The submodel that relates TB to diabetes is obtained when𝑆𝐻 = 𝑆𝑇 = 𝐸𝐻 = 𝐸𝑇 = 𝐼𝐻1 = 𝐼𝐻2 = 𝐼𝑇1 = 𝐼𝑇2 = 𝐼𝐻3 = 𝑅𝐻 = 𝐼𝑇3 = 𝑅𝑇 = 0and is given by the system:
𝑑𝑆𝐷
𝑑𝑡 = 𝑀𝐷− (𝛼𝐻 + 𝜇 + 𝜇𝐷 + 𝜔𝐷𝜆𝐷)𝑆𝐷, 𝑑𝐸𝐷
𝑑𝑡 = 𝜔𝐷𝜆𝐷(𝑆𝐷+ 𝛽1′𝑅𝐷) − (𝜂 + 𝜇 + 𝜇𝐷+ 𝛼𝐻)𝐸𝐷, 𝑑𝐼𝐷1
𝑑𝑡 = (1 − 𝛽∗)𝜖𝐷∗𝜂𝐸𝐷− (𝑙𝐷 + 𝑡𝐻𝛼𝐻 + 𝜇 + 𝜇𝐷 + 𝑑𝑇 𝐷 + 𝜂13)𝐼𝐷1, 𝑑𝐼𝐷2
𝑑𝑡 = (1 − 𝑝𝐷)𝜖𝐷∗𝛽∗𝜂𝐸𝐷+ 𝑙𝐷𝐼𝐷1 − (𝑡𝐻𝛼𝐻 + 𝑚𝐷 + 𝜇 + 𝜇𝐷 + 𝑡𝐷′𝑑𝑇 𝐷+ 𝜂16)𝐼𝐷2, 𝑑𝐼𝐷3
𝑑𝑡 = 𝑝𝐷𝛽∗𝜖𝐷∗𝜂𝐸𝐷+ 𝜂16𝐼𝐷2− (𝜂∗13+ 𝑡𝐻𝛼𝐻 + 𝜇 + 𝜇𝐷+ 𝑡𝐷∗𝑑𝑇 𝐷)𝐼𝐷3, 𝑑𝑅𝐷
𝑑𝑡 = 𝑚𝐷𝐼𝐷2 + 𝜂13𝐼𝐷1+ 𝜂∗13𝐼𝐷3− (𝛼𝐻 + 𝜇 + 𝜇𝐷+ 𝛽1′𝜔𝐷𝜆𝐷)𝑅𝐷, (2.84) with non-negative initial conditions and
𝜆𝐷 = 𝛼∗𝜖𝐷(𝐼𝐷1+ 𝐼𝐷2 + 𝐼𝐷3)
𝑁𝐷 ,
where𝑁𝐷 = 𝑆𝐷+ 𝐸𝐷+ 𝐼𝐷1 + 𝐼𝐷2+ 𝐼𝐷3 + 𝑅𝐷.
Considering biological constraints, the system (2.84) will be studied in the following region biologically feasible:
𝐷3= {
(𝑆𝐷, 𝐸𝐷, 𝐼𝐷1, 𝐼𝐷2, 𝐼𝐷3, 𝑅𝐷) ∈ ℝ6+∶ 𝑁𝐷(𝑡) ≤ 𝑀𝐷 𝜇
} .
It can be easily shown that solution(𝑆𝐷(𝑡), 𝐼𝐷1(𝑡), 𝐼𝐷2(𝑡), 𝐼𝐷3(𝑡), 𝑅𝐷(𝑡))of the system are bounded and positively invariant.
The disease-free equilibrium point, 𝜖0𝐷, is given by 𝜖0𝐷 =
(𝑆0𝐷, 0, 0, 0, 0, 0
), where 𝑆0𝐷 = 𝑀𝐷
𝜇 + 𝜇𝐷+ 𝛼𝐻.
The matrices for the new infection terms,𝐹𝐷and the other terms,𝑉𝐷are given by:
𝐹𝐷 =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0 𝑀𝐷𝛼∗𝜔𝐷𝜖𝐷
𝛼𝐻 + 𝜇 + 𝜇𝐷
𝑀𝐷𝛼∗𝜔𝐷𝜖𝐷
𝛼𝐻 + 𝜇 + 𝜇𝐷
𝑀𝐷𝛼∗𝜔𝐷𝜖𝐷
𝛼𝐻 + 𝜇 + 𝜇𝐷
0 0 0 0
0 0 0 0
0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎦ ,
𝑉𝐷 =
⎡
⎢
⎢
⎢
⎣
𝑘31 0 0 0
−(1 − 𝛽∗)𝜖𝐷∗𝜂 𝑘32 0 0
−(1 − 𝑝𝐷)𝜖𝐷∗𝛽∗𝜂 −𝑙𝐷 𝑘33 0
−𝑝𝐷𝜖𝐷∗𝛽∗𝜂 0 −𝜂16 𝑘34
⎤
⎥
⎥
⎥
⎦ ,
where𝑘31 = 𝛼𝐻 + 𝜖𝐷∗𝜂 + 𝜇 + 𝜇𝐷, 𝑘32 = 𝑙𝐷+ 𝜇 + 𝑑𝑇 𝐷 + 𝜂13+ 𝑡𝐻𝛼𝐻 + 𝜇𝐷, 𝑘33 = 𝜇 + 𝑡𝐷′𝑑𝑇 𝐷+ 𝜂16+ 𝑚𝐷+ 𝑡𝐻𝛼𝐻 + 𝜇𝐷,and𝑘34 = 𝜇 + 𝜇𝐷 + 𝑡𝐷∗𝑑𝑇 𝐷+ 𝜂∗13+ 𝑡𝐻𝛼𝐻.
The basic reproduction number is given by
ℜ𝐷0 = 𝜌(𝐹𝐻𝑉𝐻−1) = 𝛼∗𝜖𝐷𝜔𝐷𝑀𝐷((1 − 𝛽∗)𝜖𝐷∗𝜂(𝑘33𝑘34+ 𝑙𝐷(𝑘34+ 𝜂16)) + (1 − 𝑝𝐷)𝜖𝐷∗𝛽∗𝜂𝑘32(𝑘34+ 𝜂16) + 𝑘32𝑘33𝜖𝐷∗𝛽∗𝜂𝑝𝐷) 𝑁𝐷(𝛼𝐻+ 𝜇 + 𝜇𝐷)𝑘31𝑘32𝑘33𝑘34
. (2.85)
We define(𝐻1)and(𝐻2)as in the previous submodels (2.10) and (2.48) and using the same idea from the demonstration, we have the following results:
Lemma 2.3.15. The disease-free equilibrium𝜖0𝐷is asymptotically stable whenℜ𝐷0 < 1and is unstable wheneverℜ𝐷0 > 1.
Lemma 2.3.16. The fixed point 𝐸0𝐷 = (𝑆0𝐷∗, 0, 0, 0, 0) where 𝑆0𝐷∗ = (
𝑀𝐷 𝜇 + 𝛼𝐻 + 𝜇𝐷, 0
) is a globally asymptotically stable equilibrium of submodel (TB-Diabetes) if ℜ𝐷0 < 1and the assumption(𝐻1)and(𝐻2)are satisfied.
We make a procedure analogous to the previous submodels for𝑙𝐷 and𝜂16(MDR-TB and XDR-TB parameters for TB-Diabetes submodel) and we obtain the following limits:
𝑙lim𝐷→0 𝜂16→0
ℜ𝐷0 = 𝛼∗𝑀𝐷𝜔𝐷𝜖𝐷𝜖𝐷∗𝜂((1 − 𝛽∗)𝑘330𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘320𝑘34+ 𝛽∗𝑝𝐷𝑘320𝑘330) 𝑁𝐷(𝜇𝐷+ 𝛼𝐻 + 𝜇)𝑘31𝑘320𝑘330𝑘34
, (2.86) where 𝑘320 is 𝑘32 for 𝑙𝐷 = 0 and 𝑘330 is 𝑘33 for 𝜂16 = 0. Then, in practice 𝑙𝐷 → 0 and 𝜂16 → 0imply zero resistance, i.e. elimination of resistance to tuberculosis treatment. If the limit (2.86) are greater than unity, then when𝑙𝐷, 𝜂16 → 0has a negative impact on TB transmission control. That is, if we have
𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝜂𝑀𝐷
𝑁𝐷(𝛼ℎ+ 𝜇𝐷+ 𝜇) > 𝑘31𝑘320𝑘330𝑘34
(1 − 𝛽∗)𝑘330𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘320𝑘34+ 𝛽∗𝑝𝐷𝑘320𝑘330 . (2.87) Now, we study the case when𝑙𝐷 → 1and𝜂16→ 0. We have
𝑙lim𝐷→1 𝜂16→0
ℜ𝐷0 = 𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷𝜂((1 − 𝛽∗)(𝑘330𝑘34+ 𝑘34) + (1 − 𝑝𝐷)𝛽∗𝑘132𝑘34+ 𝛽∗𝑝𝐷𝑘321𝑘330)
𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇)𝑘31𝑘321𝑘330𝑘34 , (2.88) where𝑘321 is𝑘32for𝑙𝐷 = 1. Then If the limit (2.88) are greater than unity, then when𝑙𝐷 → 1 and𝜂16 → 0has a negative impact on TB transmission control. That is, if we have
𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷
𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇) > 𝑘31𝑘321𝑘330𝑘34
(1 − 𝛽∗)(𝑘330𝑘34+ 𝑘34) + (1 − 𝑝𝐷)𝛽∗𝑘321𝑘34+ 𝛽∗𝑝𝐷𝑘321𝑘330 . (2.89)
2.3 | BASIC REPRODUCTION NUMBER STUDY
In the case when𝑙𝐷 → 0and𝜂16 → 1. We have
𝑙lim𝐷→0 𝜂16→1
ℜ𝐷0 = 𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷𝜂((1 − 𝛽∗)𝑘34𝑘331 + (1 − 𝑝𝐷)𝛽∗𝑘320(𝑘34+ 1) + 𝑘320𝑘331𝛽∗𝑝𝐷)
𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇)𝑘31𝑘320𝑘331𝑘34 , (2.90) where𝑘331 is𝑘33 for𝜂16 = 1and𝑘331 = 𝑘330 + 1. If the limit (2.90) are greater than unity, then when𝑙𝐷 → 0and𝜂16 → 1has a negative impact on TB transmission control. That is, if we have
𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷
𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇) > 𝑘31𝑘320𝑘331𝑘34
(1 − 𝛽∗)𝑘34𝑘331 + (1 − 𝑝𝐷)𝛽∗𝑘320(𝑘34+ 1) + 𝑘320𝑘331𝛽∗𝑝𝐷. (2.91) For𝑙𝐷 → 1and𝜂16→ 1. We have
𝑙lim𝐷→1 𝜂16→1
ℜ𝐷0 = 𝛼∗𝑀𝐷𝜔𝐷𝜖𝐷𝜖𝐷∗𝜂((1 − 𝛽∗)(𝑘34𝑘331 + (𝑘34+ 1)) + (1 − 𝑝𝐷)𝛽∗𝑘321(𝑘34+ 1) + 𝑘321𝑘331𝛽∗𝑝𝐷) 𝑁𝐷(𝛼𝐻 + 𝜇𝐷 + 𝜇)𝑘31𝑘321𝑘331𝑘34 .
(2.92) If the limit (2.92) are greater than unity, then when𝑙𝐷 → 1and𝜂16 → 1has a negative impact on TB transmission control. That is, if we have
𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷
𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇) > 𝑘31𝑘321𝑘331𝑘34
(1 − 𝛽∗)(𝑘34𝑘331 + (𝑘34+ 1))(1 − 𝑝𝐷)𝛽∗𝑘321(𝑘34+ 1) + 𝑘321𝑘331𝛽∗𝑝𝐷. (2.93) Let us consider the following expressions:
Δ𝐷 = 𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷
𝑁𝐷(𝛼𝐷 + 𝜇𝐷 + 𝜇), (2.94)
Δ𝐷1 = 𝑘21𝑘220𝑘230𝑘24
(1 − 𝛽∗)𝑘330𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘320𝑘34+ 𝛽∗𝑝𝐷𝑘320𝑘330 , (2.95) Δ𝐷2 = 𝑘31𝑘321𝑘330𝑘34
(1 − 𝛽∗)(𝑘330𝑘34+ 𝑘34) + (1 − 𝑝𝐷)𝛽∗𝑘321𝑘34+ 𝛽∗𝑝𝐷𝑘321𝑘330 , (2.96) Δ𝐷3 = 𝑘31𝑘320𝑘331𝑘34
(1 − 𝛽∗)𝑘34𝑘331 + (1 − 𝑝𝐷)𝛽∗𝑘320(𝑘34+ 1) + 𝑘320𝑘331𝛽∗𝑝𝐷
, (2.97)
Δ𝐷4 = 𝑘31𝑘321𝑘331𝑘34
(1 − 𝛽∗)(𝑘34𝑘331 + (𝑘34+ 1))(1 − 𝑝𝐷)𝛽∗𝑘321(𝑘34+ 1) + 𝑘321𝑘331𝛽∗𝑝𝐷. (2.98) We have the following result:
Lemma 2.3.17. 1. The impact when𝑙𝐷 → 0 and 𝜂16 → 0 is positive in reducing TB transmission in this subpopulation only if Δ𝐷 < Δ𝐷1, no impact if Δ𝐻 = Δ𝐷1 and a negative impact ifΔ𝐻 > Δ𝐷1.
2. The impact when 𝑙𝐻 → 1and 𝜂16 → 0is positive in reducing TB transmission in this subpopulation only ifΔ𝐷 < Δ𝐷2, no impact ifΔ𝐷 = Δ𝐷2 and a negative impact if Δ𝐷 > Δ𝐷2.
3. The impact wheN 𝑙𝐷 → 0and 𝜂16 → 1is positive in reducing TB transmission in this subpopulation only ifΔ𝐷 < Δ𝐷3, no impact ifΔ𝐷 = Δ𝐷3 and a negative impact if
Δ𝐷 > Δ𝐷3.
4. The impact when 𝑙𝐷 → 1 and 𝜂16 → 1 is positive in reducing TB transmission in this subpopulation only ifΔ𝐷 < Δ𝐷4, no impact ifΔ𝐷 = Δ𝐷4 and a negative impact if Δ𝐷 > Δ𝐷4.
We will study the relationship between resistance and recovery parameters. First, we start with the relationship between𝑙𝐷 and𝜂13. We have:
𝑙lim𝐷→0 𝜂13→1
ℜ𝐷0 = 𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷𝜂((1 − 𝛽∗)𝑘33𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘3201(𝑘34+ 𝜂16) + 𝑝𝐷𝛽∗𝑘3201𝑘33)
𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇)𝑘31𝑘3201𝑘33𝑘34 , (2.99) where𝑘3201represents𝑘32when𝑙𝐷 → 0and𝜂13 → 1.
𝑙lim𝐷→1 𝜂13→0
ℜ𝐷0 = 𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷𝜂((1 − 𝛽∗)(𝑘33𝑘34+ (𝑘34+ 𝜂16)) + (1 − 𝑝𝐷)𝛽∗𝑘3210(𝑘34+ 𝜂16) + 𝑝𝐷𝛽∗𝑘3210𝑘33) 𝑁𝐷(𝛼𝐷+ 𝜇𝐷+ 𝜇)𝑘31𝑘3210𝑘33𝑘34
, (2.100) where𝑘3210represents𝑘32when𝑙𝐷 → 1and𝜂13 → 0.
𝑙lim𝐷→1 𝜂13→1
ℜ𝐷0 = 𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷𝜂((1 − 𝛽∗)(𝑘33𝑘34+ (𝑘34+ 𝜂16)) + (1 − 𝑝𝐷)𝛽∗𝑘3211(𝑘34+ 𝜂16) + 𝑝𝐷𝛽∗𝑘3211𝑘33) 𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇)𝑘31𝑘3211𝑘33𝑘34 ,
(2.101) where𝑘3211represents𝑘32when𝑙𝐷 → 1and𝜂13 → 1.
𝑙lim𝐷→0 𝜂13→0
ℜ𝐷0 = 𝛼∗𝜔𝐷𝜖𝐷𝜖𝐷∗𝑀𝐷𝜂((1 − 𝛽∗)𝑘33𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘3200(𝑘34+ 𝜂16) + 𝑝𝐷𝛽∗𝑘3231𝑘33) 𝑁𝐷(𝛼𝐻 + 𝜇𝐷 + 𝜇)𝑘31𝑘3200𝑘33𝑘34
, (2.102) where𝑘3200represents𝑘32when𝑙𝐷 → 0and𝜂13 → 0.
Let us consider the following expressions:
Δ𝐷5 = 𝑘31𝑘3200𝑘33𝑘34
(1 − 𝛽∗)𝑘33𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘3200𝑘34+ 𝑝𝐷𝛽∗𝑘3200𝑘33, (2.103) Δ𝐷6 = 𝑘31𝑘3201𝑘33𝑘34
(1 − 𝛽∗)𝑘33𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘3201(𝑘34+ 𝜂16) + 𝑝𝐷𝛽∗𝑘3201𝑘33, (2.104) Δ𝐷7 = 𝑘31𝑘3210𝑘33𝑘34
(1 − 𝛽∗)(𝑘33𝑘34+ (𝑘34+ 𝜂16)) + (1 − 𝑝𝐷)𝛽∗𝑘3210(𝑘34+ 𝜂16) + 𝑝𝐷𝛽∗𝑘3210𝑘33, (2.105) Δ𝐷8 = 𝑘31𝑘3211𝑘33𝑘34
(1 − 𝛽∗)(𝑘33𝑘34+ (𝑘34+ 𝜂16)) + (1 − 𝑝𝐷)𝛽∗𝑘3211(𝑘34+ 𝜂16+ 𝑝𝐷𝛽∗𝑘3211𝑘33
. (2.106)
We have the following results:
Lemma 2.3.18. 1. The impact when 𝑙𝐷 → 0and 𝜂13 → 0is positive in reducing TB transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷5, no impact ifΔ𝐷 = Δ𝐷5 and a negative impact ifΔ𝐷 > Δ𝐷5.
2. The impact when 𝑙𝐷 → 0 and 𝜂13 → 1 is positive in reducing TB transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷6, no impact ifΔ𝐷 = Δ𝐷6 and a negative
2.3 | BASIC REPRODUCTION NUMBER STUDY
impact ifΔ𝐷 > Δ𝐷6.
3. The impact when 𝑙𝐷 → 1 and𝜂13 → 0is positive in reducing TB transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷7, no impact if Δ𝐷 = Δ𝐷7 and a negative impact ifΔ𝐷 > Δ𝐷7.
4. The impact when 𝑙𝐷 → 1 and𝜂13 → 1is positive in reducing TB transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷8, no impact if Δ𝐷 = Δ𝐷8 and a negative impact ifΔ𝐷 > Δ𝐷8.
Now, we study the relationship between the XDR-TB parameter (𝜂16) and recovery (𝑚𝐷). We have:
𝜂lim16→0 𝑚𝐷→1
ℜ𝐷0 = 𝛼∗𝑀𝐷𝜔𝐷𝜖𝐷∗𝜖𝐷𝜂((1 − 𝛽∗)(𝑘3301𝑘34+ 𝑙𝐷𝑘34) + (1 − 𝑝𝐷)𝛽∗𝑘32𝑘34+ 𝑝𝐷𝛽∗𝑘32𝑘3301) 𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇)𝑘31𝑘32𝑘3301𝑘34 ,
(2.107) where𝑘3301 represents𝑘33 when𝜂16 → 0and𝑚𝐷 → 1.
𝜂lim16→1 𝑚𝐷→0
ℜ𝐷0 = 𝛼∗𝑀𝐷𝜔𝐷𝜖𝐷∗𝜖𝐷𝜂((1 − 𝛽∗)(𝑘3310𝑘34+ 𝑙𝐷(𝑘34+ 1)) + (1 − 𝑝𝐷)𝛽∗𝑘32(𝑘34+ 1) + 𝑝𝐷𝛽∗𝑘32𝑘3310) 𝑁𝐷(𝛼𝐻 + 𝜇𝐷 + 𝜇)𝑘31𝑘32𝑘3310𝑘34 ,
(2.108) where𝑘3310 represents𝑘33 when𝜂16 → 1and𝑚𝐷 → 0.
𝜂lim15→1 𝑚𝐷→1
ℜ𝐷0 = 𝛼∗𝑀𝐷𝜔𝐷𝜖𝐷∗𝜖𝐷𝜂((1 − 𝛽∗)(𝑘3311𝑘34+ 𝑙𝐷(𝑘34+ 1)) + (1 − 𝑝𝐷)𝛽∗𝑘32(𝑘34+ 1) + 𝑝𝐷𝛽∗𝑘32𝑘3311) 𝑁𝐷(𝛼𝐻 + 𝜇𝐷 + 𝜇)𝑘31𝑘32𝑘3311𝑘34 ,
(2.109) where𝑘3311 represents𝑘33 when𝜂16 → 1and𝑚𝐷 → 1.
𝜂lim15→0 𝑚𝐷→0
ℜ𝐷0 = 𝛼∗𝑀𝐷𝜔𝐷𝜖𝐷∗𝜖𝐷𝜂((1 − 𝛽∗)𝑘3300𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘32𝑘34+ 𝑝𝐷𝛽∗𝑘32𝑘3300)
𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇)𝑘31𝑘32𝑘3300𝑘34 , (2.110) where𝑘3300 represents𝑘33 when𝜂16 → 0and𝑚𝐷 → 0.
Let us define:
Δ𝐷9 = 𝑘31𝑘32𝑘3300𝑘34
(1 − 𝛽∗)𝑘3300𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘32𝑘34+ 𝑝𝐷𝛽∗𝑘32𝑘3300, (2.111) Δ𝐷10 = 𝑘31𝑘32𝑘3301𝑘34
(1 − 𝛽∗)(𝑘3301𝑘34+ 𝑙𝐷𝑘34) + (1 − 𝑝𝐷)𝛽∗𝑘32𝑘34+ 𝑝𝐷𝛽∗𝑘32𝑘3301, (2.112) Δ𝐷11 = 𝑘31𝑘32𝑘3310𝑘34
(1 − 𝛽∗)(𝑘3310𝑘34+ 𝑙𝐷(𝑘34+ 1)) + (1 − 𝑝𝐷)𝛽∗𝑘32(𝑘34+ 1) + 𝑝𝐷𝛽∗𝑘32𝑘3310, (2.113) Δ𝐷12 = 𝑘31𝑘32𝑘3311𝑘34
(1 − 𝛽∗)(𝑘3311𝑘34+ 𝑙𝐷(𝑘34+ 1)) + (1 − 𝑝𝐷)𝛽∗𝑘32(𝑘34+ 1) + 𝑝𝐷𝛽∗𝑘32𝑘3311. (2.114) We obtain the following result:
Lemma 2.3.19. 1. The impact when𝜂16 → 0and 𝑚𝐷 → 0is positive in reducing TB transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷9, no impact ifΔ𝐷 = Δ𝐷9
and a negative impact ifΔ𝐷 > Δ𝐷9.
2. The impact when𝜂16 → 0and𝑚𝐷 → 1is positive in reducing TB transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷10, no impact ifΔ𝐷 = Δ𝐷10 and a negative impact ifΔ𝐷 > Δ𝐷10.
3. The impact when𝜂16 → 1and𝑚𝐷 → 0is positive in reducing TB transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷11, no impact ifΔ𝐷 = Δ𝐷11 and a negative impact ifΔ𝐷 > Δ𝐷11.
4. The impact when 𝜂16 → 1 and 𝑚𝐷 → is positive in reducing TB transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷12, no impact ifΔ𝐷 = Δ𝐷12 and a negative impact ifΔ𝐷 > Δ𝐷12.
Studying the resistance parameters (𝑙𝐷, 𝜂16) respect to the recovery parameters (𝜂13, 𝑚𝐷).
We have:
𝑙lim𝐷→1 𝜂16→1 𝜂13→0 𝑚𝐷→0
ℜ𝐷0 = 𝛼∗𝑀𝐷𝜔𝐷𝜖𝐷∗𝜖𝐷𝜂((1 − 𝛽∗)(𝑘3310𝑘34+ (𝑘34+ 1)) + (1 − 𝑝𝐷)𝛽∗𝑘3210(𝑘34+ 1) + 𝑝𝐷𝛽∗𝑘3210𝑘3310) 𝑁𝐷(𝛼𝐻 + 𝜇𝐷 + 𝜇)𝑘31𝑘3210𝑘3310𝑘34
, (2.115)
𝑙lim𝐷→0 𝜂16→0 𝜂13→1 𝑚𝐷→1
ℜ𝐷0 = 𝛼∗𝑀𝐷𝜔𝐷𝜖𝐷∗𝜖𝐷𝜂((1 − 𝛽∗)𝑘3301𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘3201𝑘34+ 𝑝𝐷𝛽∗𝑘3201𝑘3301)
𝑁𝐷(𝛼𝐻 + 𝜇𝐷+ 𝜇)𝑘31𝑘0132𝑘3301𝑘34 . (2.116)
Let us consider:
Δ𝐷13 = 𝑘31𝑘3210𝑘3310𝑘34
(1 − 𝛽∗)(𝑘3310𝑘34+ (𝑘34+ 1)) + (1 − 𝑝𝐷)𝛽∗𝑘3210(𝑘34+ 1) + 𝑝𝐷𝛽∗𝑘3210𝑘3310, (2.117) Δ𝐷14 = 𝑘31𝑘3201𝑘3301𝑘34
(1 − 𝛽∗)𝑘3301𝑘34+ (1 − 𝑝𝐷)𝛽∗𝑘3201𝑘34+ 𝑝𝐷𝛽∗𝑘3201𝑘3301. (2.118) We obtain the following results:
Lemma 2.3.20. 1. The impact of the resistance parameters when𝑙𝐷, 𝜂16 → 1with respect to the recovery parameters when 𝜂13, 𝑚𝐷 → 0is positive in reducing tuberculosis transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷13, no impact ifΔ𝐷 = Δ𝐷13 and a negative impact ifΔ𝐷 > Δ𝐷13.
2. The impact of the recovery parameters recovery parameters when they tend to unity (𝜂13, 𝑚𝐷 → 1) with respect to the recovery parameters when they tend to zero (𝑙𝐷, 𝜂16 → 0) is positive in reducing tuberculosis transmission in TB-Diabetes subpopulation only ifΔ𝐷 < Δ𝐷14, no impact ifΔ𝐷 = Δ𝐷14 and a negative impact ifΔ𝐷 > Δ𝐷14.
Endemic Equilibrium Point
To find the endemic equilibrium point the subsystem (2.84) is transformed into the following system of equations:
2.3 | BASIC REPRODUCTION NUMBER STUDY
Analogous to the procedure applied to the previous submodel (2.10), we can obtain the following result:
Theorem 2.3.21. The Diabetes-TB submodel (2.84) has a unique endemic equilibrium point 𝜖∗𝐷,wheneverℜ𝐷0 > 1.
Persistence
In previous sections, we worked with 𝛼∗ (effective contact rate) as a constant for a given situation. In many situations we can see it as depending on the situation/region and it takes different values. In general,𝛼∗relates to the level of contagion/propagation of the disease. Now, to study persistence, we will consider𝛼∗as dependent on𝑁𝐷 (total population) and a particular case𝑡𝐻 = 1.
𝑁𝐷 and express (2.84) in these terms, as followed:
𝑁𝐷 = 𝑆𝐷+ 𝐸𝐷+ 𝐼𝐷1 + 𝐼𝐷2+ 𝐼𝐷3 + 𝑅𝐷,
𝑑𝑥2
𝑑𝑡 = 𝜔𝐷𝜖𝐷𝛼∗(𝑥3+ 𝑥4+ 𝑥5)(𝑥1+ 𝛽1′𝑥6) − (
𝑀𝐷 𝑁𝐷 − 𝜖𝐷∗𝜂
)𝑥2+ 𝑑𝑇 𝐷(𝑥3+ 𝑡𝐷′𝑥4+ 𝑡𝐷∗𝑥5)𝑥2, 𝑑𝑥3
𝑑𝑡 = (1 − 𝛽∗)𝜖𝐷∗ −
(𝑙𝐷+ 𝜂13+ 𝑀𝐷
𝑁𝐷)𝑥4+ 𝑑𝑇 𝐷(𝑥3+ 𝑡𝐷′𝑥4+ 𝑡𝐷∗𝑥5− 1)𝑥3, 𝑑𝑥4
𝑑𝑡 = (1 − 𝑝𝐷)𝛽∗𝜖𝐷∗ −
(𝑚𝐷+ 𝜂16+𝑀𝐷
𝑁𝐷)𝑥4+ 𝑑𝑇 𝐷(𝑥3+ 𝑡𝐷′𝑥4+ 𝑡𝐷∗𝑥5− 𝑡𝐷′)𝑥4, 𝑑𝑥5
𝑑𝑡 = 𝑝𝐷𝛽∗𝜖𝐷∗ −
(𝜂∗13+ 𝑀𝐷
𝑁𝐷)𝑥4+ 𝑑𝑇 𝐷(𝑥3+ 𝑡𝐷′𝑥4+ 𝑡𝐷∗𝑥5− 𝑡𝐷∗)𝑥5, 𝑑𝑥6
𝑑𝑡 = 𝑚𝐷𝑥4+ 𝜂13𝑥3+ 𝜂∗13𝑥5− 𝜔𝐷𝜖𝐷𝛼∗(𝑁𝐷)(𝑥3+ 𝑥4+ 𝑥5)𝑥6− 𝑀𝐷
𝑁𝐷𝑥6+ 𝑑𝑇 𝐷(𝑥3+ 𝑡𝐷′𝑥4+ 𝑡𝐷∗𝑥5)𝑥6. (2.120) We have that𝑥1+ 𝑥2+ 𝑥3+ 𝑥4+ 𝑥5+ 𝑥6 = 1. The manifold𝑥1+ 𝑥2+ 𝑥3+ 𝑥4+ 𝑥5+ 𝑥6= 1, 𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6 ≥ 0, is forward invariant, under solution flow of (2.120), which has a global solution that satisfies (2.84). Now, let’s find the conditions under which the disease and host subpopulation will persist.
Theorem 2.3.22. Let𝛼∗(0) = 0,𝑁 (0) > 0. Then, the population is uniformly persistent, that is,
𝑡→∞liminf 𝑁 (𝑡) ≥ 𝜖, (2.121)
where𝜖 > 0does not depend on the initial data.
Proof. We have to demonstrate that the set 𝑋1=
{
𝑁 = 0, 𝑥𝑖 ≥ 0, 𝑖 = 1, ..., 6,
6
∑
𝑖=1
𝑥𝑖 = 1 }
is uniform strong repeller for 𝑋2 =
{
𝑁 > 0, 𝑥𝑖 ≥ 0, 𝑖 = 1, ..., 6,
6
∑
𝑖=1
𝑥𝑖 = 1 }
.
The following results are presented and proved in [30,117,107] and are used to demostrate the conditions of persistence.
Theorem 2.3.23. Let𝕏be a locally compact metric space with metric𝑑. Let𝕏be the disjoint union of two sets𝑋1and𝑋2such that𝑋2is compact. Let𝜙be a continuous semiflow on𝑋1. Then𝑋2is a uniform strong repeller for𝑋1.
Theorem 2.3.24. Let𝐷 be a bounded interval inℝand𝑔 ∶ (𝑡0, ∞) × 𝐷 → ℝbe bounded and uniformly continuous. Further, let𝑥 ∶ (𝑡0, ∞) → 𝐷be a solution of ̇𝑥 = 𝑔(𝑡, 𝑥), which is defined on the whole interval(𝑡0, ∞). Then there exist sequences𝑠𝑛, 𝑡𝑛 → ∞such that
𝑛→∞lim𝑔(𝑠𝑛, 𝑥∞) = 0 = lim
𝑛→∞𝑔(𝑡𝑛, 𝑥∞). (2.122)
2.3 | BASIC REPRODUCTION NUMBER STUDY
Lemma 2.3.25. If the assumptions of Theorem (2.3.24) are satisfied, then 1.
𝑡→inflim inf 𝑔(𝑡, 𝑥∞) ≤ 0 ≤ lim
𝑡→∞sup 𝑔(𝑡, 𝑥∞). (2.123) 2.
𝑡→inflim inf 𝑔(𝑡, 𝑥∞) ≤ 0 ≤ lim
𝑡→∞sup 𝑔(𝑡, 𝑥∞). (2.124) We have that the assumptions of Theorem (2.3.23) are satisfed, it suffices to show that 𝑋2is a uniform weak repulsive for𝑋1. We define
𝑟 = 𝑥2+ 𝑥3+ 𝑥4+ 𝑥5+ 𝑥6. (2.125) Then,
𝑟′ = 𝜔𝐷𝜖𝐷𝛼∗(𝑁𝐷)(𝑥3+ 𝑥4+ 𝑥5)𝑥1− 𝑀𝐷
𝑁𝐷𝑟 + 𝑑𝑇 𝐷((𝑟 − 1)(𝑥3+ 𝑡𝐷′𝑥4+ 𝑡𝐷∗𝑥5)). (2.126) Using that𝑥1, 𝑥2, 𝑥3, 𝑥4, 𝑥5, 𝑥6 ≤ 1, we have
𝑀𝐷
𝑁𝐷∞ + 𝑑𝑇 𝐷(1 − 𝑟∞)(1 + 𝑡𝐷′ + 𝑡𝐷∗) ≤ 3𝜔𝐷𝜖𝐷𝛼∗(𝑁𝐷) (2.127)
⟹ 𝛼∗(𝑁𝐷) ≥ 𝑀𝐷
3𝜔𝐷𝜖𝐷𝑁𝐷∞ + 𝑑𝑇 𝐷(1 − 𝑟∞)(1 + 𝑡𝐷′ + 𝑡𝐷∗)
3𝜔𝐷𝜖𝐷 (2.128)
From the equation of𝑁𝐷 in (2.120), we obtain
𝑡→∞liminf 1 𝑁𝐷
𝑑𝑁𝐷
𝑑𝑡 ≥ 𝑀𝐷
𝑁𝐷∞−(𝜇 +𝑑𝑇 𝐷(𝑥3∞+𝑡𝐷′𝑥4∞+𝑡𝐷∗𝑥5∞)) ≥ 𝑀𝐷
𝑁𝐷∞+(𝜇 +𝑑𝑇 𝐷(1+𝑡𝐷′ +𝑡𝐷∗)𝑟∞). (2.129) As𝑁𝐷 increase exponentially,
𝑀𝐷
𝑁𝐷∞ ≤ 𝜇 + 𝑑𝑇 𝐷(1 + 𝑡𝐷′ + 𝑡𝐷∗)𝑟∞, (2.130) that is
1
𝑑𝑇 𝐷(1 + 𝑡𝐷′ + 𝑡𝐷∗) ( 𝑀𝐷
𝑁𝐷∞ − 𝜇) ≤ 𝑟∞. (2.131)
Combining (2.128) and (2.131), we have 𝛼∗(𝑁𝐷∞) ≥ 1
3𝜔𝐷𝜖𝐷((
𝑀𝐷
𝑁𝐷∞ − 𝜇)( 𝑀𝐷
3𝜔𝐷𝜖𝐷𝑁𝐷∞(1 + 𝑡𝐷′ + 𝑡𝐷∗) − 1) + 𝑑𝑇 𝐷(1 + 𝑡𝐷′ + 𝑡𝐷∗)). (2.132) As𝛼∗(0) = 0and𝛼∗(𝑁𝐷)is continuous at 0,𝑁𝐷∞ ≥ 𝜖 > 0whit𝜖not depending on the initial data. From (2.132) we see that we can relax𝛼∗(0) = 0and require
𝛼∗(0) < 1 3𝜔𝐷𝜖𝐷((
𝑀𝐷
𝑁𝐷∞ − 𝜇)( 𝑀𝐷
3𝜔𝐷𝜖𝐷𝑁𝐷∞(1 + 𝑡𝐷′ + 𝑡𝐷∗) − 1) + 𝑑𝑇 𝐷(1 + 𝑡𝐷′ + 𝑡𝐷∗)). (2.133)
This conclude the proof.
With this result, we proved the persistence of tuberculosis in this subpopulation.
Therefore, it is necessary to apply control strategies to reduce and eradicate the disease in the community.