For the full model (4.1), infection-free equilibrium point is
𝜖0𝐺𝛼 = (𝑆0𝑇𝛼, 𝑆0𝐻𝛼, 𝑆0𝐷𝛼, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0),
and the basic reproduction number is calculated using next-generation matrix method.
The dominant eigenvalues of the next-generation matrix areℜ𝑇0𝛼, ℜ𝐻0𝛼 andℜ𝐷0𝛼. Then, the basic reproduction number of the model (4.1) is
ℜ𝛼0 = max{ℜ𝑇0𝛼, ℜ𝐻0𝛼, ℜ𝐷0𝛼}.
Global Stability
Now, using a methodology analogous to that applied to model2.5, we prove the global stability of the infection-free equilibrium point. Following [38], we can rewrite the model (4.1) as
𝑐𝔻𝛼𝑡 𝑆 = 𝐹 (𝑆, 𝐼 ),
𝑐𝔻𝛼𝑡 𝐼 = 𝐺(𝑆, 𝐼 ), 𝐺(𝑆, 0) = 0, (4.21)
where𝑆 ∈ ℝ6+ is the vector with uninfected and recovered and 𝐼 ∈ ℝ12+ have the other compartment of the model (4.1).
The disease-free equilibrium is now denoted by𝐸0𝐺𝛼 = (𝑆0𝛼, 0), 𝑆0𝛼 = (𝑆0, 0, 0, 0),𝑆0 = (𝑆0𝑇𝛼, 𝑆0𝐻𝛼, 𝑆0𝐷𝛼)where𝑆0𝑇𝛼 = 𝑀𝑇𝛼
𝜇𝛼 + 𝛼𝐷𝛼 + 𝛼𝐻𝛼,𝑆0𝐻𝛼 = 𝑀𝐻𝛼
𝜇𝛼 + 𝜇𝐻𝛼 + 𝛼𝐻 𝐷𝛼 and𝑆0𝐷𝛼 = 𝑀𝐷𝛼 𝜇𝛼 + 𝜇𝐷𝛼 + 𝛼𝐻𝛼. The conditions(𝐻1𝛼)and(𝐻2𝛼)below must be satisfied to guarantee the global asymptotic stability of𝐸𝐺0.
(𝐻1𝛼) ∶ For 𝑐𝔻𝛼𝑡 𝑆 = 𝐹 (𝑆, 0), 𝑆0𝛼 is globally asymptotically stable,
(𝐻2𝛼) ∶ 𝐺(𝑆, 𝐼 ) = 𝐴𝐼 − 𝐺∗(𝑆, 𝐼 ), 𝐺∗(𝑆, 𝐼 ) ≥ 0, for (𝑆, 𝐼 ) ∈ Ω, (4.22) where𝐴 = 𝐷𝐼𝐺(𝑆0𝛼, 0)(𝐷𝐼𝐺(𝑆0𝛼, 0)is the Jacobian of G at (𝑆0𝛼, 0)) is a M-matrix (the off-diagonal elements of𝐴are non-negative) andΩis the biologically feasible region.
The following results show the global stability of the infection-free equilibrium point:
Theorem 4.2.10. The fixed point𝐸0𝐺𝛼 is a globally asymptotically stable equilibrium (g.a.s) of model (4.1) provided thatℜ0 < 1and that the conditions(𝐻1𝛼)and(𝐻2𝛼)are satisfied.
4.2 | MODEL FORMULATION
Since𝑆𝑇+ 𝛽1′𝑅𝑇,𝑆𝐻+ 𝛽1′𝑅𝐻 and𝑆𝐷+ 𝛽1′𝑅𝐷 are always less than or equal to𝑁 , 𝑆𝑇 + 𝛽1′𝑅𝑇 𝑁 ≤ 1, 𝑆𝐻 + 𝛽1′𝑅𝐻
𝑁 ≤ 1and 𝑆𝐷 + 𝛽1′𝑅𝐷
𝑁 ≤ 1.Thus𝐺∗(𝑆, 𝐼 ) ≥ 0for all(𝑆, 𝐼 ) ∈ 𝐷.The𝐸0𝐺𝛼 is a globally asymptotically stable.
This proof is in [54] and analogous proofs can be found in the bibliographical references [97,98].
4.3 Numerical Results
For the simulations, we use different fractional-orders (𝛼 = 0.3, 0.5, 0.7, 0.9, 1.0) to study the behavior of the basic reproduction number and the resistance compartments. The numerical results of the Caputo derivative are obtained by the predict-evaluate-correct-correct-evaluate (PECE) method of Adams-Bashforth-Moulton type [48,49,50], presented in the Subsection (1.1.2) of theorical background. Here, we use 10 years for the time horizon.
The Tables (2.2)-(2.3) show the values used as the initial conditions and parameters for these simulations.
We are going to study the basic reproduction number when we vary(𝛼∗)𝛼 and the resistance parameters in the same intervals as in the ODE study in Section (2.5) for different 𝛼−values.
When we vary(𝛼∗)𝛼 (effective contact rate) in the basic reproduction numbers and study the different𝛼-values, we get that: for lower𝛼-values we obtain lowerℜ𝑇0𝛼 ,ℜ𝐻0𝛼 andℜ𝐷0𝛼 respectively, see Table (4.1) and Figures (4.1a)-(4.1c). In particular, theℜ𝑇0𝛼 for the model with𝛼 = 1.0it reaches values greater and less than unity. For𝛼 < 1.0, we have that ℜ𝑇0𝛼 will always be less than unit. Therefore, we recommend to control the influence of the (𝛼∗)𝛼 in order to keepℜ𝑇0𝛼 always less than unity. Forℜ𝐻0𝛼 andℜ𝐷0𝛼, varying(𝛼∗)𝛼 we have that for𝛼 > 0.5we haveℜ𝐻0𝛼 < ℜ𝐷0𝛼, and for𝛼 ≤ 0.5the opposite situation occurs.
ℜ𝑇0𝛼 ℜ𝐻0𝛼 ℜ𝐷0𝛼
𝛼-value min max min max min max
0.3 2.4314𝑒 − 05 3.4892𝑒 − 05 7.4907𝑒 − 05 1.0735𝑒 − 04 3.7999𝑒 − 05 5.4545𝑒 − 05 0.5 5.3385𝑒 − 04 9.7467𝑒 − 04 5.2741𝑒 − 04 9.6292𝑒 − 04 3.7494𝑒 − 04 6.8415𝑒 − 04
0.7 0.0099 0.0232 0.0038 0.0089 0.0039 0.0090
0.9 0.1601 0.4730 0.0235 0.0695 0.03006 0.0905
1.0 0.6232 2.0773 0.0560 0.1866 0.0995 0.3317
Table 4.1:Values of the basic reproduction numbers for different𝛼-values varying(𝛼∗)𝛼.
4.3 | NUMERICAL RESULTS
Basic Reproduction Number (ℜT
0)
Basic Reproduction Number (ℜH
0)
Basic Reproduction Number (ℜD
0) numbers for the different𝛼-values, it happens analogous to when we vary(𝛼∗)𝛼, see Table (4.2) and (4.3) and Figures (4.2a)-(4.3c). But here we have that when𝛼 = 1.0,ℜ𝑇0𝛼 always takes values greater than unity, the same conditions are fulfilled for ℜ𝐻0𝛼 andℜ𝐷0 with respect to𝛼-values (for𝛼 > 0.5impliesℜ𝐷0𝛼 > ℜ𝐻0𝛼 and for𝛼 ≤ 0.5impliesℜ𝐷0𝛼 < ℜ𝐻0𝛼) and for smaller𝛼−values the basic reproduction numbers are smaller.
ℜ𝑇0𝛼 ℜ𝐻0𝛼 ℜ𝐷0𝛼
𝛼-value min max min max min max
0.3 2.1596𝑒 − 05 2.1665𝑒 − 05 9.3579𝑒 − 05 9.3805𝑒 − 05 4.7477𝑒 − 05 4.7751𝑒 − 05 0.5 6.5121𝑒 − 04 6.5682𝑒 − 04 7.6577𝑒 − 04 7.7312𝑒 − 04 6.1631𝑒 − 04 6.2922𝑒 − 04
0.7 0.0155 0.0157 0.0064 0.0065 0.0069 0.0072
0.9 0.3080 0.3120 0.0461 0.0470 0.0606 0.0639
1.0 1.3153 1.3311 0.1181 0.1205 0.2095 0.2217
Table 4.2:Values of the basic reproduction numbers for different𝛼-values varying𝑙𝑖𝛼,𝑖 = 𝑇 , 𝐻 , 𝐷.
lT
Basic Reproduction Number (ℜT
0)
Basic Reproduction Number (ℜH
0)
Basic Reproduction Number (ℜD
0)
0.7 0.0155 0.0156 0.0064 0.0065 0.0065 0.0066
0.9 0.3080 0.3086 0.0460 0.0462 0.0599 0.0602
1.0 1.3151 1.3177 0.1181 0.1184 0.2099 0.2103
Table 4.3:Values of the basic reproduction numbers for different𝛼-values varying𝜂𝛼1𝑗,𝑗 = 4, 5, 6.
In conclusion, in this case we must pay attention to the behavior of the basic reproduc-tion number due to its variareproduc-tion with respect to the different𝛼-values and how it influences the transmission of the epidemic. The Table (4.4) shows the values of the basic reproduction numbers for the fractional-orders studied in the working scenario, we can see that the basic reproduction numbers increase for higher𝛼-values. Figures (4.4a)-(4.4c) show the graphical behavior of the basic reproduction numbers when we vary the fractional-order.
We have that for𝛼 ∈ [0.9333, 1]theℜ𝑇0𝛼 ∈ [0.5043, 1.3156], which implies that from these 𝛼-values in the scenario under study we should pay attention to this behavior.
4.3 | NUMERICAL RESULTS
Basic Reproduction Number (ℜT
0)
Basic Reproduction Number (ℜH
0)
Basic Reproduction Number (ℜD
0)
Table 4.4:Values ofℜ𝛼0 for the different𝛼-values with the values of the parameters of the Table (2.3).
The following computational simulations for the resistant compartments and their discussions can be found in [54].
For MDR-TB cases in all subpopulations at the beginning of the study, we have a decrease in the number of cases reported. Initially, fewer cases are reported for the lower 𝛼-values (lower 𝛼-values implies fewer cases). At about the year of study (depending on the subpopulation) behavior changes and higher𝛼-values report lower numbers of cases, see Figures (4.5b), (4.5d) and (4.5f). Then a growth begins in all subpopulations of MDR-TB and at the end of the study period, a higher number of cases was reported for the highest𝛼-values, see Figures (4.5a), (4.5c) and (4.5e). The highest number of MDR-TB cases was reported by the TB-Only subpopulation followed by the TB-Diabetes subpopulation throughout the study period and for the different 𝛼-values. Due to these results, it is recommended to apply MDR-TB control in all subpopulations at the beginning of the
α-values (order derivative)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ℜT 0-Value
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Basic Reproduction Number (ℜT
0)
(a)Behavior ofℜ𝑇0𝛼 for𝛼 ∈ (0.00001, 1)
α-values (order derivative)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ℜH 0-Value
0 0.02 0.04 0.06 0.08 0.1 0.12
Basic Reproduction Number (ℜH
0)
(b)Behavior ofℜ𝐻0𝛼 for𝛼 ∈ (0.00001, 1)
α-values (order derivative)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ℜD 0-Value
0 0.05 0.1 0.15 0.2 0.25
Basic Reproduction Number (ℜD
0)
(c)Behavior ofℜ𝐷0𝛼 for𝛼 ∈ (0.00001, 1) Figure 4.4:Behavior ofℜ𝛼0 for𝛼 ∈ (0.00001, 1)
study period to control the growth in the number of cases [54].
The XDR-TB cases in the TB-Only subpopulation decreased throughout the study period. At the beginning of the study, fewer cases were reported in less time for lower 𝛼-values but at the end of the study period, the opposite occurred (higher𝛼-values reported a lower number of cases). We must pay attention when𝛼 = 1.0because at the end of the study there is a slight increase in the number of cases, see Figures (4.6a) and (4.6b) [54].
In the TB-HIV/AIDS subpopulation at the beginning of the study, there is a decrease in the number of cases where lower𝛼-values report fewer cases (up to approximately one year of study). Over time, the decrease in the number of cases continues, but now with higher𝛼-values the number of cases is lower. Approximately 5 years into the study, we have an increase in the number of cases for𝛼 > 0.5and at the end of the period of study, we have differentiated results for the different𝛼. For𝛼 > 0.5, higher𝛼-values reported a higher number of cases. For𝛼 ≤ 0.5, the opposite is true, at lower𝛼-values the number of reported cases is higher, see Figures (4.6c) and (4.6d). This event is important to take into account for the application of an effective control strategy in this subpopulation. We recommend applying control from the beginning of the study in order to avoid the growth of cases taking advantage of the initial decrease in these compartments [54].
In the TB-Diabetes subpopulation, there was an increase in the number of cases during the entire study period. At the beginning of the study, the highest number of cases was
4.3 | NUMERICAL RESULTS
Time
0 1 2 3 4 5 6 7 8 9 10
Number of MDR-TB cases (in thousands of people)0.5 1
5 TB-Only, MDR-TB (for different values of α) α=0.3
α=0.5 α=0.7 α=0.9 α=1
(a)MDR-TB cases in the TB-Only, study period 10 years.
Time
0 0.5 1 1.5 2 2.5 3 3.5 4
Number of MDR-TB cases (in thousands of people)0.7 0.8
1.4 TB-Only, MDR-TB (for different values of α) α=0.3
α=0.5 α=0.7 α=0.9 α=1
(b)MDR-TB cases in the TB-Only, study period 4 years.
Time
0 1 2 3 4 5 6 7 8 9 10
Number of MDR-TB cases (in thousands of people)0.1 0.2
0.8 TB-HIV/AIDS, MDR-TB (for different values of α) α=0.3
α=0.5 α=0.7 α=0.9 α=1
(c)MDR-TB cases in the TB-HIV/AIDS subpopulation, study period 10 years.
Time
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Number of MDR-TB cases (in thousands of people)0.15 0.2 0.25 0.3 0.35
0.4 TB-HIV/AIDS, MDR-TB (for different values of α) α=0.3 α=0.5 α=0.7 α=0.9 α=1
(d)MDR-TB cases in the TB-HIV/AIDS subpopulation, study period 5 years.
Time
0 1 2 3 4 5 6 7 8 9 10
Number of MDR-TB cases (in thousands of people)0.2 0.3
1.1 TB-Diabetes, MDR-TB (for different values of α) α=0.3
α=0.5 α=0.7 α=0.9 α=1
(e)MDR-TB cases in the TB-Diabetes subpopulation, study period 10 years.
Time
0 1 2 3 4 5 6
Number of MDR-TB cases (in thousands of people)0.25 0.3 0.35 0.4 0.45 0.5
0.55 TB-Diabetes, MDR-TB (for different values of α) α=0.3
α=0.5 α=0.7 α=0.9 α=1
(f)MDR-TB cases in the TB-Diabetes subpopulation, study period 6 years.
Figure 4.5:Behavior of MDR-TB cases for different𝛼-values over time.
reported for lower𝛼-values, see Figure (4.6e). At the end of the study period, it happens that higher𝛼- values report a higher number of cases, see Figure (4.6f). In general, the highest number of XDR-TB cases was reported from the TB-Diabetes subpopulation. We recommend applying an effective control strategy in this subpopulation with the objective of reducing the number of XDR-TB cases due to the growth of cases throughout the study
period for all𝛼-values [54].
During the computational experimentation, we found that the MDR-TB compartments have a decrease at the beginning of the study period and a growth at the end for the different𝛼-values, which allows us to design a control strategy with the objective of avoiding the growth of the number of cases. For XDR-TB cases, we recommend paying attention to the TB-HIV/AIDS and TB-Diabetes subpopulations due to the growing number of cases. In particular to the diabetic XDR-TB cases because they report the highest number of cases among all TB treatment resistance compartments [54].
4.3 | NUMERICAL RESULTS
Time
0 1 2 3 4 5 6 7 8 9 10
Number of XDR-TB cases (in thousands of people) 0 0.1
0.7 TB-Only, XDR-TB (for different values of α) α=0.3 α=0.5 α=0.7 α=0.9 α=1
(a)XDR-TB cases in the TB-Only, study period 10 years.
Time
0 0.5 1 1.5 2 2.5 3
Number of XDR-TB cases (in thousands of people)0.1 0.2 0.3 0.4 0.5 0.6
0.7 TB-Only, XDR-TB (for different values of α) α=0.3 α=0.5 α=0.7 α=0.9 α=1
(b)XDR-TB cases in the TB-Only, study period 3 years.
Time
0 1 2 3 4 5 6 7 8 9 10
Number of XDR-TB cases (in thousands of people)0.08 0.1
0.28 TB-HIV/AIDS, XDR-TB (for different values of α) α=0.3
α=0.5 α=0.7 α=0.9 α=1
(c)XDR-TB cases in the TB-HIV/AIDS subpopulation, study period 10 years.
Time
0 0.5 1 1.5 2 2.5 3
Number of XDR-TB cases (in thousands of people)0.08 0.1
0.22 TB-HIV/AIDS, XDR-TB (for different values of α) α=0.3
α=0.5 α=0.7 α=0.9 α=1
(d)XDR-TB cases in the TB-HIV/AIDS subpopulation, study period 3 years.
Time
0 1 2 3 4 5 6 7 8 9 10
Number of XDR-TB cases (in thousands of people) 0 2
20 TB-Diabetes, XDR-TB (for different values of α) α=0.3
α=0.5 α=0.7 α=0.9 α=1
(e)XDR-TB cases in the TB-Diabetes subpopulation, study period 10 years.
Time
0 0.5 1 1.5 2 2.5 3
Number of XDR-TB cases (in thousands of people)0.2 0.3
1 TB-Diabetes, XDR-TB (for different values of α) α=0.3
α=0.5 α=0.7 α=0.9 α=1
(f)XDR-TB cases in the TB-Diabetes subpopulation, study period 3 years.
Figure 4.6:Behavior of XDR-TB cases for different𝛼-values over time.
4.4 Partial Conclusions
• The innovative aspect of this chapter is the use of fractional-order derivatives in the Caputo sense to the model presented in Chapter (2).
• We studied MDR-TB and XDR-TB in the different subpopulations Only, TB-HIV/AIDS and TB-Diabetes and calculated and analyzed the equilibrium points of the different subpopulations (using results of fractional analysis).
• For the computational simulations, we use a predict-evaluate-correct-correct-evaluate (PECE) method in order to study the behavior of the different resistance compartments over time for different𝛼-values.
• Among the results of the simulations, we have that:
– For the MDR-TB at the beginning of the study, there is a decrease of cases in all subpopulations. Then, the number of cases begins to grow and at the end of the study period in MDR-TB, a higher number of cases is obtained for the higher 𝛼-values, see Figure (4.5). For MDR-TB control, we recommended controlling from the beginning of the study to avoid the growth in the number of cases.
– Throughout the study the number of XDR-TB cases in the TB-Only subpop-ulation decreased. At the beginning of the study (before the year of study) for lower𝛼-values, fewer cases are reported and later in the study the higher 𝛼-values report fewer cases, see Figure (4.6a).
– The XDR-TB cases in the TB-HIV/AIDS subpopulation initially have a decrease in the number of cases. Approximately 5 years into the study there is a growth in the number of cases for 𝛼 > 0.5 and at the end of the study, there is a differentiated behavior, for 𝛼 > 0.5 the higher 𝛼-values reported a higher number of cases and for𝛼 ≤ 0.5the lower the𝛼-values the higher number of cases reported, see Figure (4.6c). This factor must be taken into account to design an effective control strategy.
– In the TB-Diabetes subpopulation throughout the study period, there is an increase in the number of XDR-TB cases. At the beginning for lower𝛼-values the number of cases is higher and at the end of the period, the opposite situation occurs, see Figure (4.6e). The diabetic XDR-TB are the highest number of resistance cases compared to all TB treatment resistance compartments. We recommend paying special attention to the control in this compartment due to its growth.
• Computer simulations provide information for an effective design of a control strategy for tuberculosis (treatment resistance and transmission).