𝐻)𝛽^∗𝜖_𝐻^∗ 𝜂𝐸_𝐻 + 𝑙_𝐻𝐼_𝐻₁ − (𝑚_𝐻 + 𝜇 + 𝜇_𝐻 + 𝑡_𝐻^′ 𝑑_{𝑇 𝐻} + 𝜂₁₅+ 𝑡_{𝐻 𝐷}𝛼_{𝐻 𝐷})𝐼_𝐻₂, 𝑑𝐼_𝐻₃

𝑑𝑡 = 𝑝_𝐻𝛽^∗𝜖_𝐻^∗𝜂𝐸_𝐻 + 𝜂₁₅𝐼_𝐻₂ − (𝜂^∗₁₂+ 𝜇 + 𝜇_𝐻 + 𝑡_𝐻^∗𝑑_{𝑇 𝐻} + 𝑡_{𝐻 𝐷}𝛼_{𝐻 𝐷})𝐼_𝐻₃, 𝑑𝑅_𝐻

𝑑𝑡 = 𝑚_𝐻𝐼_𝐻₂ + 𝜂₁₂𝐼_𝐻₁ + 𝜂^∗₁₂𝐼_𝐻₃− (𝜇 + 𝜇_𝐻 + 𝛽₁^′𝜔_𝐻𝜆_𝐻 + 𝛼_{𝐻 𝐷})𝑅_𝐻, (2.48) with non-negative initial conditions and

𝜆_𝐻 = 𝛼^∗𝜖_𝐻(𝐼_𝐻₁ + 𝐼_𝐻₂+ 𝐼_𝐻₃) 𝑁𝐻

, where𝑁_𝐻 = 𝑆_𝐻 + 𝐸_𝐻 + 𝐼_𝐻₁ + 𝐼_𝐻₂ + 𝐼_𝐻₃ + 𝑅_𝐻.

Considering biological constraints, the system (2.48) will be studied in the following region:

𝐷₂= {

(𝑆_𝐻, 𝐸_𝐻, 𝐼_𝐻₁, 𝐼_𝐻₂, 𝐼_𝐻₃, 𝑅_𝐻) ∈ ℝ⁶₊∶ 𝑁_𝐻(𝑡) ≤ 𝑀_𝐻 𝜇

} .

It can be easily shown that the solution(𝑆_𝐻(𝑡), 𝐸_𝐻, 𝐼_𝐻₁(𝑡), 𝐼_𝐻₂(𝑡), 𝐼_𝐻₃(𝑡), 𝑅_𝐻(𝑡))of the sys-tem are bounded and positively invariant.

The disease-free equilibrium point, 𝜖₀^𝐻, is given by 𝜖₀^𝐻 =

(𝑆₀^𝐻, 0, 0, 0, 0, 0

), where

2.3 | BASIC REPRODUCTION NUMBER STUDY

𝑆₀^𝐻 = 𝑀𝐻

𝛼_{𝐻 𝐷}+ 𝜇 + 𝜇_𝐻.

The matrices for the new infection terms,𝐹_𝐻 and the other terms,𝑉_𝐻 are given respec-tively, by: reproduction number is given by

ℜ^𝐻₀ = 𝜌(𝐹𝐻𝑉_𝐻⁻¹) = 𝛼^∗𝜖𝐻𝜔𝐻𝑀𝐻𝜖_𝐻^∗𝜂((1 − 𝛽^∗)(𝑘23𝑘24+ 𝑙𝐻(𝑘24+ 𝜂15)) + (1 − 𝑝𝐻)𝛽^∗𝑘22(𝑘24+ 𝜂15) + 𝛽^∗𝑝𝐻𝑘22𝑘23) 𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇 + 𝜇_𝐻)𝑘₂₁𝑘₂₂𝑘₂₃𝑘₂₄ .

(2.49)

We define (𝐻₁) and (𝐻₂) as in the submodel (2.10) and using the same idea from the demonstration, we get the following results:

Lemma 2.3.8. The disease-free equilibrium𝜖₀^𝐻 is asymptotically stable whenℜ^𝐻₀ < 1and is unstable wheneverℜ^𝐻₀ > 1. globally asymptotically stable equilibrium of submodel (TB-HIV/AIDS) ifℜ^𝐻₀ < 1and the assumption(𝐻₁)and(𝐻₂)are satisfied.

We make a procedure analogous to the model (2.10) for 𝑙𝐻 and𝜂15 (MDR-TB and XDR-TB parameters for TB-HIV/AIDS submodel) and we obtain the limits:

𝑙lim𝐻→0 means zero resistance, i.e. elimination of resistance to tuberculosis treatment in this subpopulation. If the limit (2.50) is greater than unity, then when𝑙_𝐻, 𝜂₁₅ → 0, it has a negative impact on TB transmission control. That is, when

𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗𝜂𝑀_𝐻

𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇) > 𝑘₂₁𝑘₂₂⁰𝑘₂₃⁰𝑘₂₄

(1 − 𝛽^∗)𝑘₂₃⁰𝑘₂₄+ (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰𝑘₂₄+ 𝛽^∗𝑝_𝐻𝑘⁰₂₂𝑘₂₃⁰ , (2.51)

Now, we study the case when𝑙_𝐻 → 1and𝜂₁₅→ 0. We get the limit

𝑙lim𝐻→1 𝜂15→0

ℜ^𝐻₀ = 𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗𝑀_𝐻𝜂((1 − 𝛽^∗)(𝑘₂₃⁰𝑘₂₄+ 𝑘₂₄) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹𝑘₂₄+ 𝛽^∗𝑝_𝐻𝑘₂₂¹𝑘₂₃⁰)

𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂¹𝑘₂₃⁰𝑘₂₄ , (2.52) where𝑘₂₂¹ is𝑘₂₂for𝑙_𝐻 = 1. Then if the limit (2.52) is greater than unity, then when𝑙_𝐻 → 1 and𝜂₁₅ → 0it has a negative impact on TB transmission control. That is, whereas

𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗𝑀_𝐻

𝑁𝐻(𝛼𝐻 𝐷+ 𝜇𝐻 + 𝜇) > 𝑘₂₁𝑘₂₂¹𝑘₂₃⁰𝑘₂₄

(1 − 𝛽^∗)(𝑘₂₃⁰𝑘24+ 𝑘24) + (1 − 𝑝𝐻)𝛽^∗𝑘₂₂¹𝑘24+ 𝛽^∗𝑝𝐻𝑘₂₂¹𝑘₂₃⁰ . (2.53) In the case when𝑙_𝐻 → 0and𝜂₁₅ → 1. We have

𝑙𝐻lim→0 𝜂15→1

ℜ^𝐻₀ = 𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗𝑀_𝐻𝜂((1 − 𝛽^∗)𝑘₂₄𝑘₂₃¹ + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰(𝑘₂₄+ 1) + 𝑘₂₂⁰𝑘₂₃¹𝛽^∗𝑝_𝐻) 𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂⁰𝑘₂₃¹𝑘₂₄ > 0.

(2.54) where𝑘₂₃¹ is𝑘23 for𝜂15 = 1and𝑘₂₃¹ = 𝑘₂₃⁰ + 1. If the limit (2.54) is greater than unity, then 𝑙_𝐻 → 0and𝜂₁₅ → 1means a negative impact on TB transmission control. That is, if we have

𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗𝑀_𝐻

𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇) > 𝑘₂₁𝑘₂₂⁰𝑘₂₃¹𝑘₂₄

(1 − 𝛽^∗)𝑘₂₄𝑘₂₃¹ + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰(𝑘₂₄+ 1) + 𝑘₂₂⁰𝑘₂₃¹𝛽^∗𝑝_𝐻. (2.55) For𝑙_𝐻 → 1and𝜂₁₅ → 1:

𝑙lim𝐻→1 𝜂15→1

ℜ^𝐻₀ = 𝛼^∗𝑀_𝐻𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗𝜂((1 − 𝛽^∗)(𝑘₂₄𝑘₂₃¹ + (𝑘₂₄+ 1)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹(𝑘₂₄+ 1) + 𝑘₂₂¹𝑘₂₃¹𝛽^∗𝑝_𝐻) 𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂¹𝑘₂₃¹𝑘₂₄ .

(2.56) If the limit (2.56) is greater than unity, then when𝑙_𝐻 → 1and𝜂₁₅ → 1has a negative impact on TB transmission control, if

𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗𝑀_𝐻

𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇) > 𝑘₂₁𝑘₂₂¹𝑘₂₃¹𝑘₂₄

(1 − 𝛽^∗)(𝑘₂₄𝑘¹₂₃+ (𝑘₂₄+ 1)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹(𝑘₂₄+ 1) + 𝑘₂₂¹𝑘₂₃¹𝛽^∗𝑝_𝐻. (2.57) We consider

Δ_𝐻 = 𝛼^∗𝜔𝐻𝜖𝐻𝜖_𝐻^∗𝑀𝐻

𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇), (2.58)

Δ_𝐻₁ = 𝑘₂₁𝑘₂₂⁰𝑘₂₃⁰𝑘₂₄

(1 − 𝛽^∗)𝑘₂₃⁰𝑘₂₄+ (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰𝑘₂₄+ 𝛽^∗𝑝_𝐻𝑘⁰₂₂𝑘₂₃⁰ , (2.59) Δ_𝐻₂ = 𝑘₂₁𝑘₂₂¹𝑘₂₃⁰𝑘₂₄

(1 − 𝛽^∗)(𝑘₂₃⁰𝑘₂₄+ 𝑘₂₄) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹𝑘₂₄+ 𝛽^∗𝑝_𝐻𝑘₂₂¹𝑘₂₃⁰ , (2.60)

2.3 | BASIC REPRODUCTION NUMBER STUDY

Δ𝐻3 = 𝑘21𝑘⁰₂₂𝑘₂₃¹𝑘24

(1 − 𝛽^∗)𝑘₂₄𝑘₂₃¹ + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰(𝑘₂₄+ 1) + 𝑘₂₂⁰𝑘₂₃¹𝛽^∗𝑝_𝐻, (2.61) Δ_𝐻₄ = 𝑘₂₁𝑘₂₂¹𝑘¹₂₃𝑘₂₄

(1 − 𝛽^∗)(𝑘₂₄𝑘₂₃¹ + (𝑘₂₄+ 1)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹(𝑘₂₄+ 1) + 𝑘₂₂¹𝑘₂₃¹𝛽^∗𝑝_𝐻. (2.62) Then, we have the following results:

Lemma 2.3.10. 1. The impact when𝑙_𝐻 → 0and 𝜂₁₅ → 0 is positive in reducing TB transmission in this subpopulation only ifΔ_𝐻 < Δ_𝐻₁, no impact ifΔ_𝐻 = Δ_𝐻₁ and a negative impact ifΔ_𝐻 > Δ_𝐻₁.

2. The impact when𝑙_𝑇 → 1and 𝜂₁₅ → 0 is positive in reducing TB transmission in this subpopulation only ifΔ_𝐻 < Δ_𝐻₂, no impact ifΔ_𝐻 = Δ_𝐻₂ and a negative impact if Δ_𝐻 > Δ_𝐻₂.

3. The impact when 𝑙_𝐻 → 0and 𝜂₁₅ → 1is positive in reducing TB transmission in this subpopulation only ifΔ_𝐻 < Δ_𝐻₃, no impact ifΔ_𝐻 = Δ_𝐻₃ and a negative impact if Δ_𝐻 > Δ_𝐻₃.

4. The impact when 𝑙_𝐻 → 1and 𝜂₁₅ → 1is positive in reducing TB transmission in this subpopulation only ifΔ_𝐻 < Δ_𝐻₄, no impact ifΔ_𝐻 = Δ_𝐻₄ and a negative impact if Δ_𝐻 > Δ_𝐻₄.

We apply the same procedure as for the previous submodel to study the relationships for𝑙_𝐻 and𝜂₁₂. We have:

𝑙lim𝐻→0 𝜂12→1

ℜ^𝐻₀ = 𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗ 𝑀_𝐻𝜂((1 − 𝛽^∗)𝑘₂₃𝑘₂₄+ (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰¹(𝑘₂₄+ 𝜂₁₅) + 𝑝_𝐻𝛽^∗𝑘₂₂⁰¹𝑘₂₃)

𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂⁰¹𝑘₂₃𝑘₂₄ , (2.63) where𝑘₂₂⁰¹ represents𝑘₂₂ when𝑙_𝐻 → 0and𝜂₁₂ → 1.

𝑙lim𝐻→1 𝜂12→0

ℜ^𝐻₀ = 𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗ 𝑀_𝐻𝜂((1 − 𝛽^∗)(𝑘₂₃𝑘₂₄+ (𝑘₂₄+ 𝜂₁₅)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹⁰(𝑘₂₄+ 𝜂₁₅) + 𝑝_𝐻𝛽^∗𝑘₂₂¹⁰𝑘₂₃) 𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂¹⁰𝑘₂₃𝑘₂₄ ,

(2.64) where𝑘₂₂¹⁰ represents𝑘₂₂ when𝑙_𝐻 → 1and𝜂₁₂ → 0.

𝑙lim𝐻→1 𝜂12→1

ℜ^𝐻₀ = 𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗ 𝑀_𝐻𝜂((1 − 𝛽^∗)(𝑘₂₃𝑘₂₄+ (𝑘₂₄+ 𝜂₁₅)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹¹(𝑘₂₄+ 𝜂₁₅) + 𝑝_𝐻𝛽^∗𝑘₂₂¹¹𝑘₂₃) 𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂¹¹𝑘₂₃𝑘₂₄ ,

(2.65) where𝑘₂₂¹¹ represents𝑘₂₂ when𝑙_𝐻 → 1and𝜂₁₂ → 1.

𝑙lim𝐻→0 𝜂12→0

ℜ^𝐻₀ = 𝛼^∗𝜔_𝐻𝜖_𝐻𝜖_𝐻^∗ 𝑀_𝐻𝜂((1 − 𝛽^∗)𝑘₂₃𝑘₂₄+ (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰⁰(𝑘₂₄+ 𝜂₁₅) + 𝑝_𝐻𝛽^∗𝑘₂₂⁰⁰𝑘₂₃)

𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂⁰⁰𝑘₂₃𝑘₂₄ , (2.66)

where𝑘₂₂⁰⁰represents𝑘₂₂when𝑙_𝐻 → 0and𝜂₁₂ → 0. Let us define the following expressions:

Δ_𝐻₅ = 𝑘₂₁𝑘₂₂⁰⁰𝑘₂₃𝑘₂₄

(1 − 𝛽^∗)𝑘₂₃𝑘₂₄+ (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰⁰(𝑘₂₄+ 𝜂₁₅) + 𝑝_𝐻𝛽^∗𝑘₂₂⁰⁰𝑘₂₃, (2.67) Δ_𝐻₆ = 𝑘₂₁𝑘₂₂⁰¹𝑘₂₃𝑘₂₄

(1 − 𝛽^∗)𝑘23𝑘24+ (1 − 𝑝𝐻)𝛽^∗𝑘₂₂⁰¹(𝑘24+ 𝜂15) + 𝑝𝐻𝛽^∗𝑘₂₂⁰¹𝑘23

, (2.68)

Δ𝐻7 = 𝑘₂₁𝑘₂₂¹⁰𝑘₂₃𝑘₂₄

(1 − 𝛽^∗)(𝑘₂₃𝑘₂₄+ (𝑘₂₄+ 𝜂₁₅)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹⁰(𝑘₂₄+ 𝜂₁₅) + 𝑝_𝐻𝛽^∗𝑘¹⁰₂₂𝑘₂₃, (2.69) Δ_𝐻₈ = 𝑘21𝑘₂₂¹¹𝑘23𝑘24

(1 − 𝛽^∗)(𝑘₂₃𝑘₂₄+ (𝑘₂₄+ 𝜂₁₅)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹¹(𝑘₂₄+ 𝜂₁₅) + 𝑝_𝐻𝛽^∗𝑘¹¹₂₂𝑘₂₃. (2.70) We obtain the following results:

Lemma 2.3.11. 1. The impact when 𝑙_𝐻 → 0 and 𝜂₁₂ → 0 is positive in reducing TB transmission in TB-HIV/AIDS subpopulation only ifΔ_𝐻 < Δ_𝐻₅, no impact ifΔ_𝐻 = Δ_𝐻₅ and a negative impact ifΔ_𝐻 > Δ_𝐻₅.

2. The impact when 𝑙_𝐻 → 0 and𝜂₁₂ → 1 is positive in reducing TB transmission in TB-HIV/AIDS subpopulation only ifΔ𝐻 < Δ𝐻6, no impact ifΔ𝐻 = Δ𝐻6 and a negative impact ifΔ_𝐻 > Δ_𝐻₆.

3. The impact when 𝑙_𝐻 → 1 and𝜂₁₂ → 0 is positive in reducing TB transmission in TB-HIV/AIDS subpopulation only ifΔ_𝐻 < Δ_𝐻₇, no impact ifΔ_𝐻 = Δ_𝐻₇ and a negative impact ifΔ𝐻 > Δ𝐻7.

4. The impact when 𝑙_𝐻 → 1 and𝜂₁₂ → 1 is positive in reducing TB transmission in TB-HIV/AIDS subpopulation only ifΔ_𝐻 < Δ_𝐻₈, no impact ifΔ_𝐻 = Δ_𝐻₈ and a negative impact ifΔ_𝐻 > Δ_𝐻₈.

Now, we study the relationships between the parameters associated with XDR-TB (𝜂₁₅) and recovery after MDR-TB (𝑚_𝐻).

𝜂lim15→0 𝑚𝐻→1

ℜ^𝐻₀ = 𝛼^∗𝑀_𝐻𝜔_𝐻𝜖_𝐻^∗𝜖_𝐻𝜂((1 − 𝛽^∗)(𝑘₂₃⁰¹𝑘₂₄+ 𝑙_𝐻𝑘₂₄) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂𝑘₂₄+ 𝑝_𝐻𝛽^∗𝑘₂₂𝑘₂₃⁰¹) 𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂𝑘₂₃⁰¹𝑘₂₄ ,

(2.71) where𝑘₂₃⁰¹represents𝑘23when𝜂15 → 0and𝑚𝐻 → 1.

𝜂lim15→1 𝑚𝐻→0

ℜ^𝐻₀ = 𝛼^∗𝑀_𝐻𝜔_𝐻𝜖_𝐻^∗𝜖_𝐻𝜂((1 − 𝛽^∗)(𝑘₂₃¹⁰𝑘₂₄+ 𝑙_𝐻(𝑘₂₄+ 1)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂(𝑘₂₄+ 1) + 𝑝_𝐻𝛽^∗𝑘₂₂𝑘₂₃¹⁰) 𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂𝑘₂₃¹⁰𝑘₂₄ ,

(2.72) where𝑘₂₃¹⁰represents𝑘₂₃when𝜂₁₅ → 1and𝑚_𝐻 → 0.

𝜂lim15→1 𝑚𝐻→1

ℜ^𝐻₀ = 𝛼^∗𝑀_𝐻𝜔_𝐻𝜖_𝐻^∗𝜖_𝐻𝜂((1 − 𝛽^∗)(𝑘₂₃¹¹𝑘₂₄+ 𝑙_𝐻(𝑘₂₄+ 1)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂(𝑘₂₄+ 1) + 𝑝_𝐻𝛽^∗𝑘₂₂𝑘₂₃¹¹) 𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₁₁𝑘₂₂𝑘₂₃¹¹𝑘₂₄ ,

(2.73)

2.3 | BASIC REPRODUCTION NUMBER STUDY

where𝑘₂₃¹¹ represents𝑘₂₃ when𝜂₁₅ → 1and𝑚_𝐻 → 1.

𝜂lim15→0 𝑚𝐻→0

ℜ^𝐻₀ = 𝛼^∗𝑀_𝐻𝜔_𝐻𝜖_𝐻^∗𝜖_𝐻𝜂((1 − 𝛽^∗)(𝑘₂₃⁰⁰𝑘₂₄+ 𝑙_𝐻𝑘₂₄) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂𝑘₂₄+ 𝑝_𝐻𝛽^∗𝑘₂₂𝑘₂₃⁰⁰) 𝑁𝐻(𝛼𝐻 𝐷+ 𝜇𝐻 + 𝜇)𝑘11𝑘22𝑘₂₃⁰⁰𝑘24

, (2.74) where𝑘₂₃⁰⁰ represents𝑘₂₃ when𝜂₁₅ → 0and𝑚_𝐻 → 0.

We define

Δ_𝐻₉ = 𝑘₂₁𝑘₂₂𝑘₂₃⁰⁰𝑘₂₄

(1 − 𝛽^∗)(𝑘₂₃¹¹𝑘₂₄+ 𝑙_𝐻𝑘₂₄) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂𝑘₂₄+ 𝑝_𝐻𝛽^∗𝑘₂₂𝑘₂₃⁰⁰, (2.75) Δ_𝐻₁₀ = 𝑘₂₁𝑘₂₂𝑘₂₃⁰¹𝑘₂₄

(1 − 𝛽^∗)(𝑘₂₃⁰¹𝑘24+ 𝑙𝐻𝑘24) + (1 − 𝑝𝐻)𝛽^∗𝑘22𝑘24+ 𝑝𝐻𝛽^∗𝑘22𝑘₂₃⁰¹, (2.76) Δ𝐻11 = 𝑘21𝑘22𝑘₂₃¹⁰𝑘24

(1 − 𝛽^∗)(𝑘₂₃¹⁰𝑘₂₄+ 𝑙_𝐻(𝑘₂₄+ 1)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂(𝑘₂₄+ 1) + 𝑝_𝐻𝛽^∗𝑘₂₂𝑘₂₃¹⁰, (2.77) Δ_𝐻₁₂ = 𝑘₂₁𝑘₂₂𝑘₂₃¹¹𝑘₂₄

(1 − 𝛽^∗)(𝑘₂₃¹¹𝑘₂₄+ 𝑙_𝐻(𝑘₂₄+ 1)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂(𝑘₂₄+ 1) + 𝑝_𝐻𝛽^∗𝑘₂₂𝑘₂₃¹¹. (2.78) Then, we have the following result:

Lemma 2.3.12. 1. The impact when𝜂₁₅ → 0and𝑚_𝐻 → 0is positive in reducing TB transmission in TB-HIV/AIDS subpopulation only ifΔ_𝐻 < Δ_𝐻₉, no impact ifΔ_𝐻 = Δ_𝐻₉ and a negative impact ifΔ_𝐻 > Δ_𝐻₉.

2. The impact when 𝜂₁₅ → 0and𝑚_𝐻 → 1is positive in reducing TB transmission in TB-HIV/AIDS subpopulation only ifΔ𝐻 < Δ𝐻10, no impact ifΔ𝐻 = Δ𝐻10 and a negative impact ifΔ_𝐻 > Δ_𝐻₁₀.

3. The impact when 𝜂₁₅ → 1and𝑚_𝐻 → 0is positive in reducing TB transmission in TB-HIV/AIDS subpopulation only ifΔ_𝐻 < Δ_𝐻₁₁, no impact ifΔ_𝐻 = Δ_𝐻₁₁ and a negative impact ifΔ𝐻 > Δ𝐻11.

4. The impact when 𝜂15 → 1and𝑚𝐻 → 1is positive in reducing TB transmission in TB-HIV/AIDS subpopulation only ifΔ_𝐻 < Δ_𝐻₁₂, no impact ifΔ_𝐻 = Δ_𝐻₁₂ and a negative impact ifΔ_𝐻 > Δ_𝐻₁₂.

We studied the relationships between resistance (𝑙_𝐷, 𝜂₁₆) and recovered (𝜂₁₂, 𝑚_𝐻) pa-rameters. We have:

𝑙lim𝐻→1 𝜂15→1 𝜂12→0 𝑚𝐻→0

ℜ^𝐻₀ = 𝛼^∗𝑀_𝐻𝜔_𝐻𝜖_𝐻^∗𝜖_𝐻𝜂((1 − 𝛽^∗)(𝑘₂₃¹⁰𝑘₂₄+ (𝑘₂₄+ 1)) + (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂¹⁰(𝑘₂₄+ 1) + 𝑝_𝐻𝛽^∗𝑘₂₂¹⁰𝑘₂₃¹⁰) 𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂¹⁰𝑘₂₃¹⁰𝑘₂₄ ,

(2.79)

𝑙lim𝐻→0 𝜂15→0 𝜂12→1 𝑚𝐻→1

ℜ^𝐻₀ = 𝛼^∗𝑀_𝐻𝜔_𝐻𝜖_𝐻^∗𝜖_𝐻𝜂((1 − 𝛽^∗)𝑘₂₃⁰¹𝑘₂₄+ (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰¹𝑘₂₄+ 𝑝_𝐻𝛽^∗𝑘₂₂⁰¹𝑘₂₃⁰¹)

𝑁_𝐻(𝛼_{𝐻 𝐷}+ 𝜇_𝐻 + 𝜇)𝑘₂₁𝑘₂₂⁰¹𝑘₂₃⁰¹𝑘₂₄ . (2.80)

We define

Δ_𝐻₁₃ = 𝑘₂₁𝑘₂₂¹⁰𝑘₂₃¹⁰𝑘₂₄

(1 − 𝛽^∗)(𝑘₂₃¹⁰𝑘24+ (𝑘24+ 1)) + (1 − 𝑝𝐻)𝛽^∗𝑘₂₂¹⁰(𝑘24+ 1) + 𝑝𝐻𝛽^∗𝑘₂₂¹⁰𝑘₂₃¹⁰, (2.81) Δ𝐻14 = 𝑘₂₁𝑘₂₂⁰¹𝑘₂₃⁰¹𝑘₂₄

(1 − 𝛽^∗)𝑘₂₃⁰¹𝑘₂₄+ (1 − 𝑝_𝐻)𝛽^∗𝑘₂₂⁰¹𝑘₂₄+ 𝑝_𝐻𝛽^∗𝑘₂₂⁰¹𝑘₂₃⁰¹. (2.82) We obtain the following results:

Lemma 2.3.13. 1. The impact of the resistance parameters when they tend to unity (𝑙𝐻, 𝜂15→ 1) with respect to the recovery parameters when they tend to zero (𝜂12, 𝑚𝐻 → 0) is positive in reducing tuberculosis transmission in TB-HIV/AIDS subpopulation only ifΔ_𝐻 < Δ_𝐻₁₃, no impact ifΔ_𝐻 = Δ_𝐻₁₃ and a negative impact ifΔ_𝐻 > Δ_𝐻₁₃.

2. The impact of the recovery parameters recovery parameters when they tend to unity (𝜂12, 𝑚𝐻 → 1) with respect to the recovery parameters when they tend to zero (𝑙𝐻, 𝜂15 → 0) is positive in reducing tuberculosis transmission in TB-HIV/AIDS subpopulation only ifΔ_𝐻 < Δ_𝐻₁₄, no impact ifΔ_𝐻 = Δ_𝐻₁₄ and a negative impact ifΔ_𝐻 > Δ_𝐻₁₄.

Endemic Equilibrium Point

To find the endemic equilibrium point, the subsystem (2.48) is transformed into the following system of equations:

Proceeding analogously to the TB-Only submodel (2.10), we obtain the following

2.3 | BASIC REPRODUCTION NUMBER STUDY

result:

Theorem 2.3.14. The TB-HIV/AIDS submodel (2.48) has a unique endemic equilibrium point 𝜖_∗^𝐻, wheneverℜ^𝐻₀ > 1.

No documento Mathematical models for the study of adherence to tuberculosis treatment taking into account the effects of HIV/AIDS and diabetes (páginas 52-59)