Mathematical model for HIV spreads control program with ART treatment

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Mathematical model for HIV spreads control program with ART treatment To cite this article: Maimunah and Dipo Aldila 208 J. Phys.: Conf. Ser View the article online for updates and enhancements. Related content - A Mathematical Model Of Dengue- Chikungunya Co-Infection In A Closed Population Dipo Aldila and Maya Ria Agustin - Application of optimal control strategies to HIV-malaria co-infection dynamics Fatmawati, Windarto and Lathifah Hanif - Mathematical modeling of zika virus disease with nonlinear incidence and optimal control Naba Kumar Goswami, Akhil Kumar Srivastav, Mini Ghosh et al. This content was downloaded from IP address on 25/08/208 at 02:09

2 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd

3 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//

4 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 (x (t)) (x 2 (t)) (x 3 (t)) (x 4 (t)) (x 5 (t)) (x 6 (t)) (x 7 (t)) x A (µ) (x 7 ) δ δ > µ β a x 2 β c x 3, x 4, x 5 x 6 β s (x 7 ) β a > β c > β s x 2 x 3 u (x 4 ) (x 4 ) (x 5 ) ξ 4 (x 4 ) (x 5 ) (x 6 ) ρ c (x 6 ) r c r c (x 3 ) x 7 3

5 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 dx dx 2 dx 3 dx 4 dx 5 dx 6 dx 7 = A β ax x 2 β cx (θ h x 4 θ l x 5 x 3 x 6 ) β sx x 7 µx = β ax x 2 β cx (θ h x 4 θ l x 5 x 3 x 6 ) β sx x 7 γ a x 2 µx 2 = γ a x 2 ( r c ) γ b x 6 ux 3 γ c x 3 µx 3 = µx 4 ux 3 x 4 ξ 4 x 5 ξ 5 = µx 5 ρ c x 5 x 4 ξ 4 x 5 ξ 5 = ρ c x 5 ( r c ) γ b x 6 r c γ b x 6 µx 6 = r c γ b x 6 δx 7 µ x 7 γ c x 3, x i (t = 0) = x i0 i =, 2,..., 7 4

6 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 human A β a β c β s θ h β a x 4 θ l β a x 5 γ a x 2 x 3 γ b x 4 x 5 γ c r c ρ c x 5 u ξ 4 ξ 5 µ δ time u = 0 (x 4, x 5, x 6 ) dx dx 2 dx 3 dx 7 = A β ax x 2 β cx x 3 β sx x 7 µx = β ax x 2 β cx x 3 β sx x 7 γ a x 2 µx 2 = µ x 3 γ a x 2 γ c x 3 = µ x 7 δ x 7 γ c x 3, x 0, x 20, x 30 x 70 { DF E = x = A } µ, x 2 = 0, x 3 = 0, x 7 = 0. 5

7 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 EE = {x = x, x 2 = x 2, x 3 = x 3, x 7 = x 7 }. x = A R 0 µ x 2 = µ Nx(µγc)(µδ)(R 0 ) k x 3 = γa,µ Nx(µδ)(R 0 ) k 2 x 7 = γaγcµ Nx(R 0 ) k 3 k = δ µ β a δ β a γ c δ β c γ a µ 2 β a µ β a γ c µ β c γ a β s γ a γ c k 2 = δ µ β a δ β a γ c δ β c γ a µ 2 β a µ β a γ c µ β c γ a β s γ a γ c k 3 = δ µ β a δ β a γ c δ β c γ a µ 2 β a µ β a γ c µ β c γ a β s γ a γ c R 0 = A(δ µ βaδ βaγcδ βcγaµ2 β aµ β aγ cµ β cγ aβ sγ aγ c) µ (µγ a)(µγ c)(δµ) EE (x i > 0) R 0 = A ( δ µ β a δ β a γ c δ β c γ a µ 2 β a µ β a γ c µ β c γ a β s γ a γ c ) µ (µ γ a ) (µ γ c ) (δ µ) >. DF E J = µ βaa µ βca µ βsa µ 0 β aa µ γ a µ β ca µ β sa µ 0 γ a µ γ c γ c δ µ J. λ 4 h 0 λ 3 h λ 2 h 2 λ h 3 h 4 = 0. h 0 = h = δµ 4µ 2 µ γ a µ γ c Aβ a µ 6

8 International Conference on Mathematics: Pure, Applied and Computation IOP Conf. Series: Journal of Physics: Conf. Series 974 (208) doi:0.088/ /974// ( 3δµ2 Nx δµnx γa δµnx γc 6µ3 Nx 3µ2 Nx γa 3µ2 Nx γc µnx µnx γa γc Aδβa 3Aµβa Aβa γc Aβc γa ) ( 4 h3 = 4µ Nx 3Nx (δ γa γc ) µ3 (((2δ 2γc ) γa 2δγc ) Nx 3Aβa ) µ2 µnx (δnx γa γc 2A (βc γa βa (δ γc ))) µ A ((δβc βs γc ) γa δβa γc )) ( 4 h4 = µ Nx Nx (δ γa γc ) µ3 (((δ γc ) γa δγc ) Nx Aβa ) µ2 Nx (δnx γa γc A (βc γa βa (δ γc ))) µ ((δβc βs γc ) γa δβa γc ) A). h2 = Using the Routh-Hurwitz criterion [2], all eigenvalues of J will be negative if and only if : { h > 0, h3 > 0, h4 > 0 (8) h h2 h3 > h4 Next, the basic reproduction number of system 2 will be analyzed. Basic reproduction number is define as the expected number of secondary cases caused by one primary case in a closed population during one infection period [0]. The illustration of basic reproduction number R0 when it is larger than one, smaller than one, or equal to one is given in Figure 2. There are some Figure 2. Interpretation of disease spread using R0. method to calculate the basic reproduction number, such as using next-generation matrix [0], graph theory [], etc. In this article, next-generation matrix approach will be implemented to calculate the basic reproduction number of system 2. Using the next-generation matrix approach implemented to system 2, the basic reproduction number as the spectral radius of the respected next-generation matrix is given by : ( ) A δ µ βa δ βa γc δ βc γa µ2 βa µ βa γc µ βc γa βs γa γc R0 =. (9) µ Nx (µ γa ) (µ γc ) (δ µ) 000 Let us define the parameters value for system 2 as follows : A = , µ = , Nx = , βa = 000, βc = 0.75βa, βs = 0.5βa, γa = 5 365, γc = With this parameters value, 7

9 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 R 0 x = , x 2 = , x 3 = , x 7 = β a, β c, β s, γ a, γ c δ x 2, x 3 x 7 J = J γ a x x γ a, γ c, β a, β c, β s δ J γ a { DF E = x = A } µ, x 2 = 0, x 3 = 0, x 4 = 0, x 5 = 0, x 6 = 0, x 7 = 0. λ 7 a 0 λ 6 a λ 5 a 2 λ 4 a 3 λ 3 a 4 λ 2 a 5 λ a 6 a 7 = 0, a 0 =, a = Q µ, a 2 = Q 2 µ, a 3 = Q 3 µ, a 4 = Q 4 µ, a 5 = Q 5 µ, a 6 = Q 6 µ, a 7 = Q 7 Q = µ (γ a γ b γ c u 7 µ δ ξ 4 ξ 5 ρ c) Aβ a Q 2 = 2µ 3 6 (γ a γ b γ c u δ ξ 4 ξ 5 ρ c) µ 2 (( γ b γ c u δ ξ 4 ξ 5 ρ c) γ a ( γ c u δ ξ 4 ξ 5 ρ c) γ b ( γ c u ξ 4 ξ 5 ρ c) δ ( ξ 4 ξ 5 ρ c) γ µ c ( u ξ 4 ) ρ c u (ξ 4 ξ 5 )) 6Aβ a A (β cγ a β a (γ b γ c u δ ξ 4 ξ 5 ρ c)) Q 3 = 35µ 5 20 (γ a γ b γ c u δ ξ 4 ξ 5 ρ c) µ 4 (( 0γ a 0γ c 0u 0δ 0ξ 4 0ξ 5 0ρ c) γ b ( 0γ c 0u 0δ 0ξ 4 0ξ 5 0ρ c) γ a ( 0γ c 0u 0ξ 4 0ξ 5 0ρ c) δ µ3 ( 0γ c 0u 0ξ 4 ) ρ c 0 (ξ 4 ξ 5 ) (γ c u)) 20Aβ a 8

10 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 ((( 4γ c 4u 4δ 4ξ 4 4ξ 5 4ρ c) γ a ( 4γ c 4u 4ξ 4 4ξ 5 4ρ c) δ ( 4γ c 4u 4ξ 4 ) ρ c 4 (ξ 4 ξ 5 ) (γ c u)) γ b (( 4γ c 4u 4ξ 4 4ξ 5 4ρ c) δ ( 4γ c 4u 4ξ 4 ) ρ c 4 (ξ 4 ξ 5 ) (γ c u)) γ a (( 4γ c 4u 4ξ 4 ) ρ c 4 (ξ 4 ξ 5 ) (γ c u)) δ 4ξ 4 ρ c (γ c u)) N µ2 x 0 (β aγ b β cγ a β a (γ c u δ ξ 4 ξ 5 ρ c)) A (((( γ c u ξ 4 ξ 5 ρ c) δ ( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) γ a (( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (ur c γ c)) γ b ((( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (γ c u)) γ a δξ 4 ρ c (γ c u)) 4 ((β cγ a β a (γ c u δ ξ 4 ξ 5 ρ c)) γ b (uβ cθ h δβ c β cρ c β cξ 4 β cξ 5 β sγ c) γ µ a ((γ c u ξ 4 ξ 5 ρ c) δ (γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) β a) A ((uβ cθ h δβ c β cρ c β cξ 4 β cξ 5 β sγ c) γ a ((γ c u ξ 4 ξ 5 ρ c) δ (γ c u ξ 4 ) ρ c A (ξ 4 ξ 5 ) (γ c u)) β a) γ b (β c (uθ h ρ c ξ 4 ξ 5 ) δ (uβ cθ h β cξ 4 β sγ c) ρ c (uβ cθ l β sγ c) ξ 4 ξ 5 (uβ cθ h β sγ c)) γ a β a (((γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (γ c u)) Q 4 = 35 µ 5 20 (γ a γ b γ c u δ ξ 4 ξ 5 ρ c) µ 4 ( 0 δ u 0 δ γ a 0 δ γ b 0 δ γ c 0 δ ρ c 0 δ ξ 4 0 δ ξ 5 0 un xγ a 0 uγ b 0 uρ c 0 uξ 4 µ 3 0 uξ 5 0 γ aγ b 0 γ aγ c 0 γ aρ c 0 γ aξ 4 0 γ aξ 5 0 γ b γ c 0 γ b ρ c 0 γ b ξ 4 0 γ b ξ 5 0 N xγ cρ c 0 γ cξ 4 0 γ cξ 5 0 ρ cξ 4 20 Aβ a) ( 4 δ uγ a 4 δ uγ b 4 δ uρ c 4 δ uξ 4 4 δ uξ 5 4 δ γ aγ b 4 δ γ aγ c 4 δ γ aρ c 4 δ γ aξ 4 4 δ γ aξ 5 4 δ γ b γ c 4 δ γ b ρ c 4 δ γ b ξ 4 4 δ γ b ξ 5 4 δ γ cρ c 4 δ γ cξ 4 4 δ γ cξ 5 4 δ N xρ cξ 4 4 uγ aγ b 4 uγ aρ c 4 uγ aξ 4 4 uγ aξ 5 4 uγ b ρ c µ 2 4 uγ b ξ 4 4 uγ b ξ 5 4 uρ cξ 4 4 γ aγ b γ c 4 γ aγ b ρ c 4 γ aγ b ξ 4 4 γ aγ b ξ 5 4 γ aγ cρ c 4 γ aγ cξ 4 4 γ aγ cξ 5 4 γ aρ cξ 4 4 γ b γ cρ c 4 γ b γ cξ 4 4 γ b γ cξ 5 4 γ b ρ cξ 4 4 γ cρ cξ 4 0 Aδ β a 0 Auβ a 0 Aβ aγ b 0 Aβ aγ c 0 Aβ aρ c 0 Aβ aξ 4 0 Aβ aξ 5 0 Aβ cγ a) ( uγ b r cρ cξ 4 4 Auβ cγ aθ h δ uγ aγ b δ uγ aρ c δ uγ aξ 4 δ un xγ aξ 5 δ uγ b ρ c δ uγ b ξ 4 δ uγ b ξ 5 δ uρ cξ 4 δ γ aγ b γ c δ γ aγ b ρ c δ γ aγ b ξ 4 δ γ aγ b ξ 5 δ γ aγ cρ c δ γ aγ cξ 4 δ γ aγ cξ 5 δ γ aρ cξ 4 δ γ b γ cρ c δ γ b γ cξ 4 δ γ b γ cξ 5 δ γ b ρ cξ 4 δ γ cρ cξ 4 uγ aγ b ρ c uγ aγ b ξ 4 uγ aγ b ξ 5 uγ aρ cξ 4 γ aγ b γ cρ c γ aγ b γ cξ 4 γ aγ b γ cξ 5 γ aγ b ρ cξ 4 µ γ aγ cρ cξ 4 γ b γ cρ cξ 4 4 Aδ uβ a 4 Aδ β aγ b 4 Aδ β aγ c 4 Aδ β aρ c 4 Aδ β aξ 4 4 Aδ β aξ 5 4 Aδ β cγ a 4 Auβ aγ b 4 Auβ aρ c 4 Auβ aξ 4 4 Auβ aξ 5 4 Aβ aγ b γ c 4 Aβ aγ b ρ c 4 Aβ aγ b ξ 4 4 Aβ aγ b ξ 5 4 Aβ aγ cρ c 4 Aβ aγ cξ 4 4 Aβ aγ cξ 5 4 Aβ aρ cξ 4 4 Aβ cγ aγ b 4 Aβ cγ aρ c 4 Aβ cγ aξ 4 4 Aβ cγ aξ 5 4 Aβ sγ aγ c) (δ uβ cγ aθ h uβ cγ aγ b θ h uβ cγ aρ cθ h uβ cγ aθ h ξ 5 uβ cγ aθ l ξ 4 δ uβ aγ b δ uβ aρ c δ uβ aξ 4 δ uβ aξ 5 δ β aγ b γ c δ β aγ b ρ c δ β aγ b ξ 4 δ β aγ b ξ 5 δ β aγ cρ c δ β aγ cξ 4 δ β aγ cξ 5 δ β aρ cξ 4 δ β cγ aγ b A δ β cγ aρ c δ β cγ aξ 4 δ β cγ aξ 5 uβ aγ b ρ c uβ aγ b ξ 4 uβ aγ b ξ 5 uβ aρ cξ 4 β aγ b γ cρ c β aγ b γ cξ 4 β aγ b γ cξ 5 β aγ b ρ cξ 4 β aγ cρ cξ 4 β cγ aγ b ρ c β cγ aγ b ξ 4 β cγ aγ b ξ 5 β cγ aρ cξ 4 β sγ aγ b γ c β sγ aγ cρ c β sγ aγ cξ 4 β sγ aγ cξ 5 ) Q 5 = 2µ 6 5 (γ a γ b γ c u δ ξ 4 ξ 5 ρ c) µ 5 (( 0γ a 0γ c 0u 0δ 0ξ 4 0ξ 5 0ρ c) γ b ( 0γ c 0u 0δ 0ξ 4 0ξ 5 0ρ c) γ a ( 0γ c 0u 0ξ 4 0ξ 5 0ρ c) δ µ4 ( 0γ c 0u 0ξ 4 ) ρ c 0 (ξ 4 ξ 5 ) (γ c u)) 5Aβ a ((( 6γ c 6u 6δ 6ξ 4 6ξ 5 6ρ c) γ a ( 6γ c 6u 6ξ 4 6ξ 5 6ρ c) δ ( 6γ c 6u 6ξ 4 ) ρ c 6 (ξ 4 ξ 5 ) (γ c u)) γ b (( 6γ c 6u 6ξ 4 6ξ 5 6ρ c) δ ( 6γ c 6u 6ξ 4 ) ρ c µ 3 6 (ξ 4 ξ 5 ) (γ c u)) γ a (( 6γ c 6u 6ξ 4 ) ρ c 6 (ξ 4 ξ 5 ) (γ c u)) δ 6ξ 4 ρ c (γ c u)) N x 0 (β aγ b β cγ a β a (γ c u δ ξ 4 ξ 5 ρ c)) A (((( 3γ c 3u 3ξ 4 3ξ 5 3ρ c) δ ( 3γ c 3u 3ξ 4 ) ρ c 3 (ξ 4 ξ 5 ) (γ c u)) γ a (( 3γ c 3u 3ξ 4 ) ρ c 3 (ξ 4 ξ 5 ) (γ c u)) δ 3ξ 4 ρ c (ur c γ c)) γ b ((( 3γ c 3u 3ξ 4 ) ρ c 3 (ξ 4 ξ 5 ) (γ c u)) δ 3ξ 4 ρ c (γ c u)) γ a 3δξ 4 ρ c (γ c u)) 6 ((β cγ a β a (γ c u δ ξ 4 ξ 5 ρ c)) γ b (uβ cθ h δβ c β cρ c β cξ 4 β cξ 5 β sγ c) γ µ2 a ((γ c u ξ 4 ξ 5 ρ c) δ (γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) β a) A ((((( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (ur c γ c)) γ a δξ 4 ρ c (ur c γ c)) γ b γ aδξ 4 ρ c (γ c u)) 3 (((uβ cθ h δβ c β cρ c β cξ 4 β cξ 5 β sγ c) γ a ((γ c u ξ 4 ξ 5 ρ c) δ (γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) β a) γ b (β c (uθ h ρ c ξ 4 ξ 5 ) δ (uβ cθ h β cξ 4 β sγ c) ρ c (uβ cθ l β sγ c) ξ µ 4 ξ 5 (uβ cθ h β sγ c)) γ a β a (((γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (γ c u))) A (((β c (uθ h ρ c ξ 4 ξ 5 ) δ (uβ cθ h β cξ 4 β sγ c) ρ c (uβ cθ l β sγ c) ξ 4 A ξ 5 (uβ cθ h β sγ c)) γ a β a (((γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (ur c γ c))) γ b (((uθ h ξ 4 ) ρ c u (θ h ξ 5 θ l ξ 4 )) β cδ ξ 4 ρ c (uβ c β sγ c)) γ a β aδξ 4 ρ c (γ c u)) Q 6 = 7µ 7 6 (γ a γ b γ c u δ ξ 4 ξ 5 ρ c) µ 6 (( 5γ a 5γ c 5u 5δ 5ξ 4 5ξ 5 5ρ c) γ b ( 5γ c 5u 5δ 5ξ 4 5ξ 5 5ρ c) γ a ( 5γ c 5u 5ξ 4 5ξ 5 5ρ c) δ µ5 ( 5γ c 5u 5ξ 4 ) ρ c 5 (ξ 4 ξ 5 ) (γ c u)) 6Aβ a 9

11 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 ((( 4γ c 4u 4δ 4ξ 4 4ξ 5 4ρ c) γ a ( 4γ c 4u 4ξ 4 4ξ 5 4ρ c) δ ( 4γ c 4u 4ξ 4 ) ρ c 4 (ξ 4 ξ 5 ) (γ c u)) γ b (( 4γ c 4u 4ξ 4 4ξ 5 4ρ c) δ (4γ c 4u 4ξ 4 ) ρ c 4 (ξ 4 ξ 5 ) (γ c u)) γ a (( 4γ c 4u 4ξ 4 ) ρ c 4 (ξ 4 ξ 5 ) (γ c u)) δ 4ξ 4 ρ c (γ c u)) 5 (β aγ b β µ4 cγ a β a (γ c u δ ξ 4 ξ 5 ρ c)) A (((( 3γ c 3u 3ξ 4 3ξ 5 3ρ c) δ ( 3γ c 3u 3ξ 4 ) ρ c 3 (ξ 4 ξ 5 ) (γ c u)) γ a (( 3γ c 3u 3ξ 4 ) ρ c 3 (ξ 4 ξ 5 ) (γ c u)) δ 3ξ 4 ρ c (ur c γ c)) γ b ((( 3γ c 3u 3ξ 4 ) ρ c 3 (ξ 4 ξ 5 ) (γ c u)) δ 3ξ 4 ρ c (γ c u)) γ a µ 3 3δξ 4 ρ c (γ c u)) 4 ((β cγ a β a (γ c u δ ξ 4 ξ 5 ρ c)) γ b (uβ cθ h δβ c β cρ c β cξ 4 β cξ 5 β sγ c) γ a ((γ c u ξ 4 ξ 5 ρ c) δ (γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) β a) A ((((( 2γ c 2u 2ξ 4 ) ρ c 2 (ξ 4 ξ 5 ) (γ c u)) δ 2ξ 4 ρ c (ur c γ c)) γ a 2δξ 4 ρ c (ur c γ c)) γ b 2γ aδξ 4 ρ c (γ c u)) 3 (((uβ cθ h δβ c β cρ c β cξ 4 β cξ 5 β sγ c) γ a ((γ c u ξ 4 ξ 5 ρ c) δ (γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) β a) γ b µ 2 (β c (uθ h ρ c ξ 4 ξ 5 ) δ (uβ cθ h β cξ 4 β sγ c) ρ c (uβ cθ l β sγ c) ξ 4 ξ 5 (uβ cθ h β sγ c)) γ a β a (((γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (γ c u))) A γ aγ b δξ 4 ρ c (ur c γ c) 2A (((β c (uθ h ρ c ξ 4 ξ 5 ) δ (uβ cθ h β cξ 4 β sγ c) ρ c (uβ cθ l β sγ c) ξ 4 ξ 5 (uβ cθ h β sγ c)) γ a β a (((γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (ur c γ c))) γ b µ (((uθ h ξ 4 ) ρ c u (θ h ξ 5 θ l ξ 4 )) β cδ ξ 4 ρ c (uβ c β sγ c)) γ a β aδξ 4 ρ c (γ c u)) ( ) ((((uθh ξ A 4 ) ρ c u (θ h ξ 5 θ l ξ 4 )) β cδ β sξ 4 ρ c (ur c γ c)) γ a β aδξ 4 ρ c (ur c γ c)) γ b β cγ auδξ 4 ρ c Q 7 = µ 7 (γ a γ b γ c u δ ξ 4 ξ 5 ρ c) µ 6 ( ) (( γ a γ c u δ ξ 4 ξ 5 ρ c) γ b ( γ c u δ ξ 4 ξ 5 ρ c) γ a µ 5 ( γ c u ξ 4 ξ 5 ρ c) δ ( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) Aβ a ((( γ c u δ ξ 4 ξ 5 ρ c) γ a ( γ c u ξ 4 ξ 5 ρ c) δ ( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) γ b (( γ c u ξ 4 ξ 5 ρ c) δ ( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) γ a (( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (γ c u)) N µ4 x (β aγ b β cγ a β a (γ c u δ ξ 4 ξ 5 ρ c)) A (((( γ c u ξ 4 ξ 5 ρ c) δ ( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) γ a (( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (ur c γ c)) γ b ((( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (γ c u)) γ a µ 3 δξ 4 ρ c (γ c u)) ((β cγ a β a (γ c u δ ξ 4 ξ 5 ρ c)) γ b (uβ cθ h δβ c β cρ c β cξ 4 β cξ 5 β sγ c) γ a ((γ c u ξ 4 ξ 5 ρ c) δ (γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) β a) A ((((( γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (ur c γ c)) γ a δξ 4 ρ c (ur c γ c)) γ b γ aδξ 4 ρ c (γ c u)) (((uβ cθ h δβ c β cρ c β cξ 4 β cξ 5 β sγ c) γ a ((γ c u ξ 4 ξ 5 ρ c) δ (γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) β a) γ b (β c (uθ h ρ c ξ 4 ξ 5 ) δ (uβ cθ h β cξ 4 β sγ c) ρ c (uβ cθ l β sγ c) ξ µ2 4 ξ 5 (uβ cθ h β sγ c)) γ a β a (((γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (γ c u))) A γ aγ b δξ 4 ρ c (ur c γ c) A (((β c (uθ h ρ c ξ 4 ξ 5 ) δ (uβ cθ h β cξ 4 β sγ c) ρ c (uβ cθ l β sγ c) ξ 4 ξ 5 (uβ cθ h β sγ c)) γ a β a (((γ c u ξ 4 ) ρ c (ξ 4 ξ 5 ) (γ c u)) δ ξ 4 ρ c (ur c γ c))) γ b µ (((uθ h ξ 4 ) ρ c u (θ h ξ 5 θ l ξ 4 )) β cδ ξ 4 ρ c (uβ c β sγ c)) γ a β aδξ 4 ρ c (γ c u)) ( ) ((((uθh ξ A 4 ) ρ c u (θ h ξ 5 θ l ξ 4 )) β cδ β sξ 4 ρ c (ur c γ c)) γ a. β aδξ 4 ρ c (ur c γ c)) γ b β cγ auδξ 4 ρ c a 0, a, a 2, a 3, a 4, a 5, a 6, a 7 > 0 b = a a 2 a 0 a 3 a > 0, b 2 = a a 4 a 0 a 5 a c = b a 3 a b 2 b > 0, c 2 = b a 5 a b 3 b d = c b 2 b c 2 c > 0, d 2 = a 6 > 0 e = d c 2 c d 2 d > 0 f = a 6 > 0 > 0, b 3 = a 6 > 0 0

12 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 R 0 M = γ a µ M 2 = u γ c µ M γ a M ( r c ) γ b 0 0 u M H = 3 ξ ξ 4 M ρ c M γ c 0 0 r c γ b M 6, M 3 = µ ξ 4 M 4 = µ ρ c ξ 5 M 5 = ( r c ) γ b r c γ b µ M 6 = δ µ V = β ax β cx β cx θ h β cx θ l β cx β sx = T R 0 = A (P P 2 P 3 P 4 ) µ M M 6 P 5, P = um 4 M 5 M 6 β c γ a θ h um 5 M 6 β c γ a θ l ξ 4 um 6 β a γ b r c ρ c ξ 4

13 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 P 2 = uβ s γ a γ b r c ρ c ξ 4 um 6 β a γ b ρ c ξ 4 um 6 β c γ a ρ c ξ 4 P 3 = M 2 M 3 M 4 M 5 M 6 β a M 2 M 5 M 6 β a ξ 4 ξ 5 M 3 M 4 M 5 M 6 β c γ a P 4 = M 3 M 4 M 5 β s γ a γ c M 5 M 6 β c γ a ξ 4 ξ 5 M 5 β s γ a γ c ξ 4 ξ 5 P 5 = uγ b r c ρ c ξ 4 uγ b ρ c ξ 4 M 2 M 3 M 4 M 5 M 2 M 5 ξ 5 ξ 4 = [ ] T R 0 u r c A = 000/(65 365) β a = β c = β s = γ a = /(5 365) γ c = /(3 365) δ = 0 µ = /(65 365) = 000 ξ 4 = 0.75/365 ξ 5 = 0.25/365 θ h = θ l = 2 ρ c = 0./365 γ b = 0.5/365, R 0 R 0 (f(u, r c )) u r c R 0 R 0 u r c R 0 5.5, (u) R 0 (r c ) R 0 (u) A = 000/(65 365) β a = β c = β s = γ a = /(5 365) γ c = /(3 365) δ = 0 µ = /(65 365) = 000 ξ 4 = 0.75/365 ξ 5 = 0.25/365 θ h = θ l = 2 r c = 0 ρ c = 0./365 γ b = 0.5/365 u =

14 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//02035 u = 0 u 3

15 IOP Conf. Series: Journal of Physics: Conf. Series (208) doi :0.088/ /974//