A Method of Directly Defining the inverse Mapping for a HIV infection of CD4+ T-cells model

Mangalagama Dewasurendra; Ying Zhang; Noah Boyette; Ifte Islam; Kuppalapalle Vajravelu

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A Method of Directly Defining the inverse Mapping for a HIV infection of CD4+ T-cells model

Mangalagama Dewasurendra

Ying Zhang

Noah Boyette

Ifte Islam

oraz

Kuppalapalle Vajravelu

| 15 paź 2020

Applied Mathematics and Nonlinear Sciences

Tom 6 (2021): Zeszyt 2 (July 2021)

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Data publikacji: 15 paź 2020

Zakres stron: 469 - 482

Otrzymano: 07 sie 2019

Przyjęty: 06 paź 2019

DOI: https://doi.org/10.2478/amns.2020.2.00035

Słowa kluczowe
HIV infection model, directly defining inverse mapping, coupled nonlinear system, analytical method, homotopy analysis method, series solution

© 2019 Mangalagama Dewasurendra et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Plot of E(h,A) for T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 500, s = 10, Tmax = 1500. — Plot of E(h,A) for T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 500, s = 10, Tmax = 1500.

Plot of E(h,A) for T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 600, s = 10, Tmax = 1500. — Plot of E(h,A) for T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 600, s = 10, Tmax = 1500.

Plot of E(h,A) for T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.1, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 500, s = 10, Tmax = 1500. — Plot of E(h,A) for T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.1, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 500, s = 10, Tmax = 1500.

Residual Error verses Terms of approximation for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 500, s = 10, Tmax = 1500, where curve 1 has N = 500, and Curve 2 has N = 600. — Residual Error verses Terms of approximation for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 500, s = 10, Tmax = 1500, where curve 1 has N = 500, and Curve 2 has N = 600.

Comparison of the MDDiM solution of T (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results. — Comparison of the MDDiM solution of T (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results.

Comparison of the MDDiM solution of I(t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results. — Comparison of the MDDiM solution of I(t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results.

Comparison of MDDiM solution of V (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results. — Comparison of MDDiM solution of V (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results.

Comparison of MDDiM solution of T (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 600, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results. — Comparison of MDDiM solution of T (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 600, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results.

Comparison of MDDiM solution of I(t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 600, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results. — Comparison of MDDiM solution of I(t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 600, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results.

Comparison of MDDiM solution of V (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 600, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results. — Comparison of MDDiM solution of V (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.26, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 600, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results.

Comparison of MDDiM solution of T (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.1, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results. — Comparison of MDDiM solution of T (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.1, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results.

Comparison of MDDiM solution of I(t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.1, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results. — Comparison of MDDiM solution of I(t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.1, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results.

Comparison of MDDiM solution of V (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.1, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5,

k1'=2×10−5
k_1^{'} = 2 \times 10^{-5}

, N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results. — Comparison of MDDiM solution of V (t) for parameter values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μI = 0.1, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k_1^{'} = 2 \times 10^{-5} , N = 500, s = 10, Tmax = 1500, where Curve 1 is Runge-Kutta results, Curve 2 is MDDiM results.

Minimum of the squared residual error E(h,A) for three different sets of μI and N for fixed parametric values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k.1^{'} = 2 \times 10^{-5} , s = 10, Tmax = 1500.

μ_I	N	h	A	E[h,A]
0.26	500	−0.4295	0.4869	4.068 × 10⁻¹⁰
0.26	600	−0.4187	0.4731	5.682 × 10⁻¹⁰
0.1	500	−0.4311	0.4852	4.235 × 10⁻¹⁰

Numerical comparison for I(t)

t	MDDiM	OHAM	Runge-Kutta
0.0	0	0	0
0.2	3.1568 × 10⁻⁶	1.7626 × 10⁻⁷	3.1318 × 10⁻⁶
0.4	5.1445 × 10⁻⁶	2.9451 × 10⁻⁷	5.1352 × 10⁻⁶
0.6	6.5569 × 10⁻⁶	3.7344 × 10⁻⁷	6.5894 × 10⁻⁶
0.8	7.7445 × 10⁻⁶	4.2602 × 10⁻⁷	7.8026 × 10⁻⁶
1.0	8.8153 × 10⁻⁶	4.6108 × 10⁻⁷	8.9418 × 10⁻⁶

Numerical comparison for V (t)

t	MDDiM	OHAM	Runge-Kutta
0.0	0.01	0.01	0.01
0.2	6.4618 × 10⁻⁴	6.1744 × 10⁻⁴	5.5042 × 10⁻⁴
0.4	4.7805 × 10⁻⁴	3.8431 × 10⁻⁴	4.8189 × 10⁻⁴
0.6	4.0834 × 10⁻⁴	2.4258 × 10⁻⁴	4.0993 × 10⁻⁴
0.8	3.8591 × 10⁻⁴	1.5659 × 10⁻⁴	3.9044 × 10⁻⁴
1.0	3.9576 × 10⁻⁴	1.0406 × 10⁻⁴	4.0056 × 10⁻⁴

Variables and parameters for viral spread

Parameters and Variables	Meaning
DependentVariables
T	Uninfected CD⁺ T-cell population size
I	Infected CD⁺ T-cell density
V	Initial density of HIV RNA
Parameters and Constants
μ_T	Natural death rate of CD⁺ T-cell
μ_I	Blanket death rate of infected CD⁺ T-cel
μ_b	Lytic death rate for infected cells
μ_v	Death rate of free virus
k₁	Rate CD⁺ T-cel become infected with virus
$k_{1}^{'}$ k_1'	Rate infected cells becomes active
r	Growth rate of CD⁺ T-cell population
N	Number of virions produced by infected CD⁺ T-cel
T_max	Maximal population level of CD⁺ T-cel
s	Source term for uninfected CD⁺ T-cel
Derived quantities
T₀	CD⁺ T-cel population for HIV-negative persons

eISSN:: 2444-8656
Język:: Angielski

Częstotliwość wydawania:: Volume Open
Dziedziny czasopisma:: Life Sciences, other, Mathematics, Applied Mathematics, General Mathematics, Physics

Kanał RSS czasopisma

A Method of Directly Defining the inverse Mapping for a HIV infection of CD4+ T-cells model

Data publikacji: 15 paź 2020

Zakres stron: 469 - 482

Otrzymano: 07 sie 2019

Przyjęty: 06 paź 2019

DOI: https://doi.org/10.2478/amns.2020.2.00035

Słowa kluczowe
HIV infection model, directly defining inverse mapping, coupled nonlinear system, analytical method, homotopy analysis method, series solution

© 2019 Mangalagama Dewasurendra et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

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Fig. 13

Minimum of the squared residual error E(h,A) for three different sets of μI and N for fixed parametric values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k.1^{'} = 2 \times 10^{-5} , s = 10, Tmax = 1500.

Numerical comparison for I(t)

Numerical comparison for V (t)

Variables and parameters for viral spread

A Method of Directly Defining the inverse Mapping for a HIV infection of CD4+ T-cells model

Data publikacji: 15 paź 2020

Zakres stron: 469 - 482

Otrzymano: 07 sie 2019

Przyjęty: 06 paź 2019

DOI: https://doi.org/10.2478/amns.2020.2.00035

Słowa kluczoweHIV infection model, directly defining inverse mapping, coupled nonlinear system, analytical method, homotopy analysis method, series solution

© 2019 Mangalagama Dewasurendra et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Minimum of the squared residual error E(h,A) for three different sets of μI and N for fixed parametric values T0 = 1000, I0 = 0, V0 = 0.001, r = 0.03, μT = 0.02, μb = 0.24, μV = 2.4, k1 = 2.4 × 10−5, k1'=2×10−5 k.1^{'} = 2 \times 10^{-5} , s = 10, Tmax = 1500.

Numerical comparison for I(t)

Numerical comparison for V (t)

Variables and parameters for viral spread

Słowa kluczowe
HIV infection model, directly defining inverse mapping, coupled nonlinear system, analytical method, homotopy analysis method, series solution